LIBRARY 

UNIVEJtiSITY  OF  CAUFOSHIM 
PAVIS 


Digitized  by  tine  Internet  Arciiive 

in  2007  witii  funding  from 

IVIicrosoft  Corporation 


littp://www.arcliive.org/details/civilenginpocketOOtrauricli 


THE 

CIVIL  ENGINEER'S 
POCKET-BOOK 


BY 

JOHN  C.  TRAUTWINE 

CIVIL  ENGINEER 

EEVISED    BY 
JOHN  C.  TRAUTWINE,  Je. 

AND 

JOHN  C.  TRAUTWINE,  3d. 
CIVIL  ENGINEERS 


EIGHTEENTH  EDITION,  EIGHTY-SIXTH  THOUSAND 


LIBRARY 
UkiM^lTY  OF  CALIFORNIA 

iJAVIS  - 

NEW    YORK  M 

JOHN   WILEY    &    SONS 
London:  CHAPMAN  &  HALL,   Limited 
1906 

UNIVERSITY  OF  CALIFORNIA 

LIBRARY 

BRANCH  OF  THE 
COLLEGE  OF  AGRICULTURE 


CONTENTS. 


XXVll 


Air.    Atmospbere.    page 

Properties 320 

Pressure  in  Diving  Bells,  etc. .  .  321 

Dew  Point 321 

Heat  and  Cold,  Records  of 321 


Wind. 

Velocity  and  Pressure.    Table.   321 


Rain  and  Snow. 

Precipitation. 

Average    322 

Effect  of  Climate  on  — 322 

and  Stream-flow 323 

Maximum  Rates  of  — 323 

Weight  of  Snow 323 

Rain  Gauge .  324 

Precipitation,  Details  of  —   in 

U.S.,  Table 325 

IVater. 

Composition,  Properties 326 

Ice 326 

Effects  of  Water  on  Metals,  etc.  327 

Tides 328 

Evaporation,  Filtration, 

lieakag-e    329 


MECHAXICS,     FORCE     IN 
RIOII>  BODIES. 

Definitions 330 

Matter;  Body 330 


Dynamics. 

Motion,  Velocity 331 

Force 332 

Action  and  Reaction 333 

Acceleration 334 

Mass    336 

Impulse    337 

Density;  Inertia 338 

Opposite  Forces 339 

Work    341 

Power 342 

Kinetic  Energy 343 

Momentum 345 

Potential  Energy 346 

Impact   347 

Gravity,  Falling  Bodies 348 

Descent  on  Inclined  Planes  .  .  .  349 

Pendulums 350 

Center  of  Oscillation 351 

Center  of  Percussion 351 

Angular  Velocity 351 

Moment  of  Inertia 351 

Radius  of  Gyration 352 

Centrifugal  Force «...  354 


Statics.^  PAQB 

Forces 358 

Line  of  Action 359 

Stress    359 

Moments    360 

Classification  of  Forces 361 

Composition    and     Resolution 

of  Forces 362 

Force   Parallelogram 364 

Force  Triangle 367 

Rectangular  Components 369 

Inclined    Plane 369 

Stress  Components 371 

Applied  and  Imparted  Forces.  .    372 
Resolution,     etc.,     by    means 

of  Co-ordinates    372 

Force    Polygon 374 

^on-concurrentCoplanar Forces  375 

Equilibrium  of  Moments 376 

Cord  Polygon 377 

Concurrent  Non  -  coplanar 

Forces 380 

Non-concurrent     Non-coplanar 

Forces 381 

Parallel  Forces 382 

Coplanar 382 

Non-coplanar 385 

Center  of  Gravity 386 

Stable,  Unstable,  and  Indif- 
ferent Equilibrium 387 

General  Rules 387 

Special    Rules 391 

Line  of    Pressure.     Center    of 

Force  or  of  Pressure 399 

Position  of  Resultant 399 

Distribution  of  Pressure ....   400 

"Middle  Third" 402 

Couples 404 

Friction 407 

Coefficient    408 

Morin's   Laws 410 

Table  of  Coefficients 411 

Other   Experiments 412 

Rolling  Friction 414 

Lubricated  Surfaces 415 

Friction   Rollers 417 

Resistance  of  Trains 417 

WorkofOvercomingFriction  418 

Natural  Slope 419 

Friction  of  Revolving  Shaft  419 

Levers 419 

Stability 422 

Work  of  Overturning 422 

On  Inclined    Planes 424 

The  Cord 425 

Funicular  Machine 427 

Toggle  Joint 427 

Pulley 428 

Loaded  Cord  or  Chain 428 

Arches,      Dams,     etc.    Thrust 

and  Resistance  Linec;  ....   430 

Arches    430 

Graphic  Method 430 

Practical  Considerations .  .   432 

Masonry  Dam 433 

Graphic  Method 435 

Practical  Considerations .  .   436 
The  Screw 436 


XXVIU 


CONTENTS. 


PAGE 

Forces  Acting  upon  Beams  and 

Trusses 437 

Conditions  of  Equilibrium .  .  437 

End  Reactions 439 

Moments    440 

In  Cantilevers 442 

In    Beams 443 

Inclined    Beams 445 

Curved    Beams 446 

Shear    446 

Influence  Diagrams 449 

For    Moments 449 

For    Shear 450 

Relation    between    Moment 

and  Shear,. 452 


STRENOTH      OF     MATE- 
RIAIiS. 

Oeneral  Principles.  454 

Stretch,  Stress  and  Strain ....  455 

Modulus  of  Elasticity 456 

Limit  of  Elasticity 458 

Yield  Point 459 

Resilience 460 

Suddenly  Applied  Loads .....  460 

Elastic    Ratio 461 

Strengths  of  Sections 462 

Fatigue  of  Materials 465 

Transverse  Stren^tb 

Conditions  of  Equilibrium ....  466 

Neutral  Axis , 466 

Resisting   Moment 467 

Modulus  of  Rupture .  468 

Moment  of  Inertia 468 

Table    469 

Section  Modulus 473 

Loading.     Strength    473 

Table    474 

Beam  of  Unit  Dimensions.  . .  .  475 

Coefficients,  Table 476 

Weight  of  Beam  as  Load 477 

Comparison  of  Similar  Beams.  478 

Horizontal  Shear 478 

Deflections 480 

Elastic    Limit 482 

Elastic   Curve 482 

Deflection    CoeSicient 483 

Eccentric  Loads 484 

Uniform  Loads 485 

Inclined    Beams 485 

Cylindrical  Beams 485 

Maximum  Permissible — .  . .  .  485 

Suddenly  Applied  Loads .  .  .  486 

Uniform    Strength 486 

Cantilevers.     Table 487 

Beams.     Table 488 

Continuous  Beams 489 

Table    490 

Cross-shaped  Beam 492 

Plates    492 

Transverse    and    Longitudinal 

Combined 493 


PAGE 

Streng^tli  of  Pillars.  495 

Radius  of  Gyration 496 

Table 496 

Remarks 498 


Sbearinsr  Streng^tb  .  499 


Torsional  Streng^tb.    499 


Vj 


/ 


HTDROSTATICSt. 

Principles 501 

Center  of  Pressure 501 

Air  Pressure 502 

Horizontal      and       Vertical 

Components 503 

Pressure  in  Vessels 503 

Opposite  Pressures 503 

Rules  o 504 

Transmission  of  Pressure 506 

Center  of  Pressure 506 

Walls  to  Resist  Pressure 508 

Thickness  at  Base 509 

Stability 510 

Contents 510 

Liability  to  Crush 510 

Thickness  for  Cylinders 511 

Iron  Pipes 512 

Lead   Pipes 513 

Buoyancy 513 

Flotation.     Metacenter 514 

Draught  of  Vessels 515 


\/    HYBRAUIilCS. 

Flow  of  Water  tbroug-b 

Pipes 516 

Head  of  Water 516 

Velocity    Head 516 

Entry    Head 516 

Friction  Head 516 

Pressure    Head 518 

Piezometers 518 

Hydraulic  Grade  Line , .  519 

Siphon    520 

Velocity  Formulae 522 

Kutter's  Formula 523 

Weight  of  Water  in  Pipes 525 

Areas  and  Contents  of  Pipes .  .  .  526 

Total  Head  Required 527 

Table  of  Velocity  and  Friction 

Heads  and  Discharge 528 

Compound  Pipe 531 

Venturi  Meter. 

Theory 532 

Tube 534 

Register    535 

Ferris-Pitot  Meter 535 

Curves  and  Bends ...........  537 


CONTENTS. 


PAGE 

Flow  tbron^b  Orifices 

Theoretical  Velocities 539 

With  Short  Tubes 540 

Through  Thin  Partition .  .  541 

Discharge  from  One  Reservoir 

to    Another 543 

Rectangular  Openings 544 

Time  of  Emptying  Pond ....  545 

Miner's  Inch 546 


Flow  over  Weirs 

End  Contractions 547 

Measurement  of  Head 648 

Formulae 649 

Francis 550 

Table  of  Discharges 551 

Bazin    552 

Values  of  m  .^ 553 

Submerged  Weirs 554 

Velocity  of  Approach 556 

Inclined   Weirs 558 

Broad-crested  Overfall 559 

Triangular  Notch 559 

Trapezoidal  Notch 559 

Flow  in  Open  Channels 

Relations  of  Velocities 560 

Steam  Gauging 560 

Pitot  Tube,  etc 561 

Wheel  Meter 662 

Abrasion  of  Channel 563 

Theory  of  Flow 563 

Kutter's    Formula 564 

Coefficient  of  Roughness.  .  .  .  564 

Coeffs  of  Roughness.     Table  565 

Coefficient,  c,  Table 666 

To   Draw   Kutter   Diagram.  570 

Flow  in  Sewers 574 

Flow  to  Sewers 575 

Flow  in  Drain-pipes 575 

Constriction  of  Channel 576 

Scour    577 

Obstructions  in  Streams 577 

Power  of  Falling  Water 578 

Water  Wheels 678 

Hydraulic    Ram 578 

Power  of  Running  Stream ....  578 


CONSTRUCTIONS,  ETC. 

I>redgingr. 

Cost  of  Dredging 580 

Horse   Dredges 581 

Weight  of  Material 581 

Foundations. 

Foundations 582 

Borings  in  Common  Soils 582 

Unreliable  Soils 583 

Resistance  of  Soils 583 


PAGE 

Rip-rap    583 

Protection  from  Scour 583 

Timber    Cribs 584 

Caissons    585 

Coffer-dams   586 

Earth  Banks 586 

Crib    Coffer-dams 587 

Mooring  Caissons  or  Cribs 589 

Sinking  through  Soft  Soil 589 

Piles    589 

Sheet    Piles 590 

Grillage    590 

Pile  Drivers 590 

Resistance  of  Piles 592 

Penetrability  of  Soils 593 

Driving 593 

Screw  Piles • 694 

Driving  by  Water  Jet 595 

Hollow  Iron  Cylinders 696 

Pneumatic  Process 696 

Timber  Caisson 598 

Masonry  Cylinders 699 

Fascines 699 

Sand-Piles 699 

Stonework. 

Cost,  etc 600 

Retaining  Walls. 

General  Remarks 603 

Theory    606 

Surcharged    Walls. 609 

Wharf  Walls 611 

Transformation  of  profile 611 

Sliding,  etc 612 

Stone  Bridges. 

Definitions 613 

Depth  of  Keystone 613 

Pressures  on  Arch-stones 614 

Table  of  Arches 615 

Abutments 617 

Abutment  Piers 619 

Inclination  of  Courses 620 

Culverts    622 

Wing  Walls 624 

Foundations 627 

Drains     627 

Drainage  of  Roadway 628 

Contents  of  Piers 628 

Brick  Arches 629 

Centers 631 

Timber  Dams. 

Primary  Requisites 642 

Examples 642 

Abutments,     Sluices,     Ground 

Plan,  Cost 645 

Measuring    Weirs 646 

Trembling    648 

Thickness     of     Planking     Re- 
quired    648 


XXX 


CONTENTS. 


WATER  SUPPI.Y,    PAGE 

Consumption,  Use  and  Waste.   649 
Waste    Restriction ;    Water 

Meters     649 

Water  for  Fire  Protection . . .   650 

Reservoirs    650 

Leakage     through  — ,    Mud 

in  — 651 

Storage  Reservoirs 652 

Valve  Towers,  etc 652 

Compensation    653 

Distributing  Reservoirs .   653 

Water  Pipes 653 

Concretions   in  — ,     preven- 
tion of  — 655 

Weights  of  Cast  Iron  Pipes . .   656 

Wrought  Iron  Pipes 656 

Wooden  and  Other  Pipes .  . .   657 
Costs  of  Pipes  and  Laying.  .   658 

Pipe  Joints 660 

Pipe   Jointer 660 

Flexible  Joints 661 

Special  Castings 661 

Repairs  and  Connections. .  .   662 

Air  Valves 662 

Air  Vessels,  Stand-pipes 663 

Service    Pipes 664 

Tapping  Machines 664 

Anti-bursting  Device 665 

Valves,   Gates 666 

Fire    Hydrants 668 

TEST  AND   WEIali  BORIHTG. 

Test  Boring  Tools 670 

Artesian  Well  Drilling 671 

ROCK  DRI1.I.S. 

Diamond    Drills 675 

Percussion    Drills 676 

Hand  Drills 681 

Channeling 681 

Air  Compressors 681 

TRACTIOX,      ANIItlAI. 
POWER. 

On  Roads,  Canals,  etc 683 

TRUSSES. 

Introduction. 

General  Principles 689 

Loading,    Counterbracing 690 

Cross   bracing 691 

Types  of  Trusses 691 

Camber .• .  .  696 

Cantilevers     696 

Movable    Bridges 696 

Skew   Bridges 697 

Roof  Trusses 698 

Stresses      in     Trass     Mem- 
bers 

General  Principles 698 

Method  by  Sections 700 

Chord       Stresses,       Moments, 

Chord  Increments 701 


PAGE 

Shear 702 

Influence    Diagram 702 

Dead  Load  Stresses .  , .  *. 703 

Live  Load  Stresses 705 

Typical  Wheel  Loads 705 

Cooper's   706 

Live  Load  Web  Stresses ....    706 
Live  Load  Chord  Stresses.  . .    709 

Wind   Loads 710 

Impact,    etc. .  '. 711 

Maximum       and       Minimum 

Stresses 712 

Effect  of  Curves 712 

Counterbracing   713 

Stresses  in  Roof  Trusses. 713 

Weights  and  Loads 713 

Wind    Pressures   714 

Graphic  Method 715 

Timber  Roof  Trusses 716 

Deflections , 718 

Redundant  Members , .   720 

Bridge    Details    and     Con- 
struction 

General  Principles 720 

Floor  System  and  Bearings . .   720 

Design    721 

Flexible  and  Rigid  Tension 

Members     721 

Compression  Members 721 

Pin    and    Riveted    Connec- 
tions       721 

Floor  Beam  Connections.  .  .  .   721 
Tension  Members,  Detail .  . .   722 
Compression   Members,    De- 
tail       722 

End  Post  and  Portal  Bracing    723 

Joints    724 

Pin  Plates 724 

Pins 725 

Expansion    Bearings 725 

Loads,    Clearance,    etc.,    for 

Highway  Bridges 726 

Camber 726 

Examples 726 

Weights     of     Steel     Railroad 

Bridges 731 

List  of  Large  Bridges 732 

Timber  Ti;usses 732 

Joints 733 

Howe  Truss  Bridges 736 

Examples 738 

Metal  Roof  Trusses 740 

Broad  Street  Station,  Phila. .   740 
List  of  Large  Arched  Roofs.    742 

Timber  Roof  Trusses 742 

Transportation  and  Erection .  .   743 

Digests  of  Specifications  for 
Bridges  and  Buildings. 

For    Steel    Railroad    and 
Highway  Bridges. 

General    Design 745 

Material    . 751 

Loads    , . , 755 


CONTENTS. 


PAGE 

Stresses  and  Dimensions 759 

Protection    763 

Erection 763 


For    Combination   Railroad 
Bridg-es. 

General  Design 763 

Material    763 

Loads 764 

Stresses  and  Dimensions 764 

Protection    764 

For  Roofs,  Buildings,  etc. 

General  Design,  Material,  etc..   764 

SUSPENSIOX  BRIDGES. 

Data  Required 765 

Formulas    766 

Anchorages 770 

RIVETS  AHri>  RIVETING. 

Rules  and  Tables 772 

RAIEROADS. 

Curves. 

Deianitions 780 

Tables,  etc 784 

Earthwork. 

Table  of  Level  Cuttings 790 

Shrinkage  of  Embankment ....   799 
Cost  of  Earthwork 800 

Tunnels. 

Construction    812 

Trestles. 
Construction    813 

Track. 

Ballast   815 

Ties 815 

Tie  Plates 816 

Rails 817 

Spikes 818 

Rail  Joints 819 

Turnouts    824 

Equipment. 

Turntables 845 

Water  Stations 851 

Track  Tanks 853 

Track  Scales,  Fences,  etc 854 

Cost  of  Mile  of  Track 855 


Rolling:  Stock.        page 

Locomotives. 

Dimensions,   Weights,  etc...   856 

Performance    860 

Tonnage  Rating 862 

Fast  Runs 863 

Running  Expenses 864 

Cars 865 

Statistics. 

Earnings,  Expenses,  etc 867 


MATERIAES. 

Metals. 
Iron  and  Steel. 

Requirements.      International 

Ass'n  for  Testing  Materials.  870 

Cast  Iron .* 874 

Weight    875 

Weight  of  Cast  Iron  Pipes .  .  876 
Weight  of.  Wrought  Iron  and 

Steel    877 

Roofing   Iron 880 

Corrugated  Iron 881 

Wrought  Iron  Pipes  and  Fit- 
tings      882 

Screw    Threads,    Bolts,    Nuts 

and  Washers 883 

Lock-nut  Washers 885 

Buckle  Plates 885 

Bolts.     Weight  and  Strength, 

Table    886 

Wire  Gauges 887 

Circular    Measure 889 

Wire,  Table 891 

Structural  Shapes. 

I  Beams    892 

Channels 894 

Angles  and  T  Shapes 896 

Separators  for  I  Beams 900 

Z-Bar  Columns 901 

Phoenix  Segment  Columns .  .  904 

Gray    Column 905 

Strengths    of    Iron    Pillars, 

Tables     907 

Floor  Sections 914 

Chains    916 


Other  Metals. 

Tin  and  Zinc 916 

Copper,  Lead,  etc 918 

Tensile  Strengths,  Table    920 

Compressive  Strengths,  Table.  921 

Stone,  etc. 

Tensile  Strengths,  Table 922 

Compressive  Strengths,  Table  923 

Transverse  Strength,  Table...  924 


XXXll 


CONTENTS. 


Mortar,  Bricks,  etc.  page 

Lime  Mortar 925 

Bricks 927 

Cement 930 

Cement  Mortar 931 

Sand   935 

Effects  on  Metals 936 

Efflorescence 936 

Silica  Cement 937 

Rec&mmendations,  Am.  Soc. 

C.    E 937 

Tests 938 

Report  of  Board  of  U.  S.  A. 

Engineer  Officers 940 

Tests 941 

Requirements 942 

Concrete 943 

Properties    943 

Handling    946 


Explosives. 

Nitro-glycerin^  and  Dynamite.  948 

Blasting  Powders 951 

Firing   952 

Gunpowder    953 


Timber. 

Decay  and  Preservation 954 

Tensile  Strength 957 

Compressive  Strength 958 

Transverse   Strength 959 

Strength  as  Pillars 963 


Biiilding:     Materials      and 
Operations.  pagb 

Plastering     968 

Slating    969 

Shingles    971 

Painting 971 

Glass  and  Glazing 973 

Sundry  Materials. 

Rope 975 

Wire  Ropes 976 

Paper    978 

Blue  Prints,  etc 979 


Price  liist  and  Business  Di- 
rectory. 


Price  List 984 

Business  Directory 996 


Bibliog'rapby. 

List  of  Engineering  Books. 


.1008 


GLOSSARY    1025 

INDEX    1039 


PREFACE.  XV 

phia  and  Reading  Railway;  Paul  L.  Wolf  el,  Chief  Engineer, 
American  Bridge  Co.;  J.  Sterling  Deans,  Chief  Engineer,  and 
Moritz  G.  Lippert,  Assistant  Engineer,  Phoenix  Bridge  Co. ;  Ralph 
Modjeski,  Northern  Pacific  Railway;  D.  J.  Whittemore,  Chief 
Engineer,  and  C.  F.  Loweth,  Engineer  and  Superintendent  of 
Bridges  and  Buildings,  Chicago,  Milwaukee  and  St.  Paul  Railway. 

Specifications  for  Bridges  and  Buildings,  Messrs.  C.  C.  Schnei- 
der, Vice  President,  American  Bridge  Company;  J.  E.  Greiner, 
Engineer  of  Bridges  and  Buildings,  Baltimore  and  Ohio  Railroad ; 
Theodore  Cooper;  W.  K.  McFarlin,  Chief  Engineer,  Delaware, 
Lackawanna  and  Western  Railway;  Mason  B.  Strong,  Bridge 
Engineer,  Erie  Railroad;  F.  C.  Osborn,  President,  Osborn  En- 
gineering Co. ;  Wm.  A.  Pratt,  Engineer  of  Bridges,  Pennsylvania 
Railroad ;  W.  B.  Riegner,  Engineer  of  Bridges,  Philadelphia  and 
Reading  Railway;  W.  J.  Wilgus,  Chief  Engineer,  New  York 
Central  Railroad. 

Locomotives,  Baldwin  Locomotive  Work's;  Messrs.  Wilson 
Miller,  President,  Pittsburgh  Locomotive  and  Car  Works ;  Theo. 
N.  Ely,  Chief  of  Motive  Power,  Pennsylvania  Railroad;  A. 
E.  Mitchell,  C.  W.  Buchholz  and  A.  Mordecai,  of  the  Erie  Rail- 
road; Edwin  F.  Smith,  Wm.  Hunter,  A.  T.  Dice  and  Samuel  F. 
Prince,  Jr.,  of  the  Philadelphia  and  Reading  Railway;  and 
Thomas  Tait,  Manager,  Canadian  Pacific  Railway;  and  Major 
E.  T.  D.  Myers,  of  the  Richmond,  Fredericksburg  and  Potomac 
Railroad. 

Cars,  Alhson  Manufacturing  Co.,  Harlan  &  Hollingsworth  Co., 
and  Mr.  Jos.  W.  Taylor,  Secretary,  Master  Car  Builders'  Associa- 
tion. 

Railroad  Statistics,  Mr.  Edward  A.  Moseley,  Secretary,  Inter- 
state Commerce  Commission. 

Iron  and  Steel,  Mr.  Wm.  R.  Webster. 

Cement,  Mr.  Richard  L.  Humphrey. 

Concrete  Beams,  Mr.  Howard  A.  Carson,  Chief  Engineer,  Bos- 
ton Transit  Commission. 

Preservation  of  Timber,  Mr.  O.  Chanute. 

Building  Material,  Mr.  John  T.  WilHs. 

John  C.  Trautwine,  Jr., 

John  C.  Trautwine,  3d. 
Philadelphia,  October,  1902, 


Folios  xvi  to  xxiv  inclusive  are 
left  blank,  to  provide  for  future 
additions  to  prefaces. 


•     CONTENTS. 


MATHEMATICS. 

Mathematical  Symbols 33 

Greek  Alphabet 34 

Aritbinetie. 

Factors  and  Multiples 35 

Fractions    35 

Decimals 37 

Ratio  and  Proportion 38 

Progression    39 

Permutation,  Combination,  Al- 
ligation    40 

Percentage,  Interest,  Annuities  40 

Simple  Interest .  ,  . .  41 

Equation  of  Payments 42 

Compound  Interest 42 

Annuity,  Sinking  Fund,  De- 
preciation, etc 43 

Equations  and  Tables .  .  .  44-46 

Duodenal  Notation 47 

Reciprocals 48-52 

Roots  and  Powers. 
Square  and  cube. 

Tables 54 

,4           Rules 66 

Fifth  Roots  and  Powers 67 

'     Logarithms    70 

Rules 70 

Logarithmic  Chart  and  Slide 

Rule    73 

Two-page  Table 78 

Twelve-page  Table 80 


Geometry,  Mensuration, 
and  Trig-onouietry. 

liines. 

Definitions    92 


Angles. 

Definitions 92 

Construction    93 

Bisection    94 

Inscribed    94 

Complement  and  Supplement.  94 

In  a  Parallelogram 95 

Minutes  ai;id  Seconds  in  Deci- 
mals of  a  Degree,  Table  of —  95 
Approximate  Measurement  of 

Angles 96 

Sine,  Tangent,  etc 97 

Definitions 97 

Table 98 

Chords.     Table  of  — 143 


Surfaces.  paob 

Polygons. 

Regular  — ,  Tables,  etc..  of —  148 

Triangles. 

Definitions.    Properties  ....    148 

Right-angled  — 150 

Trigonometrical  Problems  .  .    150 

Parallelogram 157 

Trapezoid.     Trapezium 158 

Polygons 159 

Regular 159 

Reduction  of  Figures.  .  .159,  160 

Circle 161 

Radius,  Diameter 161 

Area,  Center,  to  Find  —   ...    161 

Problems    161,  162 

Tables  of  — . 

Diameter  in  Units,  Eighths, 

etc 163 

Diameters   in    Units    and 

Tenths 166 

Diameters    in    Units    and 
Twelfths 172 

Arc.     Circular. 

Chord,  Length 179 

Radius,  Rise,  and  Ordinates.  180 
Of  Large  Radius,  to  Draw  —  181 
Tables  of  — 182-185 

Circular    Sector,    Ring,     Zone, 
and  Lune 186 

Circular  Segment. 

Area  of  — ;  to  Find 186 

Area  of  — ;  Table 187 

Ellipse. 

Properties  of  — 189 

Ordinates  and  Circumference 

of  — ;  to  Find  — 189 

Elliptic  Arc 189 

Tables  of  Lengths  of  — . . .    190 

Area  of;  to  Find  — 190 

Construction.  Tangents.  .  .  190 
Oval  or  False  — 191 

Cyma    Recta,   Cyma    Reversa, 
Ogee    191 

Parabola, 

Properties  of  — 192 

Parabolic  Curve.  Length  of —  192 

Area    192 

Parabolic  Zone  or  Frustum .  192 
Construction    193 

Cycloid 194 


Solids. 

Regular   Bodies,    Tetiahedron, 

Hexahedron,  etc 194 

Guldinus  Theorem 194 

Paralleiopiped,  Properties  ....  195 


XXV 


CONTENTS. 


PAGE 

Prism 195 

Frustum 195 

Cylinder. 

Volume  and  Surface  of  —  .  .    196 

Volume.    Table  of  — ,  in  Cu. 
Ft.  andU.  S.  Gals 197 

Wells;    Contents    of    —  and 
Masonry  in  Walls  of  — .  .  .    198 

Cylindric  Ungula 199 

Pyramid  and  Cone 200 

Frustums    of .  .  .  .' 201 

Prismoid 202 

Wedge 203 

Sphere. 

Properties 204 

Volume,  Surface,  etc. 

Formulas  for  — 204 

Tables  of  — 205-207 

Segment  and  Zone  of  — .  .  .  .    208 

Spherical  Shell 208 

Spheroid  or  Ellipsoid 208 

Paraboloid 209 

Frustum  of  — 209 

Circular  Spindle 209 

Circular  Ring 209 


Specific  Gravity. 

Principles 210 

Table    212-215 


Weigiits  and  Measures. 

U.   S.,    British   and  Metric  — , 

Units  of  — 216 

Coins;  Foreign  and  U.  S.  — .  .  .  .  218 

Gold  and  Silver 219 

Weights;   Troy,    Apothecaries' 

and  Avoirdupois  — 220 

Long  Measure 220 

Degrees  of  Longitude.   Length.  221 
Inches  Reduced  to  Decimals  of 

a  Foot.     Table 221 

Square  or  Land  Measure 222 

Cubic  or  Solid  Measure 222 

Liquid  Measures 223 

Dry  Measure 223 

British  Imperial  Measures .....  224 

Volumes  and  Weights  of  Water  224 

Metric  Units 225 

Systeme  Usuel,  — Ancien 226 

Russian 227 

Spanish .  227 

Conversion  Tables 228 

Introduction  and  Explana- 
tion     228 

List  of  Tables 229 

Fundamental  Equivalents  .  .  230 

Abbreviations    230 

Equivalents  and  Numbers  in 

Common  Use 231 

Metric  Prefixes 231 

Tables 232 

Acres  per  Mile  and  per  100  feet. 

Table 254 


PAGE 

Grades,  Tables  of  — 365-257 

Heads      and      Pressures      of 

Water; Tables  of— 258-260 

Discharges   in .  Gals,    per   Day 

and    Cu.    Ft.    per    Second; 

Tables 261-265 

Time.    Definitions,  etc 265 

Standard  Railway  — 267 

Dialing 268 

Board  Measure.    Table 269 


Surveying. 

Tests  of  Accuracy,  Distribution 

of  Error,  etc 274 

Chaining 282 

Location  of  Meridian 284 

By  Circumpolar  Stars 284 

Definitions 284 

By  Means  of  Polaris 285 

By   Means   of  Any   Star  at 

Equal  Altitudes 287 

Times  of  Elongation  and  Cul- 
mination of  Polaris 288 

Azimuths  of  Polaris,  Table.  .  289 
Polar     Distances    and  Azi- 
muths of  Polaris,  Table .  .  290 

Engineer's  Transit 291 

Adjustment  and  Repairs.  ..  .  294 

Vernier 296 

Cross-hairs ;  to  Replace 296 

Bubble  Glass;  to  Replace.  .  .  296 

Theodolite    296 

Pocket  Sextant 297 

Compass. 

Adjustment    298 

Magnetic      Declination      and 
Variation. 

Isogenic  Chart  of  U.  S 300 

Declination 301 

Variation 301 

Demagnetization 302 

Leveling'. 

Contour  Lines 302 

Y  Level 306 

Adjustrhent   3(J7 

Forms  for  Notes 309 

Hand  Level,  Adjustment  ...  310 

Builder's  Plumb  Level 311 

Clinometer  or  Slope  Inst.  ...  311 
Leveling  by  the  Barometer 

or  Boiling  Point .''12 

Table 315 

]«^ATURAI.  PHENOMEIirA. 

Sound. 

Velocity  of 316 

Heat. 

Expansion  and  Melting  Points. 

Table 317 

Thermometer. 

Conversion  of  Scales 318 

Tables >18,  319 


THE   AUTHOR 

DEDICATES  THIS  BOOK 

TO    THE    MEMORY    OF    HIS    FRIEND, 

THE  LATE 

BENJAMIN  H.,LATROBE,Esq., 

CIVIL  ENGINEER. 


z  ZS<^S 


No  pains  have  been  spared  to  maintain  tlie  position  of  this 
as  the  foremost  Civil  Engineer's  Pocket-book,  not  only  in  the 
United  States,  but  in  the  English  language. 

JOHN  -WILEY  &  SONS, 

Scientific  Publishers, 
43  East  Nineteenth  Street,  New  Tork  City. 


PREFACE 

TO  FIRST   EDITION,  1872. 


SHOULD  experts  in  engineering  complain  that  they  do  not  find 
anything  of  interest  in  this  volume,  the  writer  would  merely 
remind  them  that  it  was  not  his  intention  that  they  should.  The 
book  has  been  prepared  for  young  members  of  the  profession  ;  and 
one  of  the  leading  objects  has  been  to  elucidate,  in  plain  English,  a 
few  important  elementary  principles  which  the  savants  have  envel- 
oped in  such  a  haze  of  mystery  as  to  render  pursuit  hopeless  to  any 
but  a  confirmed  mathematician. 

Comparatively  few  engineers  are  good  mathematicians ;  and  in 
the  writer's  opinion,  it  is  fortunate  that  such  is  the  case ;  for  nature 
rarely  combines  high  mathematical  talent,  with  that  practical  tact, 
and  observation  of  outward  things,  so  essential  to  a  successful 
engineer. 

There  have  been,  it  is  true,  brilliant  exceptions  ;  but  they  are 
very  rare.  ^  But  few  even  of  those  who  have  been  tolerable  mathe- 
maticians when  young,  can,  as  they  advance  in  years,  and  become 
engaged  in  business,  spare  the  time  necessary  for  retaining  such 
accomplishments. 

Nearly  all  the  scientific  principles  which  constitute  the  founda- 
tion of  civil  engineering  are  susceptible  of  complete  and  satis- 
factory explanation  to  any  person  who  really  possesses  only  so  much 
elementary  knowledge  of  arithmetic  and  natural  philosophy  as  is 
supposed  to  be  taught  to  boys  of  twelve  or  fourteen  in  our  public 
schools."^ 

*  Let  two  little  boys  weigh  each  other  on  a  platform  scale.  Then  when  they 
balance  each  other  on  their  board  see-saw,  let  them  see  (and  measure  for  them- 
'  selves)  that  the  lighter  one  is  farther  from  the  fence-rail  on  which  their  board  is 
placed,  in  the  same  proportion  as  the  heavier  boy  outweighs  the  lighter  one. 
They  will  then  have  learned  the  grand  principle  of  the  lever.  Then  let  them 
measure  and  see  that  the  light  one  see-saws  farther  than  the  heavy  one,  in  the 
same  proportion  ;  and  they  will  have  acquired  the  principle  of  virtual  velocities. 
Explain  to  them  that  equality  of  moments  means  nothing  more  than  that  when 

V 


VI  PREFACE. 

The  little  tliat  is  beyond  this,  might  safely  be  intrusted  to  the 
savants.  Let  them  work  out  the  results,  and  give  them  to  the  engi- 
neer in  intelligible  language.  We  could  afford  to  take  their  words 
for  it,  because  such  things  are  their  specialty ;  and  because  we 
know  that  they  are  the  best  qualified  to  investigate  them.  On  the 
same  principle  we  intrust  our  lives  to  our  physician,  or  to  the 
captain  of  the  vessel  at  sea.  Medicine  and  seamanship  are  their 
respective  specialties. 

If  there  is  any  point  in  which  the  writer  may  hope  to  meet 
the  approbation  of  proficients,  it  is  in  the  accuracy  of  the  tables. 
The  pains  taken  in  this  respect  have  been  very  great.  Most  of  the 
tables  have  been  entirely  recalculated  expressly  for  this  book  ;  and 
one  of  the  results  has  been  the  detection  of  a  great  many  errors  in 
those  in  common  use.  He  trusts  that  none  will  be  found  exceed- 
ing one,  or  sometimes  two,  in  the  last  figure  of  any  table  in  which 
great  accuracy  is  required.     There  are  many  errors  to  that  amount, 


they  seat  themselves  at  their  measured  distances  on  their  see-saw,  they  balance 
each  other.  Let  them  see  that  the  weight  of  the  heavy  boy,  when  multiplied  by 
his  distance  in  feet  from  the  fence-rail  amounts  to  just  as  much  as  the  weight  of 
the  light  one  when  multiplied*  by  his  distance.  Explain  to  them  that  each  of 
the  amounts  is  in  foot-pounds.  Tell  them  that  the  lightest  one,  because  he  see- 
saws so  much  faster  than  the  other,  will  bump  against  the  ground  just  as  hard  as 
the  heavy  one  ;  and  that  this  means  that  their  momentums  are  equal.  The  boys 
may  then  go  in  to  dinner,  and  probably  puzzle  their  big  lout  of  a  brother  who 
has  just  passed  through  college  with  high  honors.  They  will  not  forget  what 
they  have  learned,  for  they  learned  it  as  play,  withoutany  ear-pulling,  spanking, 
or  keeping  in.  Let  their  bats  and  balls,  their  marbles,  their  swings,  &c,  once 
become  their  philosophical  apparatus,  and  children  may  be  taught  {really  taught) 
many  of  the  most  important  principles  of  engineering  before  they  can  read  or 
write.  It  is  the  ignorance  of  these  principles,  so  easily  taught  even  to  children, 
that  constitutes  what  is  popularly  called  **  The  Practical  Engineer  ;  "  which, 
in  the  great  majority  of  cases,  means  simply  an  ignoramus,  who  blunders  along 
without  knowing  any  other  reason  for  what  he  does,  than  ttiat  he  has  seen  it  done 
so  before.  And  it  is  this  same  ignorance  that  causes  employers  to  prefer  this 
practical  man  to  one  who  is  conversant  with  principles.  They,  themselves,  were 
spanked,  kept  in,  &c,  when  boys,  because  they  could  not  master  leverage,  equality 
of  moments,  and  virtual  velocities,  enveloped  in  x's,  p's,  Greek  letters,  square- 
roots,  cube-roots,  &c,  and  they  naturally  set  down  any  man  as  a  fool  who  could. 
They  turn  up  their  noses  at  science,  not  dreaming  that  the  word  means  simply, 
knowing  why.  And  it  must  be  confessed  that  they  are  not  altogether  without 
reason  ;  for  the  savants  appear  to  prepare  their  books  with  the  express  object  of 
preventing  purchasers,  (they  have  but  few  readers,)  from  learning  why. 


PREFACE.  Vll 

especially  where  the  recalculation  was  very  tedious,  and  where, 
consequently,  interpolation  was  resorted  to.  They  are  too  small  to 
be  of  practical  importance.  He  knows,  however,  the  almost  impos- 
sibility of  avoiding  larger  errors  entirely ;  and  will  be  glad  to  be 
informed  of  any  that  may  be  detected,  except  the  final  ones  alluded 
to,  that  they  may  be  corrected  in  case  another  edition  should  be 
called  for.  Tables  which  are  absolutely  reliable,  possess  an  in- 
trinsic value  that  is  not  to  be  measured  by  money  alone.  "With  this 
consideration  the  volume  has  been  made  a  trifle  larger  than  would 
otherwise  have  been  necessary,  in  order  to  admit  the  stereotyped 
sines  and  tangents  from  his  book  on  railroad  curves.  These  have 
been  so  thoroughly  compared  with  standards  prepared  independ- 
ently of  each  other,  that  the  writer  believes  them  to  be  absolutely 
correct. 

In  order  to  reduce  the  volume  to  pocket-size,  smaller  type  has 
been  used  than  would  otherwise  have  been  desirable. 

Many  abbreviations  of  common  words  in  frequent  use  have  been 
introduced,  such  as  abut,  cen,  diag,  hor,  vert,  pres,  &c,  instead  of 
abutment,  center,  diagonal,  horizontal,  vertical,  pressure,  &c.  They 
can  in  no  case  lead  to  doubt ;  while  they  appreciably  reduce  the 
thickness  of  the  volume. 

Where  prices  have  been  added,  they  are  placed  in  footnotes.  They 
are  intended  merely  to  give  an  approximate  or  comparative  idea  of 
value  ;  for  constant  fluctuations  prevent  anything  farther. 

The  addresses  of  a  few  manufacturing  establishments  have  also 
been  inserted  in  notes,  in  the  belief  that  they  might  at  times  be 
found  convenient.  They  have  been  given  without  the  knowledge 
of  the  proprietors. 

The  writer  is  frequently  asked  to  name  good  elementary  books 
on  civil  engineering  ;  but  regrets  to  say  that  there  are  very  few 
such  in  our  language.  "Civil  Engineering,"  by  Prof.  Mahan  of 
West  Point ;  "  Roads  and  Railroads,"  by  the  late  Prof.  Gillespie  ; 
and  the  "  Handbook  of  Railroad  Construction,"  by  Mr.  George  L. 
Vose,  Civ.  Eng.  of  Boston,  are  the  best.  The  writer  has  reason  to 
know  that  a  new  edition  of  the  last,  now  in  press,  will  be  far 


Vlll  PREFACE. 

superior  to  all  predecessors  ;  and  better  adapted  to  the  wants  of 
the  young  engineer  than  any  book  that  has  appeared. 

Many  of  Weale's  series  are  excellent.  Some  few  of  them  are 
behind  the  times  ;  but  it  is  to  be  hoped  that  this  may  be  rectified 
in  future  editions.  Among  pocket-books,  Haswell,  Hamilton's 
Useful  Information,  Henck,  Molesworth,  Nystrom,  Weale,  &c, 
abound  in  valuable  matter. 

The  writer  does  not  include  Rankine,  Moseley,  and  Weisbach, 
because,  although  their  books  are  the  productions  of  master-minds, 
and  exhibit  a  profundity  of  knowledge  beyond  the  reach  of  ordi- 
nary men,  yet  their  language  also  is  so  profound  that  very  few 
engineers  can  read  them.  The  writer  himself,  having  long  since 
forgotten  the  little  higher  mathematics  he  once  knew,  cannot.  To 
him  they  are  but  little  more  than  striking  instances  of  how  com- 
pletely the  most  simple  facts  may  be  buried  out  of  sight  under 
heaps  of  mathematical  rubbish. 

Where  the  word  ' '  ton  ' '  is  used  in  this  volume,  it  always  means 
2240  lbs. 

There  is  no  table  of  errata,  because  no  errors  are  known  to  exist 
expept  two  or  three  of  a  single  letter  in  spelling  ;  and  which  will 

probably  escape  notice. 

John  C.  Trautwine. 

Philadelphia,  November  13th,  1871. 


PREFACE  TO  NINTH  EDITION. 

TWENTY-SECOND  THOUSAND,  1885. 


QJINCE  the  appearance  of  its  last  edition  (the  twentieth  thousand) 
^  in  1883,  the  *'  Pocket-Book  "  has  been  thoroughly  revised,  and 
many  important  additions  and  other  alterations  have  been  made. 
These  necessitated  considerable  change  in  the  places  of  the  former 
matter,  and  it  was  deemed  best  to  turn  this  necessity  to  advantage, 
and  to  make  a  thorough  re-arrangement,  putting  all  of  the  articles, 
as  far  as  possible,  in  a  rational  order. 

The  list  of  new  matter  and  of  revisions  and  extensions  is  condensed  as 
follows,  1902 : 

New  articles  on  the  steam-hammer  pile  driver,  machine  rock  drills,  air  com- 
pressors, high  explosives,  cost  of  earthwork  by  drag  and  wheel  scrapers  and  by 
steam  excavators,  iron  trestles,  track  tanks,  artesian  well-boring  and  standard 
time,  and  new  tables  of  railroad  curves  in  metric  measure,  circumferences  and 
areas  of  circles,  thermometric  scales,  and  fractions  with  their  decimal  equivalents. 

Articles  revised  and  extended,  on  circular  arcs,  thermometers,  flotation,  flow 
in  pipes,  waterworks  appliances,  velocities,  &c,  of  falling  bodies,  centrifugal 
force,  strength  of  timber,  strength  of  beams,  riveting,  riveted  girders,  trusses, 
suspension  bridges,  rail  joints,  turnouts,  turntables,  locomotives,  cars,  railroad 
statistics  and  manufactured  articles,  including  columns,  beams,'  channels,  angles 
and  tees. 

Most  of  the  new  matter  is  in  nonpareil,  the  larger  of  the  two 
types  heretofore  used.  Boldfaced  type  has  been  freely  used  ; 
but  only  for  the  purpose  of  guiding  the  reader  rapidly  to  a  desired 
division  of  a  subject.     For  emphasis,  italics  have  been  employed. 

Illustrations  which  were  lacking  in  clearness  or  neatness  have 
been  te-touched  and  re-lettered,  or  replaced  with  new  and  better 
cuts.     The  new  matter  is  very  freely  illustrated. 

New  rules  have  been  put  in  the  shai^e  of  formulae,  and  many  of 
the  old  rules  have  been  re-cast  into  the  same  form. 

ix 


X  ,  PREFACE. 

The  addition  of  new  matter,  and  a  number  of  blank  spaces 
necessarily  left  in  making  the  re-arrangement,  have  increased  the 
number  of  pages  about  one-fifth. 

The  new  index  is  in  stricter  alphabetical  order  than  that  of 
former  editions,  and  contains  more  than  twice  as  many  entries, 
although  much  repetition  has  been  avoided  by  the  free  use  of  cross- 
references,  without  which  this  part  of  the  work  might  have  been 
indefinitely  extended. 

The  selection  of  articles  of  manufacture  or  merchandise  for  illus- 
tratioD,  has  been  guided  by  no  other  consideration  than  their  fitness 
for  the  purpose,  and  the  courtesy  of  the  parties  representing  them, 
in  supplying  information. 

The  writer  gratefully  acknowledges  the  kindness  of  those  who 
have  assisted  in  furnishing  and  arranging  data. 

Philadelphia,  January,  1885.  J.  C.  T.,  Jr. 


PREFACE  TO  EIGHTEENTH  EDITION. 

(SEVENTIETH  THOUSAND,  1902.) 


IN  preparation  for  its  eighteenth  edition,  The  Civil  Engineer's 
Pocket  Book,  the  first  edition  of  which  appeared  thirty  years 
ago,  has  undergone  a  far  more  extensive  revision  than  at  any 
other  time.  More  than  370  pages  of  new  matter  have  been 
added;  and  the  new  edition  is  larger,  by  about  100  pages, 
than  its  recent  predecessors. 

Among  the  new  matter  in  this  edition  will  be  found; 

Pages 
43-    46  Annuities,  Depreciation,  etc. 
70-     72  Logarithms. 

73-     77  Logarithmic  Chart  and  SHde  Rule. 
80-     91  New  Table  of  Logarithms. 
228-  253  Conversion  Table  of  Units  of  Measiu-ement. 
300-  301  Isogonic  Chart. 
532-  535  Venturi  Meter. 

536  Ferris-Pitot  Meter. 
546  Miner's  Inch. 
649  Water  Consumption  in  Cities. 
658-  659  Cost  of  Water  Pipe  and  Laying. 
745-  764  Digests  of  Specifications  for  Bridges  and  Buildings. 

816  Tie  Plates.. 
870-  873  Digest  of  Specification  for  Iron  and  Steel. 
905-  906  Gray  Column. 

914  Trough  Floor  Sections. 
983-  995  Price  List  of  Manufactured  Articles. 
996-1007  Business  Directory. 
1008-1023  Bibhography. 

The  following  articles  have  been  almost  or  entirely  rewritten: 
New  Pages  Old  Pages 

35-  47  Arithmetic. , 33-37 

210-211  Specific  Gravity .380-381 

265-266  Time   : 395 

282-283  Chains  and  Chaining 176 

284-290  Location  of  the  Meridian 177-179 

322-325  Rain  and  Snow 220-221 

358-453  Statics „ 318  f-361,  370-375 

xi 


Xll  PREFACE. 

New  Pages  Old  Pages 

466-494  Strength  of  Beams    478-520,  528-536 

499  Shearing  Strength 476 

499-500  Torsional  Strength   476-477 

501-503  Opening  Remarks  on  Hydrostatics  .  ........  222-224 

537-538  Effect  of  Curves  and  Bends  on  Flow  in  Pipes  255-256 

689-744  Trusses  . 547-614 

856-864  Locomotives 805-810 

865-866  Cars 811-813 

867-869  Railroad  Statistics   814-818 

892-899  I  Beams,  Channels,  Angles  and  T  Shapes 521-527 

930-942  Cement 673-678 

943-947  Concrete 678-682 

954-956  Timber  Preservation    , 425-425  a 

The  articles  on  arithmetic  are  considerably  extended,  notably 
by  the  addition  of  new  matter  relating  to  interest,  annuities, 
depreciation,  etc.,  including  several  tables. 

The  new  and  greatly  enlarged  table  of  five-place  logarithms  is 
arranged  in  a  somewhat  novel  form.  In  constructing  this  table, 
the  effort  has  been  to  obviate  the  difficulty,  present  in  all  tables 
where  the  difference  between  successive  numbers  is  constant 
throughout,  that  the  differences  between  successive  logarithms 
of  the  lower  numbers  are  relatively  very  great.  In  the  new  table 
the  differences  between  logarithms  are  much  more  nearly  con- 
stant. For  convenience  in  rough  calculations,  the  old  table  of 
five-place  logarithms,  on  two  facing  pages,  is  retained. 

The  Conversion  Tables  contain  the  equivalents  of  both  English 
and  metric  units,  and  of  each  of  these  in  terms  of  the  other ;  but, 
owing  to  the  extreme  ease  with  which  one  metric  unit  may  be 
converted  into  others  of  the  same  system,  it  has  been  unnecessary 
to  burden  the  table  with  many  of  the  metric  units.  The  tables 
have  been  separately  calculated  by  at  least  two  persons,  and  their 
results  compared  and  corrected.  One  of  these  results  has  then 
been  used  by  the  compositor  in  setting  the  type,  and  the  proofs 
have  been  compared  with  the  other. 

The  new  article  on  the  location  of  the  meridian  is  much  more 
complete  than  its  predecessors,  and  a  new  table  of  azimuths  of 
Polaris,  corresponding  to  different  hour-angles,  has  been  added. 

Perhaps  the  most  radical  and  extensive  of  all  the  changes  in 
this  edition  are  those  in  the  articles  on  Statics,  on  Beams  and  on 
Trusses.  These  have  been  almost  entirely  rewritten  and  com- 
pletely modernized.  Under  Trusses,  modern  methods  of  cal- 
culating the   stresses  in    and  the  dimensions    of   the   several 


PREFACE.  Xlll 

members,  and  modern  methods  of  construction,  are  explained, 
and  several  modern  roofs  and  bridges  are  described  and  illus- 
trated. One  of  the  most  notable  features  in  the  new  article 
is  the  digest  of  prominent  modern  specifications  for  bridges 
for  steam  and  electric  railroads  and  for  highways.  The  articles 
on  the  strength  of  beams  are  greatly  sirnplified  and  brought  into 
harmony  with  modern  methods  of  dealing  with  that  subject. 

In  preparing  the  digests  of  specifications  for  iron  and  steel, 
use  has  been  made  of  the  specifications  recently  adopted 
by  the  American  Section  of  the  International  Association  for 
Testing  Materials;  while  those  of  the  American  Society  of 
Civil  Engineers  and  of  the  recent  report  of  a  Board  of  United 
States  Army  engineer  officers  have  been  similarly  used  in  con- 
nection with  cement. 

The  price  list  of  engineering  materials  and  appliances  has  been 
prepared  merely  as  a  useful  guide  in  roughly  estimating  the  ap- 
proximate costs  of  work,  and  it  is  not  to  be  supposed  that  it  can, 
in  any  important  case,  take  the  place  of  personal  inquiry  and 
correspondence  with  manufacturers  or  their  agents,  nearly  700 
of  whom  are  named  in  the  accompanying  list  of  names  and 
addresses  of  manufacturers,  etc.  From  its  first  appearance,  the 
Pocket  Book  has  undertaken  to  give  prices  of  certain  manufac- 
tured articles,  and  addresses  of  those  from  whom  they  may  be 
obtained;  but  these,  scattered  as  they  were  throughout  the 
volume,  were  necessarily  desultory,  and  limited  in  their  extent 
and  usefulness.  It  is  hoped  that  the  present  articles  will  be 
found  at  least  an  acceptable  substitute  for  them. 

As  in  preceding  editions,  all  new  work  and  all  revisions  have 
been  the  subject  of  our  personal  attention,  and  "  scissors-and- 
paste  "  methods  have  been  scrupulously  avoided.  Even  in  using 
lists  of  manufactured  articles,  etc.,  although  their  statements 
have  in  general  been  left  unchanged,  the  matter  has  in  most  or  all 
cases  been  rearranged  and  classified,  to  suit  the  requirements  of 
this  work. 

For  instance,  the  "  digests "  of  specifications  for  Cement,  for 
Steel  and  Iron,  for  Railroad  and  Highway  Bridges  and  for  Steel 
Buildings,  are  by  no  means  mere  quotations  from  the  originals; 
but,  as  their  name  implies,  the  result  of  careful  digesting  of  the 
contents  of  the  specifications  selected  for  the  purpose;  their 
several  provisions  being  carefully  studied,  in  nearly  all  cases  re- 
worded or  reduced  to  figures,  and  tabulated  in  form  convenient 


XIV  PREFACE. 

for  reference,  the  whole  being  arranged  in  such  logical  order  as  to 
facilitate  reference. 

As  in  all  cases  heretofore,  every  rule  or  formula  and  every 
description  of  methods,  etc.,  can  be  readily  understood  and  ap- 
plied by  any  one,  engineer  or  layman,  understanding  the  use  of 
common  and  decimal  fractions,  of  roots  and  powers,  of  loga- 
rithms, and  of  sines,  tangents,  etc.,  of  angles.  On  the  other  hand, 
one  who  is  not  possessed  of  this  very  meager  stock  of  mathemati- 
,  cal  knowledge  will  hardly  approach  engineering  problems,  even 
as  an  amateur;  and  we  have  therefore  followed  the  precedent, 
established  seventeen  years  ago,  of  putting  rules  in  the  shape  of 
formulas,  which  have  "  the  great  advantage  of  showing  the  whole 
operation  at  a  glance,  of  making  its  principle  more  apparent,  and 
of  being  much  more  convenient  for  reference"  (From. Preface  to 
ninth  edition,  1885). 

The  new  matter  is  very  fully  illustrated.  As  heretofore,  all 
cuts  have  been  engraved  expressly  for  this  work. 

As  in  preparing  for  the  ninth  edition  (1885),  all  the  matter 
of  the  book  has  been  rearranged.  This  has  necessitated  a  new 
paging;  and,  in  making  this,  the  lettering  of  pages,  introduced 
from  time  to  time  as  new  editions  have  appeared  in  the  past, 
has  been  eliminated.  The  rearrangement  and  the  addition  of 
so  much  new  matter  have  of  course  necessitated  the  preparation 
of  a  new  table  of  contents  and  a  new  index. 

In  this,  as  in  all  previous  editions  since  the  eighth  (1883), 
practically  all  new  matter  has  been  set  in  nonpareil,  the  larger  of 
the  two  types  hitherto  used,  and  much  of  the  old  matter  retained 
has  been  reset  in  the  larger  type. 

We  take  pleasure  in  acknowledging  our  indebtedness  to  many 
who  have  kindly  assisted  us  in  our  work,  notably  to  Messrs.  Otis 
E.  Hovey  and  Wm.  M.  White,  of  the  American  Bridge  Co.,  for 
painstaking  examination  of  the  article  on  Trusses;  to  Mr.  C. 
Robert  Grimm  and  Professor  E.  J.  McCaustland  for  similar  as- 
sistance in  connection  with.the  article  on  Statics ;  to  Misses  Laura 
Agnes  Whyte  and  Louise  C.  Hazen  for  suggestions  respecting 
mathematics  and  astronomy ;  and  to  the  following  gentlemen  for 
valuable  information  respecting  the  subjects  named : 

Isogonic  Chart,  Mr.  O.  H.  Tittmann,  Sup't,  U.  S.  Coast  and 
Geodetic  Survey. 

Trusses,  Messrs.  Wm.  A.  Pratt,  Engineer  of  Bridges,  Pennsyl- 
vania Railroad ;  W.  B.  Riegner,  Engineer  of  Bridges,  Philadel- 


MATHEMATICS. 

MATHEMATICAI.  STMBOIiS. 

+  PiUs,  p'jsitive,  add.    1.414+  means  1.414  +  other  decimals. 

—  Minus,  negative,  subtract.  

±  Plus  or  minus,  positive  or  negative.     Thus,  ^cfi  =  ±  a. 

=f  Minus  or  plus. 

X  Multiplied  by,  times.     Thus,  x  X  7  =  x.y  =xy;3X4=12. 

:  y Divided  by.    Thus,  a  -=-  b  =  a  :  b  =  a/b  =  -r-- 

:    : :  Proportion.    Thus,  a  :  6  : :  c  :  d,  as  a  is  to  &,  so  is  c  to  d. 
=  Equals,  is  equal  to. 
>  Is  greater  than.    Thus,  6  >  5. 
<  Is  less  than.    Thus,  5  <  6. 
=f^  Is  not  equal  to. 
i:^  Is  greater  or  less  than. 
:^  Is  not  greater  than. 
5^  Is  not  less  than. 
^  Is  equal  to  or  greater  than. 
^  Is  equal  to  or  less  than, 
oc  Is  proportional  to,  varies  with, 
oo  Infinity. 
J.  Is  perpendicular  to. 
^  j  Angle. 
'N^  Is  similar  to. 

II  Is  parallel  to. 
V  V'^Root  of.    Thus,  j/oor  j/a'-=  square  root  of  a,y''a  =  3d  or  cube  root  of  a, 

^4  a  ==  nih  root  of  a. 
Parenthesis.  "| 

Brackets.        !   Quantities  enclosed  or  covered  by  the  symbol  are  to  be 
I       taken  together. 

■Vinculum,    j 

•.'  Since,  because. 

.'.  Hence,  therefore. 

°  Degrees. 

'   Minutes  of  arc,*  feet. 

"   Seconds  of  arc,*  inches. 

'  "  '"  etc.      Prime,  second,  third,  etc.      Distinguishing  accents.      Thus,  a\ 
a  prime ;  a",  a  second,  etc. 

TT  Circumference  _  3  14159255+  arc  of  semicircle,  or  180°. 
Diameter 

E,  Modulus  of  elasticity. 

e  e,  Base  of  Napierian,  natural  or  hyperbolic  logarithms  =  2.718281828. 

g,  Acceleration  of  gravity  =  approximately  32.2  feet  per  second  per  second  =■ 
approximately  9.81  meters  per  second  per  second. 

♦  Minutes  and  seconds  of  time,  formerly  also  denoted  by  '  and  ",  are  now  de- 
noted by  m  and  s,  or  by  min  and  sec,  respectively. 

^  88 


il 


34 


GREEK   ALPHABET. 


THE  ORSEK   AI.PHABET. 

This  alphabet  is  inserted  for  the  benefit  of  those  who  have  occasion  to  consult 
scientific  works  in  which  Greek  letters  are  used,  and  who  find  it  inconvenient 
to  memorize  the  letters. 


Greek  letters. 

Name. 

Approximate 
equivalent. 

Commonly  used  to  designate 

Capital. 

Small. 

A 

a 

Alpha 

a 

Angles,  Coefficients. 

B 

iS 

Beta 

b 

"                « 

r 

V 

Gamma 

g 

"                "     Specific  gravity. 

A 

6 

Delta 

d 

"                "      Density,  Variation. 
/  Base  of  hyperbolic  logarithms  = 

£ 

e 

Epsilon 

e  (short) 

J     2.7182818. 
'^Eccentricity  in  conic  sections. 

Z 

c 

Zeta 

z 

Co-ordinates,  Coefficients. 

« 

V 

Eta 

e  (long) 

"                  " 

© 

e^ 

Theta 

th 

Angles. 

I 

I 

Iota 

i 

K 

K 

Kappa 

k 

A 

\ 

Lambda 

1 

Angles,  Coefficients,  Latitude. 

M 

M 

Mu 

m 

U                                  i. 

N 

V 

Nu 

n 

" 

H 

i 

Xi 

X 

Co-ordinates. 

O 

o 

Omicron 

o  (short) 

n 

IT 

Pi 

P 

Circumference  -^  radius.* 

p 

P 

Rho 

r 

Radius,  Ratio. 

2 

<rs 

Sigma 

s 

Distance  (space).t 

T 

T 

Tau 

t 

Temperature,  Time. 

Y 

V 

Upsilon 

u  or  y 

$ 

<f> 

Phi 

ph 

Angles,  Coefficients. 

X 

X 

Chi 

ch 

* 

^ 

Psi 

ps 

Angles. 

n 

(a 

Omega 

0  (lon.p;) 

Angular  velocities. 

*  The  small  letter  ir  {pi)  is  universally  employed  to  designate  the  number  of 
times  (=  3.14159265  .  .  ,)  the  diameter  of  a  circle  is  contained  in  the  circum- 
ference, or  the  radius  in  the  semi-circumference.  In  the  circular  measure  of 
angles,  an  angle  is  designated  by  the  number  of  times  the  radius  of  any  circle  is 
contained  in  an  arc  of  the  same  circle  subtending  that  angle,  n  then  stands  for 
an  angle  of  180°  (=  two  right  angles),  because,  in  any  circle,  tt  X  radius  =  the 
semi-circumference. 

The  capital  letter  IT  (pi)  is  used  by  some  mathematical  writers  to  indicate  the 
product  obtained  by  multiplying  together  the  numbers  1,  2,  3,  4,  5  .  .  .  etc.,  up  to 
any  given  point.    Thus,  n4=lX2X3X4  =  24. 

t  The  capital  letter  2  (sigma)  is  used  to  designate  a  sum.  Thus,  in  a  system 
of  parallel  forces,  if  we  call  each  of  the  forces  (irrespective  of  their  amounts)  F, 
then  their  resultant,  which  is  equal  to  the  (algebraic)  sum  of  the  forces,  may  be 
written  R  =  2  F. 


ARITHMETIC.  35 

AEITHMETIO. 

FACTORS  AND  MUIiTIPIiES. 

(1)  Factors  of  any  number,  n,  are  numbers  whose  product  is  =  n.  Thus, 
17  and  4  are  factors  of  68 ;  so  also  are  34  and  2  ;  also  17,  2,  and  2. 

(2)  A  prime  number,  or  prime,  is  a  number  which  has  no  factors, 
except  itself  and  1 ;  as  2,  3,  5,  19,  233. 

(3)  A  coiiimoii  factor,  common  divisor  or  common  measure, 
of  two  or  more  numbers,  is  a  number  which  exactly  divides  each  of  them.  Thus, 
3  is  a  common  divisor  of  6,  12,  and  18. 

(4)  The  hig-hest  common  factor  or  g-reatest  common  divisor, 
of  two  or  more  numbers,  is  called  their  H.  C.  F.  or  their  O.  C  I>.  Thus,  6  is 
the  H.  C.  F.  of  6,  12,  and  18. 

(5)  To  find  the  H.  C.  F.  of  two  or  more  numbers ;  find  the  prime  factors 
of  each,  and  multiply  together  those  factors  which  are  common  to  all,  taking 
each  factor  only  once.    Thus,  required  the  H.  C.  F.  of  78,  126,  and  234. 

78  =  2  X  3  X  13 
126  =  2X3X3X7 
234  =  2X3X3X13 
and  H.  C.  F.  =  2  X  3  =  6. 

(6)  To  find  the  H.  C  F.  of  two  large  numbers ;  divide  the  greater  by  the 
less ;  then  the  less  by  the  remainder,  A ;  A  by  the  second  remainder,  B ;  B  by 
the  third  remainder,  C  ;  and  so  on  until  there  is  no  remainder.  The  last  divisor 
is  the  Ho  C.  F.    Thus,  required  the  H.  C.  F.  of  575  and  782. 

575)782(1 
575 
A  207)575(2 
414 


C  46)161(3 
138 
D  23)46(2        H.  C.  F.  =  D  =  23. 
46 
0 

(7)  A  common  multiple  of  two  or  more  numbers  is  a  number  which  is 
exactly  divisible  by  each  of  them. 

(8)  The  least  common  multiple  of  two  or  more  numbers  is  called 
their  I*.  €.  M. 

(9)  To  find  the  L..  C.  M.  of  two  or  more  numbers ;  find  the  prime  factors 
of  each.  Multiply  the  factors  together,  taking  each  as  many  times  as  it  is  con- 
tained in  that  number  in  which  it  is  oftenest  repeated.  Thus,  required  the 
L.  C.  M.  of  7,  30,  and  48. 

7  =  7 
30  =  2  X  3  X  5 
48  =  2X2X2X2X3 
L.  C.  M.  =7X2X2X2X2X3X5  =  1680. 

(10)  To  find  the  li.  C.  M.  of  two  large  numbers;  find  the  H.  C.  F.,  as 

above ;  and,  by  means  of  it,  find  the  other  factors.     Then  find  the  product  of  the 
factors,  as  before.     Thus,   required  the  L.  C.  M.  of  575  and  782.    As  above, 

H.  C.  F.  =  23 ;  ^  =  25 ;  and  —■  =  34.    Hence, 

575  =  23  X  25 
782  =  23  X  34 
and  L.  C.  Mo  =  23  X  25  X  34  =  19,550. 

FRACTIONS. 

(1)  A  common  denominator  of  two  or  more  fractions  is  a  common 
multiple  of  their  denominators. 

(2)  The  least  common  denominator,  or  Li.  C.  D.,  of  two  or  more 
fractions  is  the  L.  C.  M.  of  their  denominators. 


36 


ARITHMETIC. 


(3)  To  reduce  to  a  common  denominator.    Let 

N  =  the  new  numerator  of  any  fraction 

n  =  its  old  numerator 

d  =  its  old  denominator 

C  =  the  common  denominator 

Then  .^         C 

N  =  n  - 
d 

Thus,  J,  1^,  |-.    C  =  L.  C.  M.  of  denominators  =  24. 

24 

3  _      ^4   ^  3X6  _  18 .    5  _  5X4  _  20 .    7  _  7X3  ^  21 

4  ""  24  ~  4  X  6  ~"  24'    6  ""  6  X  4  ~  24'    8  ""  8  X  3        24* 

If  none  of  the  denominators  have  a  common  factor,  then  C=the  product  of  all  the 
n 
denominators,  -  =  the  product,  P,  of  all  the  other  denominators,  and  N  =  P  n. 

Thus,  |,  1.,  |-.     C  =  84 

2  _  2X4X7  _  56  .  1  _  1X3X7  _  2 i .  5  _  5X3X4  _  go 

3  84  8  4  '   4  84  8  4  '   7  84  8  4' 

(4)  Addition  and  (Subtraction.  If  necessary,  reduce  the  fractions  to 
a  common  denominator,  the  lower  the  better.  Add  or  subtract  the  numerators. 
Thus, 

JL-i-i—   2_i.3.,l._4._i.3.,A=2.7.,2.0._j47:_i    1.1. 
2"^2  2  ^'4^4  4         ^'4'^9  36^36  36  ■^36» 

3,7_6,7_13_i5 

4   +  8-  -  ¥  +  ¥  -  "8"  -  ^   8-- 
3_1    _2_1.3  o_27_20__7.7_3_7  6_1 

4  4~'4~2'4  9~3  6'         36  36'8  4"~8  8~"8" 

(5)  Multiplication.  Multiply  together  the  numerators,  also  the  denomi- 
nators, cancelling  where  possible.     Thus, 

4  ""  Te" '     t  ^  t  ^  t  ^  T^ 

q4vi2_25v'^—    125_r20.     2v3_2. 
3y  X   I3   -  -y-  X  -3   -  -^1  &2"T'     3"  X    5"  -  -5-1 

Inf-LofAofl  —  -^viv-^v-I  —  1- 
_  01   2-  ot  -3  01    5-  -   ^   X   2   X   3   X  3-  -   g- . 

(6)  Division.    Invert  the  divisor  and  multiply.    Thus, 

l^l_lv2_2_i.     3.^i_3.v4_3_,. 
2     •     2~2^T~2"~-^'     4-     4~4^1~T~'' 

3     .     5_3v9   —   27_iJ7    . 


Ivl   —   1-     3vl 3_.     3.v'5v2__5 


4     .     i2_2  5     .     5_25v/3_5v3_15_ol. 

y--l-3--y--v-3---y-X-5--yXY--y--2y, 


7-  —  1  "s T~  ~  3"  ~  ~T~  X  X  —  T  ^ 

5   ^  I-  =   5   X 


4  0   _    .5 

"7 ^T- 


(7)  A  fraction  is  said  to  be  in  its  lowest  terms,  or  to  be  simplified^ 

when  its  numerator  and  denominator  have  no  common  factor.    Thus, 


-|-|-  simplified  =  -I-. 


(8)  To  reduce  to  lonrest  terms.    Divide  numerator  and  denominator 

by  their  H.  C.  F.    Thus,  required  the  lowest  terms  of  —. 

85 

H.  C.  F.  of  34  and  85  =  17  ;  and  f|  =  f f  "^  ]l  =  \. 
85   85  -T-  17   5 


ARITHMETIC. 


37 


]>£€IMAIiS. 

(9)  Multiplication.     The  product  has  as  many  decimal  places  as  the 
factors  combined.     Thus, 

Factors :  100  X  3  X  3.5  X  0.004  X  465.21  =  1953.882000 

Number  of  decimal  places:       0  +  0+1+         3+         2=  6 

(10)  Division.    The  number  of  decimal  places  in  the  quotient  =  those  in 
the  dividend  minus  those  in  the  divisor.    Thus 


5.125 


=  1.25; 


=  200; 


5  _  5^  _  .   3  _  3.00  ^  Q  7^5 .     0-42    _  0.4200 

4.1  ^  '4~4~'"^'4~'4~'      '   0.0021  ""  0.0021  '' 

When  the  divisor  is  a  fraction  or  a  mixed  number,  we  may  multiply  both 
divisor  and  dividend  by  the  least  power  of  10  which  will  make  the  divisor  a 
whole  number.     Thus, 


2.679454       26,794.54 


=-432.17. 


0.0062  62 

(11)  To  reduce  a  common  fraction  to  decimal  form ;  divid« 
the  numerator  by  the  denominator.    Thus,  ^-^  =  0.8 ;  1-|-  =  -f-  =  1.6. 

Table  1.    Decimal  equivalents  of  common  fractions. 


8th8 

16th8 

32ds 

64th8 

8ths 

leths 

32ds 

61ths 

1 

.015625 

33 

.515625 

1 

2 
3 

.03125 

.046875 

17 

34 
35 

.53125 
.546875 

1 

2 

4 
5 

.0625 

.078125 

9 

18 

36 
37 

.5625 

.578125 

3 

6 

7 

.09375 
.109375 

19 

38 
39 

.59375 
.609375 

1 

2 

4 

8 
9 

.125 
.140625 

5 

10 

20 

40 
41 

.625 
.640625 

5 

10 
11 

•  .15625 
.171875 

21 

42 
43 

'  .65625 
.671875 

3 

6 

12 
13 

.1875 
.203125 

11 

22 

44 
45 

.6875 
.703125 

7 

14 
15 

.21875 
.234375 

23 

46 
47 

.71875 
.734375 

2 

4 

8 

16 
17 

.25 
.265625 

6 

12 

24 

48 
49 

.75 

.765625 

9 

18 
19 

.28125 
.296875 

25 

50 
51 

.78125 

.796875 

5 

10 

20 
21 

.3125 
.328125 

13 

26 

52 

53 

.8125 
.828125 

11 

22 
23 

.34375 
.359375 

27 

54 

55   , 

.84375 
.859375 

3 

6 

12 

24 
25 

.375 
.390625 

7 

14 

28 

56 

57 

.875 
.890625 

13 

26 

27 

.40625 
.421875 

29 

58 
59 

.90625 
.921875 

7 

14 

28 
29 

.4375 
.453125 

15 

30 

60 
61 

.9375 
.953125 

15 

30 
31 

.46875 
.484375 

31 

62 
63 

.96875 
.984375 

4 

8 

16 

32 

.5 

8 

16 

32 

64 

1. 

(12)  To  reduce  a  dec 

imal  fractioi 

1  to  commo 

n  form 

1.  Supply 

the  denominator  (1),  and  red 

uce  the  resulting 

fraction  to  its  1( 

iwest  ter 

ns.  Thus : 

0.25    25    1 

7.1    .3 

890625 

57 

0.25  -=     =    =   : 

0.75  =  -   ==  - 

;  0.890625  = 

1 

.uo 

luO    4 ' 

1 

)0   4 

1 

OOOOOO  "" 

'  64' 

38 


ARITHMETIC. 


(13)  Recntrring,  circulating-,  or  repeating*  decimals  are  those  in 
which  certain  digits,  or  series  of  digits,  recur  indefinitely.  Thus,  i  =^  0.3333...., 
and  so  on  ;  -^-^  =  1.428571428571 ,  and  so  on.  Recurring  decimals  may  be  in- 
dicated thus :  0.3, 1.428571 ;  or  thus :  0.*3,  l.*428571. 

RATIO  AXD  PROPORTION. 

(1)  Ratio.  The  ratio  of  two  quantities,  as  A  and  B,  is  expressed  by  their 
quotient,   -  or  — .     Thus,  the  ratio  of  10  to  5  is  =       =  2 ;  the  ratio  of  5  to  10 


B 


5 
10 


-1-  -  0.5. 


Thus, 


A2 


B2 


(2)  Duplicate  ratio  is  the  ratio  of  the  squares  of  numbers, 
is  the  duplicate  ratio  of  A  and  B. 

(3)  Proportion  is  equality  of  ratios.  Thus,  -LQ-  =  -2-2.  ^  _1^2^6^6  ^  2. 
In  the  figure,  which  represents  segments,  A,  B,  C,  and  D,  between  parallel  lines ; 
A:B::C:D,or|  =  5 

(4)  The  first'and  fourth  terijos,  A  and  T>,  are  called  the  extremes,  and  the 
second  and  third,  B  and  C,  are  called  the  means.  The  first  term,  A  or  C, 
of  each  ratio,  is  called  the  antecedent,  and  the  second  term,  B  or  D,  Is  called 
the  consequent.    I)  is  called  the  fourth  proportional  of  A,  B,  and  C. 

(5)  In  a  proportion,  A  :  B  =  C  :  D,  we  have : 
Product  of  extremes  =  product  of  means.   A  D  =  B  C. 

A        B 
C  ""  D* 

A~C'    D^C'    B~A' 

^  ...  A  +  B        C+DA  +  B 

Composition.    — - —  =  — j^ —  ;    • — :^ — 


Alternation.     ^  =  f^> 


Inversion. 


-D 


D 


Division. 


A— B        C— D      A— B 


Composition  and  division. 
We  have,  also : 

m  B  ~  B  ~  T)  ~  nB  ~  tTIj*  ' 


'         B 
A  +  B 


A  — B 


m  A 

nB    '' 


C  — D 
D 

C  -hP 
C-D* 


m  C^ 
n  b'' 


=  C  =  m,  then  A  :  m  - 


(6)  If,  in  the  proportion,  A  :  B  =  C  :  D,  we  have  B 

w  :  B,  or  —  =  7-  or  m  2  =  A  D,  or  m  =  t/a  D. 
'ml) 

(7)  In  such  cases,  m  is  called  the  mean  proportional  between  A  and  D, 
and  D  is  called  the  third  proportional  of  A  and  m,. 

A  continued  proportion  is  a  series  of  equal  ratios,  as 


A:B  =  C:D  =  E:F,  etc.  ■■ 
In  continued  proportion, 


=  ^'    ^"    B  =  D  =  F'^*"-=^ 


A  -1-  C  +  E  +  etc. 
B  +  D  +  F  -f  etc. 

A 

~  B  ' 

=  ^  =  |etc,  =  R 

A 
B  ^ 

C 

"  D' 

A'        C .      A" 
'     b'  ~  D' '      B'' 

^,         A  A'  A''        C  C  C 

If 

(8)  Let  A,  B,  and  C  be  any  three  numbers.    Then 

A_AB  AAC 

^  --  ^ .  -,  and  g  -  ^  .  g. 


etc. 


*  0.*3,  l.*428571,  etc.,  standing  for  0.3 


1.428571428571....,  etc. 


ARITHMETIC.  39 

(9)  Reciprocal  or  inverse  proportion.  Two  quantities  are  said 
to  be  reciprocally  or  inversely  proportional,  when  the  ratio  -  of  two  values,  A 

and  B,  of  the  one,  is  =  the  reciprocal,  —  ,  of  the  ratio  of  the  two  corresponding 

ralues  of  the  other.     Thus,  let  A  =  a  velocity  of  2  miles  per  hour,  and  B  =  3 

miles  per  hour.     Then  the  hours  required  per  mile  are  respectively,  A'  =  —  —  -l-' 

A       ^ 

andB'  =  i  =  l.    HereA:B  =  B':A',or^  =  |'„or|  =  |  =  i  =  l-^.A;. 

(10)  If  two  variable  numbers,  A  and  B,  are  reciprocally  proportional,  so  that 
A'  :  ly  =  B"  :  A",  the  product,  A'  A'',  of  any  two  values  of  one  of  the  numbers 
is  equal  to  the  product,  B'  B"  of  the  two  corresponding  values  of  the  other. 

(11)  The  application  of  proportion  to  practical  problems  is  sometimes  called 
the  rule  of  three.  Thus  :  sing^le  rule  of  threes  If  3  men  lay  10,000 
bricks  in  a  certain  time,  how  many  could  6  men  lay  in  the  same  time? 

As  3  men  are  to  6  men,  so  are  10,000  bricks  to  20,000  bricks;  or,  10,000 
bricks  X  -|  =  20,000  bricks. 

If  3  men  require  10  hours  to  lay  a  certain  number  of  bricks,  how  many  hours 
would  6  men  require  to  lay  the  same  number? 

As  6  men  are  to  3  men,  so  are  10  hours  to  5  hours ;  or,  10  hours  X  ^  =  5  hours. 

(12)  l>ouble  rule  of  three. 

If  3  men  can  lay  4,000  bricks  in  2  days,  how  many  men  can  lay  12,000  bricks 
in  3  days?    Here  4,000  bricks  requires  men  2  days,  or  6  man-days,  and  12,000 

bricks  will  require  6  X  -jT^r^?.  =  6  X  3  =  18  man-days ;  and,  as  the  work  is  to  be 
done  in  3  days,  -i^  =  6  men  will  be  required. 

PROORESI^ilON. 

(1)  Arithmetical  Progression.  A  series  of  numbers  is  said  to  be  in 
arithmetical  progression  when  each  number  differs  from  the  preceding  one  by 
the  same  amount.  Thus,  —2,  —1,  0,  1,  2,  3,  4,  etc.,  where  difference  =  1 ;  or  4,  3, 
2, 1,  0,  —1,  —2,  etc.,  where  difference  =  —1 ;  or  — i,  —2,  0,  2,  4,  6,  8,  10,  where 
difference  =  2 ;  or  \%,  V%,  1,  %,  3^,  ^,  0,  —%,  — 3^,  etc.,  where  difference  =  —%. 

(2)  In  any  such  series  the  numbers  are  called  terms.  Let  a  be  the  first  term, 
I  the  last  term,  d  the  common  difference,  n  the  number  of  terms,  and  s  the  sum 
of  the  terms.    Then 


l  =  a  -{-  {n  —  1)  d 


Required 
I 

Given 

a  d  n 

I 

ads 

s 

a  d  71 

a 

d  I  s 

I  =  —  ^  d  ±  \/2  d  s  +  (a  —  ^  d)i 
5  =  -i-  w  [2  a  +  (w  —  1)  tZ] 
a=^d±\/{l-\-^d)^  —  2ds 


ads 


d  —  2  a  ±  y{2  a  —  d)^  +  S  d  * 
2d 


W7.  „       2l  +  d  ±  V{2l  +  dr~-8ds 

n  a  I  s  n  = ^7-3 • 

2  d 

(3)  Geometrical  Progression.  A  series  of  numbers  is  said  to  be  in 
geometrical  progression  when  each  number  stands  to  the  preceding  one  in  the 
same  ratio.  Thus:  -i,  -i,  1,  3,  9,  27,  81,  etc.,  where  ratio  =  3;  or  48,  24,  12,  6, 
3,  li,  1^,  |-,  etc.,  where  ratio  =  -J-;  or  |-^,  lii,  3-|,  6|,  13^,  27,  etc.,  where 


40  ARITHMETIC. 

(4)  Let  a  be  the  first  term,  I  the  last  term,  r  the  constant  ratio,  n  the  numbel 
of  terms,  and  s  the  sum  of  the  terms.    Then  ; 


[Hired 

Given 

I 

a  r  n 

I 

a  r  s 

I 

r  71  s 

1  = 

a  +  (r  — 

1)4 

r 

,.  «— 1 

-1 

n-\. — 

s  — 

-n-l 

yi  - 

I 

W-l  -/ 

n~l 

a  n  I 


y  I  -h  a 

PERMUTATION,  Etc. 

(1)  Permutation  shows  in  how  many  positions  any  number  of  things  can 
be  arranged  in  a  row.  To  do  this,  multiply  together  all  the  numbers  used  in 
counting  the  things.  Thus,  in  how  many  positions  in  a  row  can  9  things  be 
placed  ?    Here, 

1X2X3X4X5X6X7X8X9  =  362880  positions.     Ans. 

(2)  Combination  shows  how  many  combinations  of  .a  few  things  can  be 
made  out  of  a  greater  number  of  things.  To  do  this,  first  set  down  that  number 
which  indicates  the  greater  number  of  things ;  and  after  it  a  series  of  numbers, 
diminishing  by  1,  until  there  are  in  all  as  many  as  the  number  of  the  few  things 
that  are  to  form  each  combination.  Then  beginning  under  the  last  one,  set  down 
said  number  of  few  things;  and  going  backward,  set  down  another  series,  also 
diminishing  by  1,  until  arriving  under  the  first  of  the  upper  numbers.  Multiply 
together  all  the  upper  numbers  to  form  one  product ;  and  all  the  lower  ones  to 
form  another.     Divide  the  upper  product  by  the  lower  one. 

Ex.  How  many  combinations  of  4  figures  each,  can  be  made  from  the  9  figures 
1,  2,  3,  4,  5,  6,  7,  8,  9,  or  from  9  any  things? 

9X8X7X6       3024       ,^^         ,  .     ,.  . 

(3)  Allig^ation  shows  the  value  of  a  mixture  of  different  ingredients,  when 
the  quantity  and  value  of  each  of  these  last  is  known. 

Ex.  What  is  the  value  of  a  pound  of  a  mixture  of  20  lbs  of  sugar  worth  15  cti 
per  ft) ;  with  30  S)s  worth  25  cts  per  ft)? 
ft)S.     cts.     cts. 

20  X  15  =  300  ^^       ^        1050      „,    , 

30X25  =  750  Therefore, -^  =  21  cts.    Ans. 

50  ft)s.        1050  cts. 

PERCENTAGE,  INTEREST,  ANNUITIES. 

Pereentag-e. 

(1)  Ratio  is  often  expressed  by  means  of  the  word  "  per,"  Thus,  we  speak  of 
a  grade  of  105.6  feet  per  mile,  i.  e.,  per  5280  feet.  When  the  two  numbers  in  the 
ratio  refer  to  quantities  of  the  same  kind  and  denomination,  the  ratio  is  often 
expressed  as  a  percentage  {'per hundredage).    Thus,  a  grade  of  105.6  feet  per  mile, 

*  Equations  involving  powers  and  roots  are  conveniently  solved  by  means  of 
logarithms. 


ARITHMETIC. 


41 


or  per  5280  feet,  is  equivalent  to  a  grade  of  0.02  foot  per  foot,*  or  2  feet  per  100 
feet,  or  simply  (since  both  dimensions  are  in  feet)  2  per  100,  or  2  per  "  cent." 

(2)  Oiie-tiftieth,  or  1  per  50,  is  plainly  equal  to  two  hundredths,  or  2  per  hun' 
dred,  or  2  per  cent.  Similarly,  H^'lo  per  cent.,  %  =  3  X  25  per  cent.  =  75  per 
cent.,  etc.  Hence,  to  reduce  a  ratio  to  the  form  of  percentage,  divide  100  times* 
the  first  terra  by  the  second.  Thus,  in  a  concrete  of  1  part  cement  to  2  of  sand 
and  5  of  broken  stone,  there  are  8  parts  in  all,  and  we  have,  by  weight— f 

Cement    =  4-  =  0.125  =    12.5  per  cent,  of  the  whole. 

Sand         ^-2  =0.250==    25.0        "  " 

Stone       =  f  =  0.625  =    62.5       "  " 


(3)  Percentage  is  of  very  wide  application  in  money  matters,  payment  for 
service  in  such  matters  being  often  based  upon  the  amount  of  money  involved. 
Thus,  a  purchasing  or  selling  agent  may  be  paid  a  brokerage  or  commission 
which  forms  a  certain  percentage  of  the  money  value  of  the  goods  bought  or 
sold  ;  the  premium  paid  for  insurance  is  a  percentage  upon  the  value  of  the  goods 
insured;  etc. 

Interest. 

(4)  Interest  is  hire  or  rental  paid  for  the  loan  of  money.  The  sum  loaned  is 
called  the  principal,  and  the  number  of  cents  paid  annually  for  the  loan  of 
each  dollar,  or  of  dollars  per  hundred  dollars,  is  called  the  rate  of  interest. 
The  rate  is  always  stated  as  a  percentage. 

(5)  If  the  interest  is  paid  to  the  lender  as  it  accrues,  the  money  is  said  to  be 
at  simple  interest;  but  if  the  interest  is  periodically  added  to  the  princi- 
pal, so  that  it  also  earns  interest,  the  money  is  said  to  be  at  compound 
interest,  and  the  interest  is  said  to  be  compounded. 

Simple  Interest. 

(6)  At  the  end  of  a  year,  the  interest  on  the  principal,  P,  at  the  rate,  r,  is  =■ 
P  r,  and  the  amount.  A,  or  sum  of  principal  and  interest,  is 

A  =  P  +  P  r  =  P  (1  -i-  r). 

(7)  At  the  end  of  a  number,  w,  of  years,  the  interest  is  =  V  r  n  (see  right- 
hand  side  of  Fig.  1),  and 

A  =  P  +  P  r  71  =  P  (1  +  r  n). 
Thus,  let  P  =  $865.32,  r  =  3  per  cent.,  or  0.03,  n  =  l  year,  3  months  and  10 
days  =  1  year  and  100  days  =  ^^^-^  years  =  1.274  years.    Then  A  =  P  (1  -\-  rn) 
=  $865.32  X  (1  +  0.03  X  1.274)  =  $865.32  X  1.03822  =  $898.39. 

(8)  For  the  present  ^(vorth,  principal,  or  capitalization,  P,  of 
the  amount,  A,  we  have 

1  -\-  rn 

Thus,  for  the  sum,  P,  which,  in  1  year,  3  months,  10  days,  at  3  per  cent* 

89839 
simple  interest,  will  amount  to  $898.39,  we  have  P  = -^-^ — r-^r^  =  $865.32. 

(9)  In  commercial  business,  interest  is  commonly  calculated  approxi- 
mately by  taking  the  year  as  consisting  of  12  months  of  30  days  each.  Then, 
at  6  per  cent.,  the  interest  for  2  months,  or  60  days,  =  1  per  cent.;  1  inonth,  or  30 
days,  =  3^  per  cent.;  6  days  =  0.1  per  cent.  Thus,  required  the  interest  on 
$1264.35  for  5  months,  28  days,  at  5  per  cent. 

*K  fraction,  as  -i-,  ■^,  etc.,  or  its  decimal  equivalent,  as  0.125,  0.3125,  etc., 
is  compared  with  wm/y  or  one;  but  in  percentage  the  first  term  of  the  ratio  is 
compared  with  one  hundred  units  of  the  second  terra.  Mistakes  often  occur 
through  neglect  of  this  distinction.  Thus,  0.06  (six  per  cent,  or  six  per  hundred) 
is  sometimes  mis-read  six  one-hundredths  of  one  per  cent,  or  six  oue-hun- 
dredths  per  cent, 

fFor  proportions  by  volume,  see  pp  935  and  943. 


42 


ARITHMETIC. 


Principal .S1264.35 

Interest,  2  mos,   1  per  cent 12.64 

2mos,    1        "       12.64 

"         1  mo,     ^        "      6.32 

"       20  days,  ^        "       4.21 

"         6  days,  0.1     "      1.26 

"         2  days,  ^       "       0.42 

Interest  at  6  per  cent ^37.49 

Deduct  one-sixth 6.25 

Interest  at  5  per  cent ^31.24 

Equation  of  Payments. 

(10)  A  owes  B  $1200;  of  wliicli  $400  are  to  be  paid  in  3  months;  $500  in  4 
months ;  and  $300  in  6  months ;  all  bearing  interest  until  paid ;  but  it  has  been 
agreed  to  pay  all  at  once.    Now,  at  what  time  must  this  payment  be  made  so  that 
neither  party  shall  lose  any  interest? 
$       months. 
400    X    3    -    1200  .  ,.  5000       ,,^ 

500    X    4    =    2000  Average  time  =  —-  =  4%  months.    Ans. 

300    X    6    =    1800 
1200  5000 


Compound  Interest. 

(11)  Interest  is  usually  compounded  annually,  semi-annually,  or  quarterly. 
If  it  is  compounded  annually,  then  (see  left  side  of  Fig.  1) 
at  the  end  of  1  vear    A  =  P  (1  +  r) 
"  "     2  years  A  -  P  (1  -f  0  (1  +  r)  =  P  (1  +  r)2 

"  "     n  years  A  =  P  (1  +  r)^;  and 

A 


P  = 

(1  +  r'r 

P  =  (1  +  r)« 


=  A  (1  +  r)- 


(12)  If  the  interest  is  compounded  q  times  per  year,  we  have 

(13)  The  principal,  P,  is  sometimes  called  the  present  ivortli  or  present 

value  of  the  amount,  A.     Thus,  in  the  following  table,  $1.00  is  the  present 
worth  of  $2,191  due  in  20  years  at  4  per  cent,  compound  interest,  etc,  etc. 


^ 

^ 

"T 

I>(l  +  r)n               ^ 

^ 

?> 

.^ 

1 

Of. 

I 

^ 

^ 

'' 

••* 

'^ 

f^ 

1^^ 

'8 

1  ^ 

S 

A 

0 

t 

- 

? 

0 

i 

C) 

\ 

4    J    O    7 
Years 
Fig.  1. 


ARITHMETIC. 


43 


Table  3.    Compound  Interest. 

Amount  of  $1  at  Compound  Interest. 


3 

3K 

4 

43^ 

5 

5K 

6 

6>^ 

Years. 

per 

per 

per 

per 

per 

per 

per 

per 

ceut. 

cent. 

cent. 

cent 

cent. 

cent. 

cent. 

cent. 

1 

1.030 

1.035 

1.040 

1.045 

1.050 

1.055 

1.060 

1.065 

2 

1.061 

1.071 

1.082 

1.092 

1.103 

1.113 

1.124 

1.134 

3 

1.093 

1.109 

1.125 

1.141 

1.158 

1.174 

1.191 

1.208 

4 

1.126 

1.148 

1.170 

1.193 

1.216 

1.239 

1.262 

1.286 

5 

1.159 

1.188 

1.217 

1.246 

1.276 

1.307 

1.338 

1.370 

6 

1.194 

1.229 

1.265 

1.302 

1.340 

1.379 

1.419 

1.459 

7 

1.230 

1.272 

1.316 

1.361 

1.407 

1.455 

1.504 

1.554 

8 

1.267 

1.317 

1.369 

1.422 

1.477 

1.535 

1.594 

1.655 

9 

1.305 

1.363 

1.423 

1.486 

1.551 

1.619 

1.689 

1.763 

10 

1.344 

1.411 

1.480 

1.553 

1.629 

1.708 

1.791 

1.877 

11 

1.384 

1.460 

1.539 

1.623 

1.710 

1.802 

1.898 

1.999 

12 

1.426 

1.511 

1.601 

1.696 

1.796 

1.901 

2.012 

2.129 

13 

1.469 

1.564 

1.665 

1.772 

1.886 

2.006 

2.133 

2.267 

14 

1.513 

1.619 

1.732 

1.852 

1.980 

2.116 

2.261 

2.415 

15 

1.558 

1.675 

1.801 

1.935 

2.079 

2.232 

2.397 

2.572 

16 

1.605 

1.734 

1.873 

2.022 

2.183 

2.355 

2.540 

2.739 

17 

1.653 

1.795 

1.948 

2.113 

2.292 

2.485 

2.693 

2.917 

18 

1.702 

1.858 

2.026 

2.208 

2.407 

2.621 

2.854 

3.107 

19 

1.754 

1.923 

2.107 

2.308 

2.527 

2.766 

3.026 

8.309 

20 

1.806 

1.990 

2.191 

2.412 

2.653 

2.918 

3.207 

3.524 

21 

1.860 

2.059 

2.279 

2.520 

2.786 

3.078 

3.400 

3.753 

22 

1.916 

2.1.32 

2.370 

2.634 

2.925 

3.248 

3.604 

3.997 

23 

1.974 

2.206 

2.485 

2.752 

3.072 

3.426 

3.820 

4.256 

24 

2.033 

2.283 

2.563 

2.876 

3.225 

3.615 

4.049 

4.533 

25 

2.094 

2.363 

2.666 

3.005 

3.386 

3.813 

4.292 

4.828 

26 

2.157 

2.446 

2.772 

3.141 

3.556 

4.023 

4.549 

5.141 

27 

2.221 

2.532 

2.883 

3.282 

3.733 

4.244 

4.822 

5.476 

28 

2.288 

2.620 

2.999 

3.430 

3.920 

4.478 

5.112 

5.832 

29 

2.357 

2.712 

3.119 

■  3.584 

4.116 

4.724 

5.418 

6.211 

30 

2.427 

2.807 

3.243 

3.745 

4.322 

4.984 

5.743 

6.614 

31 

2.500 

2.905 

3.373 

3.914 

4.538 

5.258 

6.088 

7.044 

32 

2.575 

3.007 

3.508 

4.090 

4.765 

5.547 

6.453 

7.502 

33 

2.652 

3.112 

3.648 

4.274 

5.003 

5.852 

6.841 

7.990 

34 

2.732 

3.221 

3.794 

4.466 

5.253 

6.174 

7.251 

8.509 

35 

2.814 

3.334 

3.946 

4.667 

5.516 

6.514 

7.686 

9.062 

36 

2.898 

3.450 

4.104 

4.877 

5.792 

6.872 

8.147 

9.651 

37 

2.985 

3.571 

4.268 

5.097 

6.081 

7.250 

8.636 

10.279 

38 

3.075 

3.696 

4.439 

5.326 

6.385 

7.649 

9.154 

10.947 

39 

3.167 

3.825 

4.616 

5.566 

6.705 

8.069 

9.704 

11.658 

40 

3.262 

3.959 

4.801 

5.816 

7.040 

8.513 

10.286 

12.416 

Compound  interest  on  M  dollars,  at  any  rate  r  for  n  years  = 
interest  on  %\  at  same  rate,  r,  and  for  n  years. 


:  M  X  compound 


Annuity,  Sinking  Fund,  Amortization,  Depreciation. 

(14)  Under  "Interest"  we  deal  with  cases  where  a  certain  sum  or  "prin- 
cipal," P,  paid  once  for  all,  is  allowed  to  accumulate  either  simple  or  compound 
interest ;  but  in  many  cases  equal  periodical  payments  or  appropriations,  called 
annuities,  are  allowed  to  accumulate,  each  earning  its  own  interest,  usually 
compound. 


44 


ARITHMETIC. 


(15)  Thus,  a  sum  of  money  is  set  aside  annually  to  accumulate  compound 
interest  and  thus  form  a  sinking-  fund,  in  order  to  extinguish  a  debt.  In 
this  way,  the  cost  of  engineering  works  is  frequently  paid  virtually  in  instal- 
ments.   This  process  is  called  amortization. 

(16)  In  estimating  the  operating  expenses  of  engineering  works,  an  allowance 
is  made  for  depreciation.  In  calculating  this  allowance,  we  estimate  or 
assume  the  life-time,  n,  of  the  plant,  and  find  that  annuity,  p,  which,  at  an 
assumed  rate,  r,  of  compound  interest,  will,  in  the  time  n,  amount  to  the  cost  of 
the  plant,  and  thus  provide  a  fund  by  means  of  which  the  plant  may  be  replaced 
when  worn  out  or  superseded. 

(17)  The  present  wortb,  present  value,  or  capitalization,  W, 
Fig.  2,  of  an  annuity,/),  for  a  given  number,  7i,  of  years,  is  that  sum  which,  if 
now  placed  at  compound  interest  at  the  assumed  rate,  ?%  will,  at  the  end  of  that 
time,  reach  the  same  amount,  A,  as  will  be  reached  by  that  annuity. 


4    5    6    t  8    9  n 
Years 


Fig.  1. 


012S4JG789n 
Years 
Fig.  2. 


(18)  Equations  for  Compound  Interest  and  Annuities.    (See 
Figs.  1  and  2.) 

P  =  principal ;    r  =  rate  of  interest ;    n  =  number  of  years ; 
A  =  amount ;      p  =  annuity  ;  W  =  present  worth. 

The  interest  is  supposed  to  be  compounded,  and  the  annuities  to  be  set  aside, 
at  the  end  of  each  year. 

Compound  Interest. 

(1)  The  amount.  A,  of  $l,at  the  end  of  w  years,  see  (11),  isA  =  (l  +r)**. 

(2)  Since  the  present  worth  of  (1  +  ?•)'*»  <i"6  ^^  ^  years,  is  $^1,  see  (1),  it 
follows,  by  proportion,  that  tlie  present  wortb,  W,  of  ^1,  due  in  n  years, 

is  W  = ^ =  (1  +  r)-»*- 


(1  +  rr 


Annuities. 


(3)  In  n  years,  an  annuity  of  $r  will  amount  to  (1  +  r)"  —  1.*    Hence,  the 
amount.  A,  of  an  annuity  of  $1,  at  the  end  of  n  years,  is 


*In  the  case  of  compound  interest  on  SI,  the  rate,  r,  may  be  regarded  as  an 
annuity,  earning  its  interest;  and,  at  the  end  of  n  years,  the  amount  of  the 
several  annuities  (each  =  the  annual  interest,  r,  on  the  $1  principal)  with  the 
interests  earned  by  them,  is  =  the  amount,  (1  -f  r)'*,  of  $1  in  n  years  at  rate,  r, 
minus  the  ^1  principal  itself;  or,  amount  of  annuity  =  (l  -f  r)"  —  1. 


ARITHMETIC. 


45 


(4)  For  the  present  ivorth,  W,  of  an  annuity  of  $1  for  n  years, 
we  have,  from  Equations  (1)  and  (3) : 

1  — -        ^ 


(l+r)"-l. 


:  W.   Hence,  W  -- 


(l+r)"-l 
'  (1  +  r)  ^  r 


(1  +  r)  "  :  1  : 
See  Table  3 

(5)  The  annuity  for  n  years,  which  $1  will  purchase,  is 
1* 


a+r)' 


1  — - 


1 


(1-f  r)  " 
(6)  The  annuity  which,  in  n  years,  will  amount  to  $1,  is 

1                            r                                     ^t 
P'-P  —  r  =  y^  —  r  = . 


1 


(1 +0*^-1 


See  Table  4.  (!+»') 

Table  3.    Present  Value  of  Annuity  of  $1000.    See  Equation  (4). 
Rate  of  Interest  (Compound). 


2K 

3 

^K 

4 

^y^ 

5 

5K 

0 

Years. 

per 

per 

per 

per 

per 

per 

per 

per 

cent. 

cent. 

cent. 

cent. 

cent. 

cent. 

cent. 

cent. 

5 

4,646 

4,580 

4,515 

4,452 

4,390 

4,329 

4,268 

4,212 

10 

8,752 

8,530 

8,316 

8,111 

7,913 

7,722 

7,538 

7,360 

16 

12,381 

11,938 

11,517 

11,118 

10,740 

10,380 

10,037 

9,712 

20 

15,589 

14,877 

14,212 

13,590 

•13,008 

12,462 

11,950 

11,470 

25 

18,424 

17,413 

16,482 

15,622 

■  14,828 

14,094 

13,414 

12,783 

30 

20,930 

19,600 

18,392 

17,292 

16,289 

15,372 

14,534 

13,765 

35 

23,145 

21,487 

20,000 

18,664 

17,461 

16,374 

15,391 

14,498 

40 

25,103 

23,115 

21,355 

19,793 

18,401 

17,159 

16,045 

15,046 

45 

26,833 

24,519 

22,495 

20,720 

19,156 

17,774 

16,548 

15,456 

50 

28,362 

25,730 

23,456 

21,482 

19,762 

18,256 

16,932 

15,762 

100 

36,614 

31,599 

27,655 

24,505 

21,950 

19,848 

18,096 

16,618 

(19)  In  comparing  the  merits  of  proposed  systems  of  improvement,  it  is 
usual  to  add,  to  the  operating  expenses  and  to  the  cost  of  ordinary  repairs  and 
maintenance,  (1)  the  interest  on  the  cost,  (2)  an  allowance  for  depreciation,  and 
sometimes  (3)  an  annuity  to  form  a  sinking  fund  for  the  extinction  of  the  debt 
incurred  by  construction.  The  capitalization  of  the  total  annual  expense,  thus 
obtained,  is  then  regarded  as  the  true  first  cost  of  the  construction.  All  the 
elements  of  cost  are  thus  reduced  to  a  common  basis,  and  the  several  propositions 
become  properly  comparable. 

(20)  Thus,  in  estimating,  in  1899,t  the  cost  of  improving  the  water  supply  of 
Philadelphia,  the  rate,  r,  of  interest  was  assumed  at  3  per  cent,  and  depreciation 
was  assumed  as  below.  Under  "  Life"  is  given  the  assumed  life-time  of  each 
class  of  structure  or  apparatus,  and  under  "  Annuity  "  the  sum  which  must  be 
set  aside  annually  in  order  to  replace,  at  the  expiration  of  that  life,  $1,000  of  the 
corresponding  vakie. 


Present  worth      Annuity 
*  Because,    W  :      Sl.OO 

Equation  (4) 


Present  worth      Annuity 

$1.00  :        p.      Hence,  ^  = 

Equation  (5) 


W 


Annuity      Amount        Annuity  Amount 
fBecause,    r  :   (1    +   r)  "  —  1   \:  p'  :  $1.00.     Hence,  ^ 
Equation  (3)  Equation  (6) 


(1  +  r)  ^ 
J  Report  by  Rudolph  Hering,  Samuel  M.  Gray,  and  Joseph  M.  Wilson. 


46 


ARITHMETIC. 


Structures,  Apparatus,  etc.  Life, 

in  years 

Masonry  conduits,  filter  beds,  reservoirs Indefinite 

Permanent  buildings 100 

Cast  iron  pipe,  railroad  side-tracks 80 

Steel  pipe,  valves,  blovv-oflfs,  and  gates '65 

Engines  and  pumps 30 

Boilers,  electric  light  plants,  tramways  and  equipment, 

iron  fences 20 

Telephone  lines,  sand-washer,  and  regulating  apparatus....        10 


Annuity 

0.00 

1.65 

3.11 

16.54 

21.02 

37.22 

87.24 


(21)  Calculated  upon  this  basis,  two  projects,  each  designed  to  furnish  450 
million  gallons  per  day,  compared  as  follows : 

River  Water,  taken  within  City 
Limits  and  Filtered. 


Unfiltered  Water,  by  Aqueduct. 
Fi7'st  Cost. 

Storage  reservoirs 5=30,900.000 

Aqueducts  47,730,000 

Distribution  3,555,000 

Distributing  reservoir 1,000,000 

Total    §83,185,000 

Anmial. 

Interest  on  ^83,185,000 $2,495,550 

Depreciation 198,640 

Operation  and  Maintenance. 
Analyses  and  inspec- 
tion   $41,620 

Ordinary  repairs 49,150 

Pumping  and  wages  140,770 

231,540 


Filter  plants $23,174,680 

Mains  10,980,000 


Total $34,154,680 

Annual. 

Interest  on  $34,154,680 $1,024,640 

Depreciation 205,540 

Operation  and  3faintenance. 

Pumping $1,216,021 

Filtration 525,600 


1,741,621 

$2,925,730  $2,971,801 

It  will  be  noticed  that,  although  the  first  cost  of  the  filtration  project  was  much 
less  than  half  that  of  the  aqueduct  project,  its  large  proportion  of  perishable 
parts  made  its  charge  for  depreciation  somewhat  greater,  while  its  cost  for  oper- 
ation and  maintenance  was  more  than  seven  times  as  great,  and  its  total  annual 
charge  a  little  greater. 

Table  4.    Annuity  required  to  redeem  $1000.    See  Equation  (6). 


Rate  of  Interest  (Compound). 


1 

2 

2X 

3 

3M 

4 

5 

0 

Years. 

per 

per 

per 

per 

per 

per 

per 

per 

cent. 

cent. 

cent. 

cent. 

cent. 

cent. 

cent. 

cent. 

5 

196.04 

192.16 

190.24 

188.36 

186.49 

184.63 

180.98 

177.39 

10 

95.58 

91.33 

89.25 

87.23 

85.24 

83.29 

79.50 

75.87 

15 

62.12 

57.83 

55.77 

53.77 

51.82 

49.94 

46.34 

42.96 

20 

45.42 

41.15 

39.14 

37.22 

35.36 

33.58 

30.24 

27.18 

25 

35.41 

31.22 

29.27 

27.43 

25.67 

24.01 

20.95 

18.23 

30 

28.75 

24.65 

22.78 

21.02 

19.37 

17.83 

15.05 

12.65 

85 

24.00 

20.00 

18.20 

16.54 

15.00 

13.58 

11.07 

8.97 

40 

20.46 

16.55 

14.84 

13.26 

11.83 

10.52 

8.28 

6.46 

45 

17.71 

13.91 

12.27 

10.79 

9.45 

8.26 

6.26 

4.70 

60 

15.51 

11.82 

10.26 

8.87 

7.63 

6.55 

4.78 

3.44 

60 

12.24 

8.77 

7.35 

6.13 

5.09 

4.20 

2.83 

1.88 

70 

9.93 

667 

5.40 

4.34 

3.46 

2.74 

1.70 

1.03 

80 

8.22 

5.16 

4.03 

3.11 

2.38 

1.81 

1.03 

0.573 

90 

6.91 

4.05 

3.04 

2.26 

1.66 

1.21 

0.627 

0.318 

100 

5.87 

3.20 

2.31 

1.65 

1.16 

0.808 

0.383 

0.177 

ARITHMETIC.  47 

DlJODENAIi  OR  DUODENARY  STOTATIOX .  * 

(1)  111  the  Arabic  system  of  notation  10  is  taken  as  the  base,  but  in  duodenal 
notation  12,  or  "  a  dozen,"  is  the  base.  While  10  is  divisible  only  by  5,  and  (once 
only)  by  2, 12  is  divisible  twice  by  2,  and  once  by  3,  by  4,  and  by  6.  This  accounts 
for  the  popularity  of  the  dozen  as  a  basis  of  enumeration  ;  of  weights,  as  in  the 
Troy  pound  of  12  ounces ;  of  measures,  as  in  the  foot  of  12  inches  ;  the  year  of  12 
months,  and  the  half  day  of  12  hours ;  and  of  coinage,  as  in  the  British  shilling 
of  12  pence. 

(2)  The  duodenal  notation  uses  the  dozen  (12),  the  gross  (122  =  144),  and  the 
great  gross  (12^  =^  12  gross  =  1728),  as  the  decimal  system  uses  the  ten  (10),  the 
hundred  (10-  =  100),  and  the  thousand  (10^  =  10  hundred  =  1000).  Two  arbitrary 
single  characters,  such  as  T  and  E,  represent  ten  and  eleven  respectively ;  the 
symbol  10  represents  a  dozen  :  11  represents  thirteen,  and  so  on.  Thus,  the  num- 
erals of  the  two  systems  compare  as  follows ; 

Decimal  1  2  3  4  5  6  7  8  9  10  11  12  13  14  ...  20  21  22  23  24  25  36  48  60 
Duodenal  1  2  3  4  5  6  7  8  9  T  E  10  11  12  ...  18  19  IT  IE  20  21  30  40  50 
Decimal  72  84  96  99  100  108  109  110  111  112  113  117  118  119  120  121  122 
Duodenal  60  70  80  83  84  90  91  92  93  94  95  99  9T.  9E  TO  Tl  T2 
Decimal  129  130  131  132  133  138  140  141  142  143  144  288  1728  20736  etc. 
Duodenal   T9  TT  TE  EO  El   E6  E8  E9  ET  EE  100  200  1000  10000  etc. 

(3)  nnoclecimals.  Areas  of  rectangular  figures,  the  sides  of  which  are 
expressed  in  feet  and  inches,  are  still  sometimes  found  by  a  method  called 
'*  Duodecimals,"  in  which  the  products  are  in  square  feet,  in  twelfths  of  a  square 
foot  (each  equal  to  12  square  inches)  and  in  square  inches ;  but,  by  means  of  our 
table  of  '*  Inches,  reduced  to  decimals  of  a  foot,"  page  221,  the  sides  may  be  taken 
in  feet  and  decimals  of  a  foot,  and  the  multiplication  thus  more  conveniently 
performed,  after  which  the  decimal  fraction  of  a  foot  in  the  product  may,  if 
desired,  be  converted  into  square  inches  by  multiplying  by  144. 

*  See  Elements  of  Mechanics,  by  the  late  John  W.  Nystrom. 


RECIPROCALS   OF   NUMBERS. 


Table  of  Reciprocals  of  Niimbers.    See  p.  52. 


No. 

Reciprocal. 

1.000000000 

No. 

Reciprocal. 

No. 

Reciprocal. 

No. 

Reciprocal 

1 

56 

.017857143 

Ill 

.009009009 

166 

.006024096 

2 

i  0.500000000 

57 

.017M3860 

112 

.008928571 

167 

.005988024 

3 

1  .333333333 

58 

.017241379 

113 

.008849558 

168 

.005952381 

4 

:  .250000000 

59 

.016949153 

114 

.008771930 

169 

.005917160 

6 

.200000000 

60 

.016666667 

115 

.008695652 

170 

.005882353 

6 

.166666667 

61 

.016393443 

116 

.008620690 

171 

.005847953 

7 

.142857143 

62 

.016129032 

117 

.008547009 

172 

.005813953 

8 

.125000000 

63 

.015873016 

118 

.008474576 

173 

.005780347 

9 

.111111111 

64 

.015625000 

119 

.008403361 

174 

.005747126 

10 

.100000000 

65 

.015384615 

120 

.008333333 

175 

.005714286 

11 

'  .090909091 

66 

.015151515 

121 

.008264463 

176 

.005681818 

12 

.083333333 

67 

.014925373 

122 

.008196721 

177 

.005649718 

13 

.076923077 

68 

.014705882 

123 

.008130081 

178 

.005617978 

14 

.071428571 

69 

.014492754 

124 

.008064516 

179 

.005586592 

15 

.066666667 

70 

.014285714 

125 

.008000000 

180 

.005555556 

16 

.062500000 

71 

.014084507 

126 

.007936508 

181 

.005524862 

17 

.058823529 

72 

.013888889 

127 

.007874016 

182 

.005494505 

18 

.055555556 

73 

.013698630 

128 

.007812500 

183 

.005464481 

19 

.052631579 

74 

.013513514 

129 

.007751938 

184 

.005434783 

20 

.050000000 

75 

.013333333 

130 

.007692308 

185 

.005405405 

21 

.047619048 

76 

.013157895 

131 

.007633588 

186 

.005376344 

22 

.045454.545 

77 

.012987013 

132 

.007575758 

187 

.005347594 

23 

.043478261 

78 

.012820513 

133 

.007518797 

188 

.005319149 

24 

.041666667 

79 

.012658228 

134 

.007462687 

189 

.005291005 

25 

.040000000 

80 

.012500000 

135 

.007407407 

190 

.005263158 

26 

.038461538 

81 

.012345679 

136 

.007352941 

191 

.005235602 

27 

.037037037 

82 

.012195122 

137 

.007299270 

192 

.005208333 

28 

.035714286 

83 

.012048193 

138 

.007246377 

193 

.005181347 

29 

.034482759 

84 

.011904762 

139 

.007194245 

194 

.005154639 

30 

.033333333 

85 

.011764706 

140 

.007142857 

195 

.005128205 

31 

.032258065 

86 

.011627907 

141 

.007092199 

196 

.005102041 

32 

.031250000 

87 

.011494253 

142 

.007042254 

197 

.005076142 

33 

.030303030 

88 

.011363636 

143 

.006993007 

198 

.005050505 

84 

.029411765 

89 

.011235955 

144 

.006944444 

199 

.005025126 

35 

.028571429 

90 

.011111111 

145 

.006896552 

200 

.005000000 

36 

.027777778 

91 

.010989011 

146 

.006849315 

201 

.004975124 

37 

.027027027 

92 

.010869565 

147 

.006802721 

202 

.004950495 

38 

.026315789 

93 

.010752688 

148 

.006756757 

203 

.004926108 

39 

.025641026 

94 

.010638298 

149 

.006711409 

204 

.004901961 

40 

.025000000 

95 

.010526316 

150 

.006666667 

205 

.004878049 

41 

.024390244 

96 

.010416667 

151 

.006622517 

206 

.004854369 

42 

.023809524 

97 

.010309278 

152 

.006578947 

207 

.004830918 

43 

.023255814 

98 

.010204082 

153 

.006535948 

208 

,004807692 

44 

.022727273 

99  ; 

.010101010 

154 

.006493506 

209 

.004784689 

45 

.022222222 

100  ■ 

.010000000 

155 

.006451613 

210 

.004761905 

46 

.021739130 

101 

.009900990 

156 

.006410256 

211 

.004739336 

47 

.021276600 

102 

.009803922 

157 

.006369427 

212 

.004716981 

48 

.020833333 

103 

.009708738 

158 

.006329114 

213 

.004694836 

49 

.020408163 

104 

.009615385 

159 

.006289308 

214 

.004672897 

50 

.020000000 

105 

.009523810 

160 

.006250000 

215 

.004651163 

51 

.019607843 

106 

.009433962 

161 

.006211180 

216 

.004629630 

52 

.019230769 

107 

.009345794 

162 

.006172840 

217 

.004608295 

53 

.018867925 

108 

.009259259 

163 

.006134969 

218 

.004587156 

64 

.018518519 

109 

.009174312 

164 

.006097561 

219 

.004566210 

65 

.018181818 

110 

.009090909 

165 

.006060606 

220 

.004545455 

RECIPROCALS   OF   NUMBERS. 


49 


Table  of  Reciprocals  of  'SxtmherH.— {Continued.)   See  p.  52. 

No. 

Reciprocal. 

No. 

Reciprocal. 

No. 

1  Reciprocal. 

No. 

Reciprocal. 

221 

.004524887 

276 

.003623188 

331 

.003021148 

386 

.002590674 

222 

.004504.505 

277 

.003610108 

332 

.003012048 

387 

.002583979 

223 

1  .004484305 

278 

.003597122 

333 

.003003003 

388 

.002577320 

224 

.004464286 

279 

.003584229 

334 

.002994012 

389 

.002570694 

225 

.004444444 

280 

.003571429 

335 

.002985075 

890 

.002564103 

226 

.004424779 

281 

.003558719 

336 

.002976190 

391 

.002557M5 

227 

.004405286 

282 

.003546099 

337 

.002967359 

392 

.002551020 

228 

.004385965 

283 

.003533569 

338 

.002958580  • 

393 

.002544529 

229 

.0043(;6812 

2M 

.003521127 

339 

.002949853 

394 

.002538071 

230 

.004347826 

285 

.003508772 

340 

.002941176 

395 

.002531646 

231 

.004329004 

286 

.003496503 

341 

.002932551 

396 

.002525253 

232 

.004310345 

287 

.003484321 

342 

.002923977 

397 

.002518892 

233 

.004291845 

288 

.003472222 

343 

.002915452 

398 

.002512563 

234 

.00427a504 

289 

.003460208 

344 

.002906977 

399 

.002506266 

235 

.004-255319 

290 

.003448276 

345 

.002898551 

400 

.002500000 

236 

.004237288 

291 

.003436426 

346 

.002890173 

401 

.002493766 

237 

.004219409 

292 

.003424658 

347 

.002881844 

402 

.002487562 

238 

.004201681 

293 

.003412969 

348 

.002873563 

403 

.002481390 

239 

.004184100 

294 

.003401361 

349 

.002865330 

404 

.002475248 

240 

.004166667 

295 

.003389831 

350 

.002857143 

405 

.002469136 

241- 

.004149378 

296 

'  .003378378 

351 

.002849003 

406 

.002463054 

242 

.004132231 

297 

.003367003 

352 

.002840909 

407 

.002457002 

243 

.00411.5226 

298 

.003355705 

353 

.002832861 

408 

.002450980 

244 

.004098361 

299 

.003344482 

354 

.002824859 

409 

.002444988 

245 

.004081633 

300 

.003333333 

355 

.002816901 

410 

.002439024 

246 

.00406.^041 

301 

.003322259 

a56 

.002808989 

411 

.002433090 

247 

.004048583 

302 

.003811258 

357 

.002801120 

412 

.002427184 

248 

.004032258 

303 

.003300330 

358 

.002793296 

413 

.002421308 

249 

.004016064 

304 

.003289474 

359 

.002785515 

414 

.002415459 

250 

.004000000 

305 

.008278689 

360 

.002VVVVV8 

415 

.002409639 

251 

.003984064 

306 

.003267974 

361 

.002770083 

416 

.002403846 

252 

.003968254 

307 

.003257329 

362 

.002762431 

417 

.002398082 

253 

.0039.52569 

308 

.003246753 

363 

.002754821 

418 

.002392344 

254 

.003937008 

309 

.003236246 

364 

.002747253 

419 

.002386635 

255 

.003921569 

310 

.003225806 

365 

.002739726 

420 

.002380952 

256 

.0039062.50 

311 

.003215434 

366 

'  .002732240 

421 

.002375297 

257 

.003891051 

312 

.003205128 

367 

.002724796 

422 

.002369668 

258 

.003875969 

313 

.003194888 

368 

.002717391 

423 

.002364066 

259 

.003861004 

314 

.003184713 

369 

.002710027 

424 

.002358491 

260 

.003846154 

315 

.003174603 

370 

.002702703 

425 

.002352941 

261 

.003831418 

316 

.003164557 

371 

.002695418 

426 

.002347418 

262 

.003816794 

317 

.0031M574 

372 

.002688172 

427 

.002341920 

263 

.003802281 

318 

;003144654 

373 

.002680965 

428 

.002336449 

264 

.003787879 

319 

.00.3134796 

374 

.002673797 

429 

.002331002 

265 

.003773585 

320 

.003125000 

375 

.002666667 

430 

.002325581 

266 

.003759398 

821 

.00.311.5265 

376 

.002659574 

431 

.002320186 

267 

.003745318 

322 

.00310,5.590 

377 

.002652520 

432 

.002314815 

268 

.003731343 

323 

.00309.5975 

378 

.002645503 

433 

.002309469 

269 

.003717472 

324 

.003086420 

379 

.002638522 

434 

.002304147 

270 

.003703704 

325 

.003076923 

380 

.002631579 

435 

.002298851 

271 

.003690037 

326 

.003067485 

381 

.002624672 

436 

.002293578 

272 

.003676471 

327 

.003058104 

■382 

.002617801 

437 

.002288330 

273 

.003663004 

328 

-.003048780 

383 

.002610966 

438 

.002283105 

274 

.003649635 

329 

.003039514 

384 

.002604167 

439 

.002277904 

275 

.003636364 

330 

.003030303 

385 

.002597403 

440 

.002272727 

50 


RECIPROCALS   OF   NUMBERS. 


Table  of  Reciprocals  of  Numbers.— (Otm^mwe^^.)    Seep.52. 


No. 

Reciprocal. 

No. 

Reciprocal. 

No. 

Reciprocal. 

No. 

Reciprocal 

441 

.002267574 

496 

.002016129 

551 

.001814882 

606 

.00165016* 

442 

.002262443 

497 

.002012072 

552 

.001811594 

607 

.001647446 

443 

.002257336 

498 

.002008032 

553 

.001808318 

608 

.001644737 

444 

.002252252 

499 

.002004008 

554 

.001805054 

609 

.001642036 

445 

.002247191 

500 

.002000000 

555 

.001801802 

610 

.001639344 

446 

.002242152 

501 

.001996008 

556 

.001798561 

611 

.001636661 

447 

.002237136 

502 

.001992032 

557 

.001795332 

612 

.001633987 

448 

.002232143 

•  503 

.001988072 

558 

.001792115 

613 

.001631321 

449 

.002227171 

504 

.001984127 

559 

.001788909 

614 

.001628664 

450 

.002222222 

505 

.001980198 

560 

.00178.5714 

615 

.001626016 

451 

.002217295 

506 

.001976285 

561 

.001782531 

616 

.001623377 

452 

.002212389 

507 

.001972387 

562 

.001779359 

617 

.001620746 

453 

.002207506 

508 

.001968504 

563 

.001776199 

618 

.001618123 

454 

.002202643 

509 

.001964637 

564 

.001773050 

619 

.001615509 

455 

.002197802 

510 

.001960784 

565 

.001769912 

620 

.001612903 

456 

.002192982 

511 

.001956947 

566 

.001766784 

621 

.001610306 

457 

.002188184 

512 

.001953125 

567 

.001763668 

622 

.001607717 

458 

.002183406 

513 

.001949318 

568 

.001760563 

623 

.001605136 

459 

.002178649 

514 

.001945525 

569 

.001757469 

624 

.001602564 

460 

.002173913 

515 

.001941748 

570 

.001754386 

625 

.001600000 

461 

.002169197 

516 

.001937984 

571 

.001751313 

626 

.001597444 

462 

.002164502 

517 

.001934236 

572 

.001748252 

627 

.001594896 

463 

.002159827 

518 

.001930502 

573 

.001745201 

628 

.001592357 

464 

.002155172 

519 

.001926782 

574 

.001742160 

629 

.001589825 

465 

.002150538 

520 

.001923077 

575 

.001739130 

630 

.001587302 

466 

.002145923 

521 

.001919386 

576 

.001736111 

631 

.001584786 

467 

.002141328 

522 

.001915709 

■577 

.001733102 

632 

.001582278 

468 

.002136752 

523 

.001912046 

578 

.001730104 

633 

.001579779 

469 

.002132196 

524 

.001908397 

579 

.001727116 

634 

.001577287 

470 

.002127660 

525 

.001904762 

580 

'.001724138 

635 

.001574803 

471 

.002123142 

526 

.001901141 

581 

.001721170 

636 

.001572327 

472 

.002118644 

527 

.001897533 

582 

.001718213 

637 

.001569859 

473 

.002114165 

528 

.001893939 

583 

.001715266 

638 

.001567398 

474 

.002109705 

529 

.001890359 

584 

.001712329 

639 

.001564945 

475 

.002105263 

530 

.001886792 

585 

.001709402 

640 

.001562500 

476 

.002100840 

531 

.001883239 

586 

.00170648.5 

641 

.001560062 

477 

.002096436 

532 

.001879699 

587 

.001703578 

642 

.001557632 

478 

.002092050 

533 

.001876173 

588 

.001700680 

643 

.001.555210 

479 

.002087683 

534 

.001872659 

589 

.001697793 

644 

.001552795 

480 

.002083333 

535 

.001869159 

590 

.001694915 

645 

.001550388 

481 

.002079002 

536 

.001865672 

591 

.001692047 

646 

.001547938 

482 

.002074689 

537 

.001862197 

592 

.001689189 

647 

.00154,5595 

483 

.002070393 

538 

.001858736 

593 

.001686341 

648 

.001543210 

484 

.002066116 

539 

.001855288 

594 

.001683.502 

649 

.001540832 

485 

.002061856 

540 

.001851852 

595 

.001680672 

650 

.001538462 

486 

.002057613 

541 

.001848429 

596 

.0016778.52 

651 

.001536098 

487 

.002053388 

542 

.001845018 

597 

.001675042 

652 

.001533742 

488 

.002049180 

543 

.001841621 

598 

.001672241 

653 

.001531394 

489 

.002044990 

544 

.001838235 

599 

.001669449 

654 

.001529052 

490 

.002040816 

545 

.001834862 

600 

.001666667 

655 

.001526718 

491 

.002036660 

546 

.001831502 

601 

.001663894 

6.56 

.001524390 

492 

.002032520 

547 

.001828154 

602 

.001661130 

657 

.001522070 

493 

.002028398 

548 

.001824818 

603 

.001658375 

658 

.001519757 

494 

.002024291 

549 

.001821494 

604 

.001655629 

6.59 

.001517451 

495 

.002020202 

550 

.001818182 

605 

.001652893 

660 

.001515152 

RECIPROCALS   OF   NUMBERS. 


Table  of  Reciprocals  of  :Sviml»erH.—{Co7itinued.)    See  p.  52. 


No. 

Reciprocal. 

No. 

Reciprocal. 

No. 

Reciprocal. 

No. 

Reciprocal. 

661 

.001512859 

716 

.001396648 

771 

.001297017 

826 

.001210654 

662 

.001510574 

717 

.001394700 

772 

.001295337 

827 

.001209190 

663 

.001508296 

718 

.001392758 

773 

.001293661 

828 

.001207729 

664 

'  .001506024 

719 

.001390821 

774 

.001291990 

829 

.001206273 

665 

.001503759 

720 

.001388889 

775 

.001290323' 

830 

.001204819 

666 

.001501502 

721 

.001386963 

776 

.001288660 

831 

.001203369 

667 

.001499250 

722 

.001385042 

777 

.001287001 

832 

,  .001201923 

668 

.001497006 

723 

.001383126 

778 

.001285347 

833 

.001200480 

669 

.001494768 

724 

.001381215 

779 

.001283697 

834 

.001199041 

670 

.001492537 

725 

.001379310 

780 

.001282051 

835 

.001197605 

671 

.001490313 

726 

.001377410 

781 

.001280410 

836 

.001196172 

672 

.001488095 

727 

.001375516 

782 

.001278772 

837 

.001194743 

673 

.001485884 

728 

.001373626 

783 

.001277139 

838 

.001193317 

674 

.001483680 

729 

.001371742 

784 

.001275510 

839 

.001191895 

675 

.001481481 

730 

.001369863 

785 

.001273885 

840 

.001190476 

676 

.001479290 

731 

.001367989 

786 

.001272265 

841 

.001189061 

677 

.001477105 

732 

.001366120 

787 

.001270648 

842 

.001187648 

678 

.001474926 

733 

.001364256 

788 

.001269036 

843 

.001186240 

679 

.001472754 

734 

.001362398 

•789 

.001267427 

844 

.001184834 

680 

.001470588 

735 

.001360544 

790 

.001265823 

845 

001183432 

681 

.001468429 

736 

.001358696 

791 

.001264223 

846 

.001182033 

682 

.001466276 

737 

.001356852 

792 

.001262626 

847 

.001180638 

683 

-.001464129 

738 

.001355014 

793 

.001261034 

848 

.001179245 

684 

.001461988 

739 

.001353180 

794 

.001259446 

849 

.001177856 

685 

.001459854 

740 

.001351351 

795 

.001257862 

850 

.001176471 

686 

.001457726 

741 

.001349528 

796 

.001256281 

851 

.001175088 

687 

.001455604 

742 

.001347709 

797 

.001254705 

852 

.001173709 

688 

.001453488 

743 

.001345895 

798 

.001253133 

853 

.001172333 

689 

.001451379 

744 

.001344086 

799 

.001251564 

854 

.001170960 

690 

.001449275 

745 

.001342282 

800 

.001250000 

855 

.001169591 

691 

.001447178 

746 

.001340483 

801 

.001248439 

856 

.001168224 

692 

.001445087 

747 

.001338688 

802 

.001246883 

857 

.001166861 

693 

.001443001 

748 

.001336898 

803 

.001245330 

858 

.001165501 

694 

.001440922 

749 

.001335113 

804 

.001243781 

859 

.001164144 

695 

.001438849 

750 

.001333333 

805 

.001242236 

860 

.001162791 

696 

.00143^782 

751 

.001331558 

806 

.001240695 

861 

.001161440 

697 

.001434720 

752 

.001329787 

807 

.001239157 

862 

.001160093 

698 

.001432665 

753 

.001328021 

808 

.001237624 

863 

.001158749 

699 

.001430615 

754 

.001326260 

809 

.001236094 

864 

.001157407 

700 

.001428571 

755 

.001324503 

810 

.001234568 

865 

.001156069 

701 

.001426534 

756 

.001322751 

811 

.001233046 

866 

.001154734 

702 

.001424501 

757 

.001321004 

812 

.001231527 

867 

.001153403 

703 

.001422475 

758 

.001319261 

813 

.001230012 

868 

.001152074 

704 

.001420455 

759 

.001317523 

814 

.001228501 

869 

.0011o0748 

705 

.001418440 

760 

.001315789 

815 

.001226994 

870 

.001149425 

706 

.001416431 

761 

.001314060 

816 

.001225490 

871 

.001148106 

707 

.001414427 

762 

.001312336 

817 

.001223990 

872 

.001146789 

708 

.001412429 

763 

.001310616 

818 

.001222494 

873 

.001145475 

709 

.001410437 

764 

.001308901 

819 

.001221001 

874 

.001144165 

710 

.001408451 

765 

.001307190 

820 

.001219512 

875 

.001142857 

711 

.001406470 

766 

.001305483 

821 

.001218027 

876 

.001141553 

712 

.001404494 

767 

.001303781 

822 

.001216545 

877 

.001140251 

713 

.001402525 

768 

.001302083 

823 

.001215067 

878 

.001138952 

714 

.001400560 

769 

.001300390 

824 

.001213592 

879 

.001137656 

715 

.001398601 

770 

.001298701 

825 

.001212121 

880 

.001136364 

52 


RECIPROCALS  OF  NUMBERS. 


Table  of  Reciprocals  of  ^nmJ»ers.— (Continued.)    See  below. 

No. 

Reciprocal. 

No. 

Reciprocal. 

No. 

Reciprocal. 

No. 

Reciprocal. 

881 

.001135074 

911 

.001097695 

941 

.001062699 

971 

.001029866 

882 

.001133787 

912 

.001096491 

942 

.001061571 

972 

.001028807 

883 

.001132503 

913 

.001095290 

943 

.001060445 

973 

.001027749 

884 

.001131222 

914 

.001094092 

944 

.001059322 

974 

.001026694 

885 

.001129944 

915 

.001092896 

945 

.001058201 

975 

.001025641 

886 

.001128668 

916 

.001091703 

946 

.001057082 

976 

.001024590 

887 

.001127396 

917 

.001090513 

947 

.001055966 

977 

.001023541 

888 

.001126126 

918 

.001089325 

948 

.001054852 

978 

.001022495 

889 

.001124859 

919 

.001088139 

949 

.001053741 

979 

.001021450 

890 

.001123596 

920 

.001086957 

950 

.001052632 

980 

.001020408 

891 

.001122334 

921 

.001085776 

951 

.001051525 

981 

.001019368 

892 

.001121076 

922 

.001084599 

952 

.001050420 

982 

.001018330 

893 

.001119821 

923 

.001083424. 

953 

.001049318 

983 

.001017294 

894 

.001118568 

924 

.001082251 

954 

.001048218 

984 

.001016260 

895 

.001117318 

925 

.001081081 

.955 

.001047120 

985 

.001015228 

896 

.001116071 

926 

.001079914 

956 

.001046025 

986 

.001014199 

897 

.001114827 

927 

.001078749 

957 

.001044932 

987 

.001013171 

898 

.001113586 

928 

.001077586 

958 

.001043841 

988 

.001012146 

899 

.001112347 

929 

.001076426 

959 

.001042753 

989 

.001011122 

900 

.001111111 

930 

.001075269 

960 

.001041667 

990 

.001010101 

901 

.001109878 

931 

.001074114 

961 

.001040583 

991 

.001009082 

902 

.001108647  ■ 

932 

.001072961 

962 

.001039501 

992 

.001008065 

903 

.001107420 

933 

.001071811 

963 

.001038422 

993 

.001007049 

904 

.001106195 

934 

.001070664 

964 

.001037344 

994 

.001006036 

905 

.001104972 

935 

.001069519 

965 

.001036269 

995 

.001005025 

906 

.001103753 

936 

.001068376 

966 

.001035197 

996 

.001004016 

907 

.001102536 

937 

.001067236 

967 

.0010S4126 

997 

.001003009 

908 

.001101322 

938 

.001066098 

968 

.001033058 

998 

.001002004 

909 

.001100110 

'939 

.001064963 

969 

.001031992 

999 

.001001001 

910 

.001098901 

940 

.001063830 

970 

.001030928 

1000 

.001000000 

RECIPROCALS. 


(a)  The  reciprocal  of  a  nninber  is  the  quantity  obtained  by  divid- 
ing unity  or  1  by  that  number.    In  other  words,  if  n  be  any  number,  then 

Recip  n  =  —  .     Thus,  Recip  40  =  —  =  0.025 ;  Recip  0.4  =  — :  =  2.5,  etc.,  etc. 
n  40  0.4 

-T  ^.cf«&  (I  d  b        b 

Hence,  Recip  -.-  =  — ,  because  Recip  -r-  =  l-i--r-  =  lX  —  =  — • 

b        a  ^  b  b  a        a 

Thus,  since  1  yard  =  36  inches,  1  inch  =  -^-^  yard  =  MlllTil^  yard,  for  Recip 
36  =  .027777778.  Again,  1  foot  head  of  water  gives  a  pressure  of  .4335  lbs.  per 
square  inch.    Hence  a  pressure  of  1  lb.  per  square  inch  corresponds  to  a  head 

of  —^  feet  =  2.306805  feet,  for  Recip  .4335  =  2.306805.    (See  li,  below.) 

(b)  It  follows  that  if  any  number  in  the  column  headed  "  No."  be  taken  as 
the  denominator  of  a  common  fraction  whose  numerator  is  1,  the  corresponding 
reciprocal  is  the  value  of  that  fraction  expressed  in  decimals.*  Thus,  J^  =  .03125. 
Hence,  to  reduce  a  cominoii  fraction  to  decimal  form,  multiply 
the  reciprocal  of  the  denominator  by  the  numerator.  Thus,  i|^  =  .53125,  because 
Recip  32  =  .03125,  and  .03125  X  17  =-  .53125. 

(c;  Conversely,  if  the  reciprocal  of  a  number  n  be  taken  as  a  number,  then  the 

number  n  itself  becomes  the  reciprocal.    In  other  words,  Recip  —  =  n.     Thus, 
1  1  '         ^  w 

=  40 ;  Recip  2.5  =  Recip  -—  =  0.4,  etc.,  etc. 


Recip  0.025  =  Recip  —  = 


0.4 


*  The  numbers  2  and  5,  and  their  powers  and  products,  are  the  only  ones  whose 
reciprocals  can  be  exactly  expressed  in  decimals. 


RECIPROCALS  OF  NUMBERS.  53 

(d)  The  product  of  any  number  by  its  own  reciprocal  is  equal  to  unity  or  Ij 
.r,«X-'^  =  -"-  =  !. 

(e)  Any  number,  a  X  Recip  of  a  number,  n  =  «  X  —  =  —  • 

Hence,  to  avoid  the  labor  of  dividing,  we  may  multiply  by  the  recip' 
roccd  of  the  divisor.    Thus, 
200  -h  48750  ^  200  X  Recip  48750  =-  200  X.00002051282  (see  li,  below) =.004102561 

(f )  Any  number,  a  -^  Recip  of  a  number,  n  —  a^  —  =  an. 

Hence,  a  -^  Recip  a  =  a^  —  =  aXr=<*^- 

Thus,  Recip  2  =  0.5,  and  ^   ^.    ^  =-^^  =  4  =  22. 

^  '  Recip  2      0.5 

(g-)  The  numbers  in  the  foregoing  table  extend  from  1  to  1000 ;  but  the  recip- 
rocals of  multiples  of  these  nniiibers  by  10  may  be  taken  from  the 
table  by  adding  one  cipher  to  the  left  of  the  reciprocal  (after  the  decimal  point) 
for  each  cipher  added  to  the  number.    Thus, 

Recip      39§=  .002564103; 
Recip    3900  =  .0002564103; 
Recip  39000  =  .00002564103; 
and  the  reciprocals  of  numbers  containing'  decimals  may  be  taken 
from  the  table  by  shifting  the  decimal  point  in  the  tabular  reciprocal  one  place 
to  the  right  for  each  decimal  place  in  the  number.    Thus: 
Recip  227  =     .004405286 ; 

Recip     22.7       =     .04405286; 
Recip      2.27     =      .4405286; 
Recip        .227    =    4.405286; 
Recip        .0227  =  44.05286. 
(h)  The  reciprocal  of  a  number  of  more  than  three  figures  may  be 
taken  from  the  table  approximately  by  interpolation.    Thus,  to  find  Recip  236.4: 
Recip     236  =  .004237288 
Recip     237  -=  JD04219409 
Differences:        1,     .000017879,    236.4  —  236  =  0.4. 
Then,  0.4  X  .000017879  =  .000007152, 
and  Recip  236    =.004237288 

minus  .000007152 

=  Recip  236.4  =  .004230136   by  interpolation. 
The  correct  reciprocal  is  .004230118. 
(i)  The  reciprocals  of  numbers  not  in  the  table  may  be  conveniently  found 

by  means  of  logarithms.     Thus,  to  find  the  Recip  236.4  =  -xttw-t  • 

Job. 4 
Log      1     =  0.000000 
Subtract  Log  236.4  =_2^73647 

"3.626353  =  Log  0.00123012 

Recip  236.4  =  0.00423012. 
^    ^    ,  ^     .      8424         236.4 
To  find  Recip -^3^  =  -g^, 

Log    236.4  =  2.373647 
Subtract  Log  8424     =  3.925518 

"2.448129  =  Log  0.0280627. 
8424 
Recip  -^ggj  =  0.0280627. 

(j)  Position  of  the  decimal  point.  For  the  Nos.  10,  100, 1000,  etc., 
the  number  of  the  decimal  place  occupied  by  the  first  significant  figure  in  the 
reciprocal  is  equal  to  the  number  of  ciphers  in  the  No. ;  but  for  all  other  Nos.  it 
is  equal  to  the  number  of  the  digits  iu  the  integral  portion  of  the  No.  Thus: 
Recip  143.7  =  .0069..,  etc.  Here  the  number  of  digits  in  the  integral  portion 
(143)  of  the  No.  is  3,  and  the  first  significant  figure  (6)  of  the  reciprocal  occupies 
the  third  decimal  place. 


54 


SQUARE  AND  CUBE  ROOTS. 


Square  I 

loots  and  Cube  Roots  of  IViimbers  from  .1  to  28. 

No 

error*. 

No. 

Square. 

Cube. 

Sq.  Rt. 

cut. 

No. 

Sq.  Rt. 

C.Rt.     N 

o. 

Sq.  Rt. 

C.Rt. 

.1 

.01 

.001 

.316 

.464 

.7 

2.387 

1.786 

3.661 

2.375 

.15 

.0225 

.0034 

.387 

.531 

.8 

2.408 

1.797 

•      3.688 

2.387 

.2 

.04 

.008 

.447 

.585 

.9 

2.429 

1.807 

3.715 

2.399 

.25 

.0625 

.0156 

.500 

.630 

6. 

2.449 

1.817      14 

3.742 

2.410 

.3 

.09 

.027 

.548 

.669 

.1 

2.470 

1.827 

3.768 

2.422 

.35 

.1225 

.04» 

.592 

.705 

.2 

2.490 

1.837 

3.795 

2.433 

.4 

.16 

.064 

.633 

.737 

.3 

2.510 

1.847 

3.821 

2.444 

.45 

.2025 

.0911 

.671 

.766 

.4 

2.530 

1.857 

3.847 

2.455 

.5 

.25 

.125 

.707 

.794 

.5 

2.550 

1.866      15 

3.873 

2.466 

.55 

.3025 

.1664 

.742 

.819 

.6 

2.569 

1.876 

3.899 

2.477 

.6 

.36 

.216 

.775 

.843 

.7 

2.588 

1.885 

3.924 

2.488 

.65 

.4225 

.2746 

.806 

.866 

.8 

2.608 

1.895 

3.950 

2.499 

.7 

.49 

.343 

837 

.888 

.9 

2.627 

1.904 

3.975 

2.509 

.75 

.5625 

.4219 

.866 

.909 

7 

2.646 

1.913      16 

4. 

2.520 

.8 

.64 

.512 

.894 

.928 

.1 

2.665 

1.922 

4.025 

2.530 

.85 

.7225 

.6141 

.922 

.947 

.2 

2.683 

1.931 

4.050 

2.541 

,9 

.81 

.729 

.949 

.965 

.3 

2.702 

1.940 

4.074 

2.551 

.95 

.9025 

.8574 

.975 

.983 

.4 

2.720 

1.949 

4.099 

2.561 

1. 

1.000 

1.000 

1.000 

1.000 

.5 

2.739 

1.957      17 

4.123 

2.571 

.05 

1.103 

1.158 

1.025 

1.016 

.6 

2.757 

1.966 

4.147 

2.581 

1.1 

1.210 

1.331 

1.049 

1.032 

7 

2.775 

1.975 

4.171 

2.591 

.15 

1.323 

1.521 

1.072 

1.048 

.'s 

2.793 

1.983 

4.195 

2.601 

1.2 

1.440 

1.728 

1.095 

1.063 

.9 

2.811 

1.992 

4.219 

2.611 

.25 

1.563 

1.963 

1.118 

1.077 

8. 

2.828 

2.000      18 

4.243 

2.621 

1.3 

1.690 

2.197 

1.140 

1.091 

.1 

2.846 

2.008 

4.266 

2.630 

.35 

1.82.'? 

2.460 

1.162 

1.105 

.2 

2.864 

2.017 

4.290 

2.640 

1.4 

1.960 

2.744 

1.183 

1.119 

.3 

2.881 

2.025 

4.313 

2.650 

.45 

2.103 

3.049 

1.204 

1.132 

.4 

2.898 

2.0.33 

4.336 

2.659 

1.5 

2.250 

3.375 

1.225 

1.145 

.5 

2.916 

2.041       19 

4.359 

2.668 

.55 

2.403 

3.724, 

1.245 

1.157 

.6 

2.93» 

2.049 

4.382 

2.678 

1.6 

2.560 

4.096 

1.265 

1.170 

.7 

2.950 

2.057 

4.405 

2.687 

.65 

2.723 

4.492 

1.285 

1.182 

.8 

2.966 

2.065 

4.427 

2.696 

17 

2.890 

4.913 

1.304 

1.193 

.9 

2.983 

2.072 

4.450 

2.705 

.75 

3.063 

5.359 

1.323 

1.205 

9. 

3. 

2.080       20 

4.472 

2.714 

1.8 

3.240 

5.832 

1.342 

1.216 

.1 

3.017 

2.088 

4.494 

2.723 

.85 

3.423 

6.332 

1.360 

1.228 

.2 

3.033 

2.095 

4.517 

2.732 

1.9 

3.610 

6.859 

1.378 

1.239 

.3 

3.050 

2.103 

4.539 

2.741 

.95 

3.803 

7.415 

1.396 

1.249 

.4 

3.066 

2.110 

4.561 

2.750 

fL 

4.000 

8.000 

1.414 

1.260 

.5 

3.082 

2.118       21 

4.583 

2.759 

.1 

4.410 

9.261 

1.449 

1.281 

.6 

3.098 

2.125 

4.604 

2.768 

.2 

4.840 

10.65 

1.4a3 

1.301 

.7 

3.114 

2.133 

4.626 

2.776 

.3 

5.290 

12.17 

1.517 

1.320 

.8 

3.130 

2.140 

4.648 

2.785 

.4 

5.760 

13.82 

1.549 

1.339 

.9 

3.146. 

2.147 

4.669 

2.794 

.5 

6.250 

15.63 

1.581 

1.357 

10. 

3.162 

2.154       22 

4.690 

2.802 

.6 

6.760 

17.58 

1.612 

1.375 

3.178 

2.162 

4.712 

2.810 

.7 

7.290 

19.68 

1.643 

1.392 

*.2 

3.194 

2.169 

4.733 

2.819 

.8 

7.840 

21.95 

1.673 

1.409 

.3 

3.209 

2.176 

4.754 

2.827 

.9 

8.410 

24.39 

1.703 

1.426 

.4 

3.2^5 

2.183 

4.775 

2.836 

8. 

9. 

27. 

1.732 

1.442 

.5 

3.240 

2.190      23 

4.796 

2.844 

.1 

9.61 

29.79 

1.761 

1.458 

.6 

3.256 

2.197 

4.817 

2.852 

.2 

10.24 

32.77 

1.789 

1.474 

.7 

3.271 

2.204 

4.837 

2.860 

.3 

10.89 

.35.94 

1.817 

1.489 

.8 

3.286 

2.210 

4.858 

2.868 

.4 

11.56 

39.30 

1.844 

1.504 

.9 

3.302 

2.217 

4.879 

2.876 

.5 

12.25 

42.88 

1.871 

1.518 

11. 

3.317 

2.224       24 

4.899 

2.884 

.6 

12.96 

46.66 

1.897 

1.533 

3.332 

2.231 

4.919 

2.892 

.7 

13.69 

50.65 

1.924 

1.547 

'.2 

3.347 

2.2.37 

4.940 

2.900 

.8 

14.44 

54.87 

1.949 

1.560 

.3 

3.362 

2.244 

4.960 

2.908 

.9 

15.21 

59.32 

1.975 

1.574 

.4 

3.376 

2.251 

4.980 

2.916 

4. 

16. 

64. 

2. 

1.587 

.5 

3.391 

2.257      25 

5. 

2.924 

.1 

16.81 

68.92 

2.025 

1.601 

.6 

3.406 

2.264 

5.020 

2.932 

.2 

17.64 

74.09 

2.049 

1.613 

.7 

3.421 

2.270 

5.040 

2.940 

.3 

18.49 

79.51 

2.074 

1.626 

.8 

3.435 

2.277 

5.060 

2.947 

.4 

19.36 

85.18 

2.098 

1.639 

.9 

3.450 

2.283 

5.079 

2.955 

.5 

20.25 

91.13 

2.121 

1.651 

12. 

3.464 

2  289       26 

5.099 

2.962 

.6 

21.16' 

97.34 

2.145 

1.663 

.1 

3.479 

2.296 

5.119 

2.970 

.7 

22.09 

103.8 

2.168 

1.675 

.2 

3.493 

2.302 

5.138 

2.978 

.8 

23.04 

110.6 

2.191 

1.687 

.3 

3.507 

2.308 

5.158 

2.985 

.9 

24.01 

117.6 

2,214 

1.698 

.4 

3.521 

2.315 

5.177 

2.993 

5. 

25. 

125. 

2.236 

1.710 

.5 

3.536 

2.321       27 

5.196 

3.000 

.1 

26.01 

132.7 

2.258 

1.721 

.6 

3.550 

2.327 

5.215 

3.007 

.2 

27.04 

140.6 

2.280 

1.732 

.7 

3.564 

2.333 

5.235 

3.015 

.3 

28.09 

148.9 

2.302 

1.744 

.8 

3.578 

2.339 

6 

5.254 

3.022 

,4 

29.16 

157.5 

2.324 

1.754 

.9 

3.592 

2.345 

8 

5.273 

3.029 

.5 

30.25 

166.4 

2.345 

1.765 

13. 

3.606 

2..351      28 

5.292 

3.037 

-6 

31.36 

175-6 

2.366 

1.776 

.2 

3.633 

2.3b3 

2 

5.310 

3.04i 

SQUAEES,   CUBES,    AND   BOOTS. 


65 


TABIjE  of  Squares,  Cubes,  Sqnare  Roots,  and  Cube  Roots, 
of  lumbers  from  1  to  1000. 

Remark  on  the  following  Table.     Wherever  the  effect  of  a  fifth  decimal  in  the  roots  would  be  ti 
»dd  1  to  the  fourth  and  final  decimal  in  the  table,  the  addition  has  been  made.  No  errors. 


No. 

Square. 

Cube. 

Sq.  Rt. 

cut. 

No. 

Square. 

Cube. 

Sq.  Kt. 

C.  Et. 

1 

1 

1 

1.0000 

1.0000 

61 

3721 

226981 

7.8102 

3.9365 

2 

4 

8 

1.4142 

1.2599 

62 

3844 

238328 

7.8740 

3.9579 

3 

9 

27 

1.7321 

1.4422 

63 

3969 

250047 

7.9373 

3.9791 

4 

16 

64 

2.0000 

1.5874 

84 

4096 

262144 

8,0000 

4. 

5 

25 

125 

2.2361 

1.7100 

65 

4225 

274625 

8.0623 

4.0207 

6 

36 

216 

2.4495 

1.8171 

66 

4356 

287496 

8.1240 

4.0412 

7 

49 

343 

2.6458 

1.9129 

67 

4489 

300763 

8.1854 

4.0615 

8 

64 

512 

2.8284 

2.0000 

68 

4624 

314432 

8.2462 

4.0817 

9 

81 

729 

3.0000 

2.0801 

69 

4761 

328509 

8.3066 

4.1016 

10 

100 

1000 

3.1623 

2.1544 

70 

4900 

343000 

8.3666 

4.1213 

11 

121 

1331 

3.3166 

2.2240 

71 

5041 

357911 

8.4261 

4.1408 

12 

144 

1728 

3.4641 

2.2894 

72 

5184 

373248 

8.4853 

4.1602 

13 

109 

2197 

3.6056 

2.3513 

73 

5328 

389017 

8.5440 

4.1793 

14 

196 

2744 

3.7417 

2.4101 

74 

5476 

405224 

8.6023 

4.1988 

15 

225 

3375 

3.8730 

2.4662 

75 

5625 

421875 

8.6603 

4.2173 

16 

•256 

4096 

4.0000 

2.5198 

76 

5776 

438976 

8.7178 

4.2358 

17 

289 

4913 

4.1231 

2.5713 

77 

5929 

456533 

8.7750 

4.254« 

18 

324 

5832 

4.2426 

2.6207 

78 

6084 

474552 

8.8318 

4.2727 

19 

361 

6859 

4.3589 

2.6684 

79 

6241 

493039 

8.8882 

4.2908 

20 

400 

8000 

4.4721 

2.7144 

80 

6400 

612000 

8.9443 

4.3089 

21 

441 

9261 

4.5826 

2.7589 

81 

6561 

531441 

9. 

4.3267 

22 

484 

10648 

4.6904 

2.8020 

82 

6724 

551368 

9.0554 

4.3445 

23 

529 

12167 

4.7958 

2.8439 

83 

6889 

571V87 

9.1104 

4.3621 

24 

576 

13824 

4.8990 

2.8845 

84 

7056 

592704 

9.1652 

4.3795 

25 

625 

15625 

5.0000 

2.9240 

85 

7225 

614125 

9.2195 

4.3968 

26 

676 

17576 

5.0990 

2.9625 

86 

7396 

636056 

9.2736 

4.4140 

27 

729 

19683 

5.1962 

3.0000 

87 

7569 

658503 

9.3274 

4.4810 

28 

784 

21952 

5.2915 

3.0366 

.  88 

7744 

681472 

9.3808 

4.4480 

29 

841 

24389 

5.3852 

3.0723 

89 

7921 

704969 

9.4340 

4.4647 

30 

900 

27000 

5.4772 

3.1072 

90 

8100 

729000 

9.4868 

4.4814 

31 

961 

29791 

5.5678 

3.1414 

91 

8281 

753571 

9.5394 

4.4079 

32 

1024 

32768 

5.6569 

3.1748 

92 

8464 

778688 

9.5917 

4.5144 

^3 

1089 

35937 

5.7446 

3.2075 

93 

8649 

804357 

9.6437 

4.5307 

h 

1156 

39304 

5.8310 

3.2396 

94 

8836 

830584 

9.6954 

4.5468 

35 

1225 

42875 

5.9161 

3.2711 

95 

9025 

857375 

9.7468 

4.5629 

S6 

1296 

46656 

6.0000 

3.3019 

96 

9216 

884736 

9.7980 

4.5789 

37 

1369 

50653 

6.0828 

3.3322 

97 

9409 

912673 

9.8489 

4.5947 

38 

1444 

54872 

6.1644 

3.3620 

98 

9604 

941192 

9.8995 

4.6104 

39 

1521 

59319 

6.2450 

3:,VdV2 

99 

9801 

970299 

9.9499 

4.6261 

40 

1600 

64000 

6.3246 

3.4200 

100 

10000 

1000000 

10. 

4.6416 

41 

1681 

68921 

6.4031 

3.4482 

101 

10201 

1030301 

10.0499 

4.6570 

42 

1764 

74088 

6.4807 

3.4760 

102 

10404 

1061208 

10.0995 

4.6723 

43 

1849 

79507 

6.5574 

3.5034 

103 

10609 

1092727 

10.1489 

4.6875 

44 

1936 

85184 

6.6332 

3.5303 

104 

10816 

1124864 

10.1980 

4.7027 

45 

2025 

91126 

6.7082 

3.5569 

105 

11025 

1157625 

10.2470 

4.7177 

46 

2116 

97336 

6.7823 

3.58.30 

106 

11236 

1191016 

10.2956 

4.7326 

47 

2209 

103823 

6.8557 

3.6088 

107 

11449 

1225043 

10.3441 

4.7475 

48 

2304 

110592 

6.9282 

3.6342 

108 

11664 

12.59712 

10.3923 

4.7622 

49 

2401 

117649 

7.0000 

3.6593 

109 

11881 

1295029 

10.4403 

4.7769 

50 

2500 

125000 

7.0711 

3.6840 

110 

12100 

1331000 

10.4881 

4.7914 

51 

2601 

132651 

7.1414 

3.7084 

111 

12321 

1367631 

10.5357 

4.8059 

52 

2704 

140608 

7.2111 

3.7325 

112 

12544 

1404928 

10.5830 

4.8203 

53 

2809 

148877 

7.2801 

3.7563 

113 

12769 

1442897 

10.6301 

4.8346 

54 

2916 

157464 

7.3485 

3.7798 

114 

12996 

1481544 

10.6771 

4.8488 

55 

3025 

166375 

7.4162 

3.8030 

115 

13225 

1520875 

10.7238 

4.8629 

56 

3136 

175616 

7.4833 

3.8259 

116 

13456 

1560896 

10.7703 

4.8770 

57 

3249 

185193 

7.5498 

3.8485 

117 

13689 

1601613 

10.8167 

4.8910 

58 

3364 

195112 

7.6158 

3.8709 

118 

13924 

1643032. 

10.8628 

4.9049 

59 

3481 

205379 

7.6811 

3.8930 

119 

14161 

1685159 

10.9087 

4.9187 

60 

3600 

216000 

7.7460 

3.9149 

120 

14400 

1728000 

10.9545 

4.933* 

56 


SQUARES,  CUBES,  AND   ROOTS. 


TABLE  of  ISqnares,  €abes,  Square  Roots,  and  Cube  Roots 
of  lumbers  from  1  to  1000  —  (Continued.) 


No. 

Square. 

Cube. 

Sq.  Kt. 

C.Rt. 

No. 

Square. 

Cube. 

Sq.  Rt. 

C.B^- 

121 

14641 

1771561 

11. 

4.9461 

186 

34596 

6434856 

13.6382 

6.7083 

122 

14884 

1815848 

11.0454 

4.9597 

187 

34969 

6539203 

13.6748 

5.7185 

123 

15129 

1860867 

11.0905 

4.9732 

188 

35344 

6644672 

13.7113 

6.7287 

124 

15376 

1906624 

11.1355 

4.9866 

189 

35721 

675.1269 

13.7477 

5.7388 

125 

15625 

1958125 

11.1803 

5. 

190 

36100 

6859000 

13.7840 

5.748» 

126 

15876 

2000376 

11.2250 

5.0133 

191 

36481 

6967871 

13.8203 

6.7590 

127 

16129 

2048383 

11.2694 

5.0265 

192 

36864 

7077888 

13.8564 

5.7690 

128 

16384 

2097152 

11.3137 

5.0397 

193 

37249 

7189057 

13.8924 

6.7790 

129 

16641 

2146689 

11.3578 

5.0528 

194 

37636 

7301S84 

13.9284 

6.7890 

130 

16900 

2197000 

11.4018 

5.0658 

195 

38025 

7414875 

13.9642 

6.798» 

131 

17161 

2248091 

11.4455 

5.0788 

196 

38416 

7529536 

14. 

6.8088 

132 

17424 

2299968 

11.4891 

5.0916 

19T 

38809 

7645373 

14.0357 

6.8186 

133 

17689 

2352637 

11.5326 

5.1045 

198 

39204 

7762392 

14.0712 

6.8285 

134 

17958 

2406104 

11.5758 

6.1172 

199 

39601 

7880599 

14.1067 

6.838? 

13& 

18226 

2460375 

11.6190 

6.1299 

200 

40000 

8000000 

14.1421 

6.848C 

136 

18496 

2515456 

11.6619 

6.1426 

201 

40401 

8120601 

14.1774 

5.8578 

137 

18769 

2571353 

11.7047 

6.1551 

202 

40804 

8242408 

14.2127 

6.8675 

138 

19044 

2628072 

11.7473 

5.1676 

203 

41209 

8365427 

14.2478 

6.8771 

139 

19321 

2685619 

11.7898 

5.1801 

204 

41616 

8489664 

14.2829 

6.8868 

liO 

19600 

2744000 

11.8322 

6.1925 

205 

42025 

8615125 

14.3178 

6.8964 

141 

19881 

2803221 

11.8743 

5.2048 

206 

42436 

8741816 

14.3527 

5.905» 

142 

20164 

2863288 

11.9164 

6.2171 

207 

42849 

8869743 

14.3876 

5.9155 

143 

20449 

2924207 

11.9583 

6.2293 

208 

43264 

8998912 

14.4222 

6.9250 

144 

20736 

2985984 

12. 

5.2415 

209 

43681 

9129329 

14.4568 

6.9345 

145 

21025 

8048625 

12.0416 

6.2536 

210 

44100 

9261000 

14.4914 

6.9439 

146 

21316 

3112136 

12.0830 

6.2656 

211 

44521 

9393931 

14.5258 

5.953$ 

147 

21609 

3176523 

12.1244 

5.2776 

212 

44944 

9528128 

14.5602 

5.9627 

148 

21904 

3241792 

12.1655 

5.2896 

213 

45369 

9663597 

14.5945 

5.9721 

149 

22201 

3307949 

12.2066 

5.3015 

214 

45796 

9800344 

14.6287 

5.9814 

150 

22500 

3375000 

12.2474 

5.3133 

215 

46225 

9938375 

14.6629 

5.990T 

151 

22801 

3442951 

12.2882 

5.3251 

216 

46656 

10077696 

14.6969 

6. 

152 

23104 

3511803 

12.3288 

5.3368 

217 

47089 

10218313 

14.7309 

6,0092 

153 

23409 

3581577 

12.3693 

6.3485 

218 

47524 

10360232 

14.7648 

6.0185 

154 

23716 

3652264 

12.4097 

6.3601 

219 

47961 

10503459 

14.7986 

6.0277 

155 

24025 

3723875 

12.4499 

6.3717 

220 

48400 

10648000 

14.8324 

6.0368 

156 

24336 

3796416 

12.4900 

6.3832 

221 

48841 

10793861 

14.8661 

6.0459 

157 

24649 

3869893 

12.5300 

6.3947 

222 

49284 

10941048 

14.8997 

6.0550 

158 

24964 

3944312 

12.5698 

5.4061 

223 

49729 

11089567 

14.9332 

6.0641 

159 

25281 

4019679 

12.6095 

6.4175 

224 

50176 

11239424 

14.9666 

6.0732 

160 

25600 

4096000 

12.6491 

6.4288 

225 

50625 

11390625 

15. 

6.0822 

161 

25921 

4173281 

12.6886 

5.4401 

226 

61076 

11543176 

15.0333 

6.0912 

162 

26244 

4251528 

12.7279 

6.4514 

227 

51529 

11697083 

15.0665 

6.1002 

163 

26569 

4330747 

12.7671 

5.4626 

228 

51984 

11852352 

15.0997 

6.1091 

164 

26896 

4410944 

12.8062 

5.4737 

229 

62441 

12008989 

15.1327 

6.1180 

165 

27225 

4492125 

12.8452 

5.4848 

230 

52900 

12167000 

15.1658 

6.1269 

166 

27556 

4574296 

12.8841 

5.4959 

231 

63361 

12326391 

15.1987 

6.1358 

167 

27889 

4657463 

12.9228 

5.5069 

232 

53824 

12487168 

15.2315 

6.1446 

168 

28224 

4741632 

12.9615 

5.5178 

233 

54289 

12649337 

15.2643 

6.1534 

169 

28561 

4826809 

13. 

5.5288 

234 

54756 

12812904 

15.2971 

6.1622 

170 

28900 

4913000 

13.0384 

5.5397 

235 

55225 

12977875 

15.3297 

6.1710 

ITl 

29241 

5000211 

13.0767 

5.5505 

236 

55696 

13144256 

15.3623 

6.1797 

172 

29584 

5088448 

13.1149 

5.5613 

237 

56169 

13312053 

15.3948 

6.1885 

173 

29929 

5177717 

13.1529 

5.5721 

238 

56644 

13481272 

15.4272 

6.1972 

174 

30276 

5268024 

,13.1909 

5.5828 

239 

57121 

>3651919 

15.4596 

6.2058 

175 

30625 

5359375 

13.2288 

5.5934 

240 

67600 

13824000 

15.4919 

6.214& 

176 

30976 

5451776 

13.2665 

5.6041 

241 

;,8081 

13997521 " 

15.5242 

6.2231 

177 

31329 

5545233 

13.3041 

5.6147 

242 

58564 

14172488 

15.5563 

6.2317 

178 

31684 

5639752 

13.3417 

5.6252 

243 

59049 

14348907 

15.5885 

6.2403 

179 

32041 

5735339 

13.3791 

5.6357 

244 

59536 

14526784 

15.6205 

6.2488 

180 

32400 

5832000 

13.4164 

5.6462 

245 

60025 

14706125 

15.6525 

6.257a 

181 

32761 

5929741 

13.4536 

5.6567 

246 

60516 

14886936 

15.6844 

6.2658 

182 

33124 

6028568 

13.4907 

5.6671 

247 

61009 

15069223 

15.7162 

«.2743 

183 

33489 

6128487 

13.5277 

5.6774 

248 

61504 

15252992 

15.7480 

6.2828 

184 

33856 

6229504 

13.5647 

5.6877 

249 

62001 

15438249 

15.7797 

6.2912 

185 

34225 

6331625 

13.6015 

6.6980 

250 

62500 

15625000 

15.8114 

6.29^6 

SQUARES,  CUBES,  AND  BOOTS. 


67 


TABIi£  of  Squares,  Cubes,  Square  Roots,  and  Cube  Roots, 
of  Numbers  from  1  to  1000  —  (Continued.) 


No. 

Square. 

Cube. 

Sq.  Rt. 

C.  Rt. 

No. 

Square. 

Cube. 

Sq.  Rt. 

C.  Rt. 

251 

63001 

15813251 

15.8430 

6.3080 

316 

99856 

31554496 

17.7764 

6.8113 

252 

63504 

16003008 

15.8745 

6.3164 

317 

100489 

31855013 

17.8045 

6.8185 

25a 

64009 

16194277 

15.9060 

6.3247 

318 

101124 

32157432 

17.8326 

6.8256 

254 

64516 

16387064 

15.9374 

6.3330 

319 

101761 

324617.59 

17.8606 

6.8328 

255 

65025 

16581375 

15.9687 

6.3413 

320 

102400 

32768000 

17.8885 

6.839& 

256 

65536 

16777216 

16. 

6.3496 

321 

103041 

33076161 

17.9165 

6.847a 

257 

66049 

16974593 

16.0312 

6.3579 

322 

103684 

33386248 

17.9444 

6.8541 

258 

66564 

17173512 

16.0624 

6.3661 

323 

104329 

33698267 

17.9722 

6.8612: 

259 

67081 

17373979 

16.0935 

6.3743 

324 

104976 

34012224 

18. 

6.868S 

260 

67600 

17576000 

16.1245 

6.3825 

325 

105625 

34328125 

18.0278 

6.875a 

261 

68121 

17779581 

16.1555 

6.3907 

326 

106276 

34645976 

18.0555 

6.8824 

262 

68644 

17984728 

16.1864 

6.3988 

327 

106929 

34965783 

18.0831 

6.8894 

263 

69169 

18191447 

16.2173 

6.4070 

328 

107584 

35287552 

18.1108 

6.8964 

264 

69696 

18399744 

16.2481 

6.4151 

329 

108241 

35611289 

18.1384 

6.9034 

265 

70225 

18609625 

16.2788 

6.4232 

330 

108900 

35937000 

18.1659 

6.9104 

266 

70756 

18821096 

16.3095 

6.4312 

331 

109561 

36264691 

18.19.34 

6.9174 

267 

71289 

i:j034163 

16.3401 

6.4393 

332 

110224 

36594368 

18.2209 

6.9244 

268 

71824 

19248832 

16.3707 

6.4473 

333 

110889 

36926037 

18.2483 

6.9313 

269 

72361 

i;)4o5109 

16.4012 

6.4553 

334 

111556 

37259704 

18.2757 

6.9382 

270 

72900 

19683000 

16.4317 

6.4633 

335 

112225 

37595375 

18.3030 

6.9451 

271 

73441 

19902511 

16.4621 

6.4713 

336* 

11-2896 

37933056 

18.3303 

6.9521 

272 

73JW4 

20123648 

16.4924 

6.4792 

337 

113569 

38272753 

18.3576 

6.958» 

273 

74529 

20346417 

16.5227 

6.4872 

338 

114244 

38614472 

18.3848 

6.9658 

274 

75076 

20570824 

16.5529 

6.4951 

339 

114921 

38958219 

18.41':0 

6.972T 

275 

75625 

20796875 

16.5831 

6.5030 

340 

115600 

39304000 

18.4391 

6.979& 

276 

76176 

21024576 

16.6132 

6.5108 

341 

116281 

.39651821 

18.4662 

6.9864 

277 

76729 

21253933 

16.6433 

6.5187 

342 

116964 

40001688 

18.4932 

6.»93S 

278 

77284 

21484952 

16.6733 

6.5265 

343 

117649 

40353607 

18.5203 

7. 

279 

77841 

217176:^9 

16.7033 

6.5343 

344 

118336 

40707584 

18.5472 

7.0068 

280 

78400 

21952000 

16.7332 

6.5421 

345 

119025 

41063625 

18.5742 

7.0136 

281 

78961 

22188041 

16.7631 

6.5499 

346 

119716 

41421736 

18.6011 

7.020$ 

282 

79524 

22425768 

16.7929 

6.5577 

347 

120409 

41781923 

1S.6279 

7.0271 

283 

80089 

22665187 

16.8226 

6.5654 

348 

121104 

42144192 

18.6548 

7.0338 

284 

80656 

22906304 

16.8523 

6.5731 

349 

121801 

42508549 

18.6815 

7.0406 

285 

81225 

23149125 

16.8819 

fi.5808 

350 

122500 

42875000 

18.7083 

7.047a 

286 

81796 

23393656 

16.9115 

6.5885 

351 

123201 

43243551 

18.7350 

7.0540 

287 

82369 

23639903 

16.9411 

6.5962 

352 

123904 

43614208 

18.7617 

7.0607 

288 

829U 

23887872 

16.9706 

6.6039 

353 

124609 

43986977 

18.7883 

7.0674 

289 

83521 

24137569 

17. 

6.6115 

354 

125316 

44361864 

18.8149 

7.0740 

290 

84100 

24389000 

17.0294 

6.6191 

355 

126025 

44738875 

18.8414 

7.0807 

291 

84681 

24642171 

17.0587 

6.6267 

356 

126736 

45118016 

18.8680 

7.087a 

292 

85264 

24897088 

17.0880 

6.6343 

357 

127449 

45499293 

18.8944 

7.0940 

293 

85849 

25153757 

17.1172 

6.6419 

358 

128164 

45882712 

18.9209 

7.1006 

294 

86436 

25412184 

17.1464 

6.3494 

359 

128881 

46268279 

18.9473 

7.1072 

295 

87025 

25672375 

17.1756 

6.6569 

360 

129600 

18.9737 

7.1138 

2% 

87616 

25934336 

17.2047 

6.6644 

.361 

130321 

47045881 

19. 

7.1204 

297 

88209 

26198073 

17.2337 

6.6719 

362 

131044 

47437928 

19.0263 

7.126^ 

298 

88804 

26463592 

17.2627 

6.6794 

363 

131769 

47832147 

19.0526 

7.1335 

299 

89401 

26730899 

17.2916 

6.6869 

364 

132496 

48228544 

'19.0788 

7.1400 

300 

90000 

27000000 

17.3205 

6.6943 

365 

133225 

48627125 

19.1050 

7.1466 

801 

90601 

27270901 

17.3494 

6.7018 

366 

133956 

49027896 

19.1311 

7.1531 

302 

91204 

27543608 

17.3781 

6.7092 

367 

134689 

49430863 

19.1572 

7.1596 

303 

91809 

27818127 

17.4069 

6.7166 

368 

135424 

49836032 

19.1833 

7.1661 

804 

92416 

28094464 

17.4356 

6.7240 

369 

136161 

5024.3409 

19.2094 

7.1726 

805 

93025 

28372625 

17.4642 

6.7313 

370 

136900 

50653000 

19.2354 

7.1791 

306 

93636 

28652616 

17.4929 

6.7387 

371 

137641 

51064811 

19.2614 

7.1855 

307 

94249 

28934443 

175214 

6.7460 

372 

138384 

51478848 

19.2873 

7.1920 

308 

94864 

29218112 

17.5499 

6.7533 

373 

139129 

51895117 

19.31.32 

7.1984 

309 

95481 

29503629 

17.5784 

6.7606 

374 

139876 

52313624 

19.3.391 

7.2048 

310 

96100 

29791000 

17.6068 

6.7679 

375 

140625 

52734375 

19.3649 

7.2112 

311 

96721 

30080231 

17.6.352 

6.7752 

376 

141.376 

53157376 

19.3907 

7.2177 

312 

97344 

30371328 

17.6635 

6.7824 

377 

142129 

53582633 

19.4165 

7.2240 

313 

97969 

30664297 

17.6918 

6.7897 

378 

142S84 

54010152 

19.4422 

7.2304 

314 

98596 

30959144 

17.7200 

6.7969 

379 

143641 

54439939 

19.4679 

7.2368 

315 

99225 

31255875 

17.7482 

6.fc041 

380 

144400 

54872000 

19.4936 

7.2432 

58 


SQUARES,  CUBES,  AND  ROOTS. 


TABIi£  of  Squares,  Cubes,  Square  Roots,  and  Cube  Roots, 
of  JSTumbers  from  1  to  1000  — (Continued.) 


No. 

Square. 

Cube. 

.  Sq.  Rt. 

C.  Rt. 

No. 

Square. 

Cube. 

Sq.  Rt. 

C.Rt. 

381 

145161 

55306341 

19.5192 

7.2495 

446 

198916 

88716536 

21.1187 

7.6403 

382 

145924 

55742968 

19.5448 

7.2558 

447 

199809 

89314623 

21.1424 

7.6460 

383 

146689 

56181887 

19.5704 

7.2622 

448 

200704 

89915392 

21.1660 

7.6517 

384 

147456 

56623104 

19.5959 

7.2685 

449 

201601 

90518849 

21.1896 

7.6574 

385 

148225 

57066625 

19.6214 

7.2748 

450 

202500 

91125000 

21.2132 

7.6631 

386 

148996 

57512456 

19.6469 

7.2811 

451 

203401 

91733851 

21.2368 

7.6688 

387 

149769 

57960603 

19.6723 

7.2874 

452 

204304 

92345408 

21.2603 

7.6744 

388 

150544 

58411072 

19.6977 

7.2936 

453 

205209 

92959677 

21.2838 

7.6801 

389 

151321 

58863869 

19.7231 

7.2999 

454 

206116 

93576664 

21.3073 

7.6857 

390 

152100 

59319000 

19.7484 

7.3061 

455 

207025 

94196375 

21.3307 

7.6914 

391 

152881 

59776471 

19.7737 

7.3124 

456 

207936 

94818816 

21.3542 

7.6970 

392 

153664 

60236288 

19.7990 

7.3186 

457 

208849 

95443993 

21.3776 

7.7026 

393 

154448 

60698457 

19.8242 

7.3248 

458 

209764 

96071912 

21.4009 

7.7082 

394 

155236 

61162984 

19.8494 

7.3310 

459 

210681 

96702579 

21.4243 

7.7138 

395 

156025 

61629875 

19.8746 

7.3372 

460 

211600 

97336000 

21.4476 

7.7194 

396 

156816 

62099136 

19.8997 

7.3434 

461 

212521 

97972181 

21.4709 

7.7250 

397 

157609 

62570773 

19.9249 

7.3496 

462 

213444 

98611128 

21.4042 

7.7306 

398 

158404 

63044792 

19.9499 

7.3558 

463 

214369 

99252847 

21.5174 

7.7362 

399 

159201 

63521199 

19.9750 

7.3619 

464 

215296 

99897344 

21.5407 

7.7418 

400 

160000 

64000000 

20. 

7.3681 

465 

216225 

100544625 

21.5639 

7.7473 

401 

160801 

64481201 

20.0250 

7.3742 

466 

217156 

101194696 

21.5870 

7.7529 

402 

161604 

64964808 

20.0499 

7.3803 

467 

218089 

101847563 

21.6102 

7.7584 

403 

162409 

65450827 

20.0749 

7.3864 

468 

219024 

102503232 

21.6333 

7.7639 

404 

163216 

65939264 

20.0998 

7.3925 

469 

219961 

103161709 

21.6564 

7.7695 

405 

164025 

66430125 

20.1246 

7.3986 

470 

220900 

103823000 

21.67^5 

7.7750 

406 

164836 

66923416 

20.1494 

7.4047 

471 

221841 

104487111 

21.7025 

7.7805 

407 

165649 

67419143 

20.1742 

7.4108 

472 

222784 

105154048 

21.7256 

7.7869 

408 

166464 

67917312 

20.1990 

7.4169 

473 

223729 

105823817 

21.7486 

7.7915 

409 

167281 

68417929 

20.2237 

7.4229 

474 

224676 

106496424 

21.7715 

7.7970 

410 

168100 

68921000 

20.2485 

7.4290 

475 

225625 

107171875 

21.7945 

7.8025 

411 

168921 

69426531 

20.2731 

7.4350 

476 

226576 

107850176 

21.8174 

7.8079 

412 

169744 

69934528 

20.2978 

7.4410 

477 

227529 

108531333 

21.8403 

7.8134 

413 

170569 

70444997 

20.3224 

7.4470 

478 

228484 

109215352 

21.8632 

7.8188 

414 

171396 

70957944 

20.3470 

7.4530 

479 

229441 

109902239 

21.8861 

7.8243 

415 

172225 

71473375 

20.3715 

7.4590 

480 

230400 

110592000 

21.9089 

7.829T 

416 

173056 

71991296 

20.3961 

7.4650 

481 

231361 

111284641 

21.9317 

7.8362 

417 

173889 

72511713 

20.4206 

7.4710 

482 

232324 

111980168 

21.9545 

7.840« 

418 

174724 

73034632 

20.4450 

7.4770 

483 

233289 

112678587 

21.9773 

7.8460 

419 

175561 

73560059 

20.4695 

7.4829 

484 

234256 

113379904 

22. 

7.8514 

420 

176400 

74088000 

20.4939 

7.4889 

485 

235225 

114084125 

22.0227 

7.8568 

421 

177241 

74618461 

20.5ia3 

7.4948 

486 

236196 

114791256 

22.0454 

7.8622 

422 

178084 

75151448 

20.5426 

7.5007 

487 

237169 

115501303 

22.0681 

7.8676 

423 

178929 

75686967 

20.5670 

7.5067 

488 

238144 

116214272 

22.0907 

7.8730 

424 

179776 

76225024 

20.5913 

7.5126 

489 

239121 

116930169 

22.1133 

7.8784 

425 

180625 

76765625 

20.6155 

7.5185 

490 

240100 

117649000 

22.1359 

7.8837 

426 

181476 

77308776 

20.6398 

7.5244 

491 

241081 

118370771 

22.1585 

7.8891 

427 

182329 

77854483 

20.6640 

7.5302 

492 

242064 

119095488 

22.1811 

7.8944 

428 

183184 

78402752 

20.6882 

7.5361 

493 

243049 

119823157 

22.2036 

7.8998 

429 

184041 

78953589 

20.7123 

7.5420 

494 

244036 

120553784 

22.2261 

7.9051 

430 

184900 

79507000 

20.7364 

7.5478 

495 

245025 

121287376 

22.2486 

7.9105 

431 

185761 

80062991 

20.7605 

7.5537 

496 

246016 

122023935 

22.2711 

7.9158 

432 

186824 

80621568 

20.7846 

7.5595 

497 

247009 

122763473 

22.2935 

7.9211 

433 

187489 

'  81182737 

20.8087 

7.5654 

498 

248004 

123505992 

22.3159 

7.9264 

434 

188356 

81746504 

20.8327 

7.5712 

499 

249001 

124251499 

22.3383 

7.9317 

435 

189225 

82312875 

20.8567 

7.5770 

500 

250000 

125000000 

22.3607 

7.9370 

436 

190096 

82881856 

20.8806 

7.5828 

501 

251001 

125751501 

22.3830 

7.9423 

437 

190969 

83453453 

20.9045 

7.5S86 

502 

252004 

126506008 

22.4054 

7.9476 

438 

191844 

84027672 

20.9284 

7.5944 

503 

253009 

127263527 

22.4277 

7.9528 

439 

192721 

84604519 

20.9523 

7.6001 

504 

254016 

128024064 

22.4499 

7.9581 

440 

193600 

85184000 

20.9762 

7.6059 

505 

255025 

128787625 

22.4722 

7.9634 

441 

194481 

85766121 

21. 

7.6117 

506 

256036 

129554216 

22.4944 

7.9686 

442 

195364 

86350888 

21.0238 

7.6174 

507 

257049 

130323843 

22.5167 

7.9739 

443 

196249 

86938307 

21.0476 

7.6232 

508 

258064 

131096512 

22.5389 

7.9791 

444 

197136 

87528384 

21.0713 

7.6289 

509 

259081 

131872229 

22.5610 

7.9843 

445 

198025 

88121125 

21.0950 

7.6346 

510 

260100 

132651000 

22.5832 

7.9896 

59 


TABIi£  of  Sqnares,  Cubes,  ISquare  Roots,  and  Cube  Roots, 
ofKi       "  "  ' ^ 


SQUARES,  CUBES,  AND  ROOTS. 

nares.  Cubes,  Square  Roots,  am 
Bi^umbers  from  1  to  1000  —  (Continued.) 


No. 

Square. 

Cube. 

Sq.  Rt. 

C.  Rt. 

No. 

Square. 

Cube. 

Sq.  Rt. 

C.  Bt. 

511 

261121 

133432831 

22.6053 

7.9948 

576 

331776 

191102976 

24. 

8.3203 

512 

262144 

134217728 

22.6274 

8. 

577 

332929 

192100033 

24.0208 

8.3251 

613 

263169 

135005697 

22.'6495 

8.0052 

578 

334084 

193100552 

24.0416 

8.3300 

614 

264196 

135796744 

22.6716 

8.0104 

579 

335241 

194104539 

24.0624 

8.3348 

515 

265225 

136590875 

22.6936 

8.0156 

580 

336400 

195112000 

24.0832 

«.3396 

616 

266256 

137388096 

22.7156 

8.0208 

581 

337561 

196122941 

24.1039 

8.3443 

617 

267289 

138188413 

22.7376 

8.0260 

582 

338724 

197137368 

24.1247 

8.3491 

518 

268324 

138991832 

22.7596 

8.0311 

583 

339889 

198155287 

24.1454 

8.3539 

519 

269361 

139798359 

22.7816 

8.0363 

584 

341056 

199176704 

24.1661 

8.3587 

620 

270400 

140608000 

22.8035 

8.0415 

585 

342225 

200201625 

24.1868 

8.3634 

521 

271441 

141420761 

22.8254 

8.0466 

586 

343396 

2012.30056 

24.2074 

8.3682 

§22 

272484 

142236648 

22.8473 

8.0517 

587 

344569 

202262003 

24.2281 

B.3730 

523 

273529 

143055G67 

22.8692 

8.0569 

588 

345744 

203297472 

24.2487 

8.877T 

524 

274576 

143877824 

22.8910 

8.0620 

589 

346921 

204336469 

24.2693 

8.382& 

525 

275625 

144703125 

22.9129 

8.0671 

590 

348100 

205379000 

24.2899 

8.3872 

526 

276676 

145531576 

22.9347 

8.0723 

591 

349281 

206425071 

24.3105 

8.391& 

627 

277729 

146363183 

22.9565 

8.0774 

592 

350464 

207474688 

24.3311 

8.3967 

528 

278784 

147197952 

22.9783 

8.0825 

593 

851649 

208527857 

24.3516 

8.4014 

529 

279841 

148035889 

23. 

8.0876 

594 

352836 

209584584 

24.3721 

8.4061 

S30 

280900 

148877000 

23.0217 

8.0927 

595 

354025 

210644875 

24.3926 

8.4108 

631 

281961 

149721291 

23.0434 

8.0978 

596 

355216 

211708736 

24.4131 

8.4155 

632 

283024 

150568768 

23.0651 

8.1028 

597 

356409 

212776173 

24.4336 

8.4202 

533 

284089 

151419437 

23.0868 

8.1079 

598 

357604 

213847192 

24.4540 

8.4249 

634 

285156 

152273304 

23.1084 

8.1130 

599 

358801 

214921799 

24.4745 

8.4296 

636 

286226 

153130375 

23.1301 

8.1180 

600 

360000 

216000000 

24.4949 

8.4343 

536 

287296 

153990656 

23.1517 

8.1231 

601 

361201 

217061801 

24.5153 

8.4390 

637 

288369 

154854153 

23.1733 

8.1281 

602 

362404 

218167208 

24.5357 

8.443? 

638 

289444 

155720872 

2:j.l948 

8.1332 

603 

363609 

219256227 

24.5561 

8.448i 

539 

290521 

156590819 

23.2164 

8.1382 

604 

364816 

220348864 

24.5764 

8.453(1 

640 

291600 

157464000 

23.2379 

8.1433 

605 

366025 

221446125 

24.5967 

8.4677 

641 

292681 

158340421 

23.2594 

8.1483 

606 

367236 

222545016 

24.6171 

8.4623 

542 

293764 

159220088 

23.2809 

8.1533 

607 

368449 

223648543 

24.6374 

8.4670 

643 

294849 

160103007 

23.3024 

8.1583 

608 

369664 

224755712 

24.6577 

8.471« 

644 

295936 

160989184 

23.3238 

8.1633 

609 

370881 

225866529 

24.6779 

8.47G$ 

646 

297025 

161878625 

23.3452 

8.1683 

610 

372100 

226981000 

24.6982 

8.4809 

546 

298116 

162771336 

23.3666 

8.1733 

611 

373321 

228099131 

24.7184 

8.4856 

547 

299209 

163667323 

23.3880 

8.1783 

612 

374544 

229220928 

24.7386 

8.4002 

548 

300304 

164566592 

23.4094 

8.1833 

613 

375769 

230346397 

24.7588 

8.4948 

549 

301401 

165469149 

23.4307 

8.1882 

614 

376996 

231475544 

:a<.7790 

8.4994 

650 

302500 

166375000 

23.4521 

8.1932 

615 

378225 

232608375 

24.:992 

8.6040 

661 

303601 

167284151 

23.4734 

8.1982 

616 

379456 

233744896 

24.8193 

8.6066 

652 

304704 

168196608 

23.4947 

8.2031 

617 

380689 

234885113 

24.8395 

8.5132 

553 

305809 

169112377 

23.5160 

8.2081 

618 

381924 

236029032 

24.8596 

8.5178 

554 

306916 

170031464 

23.5372 

8.2130 

619 

.383161 

237176659 

24.8797 

8.5224 

555 

308025 

170953875 

23.5584 

8.2180 

620 

384400 

238328000 

24.8998 

8.5270 

656 

309136 

171879616 

23.5797 

8.2229 

621 

385641 

239483061 

24.9199 

8.5316 

557 

310249 

172808693 

23.6008 

8.2278 

622 

386884 

240641848 

24.9399 

8.5362 

558 

311364 

173741112 

23.6220 

8.2327 

623 

388129 

241804367 

24.9600 

8.5408 

559 

312481 

174676879 

23.6432 

8.2377 

624 

389376 

242970624 

24.9800 

8.5453 

560 

313600 

175616000 

23.6643 

8.2426 

625 

390625 

244140625 

25. 

8.6499 

561 

314721 

176558481 

23.6854 

8.2475 

626 

391876 

245314376 

25.0200 

8.6544 

562 

315844 

177504328 

23.7065 

8.2524 

627 

393129 

246491883 

25.0400 

8.5590 

563 

316969 

178453547 

23.7276 

8.2573 

628 

394384 

247673152 

25.0599 

8.5635 

564 

318096 

179406144 

23.7487 

8  2621 

629 

395641 

248858189 

25.0799 

8.5681 

565 

319225 

180362125 

23.7697 

8.2670 

630 

396900 

250047000 

25.0998 

8.5726 

566 

320356 

181321496 

23.7908 

8.2719 

an 

398161 

251239591 

25.1197 

8.5772 

667 

321489 

182284263 

23.8118 

8.2768 

632 

399424 

252435968 

25.1396 

8.5817 

668 

322624 

1832504:^2 

23.8328 

8.2816 

633 

400689 

253636137 

25.1595 

8.5862 

669 

323761 

184220009 

23.8537 

8.2865 

634 

401956 

2548*0104 

25.1794 

8.5907 

670 

324900 

185193000 

23.8747 

8.2913 

635 

403225 

256047875 

25.1992 

8.5952 

571 

326041 

186169411 

23.8956 

8.2962 

636 

404496 

2572594.56 

25.2190 

8.5997 

572 

327184 

187149248 

23.9165 

8.3010 

637 

405769 

258474853 

25.2389 

8.6043 

573 

328329 

18813251? 

23.9374 

8.3059 

638 

407044 

259694072 

25.2587 

8.6088 

674 

329476 

189119224 

23.9583 

8.3107 

639 

408321 

260917119 

25.2784 

8.6132 

J75 

330625 

190109375 

23.9792 

8.3155 

640 

409600 

262144000 

25.2982 

8.6171 

60 


SQUARES,  CUBES,  AND   ROOTS. 


TABIi£  of  Squares,  Cnbes,  Square  Roots,  and  Cube  Root»» 
of  JK^umbers  from  1  to  1000  —  (Continued.) 


No. 

Square. 

Cube. 

Sq.  Bt. 

C.  Rt. 

No. 

Square. 

Cube. 

Sq.  Rt. 

C.  Rt. 

641 

410881 

263374721 

25.3180 

8.6222 

706 

498436 

351895816 

26.5707 

8.9043 

642 

41216*4 

264609288 

25.3377 

8.6267 

707 

499849 

353393243 

26.5895 

8.908& 

643 

413449 

265847707 

25.3574 

8.6312 

708 

501264 

354894912 

26.6083 

8.912T 

644 

414736 

267089984 

25.3772 

8.6357 

709 

502681 

356400829 

26.6271 

8.916* 

645 

416025 

268336125 

25.3969 

8.6401 

710 

504100 

357911000 

26.6458 

8.9211 

646 

417316 

269586136 

25.4165 

8.6446 

711 

505521 

359425431 

26.6646 

8.925a 

647 

418609 

270840023 

25.4362 

8.6490 

712 

506944 

360944128 

26.6833 

8.9295 

648 

419904 

272097792 

25.4558 

8.6535 

713 

508369 

362467097 

26.7021 

8.9337 

649 

421201 

273359449 

25.4755 

8.6579 

714 

509796 

363994344 

26.7208 

8.9378 

660 

422500 

274625000 

25.4951 

8.6624 

715 

511225 

365525875 

26.7395 

8.9420 

651 

423801 

275894451 

25.5147 

8.6668 

716 

512656 

367061696 

26.7582 

8.9462 

652 

425104 

277167808 

25.5343 

8.6713 

717 

514089 

368601813 

26.7769 

8.950a 

653 

426409 

278445077 

25.5539 

8.6757 

718 

515524 

370146232 

26.7955 

8.954S 

654 

427716 

279726264 

25.5734 

8.6801 

719 

516961 

371694959 

26.8142 

8.9587 

655 

429025 

281011375 

25.5930 

8.6845 

720 

518400 

373248000 

26.8328 

8.9628 

656 

430336 

282300416 

25.6125 

8.6890 

721 

519841 

374805361 

26.8514 

8.9670 

657 

431649 

283593393 

25.6320 

8.6934 

722 

521284 

376367048 

26.8701 

8.9711 

658 

432964 

284890312 

25.6515 

8.6978 

723 

522729 

377933067 

26.8887 

8.975? 

659 

434281 

286191179 

25.6710 

8,7022 

724 

524176 

379503424 

26.9072 

8.9794 

660 

435600 

287496000 

25.6905 

8.7066 

725 

525625 

381078125 

26.9258 

8.9835 

661 

436921 

288804781 

25.7099 

8.7110 

726 

527076 

382657176 

26.9444 

8.9876 

662 

438244 

290117528 

25.7294 

8.7154 

727 

528529 

384240583 

26.9629 

8.9918 

663 

439369 

291434247 

25.7488 

8.7198 

728 

529984 

385828352 

26.9815 

8.995» 

664 

440896 

292754944 

25.7682 

8.7241  • 

729 

531441 

387420489 

27. 

9. 

665 

442225 

294079625 

25.7876 

8.7285 

730 

532900 

389017000 

27.0185 

9.0041 

666 

443556 

295408296 

25.8070 

8.7329 

731 

534361 

390617891 

27.0370 

9.008* 

667 

444889 

296740963 

25.8263 

8.7373 

732 

535824 

392223168 

27.0555 

9.0123 

668 

446224 

298077632 

25.8457 

8.7416 

733 

537289 

393832837 

27.0740 

9.0164' 

669 

447*61 

299418309 

25.8650 

8.7460 

734 

538756 

395446904 

27.0924 

9.0205 

mo 

448900 

600763000 

25.8844 

8.7503 

735 

540225 

397065375 

27.1109 

9.0246 

671 

450241 

302111711 

25.9037 

8.7547 

736 

541696 

398688256 

27.1293 

9.0287 

672 

451584 

303464448 

25.9230 

8.7590 

737 

543169 

400315553 

27.1477 

9.0328 

673 

452929 

304821217 

25.9422 

8.7634 

738 

544644 

401947272 

27.1662 

9.036* 

674 

454276 

306182024 

25.9615 

8.7677 

739 

546121 

403583419 

27.1816 

9.0410 

676 

455625 

307546875 

25.9808 

8.7721 

740 

547600 

405224000 

27.2029 

9.0450 

676 

456976 

308915776 

26. 

8.7764 

741 

549081 

406869021 

27.2213 

9.0491 

677 

458329 

310288733 

26.0192 

8.7807 

742 

550564 

408518488 

27.2397 

9.0632 

678 

459684 

311665752 

26.0384 

8.7850 

743 

552049 

410172407 

27.2580 

9.057? 

679. 

461041 

313046839 

26.0576 

8.7893 

744 

553536 

411830784 

27.2764 

9.0613 

680 

462400 

314432000 

26.0768 

8.7937 

745 

555025 

413493625 

27.2947 

9.0654 

681 

463761 

315821241 

26.0960 

8.7980 

746 

556516 

415160936 

27.3130 

9.0694 

682 

465124 

317214568 

26.1151 

8.8023 

747 

558009 

416832723 

27.3313 

9.073* 

683 

466489 

318611987 

26.1343 

8.8066 

748 

559504 

418508992 

27.3496 

9.0775 

684 

467856 

320013504 

26.1534 

8.8109 

749 

^1001 
562500 

420189749 

27.3679 

9.0816 

685 

469225 

321419125 

26.1725 

8.8152 

750 

421875000 

27.3861 

9.0866 

686 

470596 

322828856 

26.1916 

8.8194 

751 

564001 

423564751 

27.4044 

9.0896 

687 

471969 

324242703 

26.2107 

8.8237 

752 

565504 

425259008 

27.4226 

9.0937 

688 

473344 

325660672 

26.2298 

8.8280 

753 

567009 

426957777 

27.4408 

9.097T 

689 

474721 

327082769 

26.2488 

8.8323 

754 

568516 

428661064 

27.4591 

9.1017 

690 

476100 

328509000 

26.2679 

8.8366 

755 

570025 

430368875 

27.4773 

9.1057 

891 

477481 

329939371 

26.2869 

8.8408 

756 

571536 

432081216 

27.4955 

9.1098 

692 

478864 

331-573888 

26.3059 

8.8451 

757 

573049 

433798093 

27.5136 

9.1138 

693 

480249 

332812557 

26.3249 

8.8493 

758 

574564 

435519512 

27.5318 

9.1178 

694 

481636 

334255384 

26.3439 

8.8536 

759 

576081 

437245479 

27.5500 

9.1218 

695 

483025 

S35702375 

26.3629 

8.8578 

760 

577600 

438976000 

27.5681 

9.1268 

696 

484416 

337153536 

26.3818 

8.8621 

761 

579121 

440711081 

27.5862 

9.1298 

697 

485809 

338608878 

26.4008 

8.8663 

762 

580644 

442450728 

27.6043 

9.1338 

698 

487204 

340068392 

26.4197 

8.8706 

763 

582169 

444194947 

27.6225 

9.1378 

699 

488601 

341532099 

26.4386 

8.8748 

764 

583696 

445943744 

27.6405 

9.1418 

700 

490000 

343000000 

26.4575 

8.8790 

765 

585225 

447697125 

27.6586 

9.1458 

701 

491401 

344472101 

26.4761 

8.8833 

766 

586756 

449455096 

27.6767 

9.1498 

702 

492804 

345948408 

26.4953 

8.8875 

767 

588289 

451217663 

27.6948 

9.1537 

703 

494209 

347428927 

26.5141 

8.8917 

768 

589824 

452984832 

27.7128 

9.157T 

t04 

495616 

348913664 

26.5330 

8.8959 

769 

591361 

454756609 

27.7308 

9.161T 

t05 

497025 

350402625 

26.5518 

8.9001 

770 

592900 

456533000 

27.7489 

9.1657 

SQUARES,  CUBES,  AND  BOOTS. 


61 


TABL.E  of  Squares,  Cubes,  Square  Roots,  and  €nbe  Roots, 
of  J^umbers  from  1  to  1000  —  (Continued.) 


Square. 


Sq.  Ht. 


Square. 


Sq.  Rt. 


594441 
595984 
597529 
599076 
600625 

602176 
603729 
605284 
606841 
608400 


617796 
619369 
620944 
622521 
624100 


627264 
628849 
630436 
632025 

633616 
635209 
636804 
638401 
640000 

641601 
643204 
644809 
646416 
648025 

649636 
651249 
652864 
654481 
656100 

657721 
659344 
660969 
662596 
664225 


669124 
670761 
672400 

674041 
675684 
677329 
678976 
680625 

682276 
683929 
685584 
687241 


690561 
692224 
693889 
695556 
697225 


458314011 
460099648 
461889917 
463684824 
465484375 

467288576 
469097433 
470910952 
472729139 
474552000 

476379541 

478211768 
480048687 
481890304 
483736625 

485587656 
487443403 
489303872 
491169069 
493039000 

494913671 
496793088 
498677257 
500566184 
502459875 

504358336 
506261573 
508169592 


512000000 

513922401 
515849608 
517781627 
519718464 
521660125 


525557943 
527514112 
529475129 
531441000 

533411731 
535387328 
537367797 
539353144 
54134:3375 

543338496 
545338513 
547343432 
549353259 
551368000 

553387661 
555412248 
557441767 
559476224 
561515625 

563559976 
565609283 
567663552 
569722789 
571787000 

573856191 
575930368 
578009537 
580093704 
582182875 


27.7669 
27.7849 
27.8029 
27.8209 
27.8388 

27.8568 
27.8747 
27.8927 
27.9106 
27.9285 

27.9464 
27.9643 
27.9821 


28.0357 
28.0535 
28.0713 


28.1247 
28.1425 
28.1603 
28.1780 
28.1957 

28.2135 
28.2312 
28.2489 
28.2666 
28.2843 

28.3019 
28.3196 
28.3373 
28.3549 
28.3725 

28..3901 
28.4077 
28.4253 
28.4429 
28.4605 

28.4781 
28.4956 
28.5182 
28.5307 

28.5482 

28.5657 
28.5832 
28.6007 
28.6182 
28.6356 

28.6531 
28.6705 

28.6880 
28.7054 
28.7228 

28.7402 
28.7576 
28.7750 
28.7924 
28.8097 

28.8271 
28.8444 
28.8617 
28.8791 


9.1736 
9.1775 
9.1815 
9.1855 

9.1894 
9.1933 
9.1973 
9.2012 
9.2052 

9.2091 
9.2130 
9.2170 
9.2209 
9.2248 

9.2287 
9.2326 
9.2365 
9.2404 
9.2443 

9.2482 
9.2521 
9.2560 
9.2599 
9.2638 

9.2677 
9.27ie 
9.2754 
9.2793 
9.2832 

9.2870 
9.2909 
9.2948 
9.2986 
9.3025 

9.3063 
9.3102 
9.3140 
9.3179 
9.3217 

9.3255 
9.3294 
9.3332 
9.3370 
9.3408 

9.3447 
9.3485 
9.3523 
9.3561 
9.3599 

9.3637 
9.3675 
9.3713 
9.3751. 
9.3789 

9.3827 
9.3865 
9.3902 
9.3940 


9.4016 
9.4053 
9.4091 
9.4129 
9.4166 


700569 
702244 
703921 
705600 

707281 
708964 
710649 
712336 
714025 

715716 
717409 
719104 
720801 
722500 

724201 
725904 
727609 
729316 
731025 

732736 
734449 
736164 
737881 
739600 

741321 
743044 
744769 
746496 
748225 

749956 
751689 
753424 
755161 
756900 

758641 
760384 
7621 29 
763876 
765625 

767376 


584277056 
58637S253 

588480472 


770884 
772641 
774400 

776161 
777924 
779689 
781456 
783225 

784996 
786769 
788544 
790321 
792100 

793881 
795664 
797449 
799236 
801025 

802816 
804609 
806404 
808201 
810000 


599077107 
601211584 
603351125 

605495736 
607645423 
609800192 
611960049 
614125000 

616295051 
618470208 
620650477 
622835864 
625026375 

627222016 
629422793 
631628712 
633839779 
636056000 

638277381 
640503928 
642735647 
644972544 
647214625 

649461896 
651714363 
653972032 
656234909 
658503000 

660776311 
663054848 
6653.38617 
667627624 


672221376 
674526133 
676836152 
6791514.39 
681472000 

683797841 
686128968 
688465387 
690807104 
693154125 

695506456 
697864103 
700227072 
702595369 
704969000 

707347971 
709732288 
712121957 
714516984 
716917375 

719323136 
721734273 
724150792 
726572699 
729000000 


28.9137 
28.9310 
28.9482 
28.9655 

28.9828 


29.0172 
29.0345 
29.0517 
29.0689 

S9.0861 
29.1033 
29.1204 
29.1376 
29.1548 

29.1719 
29.1890 
29.2062 
29.2233 
29.2404 

29.2575 
29.2746 
29.2916 
29.3087 
29.3258 

29.3428 
29.3598 
29.3769 
29.3939 
29.4109 

29.4279 
29.4449 
29.4618 

29.4788 
29.4958 

29.5127 
29.5296 
29.5466 
29.5635 
29.5804 

29.5973 
29.6142 
29.6311 
29.6479 
29.6648 

29.6816 
29.6985 
29.7153 
29.7321 
29.7489 

29.7658 
29.7825 
29.7993 
29.8161 
29.8329 

29.8496 
29.8664 
29.8831 


29.9500 
29.9666 
29.9833 
30. 


62 


SQUARES,  CUBES,  AND  ROOTS. 


TABIiE  of  Squares,  Cnbes,  Sqnare  Roots,  and  Cube  Roots, 
of  3f^uiiiii>ers  from  1  to  1000  — (Continued.) 


No. 

Square. 

Cube. 

Sq.  Rt. 

cut. 

No. 

Square. 

Cube. 

Sq.  Rt. 

C.Rt. 

901 

811801 

731432701 

30.0167 

9.6585 

951 

904401 

860085351 

30.8383 

9.8339 

902 

813604 

73387080*: 

30.0333 

9.6620 

952 

906304 

862801408 

30.8545 

9.8374 

903 

815409 

736314327 

30.0500 

9.6656 

953 

908209 

865523177 

30.8707 

9.8408 

904 

817216 

738763264 

30.0666 

9.6692 

954 

910116 

868250664 

30.8869 

9.8443 

905 

819025 

741217625 

30.0832 

9.6727 

955 

912025 

870983875 

30.9031 

9.847T 

906 

820836 

743677416 

30.0998 

9.6763 

956 

913936 

873722816 

30.9192 

9.8511 

907 

822649 

.  746142643 

30.1164 

9.6799 

957 

915849 

876467493 

30.9354 

9.8546 

908 

824464 

748613312 

30.1330 

9.6834 

958 

917764 

879217912 

30.9516 

9.8580 

909 

826281 

751089429 

30.1496 

9.6870 

959 

919681 

881974079 

30.9677 

9.8614 

910 

828100 

753571000 

30.1662 

9.6905 

960 

921600 

884736000 

30.9839 

9.8648 

911 

829921 

756058031 

30.1828 

9.6941 

961 

923521 

887503681 

31. 

9.8683 

912 

831744 

758550528 

30.1993 

9.6976 

962 

925444 

890277128 

31.0161 

9.871T 

913 

833569 

761048497 

30.2159 

9.7012 

963 

927369 

893056347 

31.0322 

9.8751 

914 

835396 

763551944 

30.2324 

9.7047 

964 

929296 

895841344 

31.0483 

9.8785 

915 

837225 

766060875 

30.2490 

9.7082 

965 

931225 

898632125 

31.0644 

9.8819 

916 

839056 

768575296 

30.2655 

9.7118 

966 

933156 

901428696 

31.0805 

9.8854 

917 

840889 

771095213 

30.2820 

9.7153 

967 

935089 

904231063 

31.0966 

9.8888 

918 

842724 

773620632 

30.2985 

9.7188 

968 

937024 

907039232 

31.1127 

9.8922 

919 

844561 

776151559 

30.3150 

9.7224 

969 

938961 

909853209 

31.1288 

9.8956 

920 

846400 

778688000 

30.3315 

9.7259 

970 

940900 

912673000 

31.1448 

9.889C 

921 

848241 

781229961 

30.3480 

9.7294 

971 

942841 

915498611 

31.1609 

9.9024 

922 

850084 

783777448 

30.3645 

9.7329 

972 

944784 

918330048 

31.1769 

9.9058 

923 

851929 

786330467 

30.3809 

9.7364 

973 

946729 

921167317 

31.1929 

9.909e 

924 

853776 

788889024 

30.3974 

9.7400 

974 

948676 

924010424 

31.2090 

9.9126 

925 

855625 

791453125 

30.4138 

9.7435 

975 

950625 

926859375 

31.2250 

9.9160 

926 

857476 

794022776 

30.4302 

9.7470 

976 

952576 

929714176 

31.2410 

9.9194 

927 

859329 

796597983 

30.4467 

9.7505 

977 

954529 

932574833 

31.2570 

9.922T 

928 

861184 

799178752 

30.4631 

9.7540 

978 

956484 

935441.352 

31.2730 

9.9261 

929 

863041 

801765089 

30.4795 

9.7575 

979 

958441 

938313739 

31.2890 

9.9295 

930 

864900 

804357000 

30.4959 

9.7610 

980 

960400 

941192000 

31.3050 

9.9329 

931 

866761 

806954491 

30.5123 

9.76i5 

981 

962361 

944076141 

31.3209 

9.9363 

932 

868624 

809557568 

30.5287 

9.7680 

982 

964324 

946966168 

31.3369 

9.9396 

933 

870489 

812166237 

30.5450 

9.7715 

983 

966289 

949862087 

31.3528 

9.9430 

934 

872356 

814780504 

30.5614 

9.7750 

984 

968256 

952763904 

31.3688 

9.9464 

935 

874225 

817400375 

30.5778 

9.7785 

985 

970225 

955671625 

31.3847 

9.9497 

936 

876096 

820025856 

30.5941 

9.7819 

986 

972196 

958585256 

31.4006 

9.9531 

937 

877969 

822656953 

30.6105 

9.7854 

987 

974169 

961504803 

31.4166 

9.9665 

938 

879844 

825293672 

30.6268 

9.7889 

988 

976144 

964430272 

31.4325 

9.9598 

939 

881721 

827936019 

30.6431 

9.7924 

989 

978121 

967361669 

31.4484 

9.9632 

940 

S83600 

830584000 

30.6594 

9.7959 

990 

980100 

970299000 

31.4643 

9.9666 

941 

885481 

833237621 

30.6757 

9.7993 

991 

982081 

973242271 

31.4802 

9.9699 

942 

887364 

835896888 

30.6920 

9.8028 

992 

984064 

976191488 

31.4960 

9.9733 

943 

889249 

838561807 

30.7083 

9.8063 

993 

986049 

979146657 

31.5119 

9.9766 

944 

891136 

841232384 

30.7246 

9.8097 

994 

988036 

982107784 

31.5278 

9.9800 

945 

893025 

843908625 

30.7409 

9.8132 

995 

990025 

985074875 

31.5436 

9.9833 

946 

894916 

846590536 

30.7571 

9.8167 

996 

992016 

988047936 

31.5595 

9.9866 

947 

896809 

849278123 

30.7734 

9.8201 

997 

994009 

991026973 

31.5753 

9.9900 

948 

898704 

851971392 

30.7896 

9.8236 

998 

996004 

994011992 

31.5911 

9.993S 

949 

900601 

854670349 

30.8058 

9.8270 

999 

998001 

997002999 

31.6070 

9.9967 

950 

902500 

857375000 

30.8221 

9.8305 

1000 

1000000 

1000000000 

31.6228 

10. 

To  find  tlie  square  or  cube  of  any  \¥hole  number  ending; 
with  cipliers.    First,  omit  all  the  final  ciphers.    Take  from  the  table  the 

square  or  cube  (as  the  case  may  be)  of  the  rest  of  the  number.  To  this  square  add  twice  as  many 
ciphers  as  there  were  final  ciphers  in  the  original  number.  To  the  cube  add  three  times  as  many  as 
m  the  ori^ginal  number.  Thus,  for  905002;  9052  =  819025.  Add  twice  2  ciphers,  obtaining  819025C0OO, 
For  905003,  9053  =:  741217625.     Add  3  times  2  ciphers,  obtaining  741217625000000. 


SQUARE  AND  CUBE  ROOTS. 


63 


Square  Roots  aii<l  Cube  Roots  of  Xumbers  from  1000  to  10000. 

Noerrors. 


Num. 

Sq.  Rt 

Cu.  Rt. 

Num. 

Sq.  Rt. 

Cu.Rt. 

Num. 

Sq.  Rt. 

Cu.  Rt. 

Num. 

Sq.  Rt. 

Cu.  Rt. 

1005 

31.70 

10.02 

1405 

37.48 

11.20 

1805 

42.49 

12.18 

2205 

46.96 

13.02 

1010 

31.78 

10.03 

1410 

37.55 

11.21 

1810 

42.64 

12.19 

2210 

47.01 

13.03 

1015 

31.86 

10.05 

1415 

37.62 

11.23 

1815 

42.60 

12.20 

2215 

47.06 

13.04 

1020 

31.94 

10.07 

1420 

37.68 

11.24 

1820 

42.66 

12.21 

2220 

47.12 

13.05 

1025 

32.02 

10.08 

1425 

37.75 

11.25 

1825 

42.72 

12.22 

2225 

47.17 

13.05 

1030 

32.09 

10.10 

1430 

37.82 

11.27 

1830 

42.78 

12.23 

2230 

47.22 

13.06 

1035 

32.17 

10.12 

1435 

37.88 

11.28 

1835 

42.84 

12.24 

2235 

47. 28 

13.07 

1040 

32.25 

10.13 

1440 

37.95 

11.29 

1840 

42.90 

12.25 

2240 

47.33 

13.08 

1045 

32.33 

10.15 

1445 

38.01 

11.31 

1845 

42.95 

12.26. 

2245 

47.38 

13.09 

1050 

32.40 

10.16 

1450 

38.08 

11.32 

1850 

43.01 

12.28 

2250 

47.43 

13.10 

1055 

32.48 

10.18 

1455 

38.14 

11.33 

1855 

43.07 

12.29 

2255 

47.49 

13.11 

1060 

32.56 

10.20 

1460 

38.21 

11.34 

1860 

43.13 

12.30 

2260 

47.54 

13.12 

1065 

32.63 

10.21 

1465 

38.28 

11.36 

1865 

43.19 

12.31 

2265 

47.59 

13.13 

1070 

32.71 

10.23 

1470 

38.34 

11.37 

1870 

43.24 

12.32 

2270 

47.64 

13.14 

1075 

32.79 

10.24 

1475 

38.41 

11.38 

1875 

43.30 

12.33 

2275 

47.70 

13.15 

1080 

32.86 

10.26 

1480 

38.47 

11.40 

1880 

43.36 

12.34 

2280 

47.75 

13.16 

1085 

32.94 

10.28 

1485 

38.54 

11.41 

1885 

43.42 

12.35 

2285 

.  47.80 

13.17 

1090 

33.02 

10.29 

1490 

38.60 

11.42 

1890 

43.47 

12.36 

2290 

47.85 

13.18 

1095 

33.09 

10.31 

1495 

38.67 

11.13 

1895 

43.53 

12.37 

2295 

47.91 

13.19 

1100 

33.17 

10.32 

1500 

38.73 

11.45 

1900 

43.59 

12.39 

2300 

47.96 

13.20 

1105 

33.24 

10.34 

1505 

38.79 

11.46 

1905 

43.65 

12.40 

2305 

48.01 

13.21 

1110 

33.32 

10.35 

1510 

38.86 

11.47 

1910 

43.70 

12.41 

2310 

48.06 

13.22 

1115 

33.39 

10.37 

1515 

38.92 

11.49 

1915 

43.76 

12.42 

2315 

48.11 

13.23 

1120 

33.47 

10.38 

1520 

38.99 

11.50 

1920 

43.82 

12.43 

2320 

48.17 

13.24 

1125 

33.54 

10.40 

1525 

39.05 

11.51 

1925 

43.87 

12.44 

2325 

48.22 

13.25 

1130 

33.62 

10.42 

•  1530 

39.12 

11.52 

1930 

43.93 

12.45 

2330 

48.27 

13.26 

1135 

33.69 

10.43 

1535 

39.18 

11.54 

1935 

43.99 

12.46 

2335 

48.32 

13.27 

1140 

33.76 

10.45 

1540 

39.24 

11.55 

1940 

44.05 

12.47 

2340 

48-37 

13.36 

1145 

33.84 

10.46 

1.545 

39.31 

11.56 

1945 

44.10 

12,48 

2345 

48.43 

13.29 

1150 

33.91 

10.48 

1550 

39.37 

11.57 

195P 

44.16 

12.49 

2350 

48.48 

13.30 

1155 

33.99 

10.49 

1555 

39.43 

11.59 

1955 

44.22 

12.50 

2355 

48.53 

13.30 

1160 

34.06 

10.51 

1560 

39.50 

11.60 

1960 

44.27 

12.51 

2360 

48.58 

13.31 

1165 

34.13 

10.52 

1565 

39.56 

11.61 

1965 

44.33 

12.53 

2365 

48.63 

13.32 

1170 

34.31 

10.54 

1570 

39.62 

11.62 

1970 

44.38 

12.54 

2370 

48.68 

13.33 

1175 

34.28 

10.55 

1575 

39.69 

1 1 .63 

1975 

44.44 

12.55 

2375 

48.73 

13.34 

1180 

34.35 

10.57 

1580 

39.75 

11.65 

1980 

44.50 

12.56 

2380 

48.79 

13.35 

1185 

34.42 

10.58 

1585 

39.81 

11.66 

1985 

44.55 

12.57 

2385 

48.84 

13.36 

1190 

34.50 

10.60 

1590 

39.87 

11.67 

1990 

44.61 

12.58 

2390 

48.89 

13.37 

1195 

34.57 

10.61 

1595 

39.94 

11. 08 

1995 

44.67 

12.59 

2395 

48.94 

13.38 

1200 

34.64 

10.63 

1600 

40.00 

11.70 

2000 

44.72 

12.60 

2400 

48.99 

13.39 

1205 

34.71 

10.64 

1605 

40.06 

11.71 

2005 

44.78 

12.61 

2405 

49.04 

13.40 

1210 

34.79 

10.66 

1610 

40.12 

11.72 

2010 

44.83 

12.62 

2410 

49.09 

13.41 

1215 

34.86 

10.67 

1615 

40.19 

11.73 

2015 

44.89 

12.63 

2415 

49.14 

13.42 

1220 

34.93 

10.69 

1620 

40.25 

11.74 

2020 

44.94 

12.64 

2420 

49.19 

13.43 

1225 

35.00 

10.70 

1625 

40.31 

11.76 

2025 

45.00 

12.65 

2425 

49.24 

13.43 

1230 

35.07 

10.71 

16.30 

40.37 

11.77 

2030 

45.06 

12.66 

2430 

49.30 

13.44 

1235 

35.14 

10.73 

1635 

40.44 

11.78 

2035 

45.11 

12.67 

2435 

49.35 

13.45 

1240 

35.21 

10.74 

1640 

40.50 

11.79 

2040 

45.17 

12.68 

2440 

49.40 

13.46 

1245 

35.28 

10.76 

1645 

40.56 

11.80 

2045 

45.22 

12.69 

2445 

49.45 

13.47 

1250 

35.36 

10.77 

1650 

40.62 

11.82 

2050 

45.28 

12.70 

2450 

49.50 

13.48 

1255 

35.43 

10.79 

1655 

40.68 

11.83 

2055 

45.33 

12.71 

2460 

49.60 

13.50 

1260 

35.50 

10.80 

1660 

40.74 

11.84 

2060 

45.39 

12.72 

2470 

49.70 

13.52 

1265 

35.57 

10.82 

1665 

40.80 

11.85 

2065 

45.44 

12.73 

2480 

49.80 

13.54 

1270 

35.64 

10.83 

1670 

40.87 

11.86 

2070 

45.50 

12.74 

2490 

49.90 

13.55 

1275 

35.71 

10.84 

1675 

40.93 

11.88 

2075 

45.55 

12.75 

2500 

50.00 

13.57 

1280 

35.78 

10.86 

1680 

40.99 

11.89 

2080 

45.61 

12.77 

2510 

50.10 

13.59 

1285 

35.85 

10.87 

1685 

41.05 

11.90 

2085 

45.66 

12.78 

2520 

50.20 

13.61 

1290 

35.92 

10.89 

1690 

41.11 

11.91 

2090 

45.72 

12.79 

2530 

50.30 

13.63 

1295 

35.99 

10.90 

1695 

41.17 

11.92 

2095 

45.77 

12.80 

2540 

50.40 

18.64 

1300 

36.06 

10.91 

1700 

41.23 

11.93 

2100 

45.83 

12.81 

2550 

50.50 

1S.66 

1305 

36.12 

10.93 

1705 

41.29 

11.95 

2105 

45.88 

12.82 

2560 

50.60 

13.68 

1310 

36.19 

10.94 

1710 

41.35 

11.96 

2110 

45.93 

12.83 

2570 

50.70 

13.70 

1315 

36.26 

10.96 

1715 

41.41 

11.97 

2115 

45.99 

12.84 

2580 

50.79 

13.72 

1320 

16.33 

10.97 

1720 

41.47 

11.98 

2120 

46.04 

12.85 

2590 

50.89 

13.73 

1325 

36.40 

10.98 

1725 

41.53 

11.99 

2125 

46.10 

12.86 

2600 

50.99 

13.75 

1330 

36.47 

11.00 

1730 

41. 5» 

12.00 

2130 

46.15 

12.87 

2610 

51.09 

13.77 

1335 

36.54 

11.01 

1735 

41.65 

12.02 

2135 

46.21 

12.88 

2620 

51.19 

13.79 

1340 

36.61 

11.02 

1740 

41.71 

12.03 

2140 

46.26 

12.89 

2630 

51.28 

13.80 

1345 

36.67 

11.04 

1745 

41.77 

12.04 

2145 

46.31 

12.90 

2640 

51.38 

13.82 

1350 

36.74 

11.05 

1750 

41.83 

12.05 

2150 

46.37 

12.91 

2650 

51.48 

13.84 

1355 

36.81 

11.07 

1755 

41.89 

12.06 

2155 

46.42 

12.92 

2660 

51.58 

13.86 

1360 

36.88 

11.08 

1760 

41.95 

12.07 

2160 

46.48 

12.93 

2670 

51.67 

13.87 

1365 

36.95 

11.09 

1765 

42.01 

12.09 

2165 

46.53 

12.94 

2680 

51.77 

13.89 

1370 

37.01 

11.11 

1770 

42.07 

12.10 

2170 

46.58 

12.95 

2690  . 

51.87 

13.91 

1375 

37.08 

11.12 

1775 

42.13 

12.11 

2175 

46.64 

12.96 

2700 

51.96 

13.92 

1380 

37.15 

11.13 

1780 

42.19 

12.12 

2180 

46.69 

12.97 

2710 

52.06 

13.94 

13B5 

37.22 

11.15 

1785 

42.25 

12.13 

2185 

46.74 

12.98 

2720 

52.15 

13.96 

1390 

37.28 

11.16 

1790 

42.31 

12.14 

2190 

46.80 

12.99 

2730 

52.25 

13.98 

1395 

37.35 

11.17 

1795 

42.37 

12.15 

2195 

46.85 

13.00 

2740 

52.. 35 

13.99 

1400 

37.42 

11.19 

1800 

42.43 

12.16 

2200 

46.90 

13.01 

2750 

52.44 

14.01 

64 


SQUARE  AND  CUBE  ROOTS. 


Sqiiare  Roots  and  Cube  Roots  of  Numbers  from  1000  to  10000 

— (Continued.) 


Num. 

Sq.  Rt. 

Cu.  Rt. 

Num. 

Sq.  Rt. 

Cu.  Rt. 

Num. 

Sq.  Rt. 

Cu.Rt. 

Num. 

Sq,  Rt. 

Cu.  Rt. 

2760 

52.54 

14.03 

3550 

59.58 

15.25 

4340 

65.88 

16.31 

5130 

71,62 

17.25 

2770 

52.63 

14.04 

3560 

59.67 

15.27 

4350 

65.95 

16.32 

5140 

71.69 

17.26 

2780 

52.73 

14.06 

3570 

59.75 

15.28 

4360 

66.03 

16.34 

5150 

71.76 

17.27 

3790 

52.82 

14.08 

3580 

59.83 

15.30 

4370 

66.11 

16.35 

5160 

71.83 

17.28 

2800 

52.92 

14.09 

3590 

59.92 

15.31 

4380 

66.18 

16.36 

5170 

71.90 

17.29 

2810 

53,01 

14.11 

3600 

60.00 

15.33 

4390 

66.26 

16.37 

5180 

71.97 

17.30 

2820 

53.10 

14.13 

3610 

60.08 

15.34 

4400 

66.33 

16.39 

5190 

72.04 

17.31 

2830 

53.20 

14.14 

3620 

60.17 

15.35 

4410 

66.41 

16.40 

5200 

72.11 

17.32 

2840 

53.29 

14.16 

3630 

60.25 

15.37 

4420 

66.48 

16.41 

5210 

72.18 

17.34 

2850 

S3.39 

14.18 

3640 

60.33 

15.38 

4430 

66.56 

16.42 

5220 

72.25 

17.35 

2860 

53.48 

14.19 

3650 

60.42 

15.40 

4440 

66.63 

16.44 

5230 

72.32 

17.36 

2870 

53.57 

14.21 

3660 

60.50 

15.41 

4450 

66.71 

16.45 

5240 

72.39 

17.37 

2880 

53.67 

14.23 

3670 

60.58 

15.42 

4460 

66.78 

16.46 

5250 

72.46 

17.38 

2890 

53.76 

14.24 

3680 

60.66 

15.44 

4470 

66.86 

16.47 

5260 

72.53 

17.39 

2900 

53.85 

14.26 

3690 

60.75 

15.45 

4480 

66.93 

16.49 

5270 

72.59 

17.40 

2910 

53.94 

14.28 

3700 

00.83 

15.47 

4490 

67.01 

16.50 

5280 

72.66 

17.41 

2920 

54.04 

14.29 

3710 

60.91 

15.48 

4500 

67.08 

16.51 

5290 

72.73 

17.42 

2930 

54.13 

14.31 

3720 

60.99 

15.49 

4510 

67.16 

16.52 

5300 

72.80 

17.44 

2940 

54.22 

14.33 

3730 

61.07 

15.51 

4520 

67.23 

16.53 

5310 

72.87 

17.45 

2950 

54.31 

14.34 

3740 

61.16 

15.52 

4530 

67.31 

16.55 

5320 

72.94 

17,46 

2960 

54.41 

14.36 

3750 

61.24 

15.54 

4540 

67.38 

16.56 

5330 

73.01 

17.47 

2970 

54.50 

14.37 

3760 

61.32 

15.55 

4550 

67.45 

16.57 

5340 

73.08 

17.48 

2980 

54.59 

14.39 

3770 

61.40 

15.56 

4560 

67.53 

16.58 

5350 

73.14 

17.49 

2990 

54.68 

14.41 

3780 

61.48 

15.58 

4570 

67.60 

16.59 

5360 

73.21 

17.50 

3000 

54.77 

14.42 

3790 

61.56 

15.59 

4580 

67.68 

16.61 

.  5370 

73.28 

17.51 

3010 

54.86 

14.44 

3800 

61.64 

15.60 

4590 

67.75 

16.62 

5380 

73.35 

17.52 

3020 

54.95 

14.45 

3810 

61.73 

15.62 

4600 

67.82 

16.63 

5390 

73.42 

17.5$ 

3030 

55.05 

14.47 

3820 

61.81 

15.63 

4610 

67.90 

16.64 

5400 

73.48 

17.54 

8040 

55.14 

14.49 

3830 

61.89 

15.65 

4620 

67.97 

16.66 

5410 

73.55 

17.55 

3050 

55.23 

14.50 

3840 

61.97 

15.66 

4630 

68.04 

16.67 

5420 

73.62 

17.57 

3060 

55.32 

14.52 

3850 

62.05 

15.67 

4640 

68.12 

16.68 

5430 

73.69 

17.58 

3070 

55.41 

14.53 

3860 

62.13 

15.69 

4650 

68.19 

16.69 

5440 

73.76 

17.59 

3080 

55.50 

14.55 

3870 

62.21 

15.70 

4660 

68.26 

16.70 

5450 

73.82 

17.60 

3090 

55.59 

14.57 

3880 

62.29 

15.71 

4670 

68.34 

16.71 

5460 

73.89 

17.61 

8100 

55.68 

14.58 

3890 

62.37 

15.73 

4680 

68.41 

16.73 

5470 

73.96 

17,62 

3110 

55.77 

14.60 

3900 

62.45 

15.74 

4690 

68.48 

16.74 

5480 

74.03 

17.63 

3180 

55.86 

14.61 

3910 

62.53 

15.75 

4700 

68.56 

16.75 

5490 

74.09 

17.64 

3130 

55.95 

14.63 

3920 

62.61 

15.77 

4710 

68.63 

16.76 

5500 

74.16 

17,65 

3140 

56.04 

14.64 

3930 

62.69 

15.78 

4720 

68.70 

16.77 

5510 

74.23 

17.66 

3150 

56.12 

14.66 

3940 

62.77 

15.79 

4730 

68.77 

16.79 

5520 

74.30 

17.67 

3160 

56.21 

14.67 

3950 

62.85 

15.81 

4740 

68.85 

16.80 

5530 

74.36 

17.68 

3170 

56.30 

14.69 

3960 

62.93 

15.82 

4750 

68.92 

16.81 

5540 

74.43 

17.69 

3180 

56.39 

14.71 

3970 

63.01 

15.83 

4760 

68.99 

16.82 

5550 

74.50 

17.71 

3190 

56.48 

14.72 

3980 

63.09 

15.85 

4770 

69.07 

16.83 

5560 

74.57 

17.72 

3200 

56.57 

14.74 

3990 

63.17 

15.86 

4780 

69.14 

16.85 

5570 

74.63 

17.73 

3210 

56.66 

14.75 

4000 

63.25 

15.87 

4790 

69  21 

16.86 

5580 

74.70 

17,74 

3220 

.56.75 

14.77 

4010 

63.32 

15.89 

4800 

69.28 

16.87 

5590 

74.77 

17,75 

3230 

56.83 

14.78 

4020 

63.40 

15.90 

4810 

69.35 

16.88 

5600 

74.83 

17.76 

3240 

56.92 

14.80 

4030 

63.48 

15.91 

4820 

69.43 

16  89 

5610 

74.90 

17.77 

3250 

57.01 

14.81 

4040 

63.56 

15.93 

4830 

69.50 

16.90 

5620 

74.97 

17.78 

3260 

57.10 

14.83 

4050 

63.64 

15.94 

4840 

69.57 

16.92 

5630 

75.03 

17.79 

3270 

57.18 

14.fe4 

4060 

63.72 

15.95 

4850 

69.64 

16.93 

5640 

75.10 

17.80 

3280 

57.27 

14.86 

4070 

63.80 

15.97 

4860 

69.71 

16.94 

5650 

75.17 

17.81 

3290  ' 

57.36 

14.87 

4080 

63.87 

15.98 

4870 

69.79 

16.95 

5660 

75.23 

17.82 

3300 

57.45 

14.89 

4090 

63.95 

15.99 

4880 

69.86 

16.96 

5670 

75.30 

17.83 

3310 

57.53 

14.90 

4100 

64.03 

16.01 

4890 

69.93 

16.97 

5680 

75.37 

17.84 

3320 

57.62 

14.92 

4110 

64.11 

16.02 

4900 

70.00 

16.98 

5690 

75.43 

17.85 

3330 

57.71 

14.93 

4120 

64.19 

16.03 

4910 

70.07 

17.00 

5700 

75.50 

17.86 

3340 

57.79 

14.95 

4130 

64.27 

16.04 

4920 

70.14 

17.01 

5710 

75.56 

17.87 

3350 

57.88 

14.96 

4140 

64.34 

16.06 

4930 

70.21 

17.02 

5720 

75.63 

17.88 

3360 

57.97 

14.98 

4150 

64.42 

16.07 

4940 

70.29 

17.03 

5730 

75.70 

17.89 

3370 

58.05 

14.99 

4160 

64.50 

16.08 

4950 

70.36 

17.04 

5740 

75.76 

17.90 

3380 

58.14 

15,01 

4170 

64.58 

16.10 

4960 

70.43 

17.05 

5750 

75.83 

17.92 

3390 

58.22 

15.02 

4180 

64.65 

16.11 

4970 

70.50 

17.07 

5760 

75.89 

17.93 

3400 

58.31 

15.04 

4190 

64.73 

16.12 

4980 

70.57 

17.08 

5770 

75.96 

17.94 

3410 

58.40 

15.05 

4200 

64.81 

16.13 

4990 

70.64 

17.09 

5780 

76.03 

17.95 

3420 

58.48 

15.07 

4210 

64.88 

16.15 

5000 

70.71 

17.10 

5790 

76.09 

17.96 

3430 

58.57 

15.08 

4220 

64.96 

16.16 

5010 

70.78 

17.11 

5800 

76.16 

17,97 

3440 

58.65 

15.10 

42.30 

65.04 

16.17 

5020 

70.85 

17.12 

5810 

76.22 

17,98 

3450 

58.74 

15.11 

4240 

65.12 

16.19 

5030 

70.92 

17.13 

5820 

76.29 

17.99 

3460 

58.82 

15.12 

4250 

65.19 

16.20 

5040 

70.99 

17.15 

5830 

76.35 

18.00 

3470 

58.91 

15.14 

4260 

65.27 

16.21 

5050 

71.06 

17.16 

5840 

76.42 

18.01 

3480 

58.99 

15.15 

4270 

66.35 

16.22 

5060 

71.13 

17.17 

5850 

76.49 

18.02 

3490 

59.08 

15.17 

4280 

65.42 

16.24 

■  5070 

71.20 

17.18 

^860 

76.55 

18.03 

3500 

59.16 

15.18 

4290 

65.50 

16.25 

5080 

71.27 

17.19 

5870 

76.62 

18.04 

3510 

59.25 

15.20 

4300 

6557 

16.26 

5090 

71.34 

17.20 

5880 

76.68 

18.05 

3520 

59.33 

15.21 

4310 

65.65 

16.27 

5100 

71.41 

17.21 

5890 

76.75 

18.06 

3530 

59.41 

15.23 

4320 

65.73 

16.29 

5110 

71.48 

17.22 

5900 

76.81 

18,07 

3540 

59.50 

15.24 

4330 

65.80 

16,30 

5120 

71.55 

17.24 

5910 

76.88 

18.08 

SQUARE  AND  CUBE  ROOTS. 


65 


•qnare  Roots  and  €abe  Roots  of  Numbers  from  1000  to  lOOOO 

—  (COXTINUED.) 


Num. 

Sq.  Rt. 

Cu.  Et. 

Num. 

Sq.  Rt. 

Cu.Rt. 

Num. 

Sq.  Rt. 

Cu.  Rt. 

Num. 

Sq.  Rt. 

Cu.R 

5920 

76.94 

18.09 

671(^ 

81.91 

18.86 

7500 

86.60 

19.57 

8290 

91.05 

20.24 

5930 

77.01 

18.10 

6720^ 

81.98 

18.87 

7510 

86.66 

19.58 

8J00 

91.10 

20.25 

5940 

77.07 

18.11 

6730 

82.04 

18.88 

7520 

86.72 

19.59 

8310 

91.16 

20.26 

5950 

77.14 

18.12 

6740 

82.10 

18.89 

7530 

86.78 

19.60 

8;;20 

91.21 

20.26 

6960 

77.20 

18.13 

6750 

82.16 

18.90 

7540 

86.83 

19.61 

8330 

91.27 

20.27 

5970 

77.27 

18.14 

6760 

82.22 

18.91 

7550 

86.89 

19.62 

8340 

91.32 

20.28 

6980 

77.33 

18.15 

6770 

82.28 

18.92 

7560 

86.95 

19.63 

8350 

91.38 

20.29 

5990 

77.40 

18.16 

6780 

82.34 

18.93 

7570 

87.01 

19.64 

8360 

91.43 

20.30 

6000 

77.46 

18.17 

6790 

82.40 

18.94 

7580 

87.06 

19.64 

8370 

91.49 

20.30 

6010 

77.52 

18.18 

6800 

82.46 

18.95 

7590 

87.12 

19.65 

8380 

91.54 

20.31 

6020 

77.59 

18.19 

6810 

82.52 

18.95 

7600 

87.18 

19.66 

8390 

91.60 

20.32 

6030 

77.65 

18.20 

6820 

82.58 

18.96 

7610 

87.24 

19.67 

8400 

91.65 

20.33 

6040 

77.72 

18.21 

6830 

82.64 

18.97 

7620 

87.29 

19.68 

8410 

91.71 

20.34 

6050 

77.78 

18.22 

6840 

82.70 

18.98 

7630 

87.35 

19.69 

8420 

91.76 

20.34 

«060 

77.85 

18.23 

6850 

82.76 

18.99 

7640 

87.41 

19.70 

8430 

91.82 

20.35 

6070 

77.91 

18.24 

6860 

82.83 

19.00 

V650 

87.46 

19.70 

8440 

91.87 

20.36 

6080 

77.97 

18.25 

6870 

82.89 

19.01 

7660 

87.52 

19.71 

8450 

91.92 

20.37 

6090 

78.04 

18.26 

6880 

82.95 

19.02 

7670 

87.58 

19.72 

8460 

91.98 

20.38 

6100 

78.10 

18.27 

6890 

83.01 

19.03 

7680 

87.64 

19.73 

8470 

92.03 

20.38 

6110 

78.17 

18.28 

6900 

83.07 

19.04 

7690 

87.69 

19.74 

8480 

92.09 

20.39 

6120 

78.23 

18.29 

6910 

83.13 

19.05 

77M 

87.75 

19.75 

8490 

92.14 

20.40 

6130 

78.29 

18.30 

6920 

83.19 

19.06 

7710 

87.81 

19.76 

8500 

92.20 

20.41 

6140 

78.36 

18.31 

6930 

83.25 

19.07 

7730 

87.8b 

19.76 

8510 

92.25 

20.42 

6150 

78.42 

18.32 

6940 

83.31 

19.07 

7730 

87.92 

19.77 

8520 

92.30 

20.42 

6160 

78.49 

18.33 

6950 

83.37 

19.08 

7740 

87.98 

19.78 

8530 

92.36 

20.43 

6170 

78.55 

18.34 

6960 

83.43 

19.09 

7750 

88.03 

19.79 

8540 

92.41 

20.44 

6180 

78.61 

18.35 

6970 

83.49 

19.10 

7760 

88.09 

19.80 

8550 

92.47 

20.45 

6190 

78.68 

18.36 

6980 

83.55 

19.11 

7770 

88.15 

19.81 

8560 

92.52 

20.46 

6200 

78.74 

18.37 

6990 

83.61 

19.12 

7780 

88,20 

19.81 

8570 

92.57 

20.46 

8210 

78.80 

18.38 

7000 

83.67 

19.13 

7790 

88.26 

19.82 

8580 

92.63 

20.47 

6220 

78.87 

18.39 

7010 

83.73 

19.14 

7800 

88.32 

19.83 

8590 

92.68 

28.48 

6230 

78.93 

18.40 

7020 

83.79 

19.15 

7810 

88.37 

19.84 

8600 

92.74 

20.49 

624C 

78.99 

18.41 

7030 

83.85 

19.16 

7820 

88.43 

19.85 

8610 

92.79 

20.50 

6250 

79.06 

18.42 

7040 

83.90 

19.17 

7830 

88.49 

19.86 

8620 

92.84 

20.50 

6260 

79.12 

18.43 

7050 

83.96 

19.17 

7840 

88.54 

19.87 

8630 

92.90 

20.51 

6270 

79.18 

18.44 

7060 

84.02 

19.18 

7850 

88.60 

19.87 

8640 

92.95 

20.52 

6280 

79.25 

18.45 

7070 

84.08 

19.19 

7860 

88.66 

19.88 

8650 

93.01 

20.5S 

6290 

79.31 

18.46 

7080 

84.14 

19.20 

7870 

88.71 

19.89 

8660 

93.06 

20.54 

6300 

79.37 

18.47 

7090 

84.20 

19.21 

7880 

88.77 

19.90 

8670 

93.11 

20.54 

6310 

79.44 

18.48 

7100 

84.26 

19.22 

7890 

88.83 

19.91 

8680 

93.17 

20.55 

6320 

79.50 

18.49 

7110 

84.32 

19.23 

7900 

88.88 

19.92 

8690 

93.22 

20.56 

6330 

79.56 

18.50 

7120 

84.38 

19.24 

7910 

88.94 

19.92 

8700 

93.27 

20.57 

6340 

79.62 

18.51 

7130 

84.44 

19.25 

7920 

88.99 

19.93 

8710 

93.33 

20.57 

6350 

79.69 

18.52 

7140 

84.50 

19.26 

7930 

89.05 

19.94 

8720 

93.38 

20.58 

6360 

79.75 

18.53 

7150 

84.56 

19.26 

7940 

89.11 

19.95 

8730 

93.43 

20.59 

6370 

79.81 

18.54 

7160 

84.62 

19.27 

7950 

89.16 

19.96 

8740 

93.49 

20.60 

6380 

79.87 

18.55 

7170 

84.68 

19.28 

7960 

89.22 

19.97 

8750 

93.54 

20.61 

6390 

79.94 

18.56 

7180 

84.73 

19.29 

7970 

89.27 

19.97 

8760 

93.59 

20.61 

6400 

80.00 

18.57 

7190 

84.79 

19.30 

7980 

89.33 

19.98 

8770 

93.65 

20.62 

6410 

80.06 

18.58 

7200 

84.85 

19.31 

7990 

89.39 

19.99 

8780 

93.70 

20.63 

6420 

80.12 

18.59 

7210 

84.91 

19.32 

8000 

89.44 

20.00 

8790 

93.75 

20.64 

6430 

80.19 

18.60 

7220 

84.97 

19.33 

8010 

89.50 

20.01 

8800 

93.81 

20.65 

6440 

80.25 

18.60 

7230 

85.03 

19.34 

8020 

89.55 

20.02 

8810 

93.86 

20.65 

6450 

80.31 

18.61 

•7240 

85.09 

19.35 

8030 

89.61 

20.02 

8820 

93.91 

20.66 

6460 

80.37 

18.62 

7250 

85.15 

19.35 

8040 

89.67 

20.03 

8830 

93.97 

20.67 

6470 

80.44 

18.63 

7260 

85.21 

19.36 

8050 

89.72 

20.04 

8840 

94.02 

20.68 

6480 

80.50 

18.64 

7270 

85.26 

19.37 

8060 

89.78 

20.05 

8850 

94.07 

20.68 

6490 

80.56 

18.65 

7280 

85.32 

19.38 

8070 

89.83 

20.06 

8860 

94.13 

20.69 

«500 

80.62 

18.66 

7290 

85.38 

19.39 

8080 

89.89 

20.07 

8870 

94.18 

20.70 

6510 

80.68 

18.67 

7300 

85.44 

19.40 

8090 

89.94 

20.07 

8880 

94.23 

20.71 

6520 

80.75 

18.68 

7310 

85.50 

19.41 

8100 

90.00 

20.08 

8890 

94.29 

20.72 

€530 

80.81 

18.69 

7320 

85.56 

19.42 

8110 

90.06 

20.09 

8900 

94.34 

20.72 

6540 

80.87 

18.70 

7330 

85.62 

19.43 

8120 

90.11 

20.10 

8910 

94.39 

20.73 

6550 

80.93 

18.71 

7340 

85.67 

19.43 

8130 

90.17 

20.11 

8920 

94.45 

20.74 

6560 

80.99 

18.72 

7350 

85.73 

19.44 

8140 

90.22 

20.12 

8930 

94.50 

20.75 

6570 

81.06 

18.73 

7360 

85.79 

19.45 

8150 

90.28 

20.12 

8940 

94.55 

20.75 

6680 

81.12 

18.74 

7370 

85.85 

19.46 

8160 

90.33 

20.13 

8950 

94.60 

20.76 

6690 

81.18 

18.75 

7380 

85.91 

19.47 

8170 

90.39 

20.14 

8960 

94.66 

20.77 

6600 

81.24 

18.76 

7390 

85.97 

19.48 

8180 

90.44 

20.15 

8970 

94.71 

20.78 

6610 

81.30 

18.77 

7400 

86.02 

19.49 

8190 

90.50 

20.16 

8980 

94.76 

20.79 

6620 

81.36 

18.78 

7410 

86.08 

19.50 

8200 

90.55 

20.17 

8990 

94.82 

20.79 

6630 

81.42 

18.79 

7420 

86.14 

19.50 

8210 

90.61 

20.17 

9000 

94.87 

20.80 

6640 

81.49 

18.80 

7430 

86.20 

19.51 

8220 

90.66 

20.18 

9010 

94.92 

20.81 

6650 

81.55 

18.81 

7440 

86.26 

19.52 

8230 

90.72 

20.19 

9020 

94.97 

20.82 

6660 

81.61 

18.81 

7450 

86.31 

19.53 

8240 

90.77 

20.20 

9030 

95.03 

20.82 

6670 

81.67 

18.82 

7460 

86.37 

19.54 

8250 

90.83 

20.21 

9040 

95.08 

20.83 

6680 

81.73 

18.83 

7470 

86.43 

19.55 

8260 

90.88 

20.21 

9050 

95.13 

20.84 

6690 

81.79 

18.84 

7480 

86.49 

19.56 

8270 

90.94 

20.22 

9060 

»5.18 

20.85 

6700 

W.85 

18.85 

7490 

86.54 

19.57 

8280 

SM).99 

20.23 

9070 

96.24 

20.86 

66 


SQUARE  AND  CUBE  ROOTS. 


Square  Roots  and  €abe  Roots  of  Numbers  from  1000  to  lOOOf 

—  (Continued.) 


Num. 

Sq.  Rt. 

Cu.  Rt. 

Num. 

Sq.  Rt. 

Cu.  Rt. 

Num. 

Sq.  Rt. 

Cu.Rt. 

Num. 

Sq.  Rt. 

Cu.Bfc 

9080 

95.29 

20.86 

9320 

96.54 

21.04 

9550 

97.72 

^1.22 

9780 

98.89 

21.39 

9090 

95.34 

20.87 

9330 

96.59 

21.05 

9560 

97.78  < 

^21.22 

9790 

98.94 

21.39 

9100 

95.39 

20.88 

9340 

96.64 

21.06 

9570 

97.83 

21.23 

9800 

98.99 

21.40 

9110 

95.45 

20.89 

9350 

96.70 

21.07 

9580 

97.88 

21.24 

9810 

99.05 

21.41 

9120 

95.50 

20.89 

9360 

96.75 

21.07 

9590 

97.93 

21.25 

9820 

99.10 

21.41 

9130 

95.55 

20.90 

9370 

96.80 

21.08 

9600 

97.98 

21.25 

9830 

99.15 

21.42 

9140 

96.60 

20.91 

9380 

96.85 

21.09 

9610 

98.03 

21.26 

9840 

99.20 

21.4» 

9150 

95.66 

20.92 

9390 

96.90 

21.10 

9620 

98.08 

21.27 

9850 

99.25 

21.44 

9160 

95.71 

20.92 

9400 

96.95 

21.10 

9630 

98.13 

21.28 

9860 

99.30 

21.44 

9170 

95.76 

20.93 

9410 

97.01 

21.11 

9640 

98.18 

21.28 

9870 

99.35 

21.45 

9180 

95.81 

20.94 

9420 

97.06 

21.12 

9650 

98.23 

21.29 

9880 

99.40 

21.46 

9190 

95.86 

20.95 

9430 

97.11 

21.13 

9660 

98.29 

21.30 

9890 

99.45 

21.47 

9200 

95.92 

20.95 

9440 

97.16 

21.13 

9670 

98.34 

21.30 

9900 

99.50 

21.47 

9210 

95.97 

20.96 

9450 

97.21 

21.14 

9680 

98.39 

21.31 

9910 

99.55 

21.48 

9220 

96.02 

20.97 

9460 

97.26 

21.15 

9690 

98.44 

21.32 

9920 

99.60 

21.49 

9230 

96.07 

20.98 

9470 

97.31 

21.16 

9700 

98.49 

21.33 

9930 

99.65 

21.4» 

9240 

96.12 

20.98 

9480 

97.37 

21.16 

9710 

98.54 

21.33 

9940 

99.70 

21.5a 

9250 

96.18 

20.99 

9490 

97.42 

21.17 

9720, 

98.59 

21.34 

9950 

99.75 

21.51 

9260 

96.23 

21.00 

9500 

97.47 

21.18 

9730 

98.64 

21.35 

9960 

99.80 

21.52 

9270 

96.28 

21.01 

9510 

97.52 

21.19 

9740 

98.69 

21.36 

9970 

99.85 

21.52 

9280 

96.33 

21.01 

9520 

97.57 

21.19 

9750 

98.74 

21.36 

9980 

99.90 

21.53 

9290 

96.38 

21.02 

9530 

97.62 

21.20 

9760 

98.79 

21.37 

9990 

99.95 

21.54 

9800 

96.44 

21.03 

9540 

97.67 

21.21 

9770 

98.84 

21.38 

10000 

100.00 

21.54 

9310 

96.49 

21.04 

To  find  Square    or  Cube  Roots  of  larg>e  numbers  not  con- 
tained in  tbe  column  of  numbers  of  tbe  table. 

Such  roots  may  sometimes  be  taken  at  once  from  the  table,  by  merely  regarding  the  columns  of 
powers  as  being  columns  of  numbers;  and  those  of  numbers  as  being  those  of  roots.  Thus,  if  the 
gq  rt  of  25281  is  reqd,  first  find  that  number  in  the  column  of  squares ;  and  opposite  to  it,  in  the 
column  of  numbers,  is  its  sq  rt  159.  For  the  cube  rt  of  857375,  find  that  number  in  the  column  of 
cubes ;  and  opposite  to  it,  in  the  col  of  numbers,  is  its  cube  rt  95.  When  the  exact  number  is  not  con- 
tained in  the  column  of  squares,  or  cubes,  as  the  case  may  be,  we  may  use  instead  the  number  nearest 
to  it,  if  no  great  accuracy  is  reqd.  But  when  a  considerable  degree  of  accuracy  is  necessary,  tb« 
following  very  correct  methods  may  be  used. 

For  tbe  square  root. 

This  rule  applies  both  to  whole  numbers,  and  to  those  which  are  partly  (not  wholly)  decimal.  First, 
i>  the  foregoing  manner,  take  out  the  tabular  number,  which  is  nearest  to  the  given  one  ;  and  also  itt 
tabular  sq  rt.  Mult  this  tabular  number  by  3  ;  to  the  prod  add  the  given  number.  Call  the  sum  Ju 
Then  mult  the  given  number  by  3 ;  to  the  prod  add  the  tabular  number.  Call  the  sum  B.  Then 
A  :  B  :  :  Tabular  root  :  Reqd  root. 
Ex.  Let  the  given  number  be  946.53.  Here  we  find  the  nearest  tabular  number  to  be  947 ;  and  Itl 
tabular  sq  rt  30.7734.     Hence, 

947  =  tab  num  'J  r  946.53  =  given  num. 

3  3 


2841 
946.53  =  giTen  num. 

J787,53  =  A. 


and      J,  28.39.59 

947      =r  tab  num. 

I3786.59  =  B. 


A.  B.  Tab  root.     Reqd  root. 

Then  3787.53    :    3786.59    :  :    30.7734    :    30.7657  -[-■ 

The  root  as  found  by  actual  mathematical  process  is  also  30.7657  -f-. 

For  the  cube  root. 

This  rule  applies  both  to  whole  numbers,  and  to  those  which  ure  partly  decimal.  First  take  out  ttJe 
tabular  number  which  is  nearest  to  the  given  one;  and  also  its  tabular  cube  rt.  Mult  this  tabular 
number  by  2  ;  and  to  the  prod  add  the  given  number.  Call  the  sum  A.  Then  mult  the  given  number 
by  2 ;  and  to  the  prod  add  the  tabular  number.  Call  the  sum  B.  Then 
A  :  B  :  :  Tabular  root  :  Reqd  root. 
Ex.  Let  the  given  number  be  7368.  Here  we  fiuu  tne  nearest  tabular  number  (in  the  column  ot 
cu6e«)  to  be  6859;  and  its  tabular  cube  rt  19.     Hence, 

6859  =  tab  num.       >  f     7368  =  given  num. 

—  — 

13718  y  and       ■{    147.36 

7368  =  given  num.  6859  =  tab  num. 


A.  B.  Tab  Root.    Reqd  Rt. 

Then,  as  21086      :      21595      :  :       19      ;      19.4585 
The  root  as  found  by  correct  mathematical  process  is  19.4588.    The  engineer  rarely  require*  « 


SQTJAEE  AND  CUBE  BOOTS. 


67 


this  degree  of  accuracy ;  for  his  purposes,  therefore,  this  process  is  greatly  preferable  to  the  ordinary 
laborious  one. 

To  find  the  square  root  of  a  number  mrbieli  is  wholly 
decimal. 

Very  simple,  and  correct  to  the  third  numeral  figure  inclusive.  If  the  number  does  not  contain  at 
least  five  figures,  counting  from  the  first  numeral,  and  including  it,  add  one  or  more  ciphers  to  make 
five.  If,  after  thai,  the  whole  number  is  not  separable  into  twos,  add  another  cipher  to  make  it  bo. 
Then  beginning  at  the  first  numeral  figure,  and  including  it,  assume  the  number  to  be  a  whole  one. 
In  the  table  find  the  number  nearest  to  this  assumed  one  ;  take  out  its  tabular  sq  rt;  move  the  deci- 
mal point  of  this  tabular  root  to  the  left,  half  a,a  many  places  as  the  finally  modified  decimal  number 
has  figures. 

Ex.  vvbat  is  the  sq  rt  of  the  decimal  .002?  Here,  in  order  to  have  at  least  five  decimal  figures, 
counting  from  the  first  numeral  (2),  and  including  it.  add  ciphers  thus,  .00.20.00.0.  But.  as  it  is  not 
now  separable  into  twos,  add  ano.ther  cipher,  thus,  .00,20,00,00.  Then  beginning  at  the  first  numeral 
(2),  assume  this  decimal  to  be  the  whole  number  200000.  The  nearest  to  this  in  the  table  is  199809; 
and  the  sq  rt  of  this  is  447.  Now.  the  decimal  number  as  finally  modihed,  namely,  .00.20,00.00,  has 
eight  figures  ;  one  half  of  which  is  4;  therefore,  move  the  decimal  point  of  the  root  447,  four  places  to 
the  left;  making  it  .0447.     This  is  the  reqd  sq  rt  of  .002,  correct  to  the  third  numeral  7  included. 

To  find  the  cube  root  of  a  number  nrhich  is  wholly  decimal. 

Very  simple,  and  correct  to  the  third  numeral  inclusive. 

If  the  number  does  not  contain  at  least  five  figures,  counting  from  the  first  numeral,  and  including 
It,  add  one  or  more  ciphers  to  make  five.  If.  after  that,  the  number  is  not  separable  into  threes,  add 
one  or  more  ciphers  to  make  it  so.  Then  beginning  at  the  first  numeral,  and  including  it.  assume 
the  number  to  be  a  whole  one.  In  the  table  find  the  number  nearest  to  this  assumed  one,  and  take 
out  its  tabular  cub  rt.  Move  the  decimal  point  of  this  rt  to  the  left,  one-third  as  many  places  as  the 
finally  modified  decimal  number  has  figures. 

Ex.  What  is  the  cube  rt  of  the  decimal  .002  ?  Here,  in  order  to  have  at  least  five  figures,  counting 
from  the  first  numeral  (2),  and  including  it,  add  ciphers  thus.  .002.000,0.  But  as  it  is  not  now  separ 
able  into,  threes,  add  two  more  ciphers  to  make  it  so:  thus,  .002,000.000.  Then  beginning  witn  the 
first  numeral  (2),  assume  the  decimal  to  he  the  whole  number  2000000.  The  nearest  cube  to  this  in 
the  table  in  the  column  of  cubes,  is  2000.^76;  and  its  tabular  cube  rt  as  found  in  the  col  of  numbers, 
is  126.  Now,  the  decimal  number  as  finally  modified,  namely,  .002  000  000.  has  nine  figures ;  one-third 
of  which  is  3;  therefore,  move  the  decimal  point  of  the  root  126,  three  places  to  the  left,  making  it 
.  126.    This  i»  the  reqd  cube  rt  of  the  decimal  .002,  correct  to  the  third  numeral  6  included. 


Fifth  roots  and  fifth  powers. 


Power. 

No.  or 
Root. 

Power. 

No.  or 
Root. 

Power. 

No.  or 
Root. 

Power. 

No.  or 
Root. 

Power. 

No.  or 
Root. 

Power. 

No.  or 
Root. 

.0000100 

.1 

.000142 

.170 

.004219 

.335 

.077760 

.60 

.695688 

.93 

8.11368 

1.52 

(J00164 

.175 

.004544 

.340 

.084460 

.61 

.733904 

.94 

8.66171 

1.54 

.0000110 

.102 

.000189 

.180 

.004888 

.345 

.091613 

.62 

.773781 

.95 

9.23896 

1.56 

.000217 

.185 

.005252 

.350 

.099244 

.63 

.815373 

.96 

9.8465« 

1.58 

.0000122 

.104 

.000248 

.190 

.005638 

.355 

.107374 

.64 

.858734 

.97 

10.4858 

1.60 

.000282 

.195 

.006047 

.360 

.116029 

.65 

.903921 

.98 

11.1577 

1.62 

.0000134 

.106 

.000:520 

.200 

.006478 

.365 

.125233 

.66 

.950990 

.99 

11.8637- 

1.64 

.0003o2 

.205 

.006934 

.370 

.135012 

.67 

1. 

1. 

12.6049 

1.66 

.0000147 

.108 

000408 

.210 

.007416 

.375 

.145393 

.68 

1.10408 

1.02 

13.. 3828 

1.68 

.0000161 

.110 

.000459 

.215 

.007924 

.3S0 

.156403 

.69 

1.21665 

1.04 

14.1986 

l.TO 

.0000176 

.112 

.000515 

.220 

.008459 

.385 

.168070 

.70 

1.3.3823 

1.06 

15.0537 

1.72 

.0000193 

.114 

.000577 

.225 

.009022 

.390 

.180423 

.71 

1.46933 

1.08 

15.9495 

1.74 

.0000210 

.116 

.000644 

.230 

.009616 

.395 

.193492 

.72 

1.61051 

1.10 

16.8874 

1.76 

.0000229 

.118 

.000717 

.235 

.010240 

.400 

.207307 

.73 

1.76234 

1.12 

17.8690 

1.78 

.0000249 

.120 

.000796 

.240 

.011586 

.41 

.221901 

.74 

1.92541 

1.14 

18.8957 

1.80 

.0000270 

.122 

.000883 

.245 

.013069 

.42 

.237.305 

.75 

2.10034 

1.16 

19.9690 

1.82 

.0000293 

.124 

.000977 

.250 

.014701 

.43 

.253553 

.76 

2.28775 

1.18 

21.0906 

1.84 

.0000318 

.126 

.001078 

.255 

.016492 

.44 

.270678 

.77 

2.48832 

1.20 

22.2620 

1.86 

.0000344 

.128 

.001188 

.260 

.018453 

.45 

.288717 

.78 

2.70271 

1.22 

23.4849 

1.88 

.0000371 

.130 

.001307 

.265 

.020596 

.46 

.307706 

.79 

2.93163 

1.24 

24.7610 

1.90 

.0000401 

.132 

.001435 

.270 

.022935 

.47 

.327680 

.80 

3.17580 

1.26 

260919 

1.92 

.0000432 

.134 

.001573 

.275 

•025480 

.48 

.348678 

.81 

3.43597 

1.28 

27.4795 

1.94 

.0000465 

.136 

.001721 

.280 

.028248 

.49 

.370740 

.82 

3.71293 

1.30 

28.9255 

1.96 

.0000500 

.138 

.001880 

.285 

.031250 

.50 

.393904 

.83 

4.00746 

1.32 

30.4317 

1.98 

.0000538 

.140 

.002051 

.290 

.a34.503 

.51 

.418212 

.84 

4.32040 

1.34 

32.0000 

2.00 

.0000577 

.142 

.002234 

.295 

.0.38020 

.52 

.443705 

.85 

4.65259 

1.36 

36.2051 

2.05 

.0000619 

.144 

0024.30 

.300 

.041820 

.53 

.470427 

.86 

5.00490 

1.38 

40.8410 

2.10 

.0000663 

.146 

.0026.'i9 

.305 

.045917 

.54 

.498421 

.87 

5.37824 

1.40 

45.9401 

2.15 

.0000710 

.148 

.002863 

.310 

.050328 

.55 

..527732 

.88 

5.77353 

1.42 

51.5363 

2.20 

.0000754 

.150 

.003101 

.315 

.055073 

.56 

.558406 

.89 

6.19174 

1.44 

57.6650 

2.25 

.0000895 

.155 

.00.3355 

.320 

.060169 

.57 

..590490 

.90 

6.6.3383 

1.46 

64.3634 

2.30 

.000105 

.160 

.003626 

.325 

.06.56.36 

.58 

.624032 

.91 

7.10082 

1.48 

71.6703 

2.35 

.000122 

.165 

.003914 

.330 

.071492 

.59 

.659082 

.92 

7.59375 

1.50 

79.6262 

2.40 

ROOTS   AND   POWERS. 


Fiftb  roots  and  fiftb  powers— (Continued). 


88.2735 
97.6562 
107.820 
118.814 
130.686 
143.489 
157.276 
172.104 
188.029 
205.111 
223.414 
243.000 
263.936 
286.292 
310.136 
335.544 
362.591 
391.354 
421.419 
454.354 
488.760 
525.219 
563.822 
604.662 
647.835 
693.440 
741.577 
792.352 
845.870 
902.242 
961.580 
1024.00 
1089.62 
1158.56 
1230.95 
1306.91 
1386.58 
1470.08 
1557.57 
1649.16 
1745.02 
1845.28 
1950.10 
2059.63 
2174.03 
2293.45 
2418.07 
2548.04 
2683.54 


No.  or 
Root. 


3.00 
3.05 
3.10 
3.15 
3.20 
3.25 
3.30 
3.35 
3.40 
3.45 
3.50 
3.55 
3.60 
3.65 
3.70 
3.75 
3.80 
3.85 
3.90 
3.95 
4.00 
4.05 
4.10 
4.15 
4.20 
4.25 
4.30 
4.35 
4.40 
4.45 
4.50 
4.55 
4.60 
4.65 
4.70 


No.  or 
Root. 


2824.75 
2971.84 
3125.00 
3450.25 
3802.04 
4181.95 
4591.65 
5032.84 
5507.32 
6016.92 
6563.57 
7149.24 
7776.00 
8445.96 
9161.33 
9924.37 

10737 

11603 

12523 

l;5501 

14539 

15640 

16807 

18042 

19349 

20731 
2190 

23730 

25355 

27068 

28872 

30771 

32768 

34868 

37074 

;59390 

41821 

44371 

47043 

49842 

52773 

55841 

59049 

62403 

65908 

69569 

73390 
7378 

81537 


85873 
90392 
95099 
100000 
110408 
121665 
133823 
146933 
161051 
176234 
192541 
210034 
228776 
248832 
270271 
2931G3 
317580 
343597 
371293 
400746 
432040 
465259 
500490 
537824 
577353 
619174 
663383 
710082 
759375 
811368 
866171 
923896 
984658 
1048576 
1115771 
1186367 
1260493 
1338278 
1419857 
1505366 
1594947 
1688742 
1 786899 
1889568 
1996903 
2109061 
2226203 
2348493 
2476099 


No.  or 

Root. 


2609193 
2747949 
2892547 
3043168 
3200000 
3363232 
3533059 
3709677 
3893289 
4084101 
4282322 
4488166 
4701850 
4923597 
5153632 
5392186 
5639493 
5895793 
6161327 
6436343 
6721093 
7015834 
7320825 
7636332 
7962624 
8299976 


9008978 
9381200 
9765625 
10162.550 
18572278 
10995116 
11431377 
11881376 
12345437 
12823886 
13317055 
13825281 
14348907 
14888280 
15443752 
16015681 
16604430 
17210368 
17833868 
18475309 
19135075 
19813557 


No.  or 
Boot. 


No.  or 

Root. 


20511149 
21228253 
21965275 
22722628 
23500728 
24300000 
26393634 
28629151 
31013642 
33554432 
36259082 
39135393 
42191410 
45435424 
48875980 
52521875 
5638216: 
604661 7( 
6478348^ 
6934395': 
74157715 
79235168 
84587005 
90224199 
96158012 
102400000 
108962013 
115856201 
123095020 
130691232 
138657910 
147008443 
155756538 
164916224 
174501858 
184528125 
195010045 
205962976 
217402615 
229345007 
241806543) 
254803968 
268354383 
282475249 
297184391 
312500000 
345025251 
380204032 
418195493 


29.0 
29.2 
29.4 
29.6 
29.8 
30.0 
30.5 
31.0 
31.5 
32.0 
32.5 
33.0 
33.5 
34,0 
34.5 
35.0 
35.5 
36  0 
36  5 
37.0 
37.5 
38.0 
38.5 
39.0 
39.5 
40.0 
40.5 
41.0 
41.5 
42.0 
42.5 
43.0 
43.5 
440 
44.5 
45.0 
45.5 
46.0 
46.5 
47.0 
47.5 
48.0 
48.5 
49.0 
49.5 
50.0 


459165024 

503284375 

550731776 

601692057 

656356768 

714924299 

777600000 

844596301 

916132832 

992436543 

1073741824 

1160290625 

1252332576 

1350125107 

1453933568 

1564031349 

1680700000 

1804229351 

1934917632 

2073071593 

2219006624 

2373046875 

25355253' 

2706784157 

2887174368 

3077056399 

3276800000 

3486784401 

07398432 

3939040643 

4182119424 

4437053125 

4704270176 

4984209207 

(319168 

5584059449 

5904900000 

6240321451 

6590815232 

6956883693 

7339040224 

7-73780J 

8153726976 

8587340257 

9039207968 

9509900499 


Square  roots  of  fifth  powers  of  numbers,  l/n^, 
or  %  powers  of  numbers,  n%. 

See  table,  page  69. 

The  column  headed  "12  n"  facilitates  the  use  of  the  table  in  cases  where, 
for  instance,  the  quantity  is  given  in  inches^  atid  where  it  is  desired  to  obtain 
the  %  power  of  the  same  quantity  in  feet.  Thus,  suppose  we  have  a  i,^  inch 
pipe,  and  we  require  the  %  power  of  the  diameter  in  feet.  Find  i^  (the 
diameter,  in  inches)  in  the  column  headed  "  12  n,"  opposite  which,  in  the  column 
headed  "n,"  is  0.041666  (the  diameter,  in  feet),  and,  in  column  headed  "  n%," 
0.00035  (the  %  power  of  the  diameter,  0.041666,  in  feet). 

Values  of  n,  ending  in  0  or  in  5,  are  exact  values.  All  others  end  in  repeat- 
ing decimals.     Thus:  n  =  0.052083  signifies  n  =  0.052083333 


ROOTS  AND   POWERS. 


69 


Square  roots  of  fifth  powers  of  numbers.    See  page  6 


12  11 

n 

n^ 

12  u 

n 

ni 

12  n 

n 

nt 

12  n 

n 
39 

ni 

Va 

0.020833 

0.00006 

22 

1.8333 

4.551 

84 

7.000 

129.64 

468 

9499 

3|0.031250 

0.00017 

23 

1.9166 

5.086 

85 

7.083 

133.53 

480 

40 

10119 

y' 

0.041666 

0.00035 

24 

2.0000 

5.657 

86 

7.166 

137,50 

492 

41 

10764 

0.052083 

0.00062 

25 

2.0833 

6.265 

87 

7.250 

141.53 

504 

42 

11432 

0.06250010.00098 

26 

2.1666 

6.910 

88 

7,333 

145.63 

516 

43 

12125 

0.07291610.00144 

27 

2.2500 

7.594 

89 

7.416 

149.80 

528 

44 

12842 

1 

0.083333 

0.0020 

28 

2.3333 

8.317 

90 

7.500 

154.05 

540 

45 

13584 

Ys 

0.093750 

0.0027 

29 

2.4166 

9.079 

91 

7.583 

158.36 

552 

46 

14351 

i 

0.104166 

0.0035 

30 

2.5000 

9.882 

92 

7.666 

162.75 

564 

47 

15144 

0.114583 

0.0044 

31 

2.5833 

10.73 

93 

7.750 

167.21 

576 

48 

1596a 

0.125000 

0.0055 

32 

2.6666 

11.61 

94 

7.833 

171.74 

588 

49 

16807 

^'0.135416 

0.0067 

33 

2.7500 

12.54 

95 

7.916 

176.34 

600 

50 

17678 

% 

0.145833 

0.0081 

34 

2.8333 

13.51 

96 

8.000 

181.02 

612 

51 

18575 

% 

0.156250 

0.0097 

35 

2.9166 

14.53 

97 

8.083 

185.77 

624 

52 

19499 

2 

0.166666 

0.0113 

36 

3.0000 

15.59 

98 

8.166 

190.60 

636 

53 

20450 

^ 

0.187500 

0.0152 

37 

3.0833 

16.69 

99 

8.250 

195.49 

648 

64 

21428 

0.208333 

0.0198 

38 

3.1666 

17.85 

100 

8.333 

200.47 

660 

55 

22434 

0.229166 

0.0251 

39 

3.2500 

19.04 

102 

8.50 

210.64 

672 

56 

23468 

3^ 

0.250000 

0.0313 

40 

3.3333 

20.29 

105 

8.75 

226.47 

684 

57 

24529 

M 

0.270833 

0.0382 

41 

3.4166 

21.58 

108 

9.00 

243.00 

696 

58 

25619 

H 

0.291666 

0.0459 

42 

3.5000 

22.92 

111 

9.25 

260.23 

708 

59 

26738 

0.312500 

0.0546 

43 

3.5833 

24.31 

114 

9.50 

278.17 

720 

60 

27885 

4 

0.333333 

0.0642 

44 

3.6666 

25.74 

117 

9.75 

296.83 

732 

61 

29062 

0.354166 

0.0746 

45 

3.7500 

27.23 

120 

10.0 

316.23 

744 

62 

30268 

1^ 

0.375000 

0.0861 

46 

3.8333 

28.77 

126 

10.5 

357.25 

756 

63 

31508 

% 

0.395833 

0.0986 

47 

3.9166 

30.36 

132 

11.0 

401.31 

768 

64 

32768 

5 

0.416666 

0.1121 

48 

4.0000 

32.00 

138 

11.5 

448.48 

780 

65 

34068 

^ 

0.437500 

0.1266 

49 

4.0833 

33.69 

144 

12.0 

498.83 

792 

66 

35388 

0.458333 

0.1422 

50 

4.1666 

35.44 

150 

12.5 

552.43 

804 

67 

36744 

3^0.479166 

0.1589 

51 

4.2500 

37.24 

156 

13.0 

609.34 

816 

68 

3813() 

6^^ 

0.500000 

0.1768 

52 

4.3333 

39.09 

162 

13.5 

669.63 

828 

69 

39548 

y^ 

0.541666 

0.2159 

53 

4.4166 

41.00 

168 

14.0 

733.36 

840 

70 

4099& 

7 

0.583333 

0.2599 

54 

4.5000 

42.96 

174 

14.5 

800.61 

852 

71 

42476 

3^ 

0.625000 

0.3088 

55 

4.5833 

44.97 

180 

15.0 

871.42 

864 

72 

43988 

8 

0.666666 

0.3629 

56 

4.6666 

47.05 

186 

15.5 

945.87 

876 

73 

45531 

y-i 

0.708333 

0.4223 

57 

4.7ri00 

49.17 

192 

16.0 

1024.00 

888 

74 

47106 

9 

0.750000 

0.4871 

58 

4.8333 

51.36 

198 

16.5 

1105.9 

900 

75 

48714 

K 

0.791666 

0.5576 

59 

4.9166 

53.60 

204 

17.0 

1191.6 

912 

76 

50354 

10^ 

0.833333 

0.6339 

60 

5.0000 

55.90 

210 

17.5 

1281.1 

924 

77 

52027 

y<i 

0.875000 

0.7162 

61 

5.0833 

58.26 

216 

18.0 

1374.6 

936 

78 

53732 

11  ^ 

0.916666 

0.8045 

62 

5.1666 

60.677 

222 

18.5 

1472.1 

948 

79 

55471 

K 

0.958333 

0.8991 

63 

5.2500 

63.154 

228 

19.0 

1573.6 

960 

80 

57248 

12 

1.000000 

1.0000 

64 

5.3333 

65.690 

234 

19.5 

1679.1 

972 

81 

59049 

3^  1.041666 

1.107 

65 

5.4166 

68.286 

240 

20 

1788.9 

984 

82 

60888 

13 

1.083333 

1.222 

66 

5.5000 

70.943 

252 

21 

2020.9 

996 

83 

62762 

% 

1.125000 

1.342 

67 

5.5833 

73.660 

264 

22 

2270.2 

1008 

84 

64669 

14 

1.166666 

1.470 

68 

5.6666 

76.440 

276 

23 

2537.0 

1020 

85 

66611 

3^ 

1.208333 

1.605 

69 

5.7500 

79.281 

288 

24 

2821.8 

1032 

86 

68588 

15 

1.250000 

1.747 

70 

5.8333 

82.185 

300 

25 

3125.0 

1044 

87 

70599 

K 

1.291665 

1.896 

71 

5.9166 

85.152 

312 

26 

3446.9 

1056 

88 

72645 

16  ^ 

1.333333 

2.053 

72 

6.0000 

88.182 

324 

27 

3788.0 

1068 

89 

74727 

K 

1.375000 

2.217 

73 

6.0833 

91.276 

336 

28 

4148.5 

1080 

90 

76848 

17 

1.416666 

2.389 

74 

6.1666 

94.434 

348 

29 

4528.9 

1092 

91 

78996 

3^ 

1.458333 

2.568 

75 

6.2500 

97.656 

360 

30 

4929.5 

1104 

92 

81184 

18 

1.500000 

2.756 

76 

6.3333 

100.94 

372 

31 

5350.6 

1116 

93 

83408 

3^ 

1.541666 

2.951 

77 

6.4166 

104.30 

384 

32 

5792.6 

1128 

94 

85668 

19^^ 

1.583333 

3.155 

78 

6.5000 

107.72 

396 

83 

6255.8 

1140 

95 

•87965 

K 

1.625000 

3.366 

79 

6.5833 

111.20 

408 

34 

6740.6 

1152 

96 

90298 

20  11.666666 

3.586 

80 

6.6666 

114.76 

420 

35 

7247.2 

1164 

97 

92668 

341.708333 

3.814 

81 

6.7500 

118.38 

432 

36 

7776.0 

1176 

98 

95075 

21   1.750000 

4.051 

82 

6.8333 

122.06 

444 

37 

8327.3 

1188 

99 

97519 

K  l.")  91666 

4.297 

83 

6.9166 

125.82 

456 

38 

8901.4 

1200 

100 

100000 

70  LOGARITHMS. 

liOg^arithius. 

(1)  Tables  of  logarithmsgreatly  facilitate  multiplication  and  division  and 
the  finding  of  powers  and  roots  of  numbers.* 

(2)  The  tables  pp.  78  to  91  contain  the  common,  decimal  or  Brig-gs 

logpari thins  of  numbers.  The  common  logarithm  of  a  number  is  the  ex- 
ponent or  index  of  that  number  as  a  power  of  10.  See  (18).  Thus :  1000  ^  108, 
and  log  1000  (logarithm  of  1000)  -=  3.00  000.  Similarly,  28.7  =  10  1.45  788,  and 
log  28.7  =  1.45  788. 

(3)  In  general,  let  A  and  B  be  any  two  numbers,  and  k  any  exponent. 
Then,  from  the  principle  of  powers  and  roots,  we  have 

(1)  log  AB  =  log  A  +  log  B  ;        (2)  log  |  =  log  A  -  log  B ; 

(3)  log  A*  =  A;  (log  A) ;  .        (4)  log  ^/^  =  --|^ 

In  other  words  :  — 

(1)  Log  of  product  =  sum  of  lo^  of  factors. 

(2)  Log  of  quotient  -^  log  of  dividend  —  log  of  divisor. 

Or  log  of  fraction  -=  log  of  numerator  —  log  of  denominator. 
^3)  Log  of  power  ^  log  of  number,  multiplied  by  exponent. 

(4)  Log  of  root     =  log  of  number,  divided  by  exponent. 

(4)  From  what  has  been  said,  it  follows  that 

=--  log  10-1  ^  1.00  coot 

=    log  10-2   =    2.00  000 
=    log  10-3   =    3.00  000 

(5)  Each  common  logarithm  is  a  mixed  number,  consisting  of  an  integral 
portion,  called  the  cliaracteristic  or  index  {preceding  the  decimal 
point),  ^nd  a,  fractional  portion,  called  the  mantissa  (/oZZommgr  the  decimal 
point).  The  table  gives  only  the  mantissa  of  each  log,  the  characteristic 
being  found  as  explained  below.  The  mantissa  is  always  positive.  The 
characteristic,  in  the  log  of  any  whole  or  mixed  number,  is  positive,  and  ia 
equal  to  the  number  of  digits  in  the  whole  number,  minus  1;  while  the 
characteristic  of  a  ivholly  decimal  number  is  negative,  and  is  numerically 
equal  to  1  plus  the  number  of  ciphers  immediately  following  the  decimal 
point.    Thus: 


Log 

100 

-       log  102       = 

=      2.00  000 

Log  0.1 

Log 

10 

==     log  101      = 

=     1.00  000 

Log  0.01 

Log 

1 

=     log  100     - 

=     0.00  ooot 

Log  0.001 

log  2870       =  3.45  788 

log  0.287       =1.45-788t 

"287       =  2.45  788 

"    0.0287      =  2.45  788 

"       28.7    =r  1.45  788 

"    0.00287    -  3.45  788 

2.87  =  0.45  788 

•'    0.000287  =  4.45  788 

It  will  be  noticed  that  the  mantissa  remains  constant  for  any  given  com- 
bination of  significant  figures  in  a  number,  wherever  the  decimal  point  in 
the  number  be  placed  ;  while  the  characteristic  depends  solely  upon  the 
position  of  the  decimal  point  in  the  number. 

(6)  Let  the  number  be  resolved  into  two  factors,  one  of  which  is  an 
Integer  power  of  10,  whiles  the  other  is  greater  than  1  and  less  than  10.  Then 
the  index  of  the  power  of  10  is  the  characteristic  of  the  logarithm,  and  the 
logarithm  of  the  other  factor  is  the  mantissa.  Thus,  2870  =  1000  X  2.87  = 
103  X  2.87,  and  the  logarithm  of  2870  (3.45  788)  is  the  sum  of  the  exponent  3 
(or  3.00  000)  and  the  log  (0.45  788)  of  2.87.t 

*  Logarithms  not  being  exact  quantities,  operations  performed  with  them 
are  subject  to  some  inaccuracy,  especially  where  a  logarithm  is  multiplied 
by  a  large  number,  the  existing  error  being  thus  magnified.  Logarithms  of 
only  five  places  in  the  mantissa  usually  suffice  for  calculations  with  num- 
bers of  four  or  five  places.  Greater  accuracy  is  obtained  by  the  use  of 
tables  of  logarithms  carried  out  to  seven  places. 

t  Log  1  =  log  \%  =--  log  10— log  10  :=  1—1  =  0 ;  or  1  =  IQO. 
Log  0.1  =  log  T^^  =  log  1  —  log  10  =  0  —  1  =  1.0 ;  or  0.1  =  lO-i. 

X  0.287  =  2.87  -^  10.  Hence,  log  0.287  =  log  2.87  -  log  10  =  0.45  788  -  1, 
which,  for  convenience,  is  written  1^45  788.  See  (16).  Similarly,  log  0.0287 
=  log  2.87  —  log  100  =  0.45  788  —  2  =  2.45  788. 


LOGARITHMS. 


71 


(7)  To  find  the  logaritbin  of  a  number.  The  short  table  on  pages 
78,  79  gives  logs  of  numbers  up  to  1000.  The  longer  table,  pages  80  to  91, 
gives 

(1)  The  mantissa  for  each  number  from  1000   to    1750 

(2)  The  mantissa  for  each  even  number  from  1750   to    3750 

(3)  The  mantissa  for  each  fijlh  number  from  3750    to  10000 

(8)  Logs  of  numbers  intermediate  of  those  given  in  the  tables  are 
found  by  simple  proportion.  The  procedure  necessary  in  these  cases  is 
explained  in  the  examples  given  in  connection  with  the  tables,  but  it  will 
often  be  found  sufficiently  accurate  to  use  the  log  of  the  nearest  numiber 
given  in  the  table,  neglecting  interpolation. 

The  antilog'aritiim  or  num  log-  {numerus  logarithmi)  is  the  num- 
ber corresponding-  to  a  given  logarithm.  Thus,  log.  2  =  0.30  103,  and 
antilog  0.30  103  ==  2. 

(9)  Multiplication.  To  multiply  together  two  or  more  numbers,  add 
together  their  logs  and  find  the  antilog  of  their  sum.  See  Proportion 
(11)  below. 

(10)  Division.  Subtract  the  log  of  the  divisor  from  that  of  the  dividend, 
and  find  the  antilog  of  the  remainder.    See  Proportion  (11)  below. 

The  reciprocal  of  any  number,  n,=    .    Seepage  52.      Thus,  recip  2  = 
-  =  0,5.    Hence,  log  recip  n  =  log     =  log  1  —  log  w  =  0  —  log  n. 
Similarly,   log  recip  —  =  log  -^ =0  —  log  -  . 

'T'^"^'       '°s  '^?  = » -  i«s  fi^  = "  -  ^-"^  "^  =  *-*3  ««^- 

Since  w2— i  =  n^  ^  — ,  n^— i  =  n"  =  -  =  1,  n^— ^  =n-i  =     ,  and  n^— 2  =  71-* 

n  ^n 

r  -  =  log  recip  n  ;  log  w— 2  =  log  — 


■-  -j  it  follows  that  log  n-i  =  log  -  =  log  recip  n  ;  log  w— 2  =  log  —   =  log 


recip  n-,  etc, 
(11)  Proportion.    Examplfe.  6,3023  :  290,19  =  1260.7  :  ? 

TV,    !*•   1     XT  (  Log   290.19      -2.46  269 

Multiply  Nos,      J  -     1260,7        =  3.10  062 

Add  Logs.  (^   ^^^   290.19  X  1260.7        =  5.56  331 

Divide  Nos,         /  Log         6.3023  =  0.79  950 

Subtract  Log.      t  Log  58051  ^  4.76  381 

The  true  value  is  58049.05  + 

(13)  Instead  of  subtracting  the  log  of  the  divisor,  we  may  add  its  eologa- 
ritbm  or  arithmetical  complement,  which  is  log  of  reciprocal 
of  divisor,  —  0  —  log  divisor  =  10  —  log  divisor  —  10.    Thus  : 

1523         _ 

3,332  X  8,655 

'    Log      1523  =    3.18  270 

Colog  3.332  -=  10  —  log  3.332  —  10  =  10  —  0.52  270  —  10  =    9.47  730  —  10 

Colog  8.G55  -  10  —  log  8,655  —  10  =  10  —  0.93  727  —  10  =    9.06  273  —  10 

Sum  of  logs  and  colbgs  =  21.72  273  —  20, 

=  Log  52.813  =    1.72  273 
The  true  value  is  52.8114  + 

(13)  luTolution,  or  finding  powers  of  numbers.  Multiply  log  of 
given  number  bv  the  exponent  of  the  required  power,  and  find  the  anti- 
log  of  the  product.    Thus :  363  =  ? 

Log  36  =  1.55  630.    1,55  630  X  3  =  4.66  890.     Antilog  4.66  890  =  46656. 

(14)  Evolution,  or  finding  roots  of  numbers.  Divide  log  of  given 
number  by  exponent  of  required  root,  and  find  antilog  of  quotient.    Thus : 

1/46656  =  ?    Log  46656  =  4.66  890.  4,66  890  -r-  3  =  1.55  630,   Antilog  1,55  630  =  36. 

(15)  In  finding  roots  of  numbers,  if  the  given  number  is  a  whole  or  mixed 


72 


LOGARITHMS. 


number,  the  division  of  the  log  is  performed  in  the  nsual  way,  as  in  the 
preceding  example,  even  where,  as  in  that  example,  the  characteristic  is 
not  exactly  divisible  by  the  exponent  of  the  required  root.  But  if  tlie 
number  is  a  fraction,  and  the  characteristic  of  its  log  therefore  nega- 
tive, and  if  the  characteristic  is  not  exactly  divisible  by  the  exponent,, 
division  in  the  usual  way  would  give  erroneous  results,  in  such  cases  we 
may  add  a  suitable  number  to  the  mantissa  and  deduct  the  same  number 


from  the  characteristic,  thusj  to  find  Vo.00048.  Log  0.00048  =  4.68  124  = 
0.68  124  —  4  =  2.68  124  —  6  =  6  -f  2.68  124,  which,  divided  by  3,  =  2  -f  0.89  375 
=  2.89  375  =  log  0.0783.     Or,  see  (16)  and  (17). 

(16)  To  avoid  inconvenience  from  the  use  of  neg-ative  cliaracter- 
istics,  it  is  customary  to  modify  them  by  adding  10  to  them,  afterward 
deducting  each  such  10  from  tjie  sum,  etc.,  of  the  logarithms.  Thus :  in 
multiplying  or  dividing  7425  by  0  25,  we  have 


either 


Multiplying, 
log  7425       =    3.87  070 

Dividing. 

-  3.87  070 

log       0.25  -    1.39  794 

=  1.39  794 

3.26  864 

4.47  276 

log  7425       -    3.87  070 
ified  log       0.25  =    9.89  794  - 

10 

=  3.87  070 

=  9.39  794  —  10 

13.26  864  — 
=    3.26  864 

10 

6.47  276  +  10 
=  4.47  276 

In  most  cases  the  actual  process  of  deducting  the  added  tens  may  be 
neglected,  the  nature  of  the  work  usually  being  such  that  an  error  so  great 
as  that  arising  from  such  neglect  could  hardly  pass  unnoticed. 

(17)  To  divide  a  modified  logaritlim,  add  to  it  such  a  multiple  of 
10  as  will  make  the  sum  exceed  the  true'log  by  10  times  the  divisor.  Thus : 
to  divide  log  0.00048  by  3.  Log  0.00048  -=  4.68  124,  which,  divided  by  3,  = 
2.89  375.    See  (15). 

Log  0.00048=    4.68  124 
Modified  log  0.00048  =    6.68  124  —  10 

Add  2  X  10      20 —  20 

•  Dividing  by  3)  26.68  124  —  30 

we  obtain  8.89  375  —  10,  which  is  2.89  375  modified. 

(18)  Except  1,  any  number  can  (like  10)  be  made  the  base  of  a  system  of 
logarithms.  The  base  of  the  hyperbolic,  Napierian,  or  natural 
logarithms,  much  used  in  steam  engineering,  is 

i  +  i  +  r^2  +  ixks  +  1x^x1  +  •  •  ■  •  ="'  «2«  + 

and  is  called  e  (epsilon)  or  c. 

M  =  logi  oe  (common  log  e)  =  0.43  429 ;  ^  =log  e  10  (hyperbolic  log  10) =2.30  259. 
For  any  number,  n, 
loge  n  =  ~^^--  =  2.30259  log^o  n ;  logio  7i  =  M  loge  n  =  0.43429  loge  n 

(19)  Whatever  may  be  the  base  chosen  for  a  system  of  logs,  the  man- 
tissas of  the  logs  of  any  given  numbers  bear  a  constant  ratio  to  each 

other.     Thus,  in  any  system  of  logs,  log  4  is  always  =  2  X  log  2,  and 
=■■  %X  log  8,  etc.,  etc. 

(20)  liOg-arithmic  sines,  tangents,  etc.  of  angles  are  the  logs  of 
the  sines,  tangents,  etc.  of  those  angles.  Thus,  sin  30°  =  0.5000000,  and  log 
sin  30°  =  log.  0.5  =  1.69  897,  usually  written  9.69  897  —  10,  or  simply  9.69  897. 

(31)  Since  no  power  of  a  positive  number  can  be  negative,  negative  num- 
bers properly  have  no  logs;  but  operations  with  nejfatfive  num- 
bers can  nevertheless  be  performed  by  means  of  logs,  by  treating  all  the 
numbers  as  positive  and  taking  care  to  use  the  proper  sign  ,+  or  — ,  in  the 
result. 


LOGARITHMIC    CHART    AND    SLIDE    RULE. 


73 


1.1- 

Liogs. 
l.O- 

0.9- 

0.8- 

0.7- 

O.G- 

O.S 

0.4-] 

0.3 

0.2 


zi 1 1 1 1 1 1 \ \ \ \ 1 r 

Logs.  1.0     0.0      0.1       0.2      0.3      0.4      0.5      O.G      0.7      0.8       0.9      1.0       1.1 


Zoff9. 

1.8     O.O      0.2    0.4:      0.6     0.8      l.O     1.2      1.4      1.6     1.8      2.0     2.2 


2fos.      U 1 


3     4   5  67891 


3     4    4t   67S91M 


Bl 

C  1 


2        3     4    5  0  7  SOI 
1.5  2  3 


2        3     4   5  67  891 
4         5       6      7     8    9    Ic 


No8.    [nx 


.      .      .  -i— r^  , 

6      7     S    O    ID] 


i.O     O.O      0.1      0.2     0.3     0.4      0.5      O.G     0.7      0:8     O.O      l.O     1.1 
Logs,  , 

The  liO^arithmie  Chart  and  the  Slide  Role. 

(1)  By  means  of  a  logarithmic  chart  or  diasrram  (often  miscalled  loga- 
rithmic cross-section  paper)  losrarithmic  operations  are  performed  graphi- 
cally, and  by  means  of  the  slide  rule  mechanically,  without  reference 
to  the  logarithms  themselves  *.  But  see  t.  P  76.  Their  use  greatly  facili- 
tates many  hydraulic  and  other  engineering  computations. 


(*)  The  ratio  between  the  mantissas  of  the  logs  of  any  given  numbers 
being  constant  for  all  systems  of  logs,  the  ratio  between  the  distances  laid 
off  on  the  chart  or  slide  rule  is  the  same  for  all  systems,  and  the  use  of  the 
chart  or  rule  is  independent  of  tKe  system  of  logs  used. 


74 


LOGARITHMIC  CHART  AND  SLIDE  RULE. 


(2)  Tli*»  lotf-arithmic  cbart  consists  primarily  of  a  square,*  on  the 
sides  of  which  the  distances  marked  1-2,  1-3,  etc.,  are  laid  off  by  scale 
according  to  the  logs  (0.30  103,  0.47  712,  etc.)  of  2,  3,  etc.  Ordinary 
"  squared  "  or  cross  section  paper  may  of  course  be  used  for  loga- 
rithmic plotting,  by  plotting  on  it  the  logs  instead  of  their  Nos.  Lines 
representing  Nos.  may  be  drawn  in  their  proper  places  as  desired. 

(3)  As  ordinarily  constructed,!  the  sli<le  rule  consists  essentially  of 
four  scales,  A,  B,  C,  and  D.  see  (17),  scales  A  and  D  being  placed  ori  the 
"  rule,"  while  B  and  C  are  placed  upon  the  sliding  piece,  or  "  slide."  As 
in  the  logarithmic  chart,  see  (2),  the  scales  are  divided  logarithmically 
(see  figure),  but  marked  with  the  numbers  corresponding  to  the  logs.  Scales 
A  and  B  are  equal,  as  are  also  scales  C  and  D,  but  a  given  length  on  A  or  B 
represents  a  logarithm  twice  as  great  as  on  C  or  D.  See  (4).  Hence,  each 
number  marked  on  A  is  the  square  of  the  coinciding  number  marked  on  D. 

(4)  A  single  logarithmic  scale  is  usually  numbered  from  1  to  10,  or  from 
10  to  100;  but  it  may  be  taken  as  representing  any  series  embracing  the 
numbers  from  10"  to  10'*^+  ^;  as  from  0.1  to  1.0  (n  =  —1);  or  from  1.0  to 
10.0  (n  =  0);  or  from  10.0  to  100.0  (n  =  1);  or— etc.,  etc.  Here  n  and 
n  +  1  are  the  characteristics  of  the  corresponding  logarithms. 

A  single  scale  would  therefore  serve  for  all  values,  from  0  to  infinity; 
but  for  convenience  several  contiguous  scales  are  sometimes  added,  as  in 
the  log  chart*. 

When  a  line  reaches  the  limit  of  a  square,  the  next  square  may  be 
entered*  or  the  same  square  may  be  re-entered  at  a  point  directly  opposite. 
Thus,  in  the  case  of  line  x%  (=  l^x^). 


Line  marked 

between 

corresponds  to  values  of 

a;  from 

x%  from 

(1) 
(2) 
(3) 
(4) 

1    and  S 
Sj  and  Sa 
S3  and  S4 
Se  and  H 

Ito      10 
10  to      31.62 
31.62  to    100 
100       to  1000 

1      to      4.64 
4.64  to    10 
10      to    21.54 
21.54  to  100 

Note  that  the  numbers,  marked  on  any  given  scale,  must  be  taken  as  10 
times  the  corresponding  numbers  marked  in  the  next  scale  preceding,  and 
the  characteristics  therefore  as  being  greater  by  1,  and  vice  versa.  Thus,  in 
our  figure,  log  1.5  -f  log  2  =  1-1.5  +  1-2  =  log  3  =  distance  1-M.  But 
log  15  +  log  20  =  (1-1.5  +  1-10)  +  (1-2  4- 1-10),  so  that  the  characteristic 
of  the  resulting  log  is  greater  by  2,  and  the  3  representing  the  product  of  15 
and  20  is  really  in  the  second  square  to  the  right  of  that  shown.  In  finding 
powers  and  roots,  remember  that  multiplying  or  dividing  the  number  by 
0.1, 10, 100,  etc.  a.  c.,. changing  the  characteristic  of  its  log),  changes  also  the 
mantissa  of  the  log  of  its  power  or  root.  Thus,  1^277  =  1.39 . . ,  (log  =  0.14  379) ; 
but  i>'27  =  3,  (log  =  0.47  712)  and  1^270  =  6.46  .  .  ,  (log  =  0.81  023).  The 
chart  or  rule  gives  all  such  possible  roots,  and  care  must  be  taken  to  select 
the  proper  one.  Most  operations  exceed  the  limits  of  one  scale,  and  facility 
in  using  either  instrument  depends  largely  upon  the  ability  to  pass  readily 
and  correctly  from  one  scale  to  another.  This  ability  is  best  gained  by  prac- 
tice, aided  by  a  thorough  grasp  of  the  principles  involved.  Where  several 
successive  operations  are  to  be  performed,  a  sliding  runner  or  marker 
(furnished  with  each  slide  rule)  is  used,  in  order  to  avoid  error  in  shifting 
the  slide.    Petailed  instructions  are  usually  furnished  with  the  slide  rule. 

(*)  A  common  form  of  chart  has  four  or  more  similar  squares  joined 
together.  See  (4).  Our  figure  represents  one  complete  square,  with  por- 
tions of  adjoining  squares.  For  actual  use,  both  charts  and  slide  rules 
are,  of  course,  much  more  finely  subdivided  than  in  our  figures,  which  are 
given  merely  to  illustrate  the  principles.  Carefully  engraved  charts  are 
published  by  Mr.  John  R.  Freeman,  Providence.  R.  I. 

(t)  Other  forms  embodying  the  same  principle  are  :  The  "  Reaction  Scale 
and  General  Slide  Rule,"  by  W.  H.  Breithaupt,  M.  Am.  Soc.  C.  E. ;  Sexton's 
Omnimeter  or  Circular  Slide  Rule,  bv  Thaddeus  Norris  :  The  Goodchild 
Computing  Chart;  The  Thacher  Calculating  Machine  or  Cylindrical  Slide 
Rule  ;  The  Cox  Computers,  designed  for  special  formulas ;  and  the  Pocket 
Calculator,  issued  by  "  The  Mechanical  Engineer,"  London. 


LOGARITHMIC  CHART  AND  SLIDE  RULE. 


75 


(5)  Multiplicatioii  and  division.  For  example,  2  X  1.5=-  ?  On 
1-X,  in  the  chart,  or  on  C  or  D,  in  the  sUde  rule,  the  distance  1-1.5  repre- 
sents by  scale  the  logarithm  (0.17  609)  of  1.5,  and  1-2  represents  the 
logarithm  (0.30  103)  of  2.  If  now  we  add  these  two  distances  together, 
by  laying  off  1-2  from  1.5  on  1-X  of  the  chart,  or  by  placing  the  slide  as 
in  the  figure,  we  obtain  the  distance  1-3  =  .47  712  =  the  mantissa  of  log  3 
or  of  log  (2  X  1.5).*  Conversely,  to  divide  3  by  2,  we  graphically  or  mechani- 
cally subtract  1-2  from  1-3. 


1.1- 

Logs. 
1.0- 

0.0- 

0.8- 

0.7- 

o.o- 

0.4- 
0.3- 
0.2- 

o.r 

0.0- 


d 1 — "T 1 1 1 \ 1 1 1 1 1 r 

Logs.  1.9      O.O      O.l       0.2      0.3      0.4      O.J      O.G      0.7      0.8       O/J      l.O       1.1 


.  (6)  In  tlie  lo^arittimie  chart,  the  scales  of  both  axes,  1-X  and 
1-Y,  being  equal,  a  line  1-H,  marked  x,  bisecting  the  square  and  form- 
ing an  angle  of  45°  witR  each  axis  (tan  45°  =  l),t  will  bisect  also  the  inter- 
sections of  all  equal  co-ordinates.  Thus,  points  in  the  line  x,  immediately 
over  2,  3,  4,  etc.,  in  1-X,  are  also  opposite  2,  3,  4,  etc.,  respectively,  on 
1-Y.    See  (4). 

(7)  If  lines  2- A,  3-K,  etc.  (marked  2a:,  3a:,  etc.\  parallel  to  and  above 
1-H,  be  drawn  through  2,  3,  etc.,  on  1-Y,  then  points  in  such  lines,  im- 
mediately over  any  number,  x,  in  1-X,  will  be  respectively  opposite  the 


(*)  In  the  slide  rule,  with  the  slide  as  shown,  each  number  on  D  is  = 
1.5  X  the  coinciding  number  on  C. 

(t)  In  discussing  tangents  of  angles  on  log  chart,  we  refer  to  the  actual 
measured  distances,  as  shown  on  the  equally  divided  scales  of  logs  in  our 
figures,  and  not  to  the  numbers,  which,  for  mere  convenience,  are  marked 

/"<   T>  -|  n  in 

on  the  chart.    Thus,  in  line  1-B,  tan  C  1  B  =  =  r-^,  not 


2.15 


76  LOGARITHMIC    CHART    AND    SLIDE    RULE. 

numbers  giving  the  products  22;,  Zz,  etc.,  on  1-Y;  while  similar  lines, 
drawn  helow  1-H  and  through  2,  3,  etc.,  on  1-X,  give  values  of  ^,   ?  etc., 

respectively.    If  these  lines  |,  |,  etc.,  be  produced  downward,  they  will 

^^/U~\iP^°^^^^^)  ^i  ^-^  (=  ^)'  0-^3  •  •  (=  ^)'  etc.,  respectively  *    See  (4) 

(8)  Powers  and  roots.    If  a  line  x2  be  drawn  through  1,  at  an  angle 

S2  1-X,  whose  tangent,  "^2  jg  2^  jt  will  give  values  of  x''.  Thus,  the  ver- 
tical through  3,  on  1-X,  cuts  the  line  x^  ppposite  9  (=  3^)  on  1-Y.  Simi- 
larly, line  x^  (tangent  =  3)  gives  values  of  x^  ;  and  line  i>x  (tangent  =  y^) 
gives  values  of  x^  or  ^^    g^^  (4-)^ 

(9)  Any  equation  of  the  form  y  =  Q.x^  in  which  log  w  =  log-  C  +  w  loe  x 
(such  as  :  area  of  circle  =  -n  radius^),  is  represented,  on  a  logarithmic  chirt' 
by  a  straight  line  so  drawn  that  the  tangent  T  of  its  angle  with  1-X  is  =  n 
and  intersecting  1-Y  at  that  point  which  represents  the  value  C  Thus' 
th?miSh  ^^^1  ''^^I  f '  ^^T^"^}  ^  ^^  ^'  f  ^"'^  ?^  squares,  and,  being  drawn 
i«H?,\^o  ^r  5,-^- •  \^'^}r^^  '^  ^'^^^  ^'^'l^es  of  TT  a;2.  Thus,  for  a  circle  of 
radius  2,  we  find  m  the  line  tt  x^  over  2,  a  point  L  opposite  E,  or  12  57         the 

t'hTd^Vram,  ?adts^=^'"''"^^^  '  ^"^^"^  "^^"  ^  ^'•^^-  '  ' '  "«  obtain,"  from 

(10)  If  a  chart  is  to  be  used  for  solving  many  equations  of  a  single 
kind,  such  as  y=  C  a:«  where  C  is  a  variable  coefficient,  and  n  a  constant 
exponent  parallel  lines,  forming  the  proper  angle  with  1-X,  should  be  perma- 
nently ruled  across  the  sheet  at  short  intervals. 

^Vl  ^^J  ^^y  ^o^'  ^,^  ^"^  (=  ^^S  3),  we  may  substitute  its  equal,  M-N 
or  3-N,  extending  to  the  central  diagonal  line  1-H,  marked  x-  and  then 
Ja^'^f'^fT  "^^t^PC^'  1-1.2  ==N-Q,  1-3  =  N-K,  etc.,  we  mav  add  anv  log 
(as  1-3  by  moving  upward  from  line  x  (as  from  N  to  K)  or  to  the  "nW 
and  subtract  any  log  (as  1-1.2)  by  moving  downward  (as  from  N  to  Q)  or  to 
the  left.    This  facilitates  the  performance  of  a  series  of  operations 

Thus : 

To  multiply  1.5  by  2  (-  3),  by  3  (=  9),  and  divide  by  2  (=  4.5). 
F-G  =  1-F  =  log  1.5.    Add  G-J  =  1-2  =  log  2 ;    sum  =  F-J  =  log  3  =  1-3  = 
M-N.    Add  N-K  -=  1-3  =  log  3;  sum  =  M-K  =  log  9  =  1-9  =  9-R.    Subtract 
R-T  =  1-2  ==  log  2 ;  remainder  =  9-T  =  log  4.5. 

For  an  example  of  the  application  of  this  principle  to  engineering  prob- 
lems, see  "  Diagrams  for  proportioning  wooden  beams  and  posts,"  by  Carl 
S.  Fogh,  "  Engineering  News  ",  Sept.  27, 1894. 

(13)  Negative  exponents.  If  ar  is  in  the  dmsor,  the  line  will  slope 
in  the  opposite  direction,  or  downward  from  left  to  right.  Thus,  line  4-2 
leaving  1-Y,  at  4,  and  forming,  with  1-X,  the  angle  X,  2,  4,  with  tangent 

=  — —  '  '  '  .  =  —  2,  represents  the  equation :  y  =  -  ^  =  4  xr'. 

(13)  If  the  lines  of  products,  powers,  and  roots,  C  x,  x^,  and  -//^  etc., 
be  drawn  at  angles  whose  tangents  are  less  by  1  than  those  of  the  angles 
formed  by  the  correspondinfi:  lines  in  our  figure,  the  results  may  be  read 
directly  from  oblique  lines  drawn  parallel  to  2-2.  Lines  (C  a;)  giving  multi-  . 
pies  and  sub-multiples  of  the  first  power  of  x  then  become  horizontal  lines 
(tan  =  0).t 

(14)  Powers  and  roots  bT  the  slide  rule.  Scales  C  and  D  being 
twice  as  large  as  scales  A  and  B,  these  scales,  with  their  ends  coinciding, 
form  a  table  of  squares  and  of  square  roots.  See  (3).  By  moving  the  slide 
we  solve  equations  of  the  forms  y  =  {C  x)^  and  2/  =  C  a^^.    Thus,  with  the 


(*)  In  each  of  these  lines,  the  product  of  the  two  numbers  at  its  ends  is 
=  10.  Thus,  in  line  2-A.  2  X  5  =  10 ;  in  3-K,  3  X  3.33  .  .  .  =  10,  etc.  The 
chart  thus  furnishes  a  table  of  reciprocals. 

(f)  Even  with  full-size  charts  and  slide  rules  for  actual  use,  accuracy  is 
not  to  be  expected  beyond  the  third  or  fourth  significant  figure. 

(1)  A  chart  of  this  kind,  prepared  by  Major  Wm.  H.  Bixby,  U.  S.  A., 
after,  the  method  of  Leon  Lnlanne.  Corps  de  Fonts  et  Chauss^es,  France, 
is  published  by  Messrs.  John  Wiley  &  Sons,  New  York.    Price,  25  cents. 


LOGARITHMIC    CHART    AND    SLIDE   RULE. 


77 


slide  as  shown,  each  number  on  A  is  =  the  square  of  (1.5  X  the  coinciding 
number  on  C) ;  while,  with  1  on  B  opposite  1.5  on  A,  each  number  on  A  is  = 
1.5  X  the  square  of  the  coinciding  number  on  C. 

(15)  Since  x'-^  =  x^  X  x,  we  find  cubes  or  third  powers  by  placing  the 
slide  with  1  on  B  opposite  x^  on  A  {i.  e.,  opposite  x  on  D),  see  (3),  and  read- 
ing x'^  from  A  opposite  x  on  B.  Thus,  1.5"^  =  ?.  Place  1  on  B  opposite  1.5  on 
D;  i,  e.,  opposite  1.5^  (=  2.25)  on  A.  Then,  on  A,  opposite  1.5  on  B,  find 
3.375  =  1.5=^.  Or,  turn  the  slide  end  for  end.  Place  1.5  on  B  opposite  1.5 
on  D,  i.  e.,  opposite  1.5*  =  2.25  on  A.  Then,  adding  log  1.5  (on  B)  to  log  2.25 
on  A,  we  find  3.375  (=  1.5')  on  A  opposite  1  on  B. 

(16)  Conversely,  to  find  i/x7  we  shift  the  slide  (in  its  normal  position) 
until  we  find,  on  B,  opposite  x  on  A,  the  same  number  as  we  have  on  D  op- 
posite 1  on  C,  and  this  number  will  be  =  \/x^  .  Or,  turn  the  slide  end 
for  end,*  place  1  on  C  opposite  x  on  A,  and  find,  on  B,  a  number  which 
coincides  with  its  equal  on  D.    This  number  is=  i/'xT  See  also  (17),  (18). 

(17)  On  the  back  of  the  slide  is  usually  placed  a  scale  of  logs  (see  scale 
shown  below  the  rule  in  figure)  and  two  scales  of  angles,  marked  "  S  "  and 
"  T  "  respectively,  for  finding  sines  of  angles  greater  than  0°  34' . .  . ",  and 
tangents  of  angles  between  5°  42' .  .  ."  and  45°. 

(18)  Placing  1  on  C  opposite  any  number  a:  on  D  (with  slide  in  its  normal 

Eosition),  log  x  is  read  from  the  scale  of  logs  by  means  of  an  index  on  the 
ack  of  the  rule.    The  logs  may  be  used  in  finding  powers  and  roots. 

Logs. 

1.8     O.O      0.2     0.4      O.e     0.8      l.O     1.2      1.4      1.0     1.8      2.0     2.2 


:sos. 

kf 

2        3 

1          1 

4   5  erSOl               2        345  07SVlJ\ 

1      1     1    1    1  1  I                  1          1        1      1     1    1   1   1  1      1 

Bl 

i          1        1      1     1    1   1   1  1                  1           1        1      1     1    1   1  1       1 
2        3     4    5  0  7891               2        3     4   5G7S91S 

A^           f                   3             4         5       G      7     8    9    lc\ 

Nos. 

U'i 

1.5 

2                   3             4          5       G      7     8    9   ID] 

1.9     O.O      O.l      0.2     0.3     0.4      0.5      O.G     0.7      0:8     0.9      l.O     1.1 

Logs. 

* 

(19)  To  find  the  sine  or  tang:eiit  of  an  angle  a ;  bring  a,  on  scale  S  or 
T,  as  the  case  may  be,  opposite  the  index  on  back,  and  read  the  natural 
(not  logarithmic)  sine  or  tangent  opposite  10  at  the  end  of  A  or  D  :  sines  on 
B,  and  tangents  on  C.  Or,  invert  the  slide,  placing  S  under  A,  and  T  over 
D,  with  the  ends  of  the  scales  coinciding.  Then  the  numbers  on  A  and  D 
are  the  sines  and  tangents,  respectively,  of  the  angles  on  S  and  T. 

Caution.    Sines         of  angles  less  than   5°  45'  .  .  .  "  are  less  than  0.1. 

"      90°  "      "        "       1.0. 

Tangents  '*       "  betw.5°  42' .  .  .  "  and45°arebetw.  O.land  1.0. 

(20)  On  the  back  of  the  rule  is  usually  printed  a  table  of  ratios  of  num- 
bers in  common  use,  for  convenience  in  operating  with  thesliderule.  Thus: 

diameter  113    U.  S.  gallons         3    ,.  .  ,.,       ^       ^     , 

~: 7. =  i^  ; ^~, —   =  ^^  (in  a  given  quantity  of  water). 

circumference        355        pounds  20®^  ^  ' 

(31)  Soaping  the  edges  of  the  slide  and  the  groove  in  which  it  runs,  will 
often  cure  sticking,  which  is  apt  to  be  very  annoying.  If  the  slide  is  too 
loose,  the  groove  may  be  deepened,  and  small  springs,  cut  from  narrow 
steel  tape,  inserted  between  it  and  the  edge  of  the  slide. 

(*)  With  the  slide  thus  reversed,  and  with  the  ends  of  the  scales  coin- 
ciding, the  numbers  on  A  and  B  are  reciprocals  (page  52),  as  are  also 
those  on  C  and  D, 


78 


TABLE   OF   LOGARITHMS. 


Common  or 

Brig^g^s  I,og^aritbms 

,  Base  =  10. 

No. 

0 

1 

2 

3 

4 

5 

6 

7 

8    9 

Prop. 

00000 

30103 

47712 

60206 

69897 

77815 

84510 

90309  95424 

10 

00000 

00432 

00860 

01283 

01703 

02118 

02530 

02938 

03342  03742 

415 

11 

04139 

04532 

04921 

05307 

05690 

06069 

06445 

06818 

07188  t>75o4 

379 

12 

07918 

08278 

08636 

08990 

09342 

09691 

10037 

10380 

10721 

11059 

349 

V.i 

11394 

11727 

12057 

12385 

12710 

13033 

13353 

13672 

13987 

14301 

323 

U 

14613 

14921 

15228 

15533 

15836 

16136 

16435 

16731 

17026 

17318 

300 

15 

17609 

17897 

18184 

18469 

18752 

19033 

19312 

19590 

19865 

20139 

281 

16 

20412 

20682 

20951 

21218 

21484 

21748 

22010 

22271 

22530 

22788 

264 

17 

23045 

23299 

23552 

23804 

24054 

24303 

24551 

24797 

25042 

25285 

249 

18 

25527 

25767 

26007 

26245 

26481 

26717 

26951 

27184 

27415 

27646 

236 

19 

27875 

28103 

28330 

28555 

28780 

29003 

•-'9225 

29446 

29666 

29885 

223 

20 

30103 

30319 

30535 

30749 

30963 

31175 

31386 

31597 

31806 

32014 

212 

21 

32222 

32428 

32633 

32S38 

33041 

33243 

3:5445 

33646 

33845 

34044 

202 

22 

34242 

34439 

34635 

34830 

35024 

35218 

35410 

35602 

35793 

35983 

194 

23 

36173 

36361 

36548 

36735 

36921 

37106 

37291 

37474 

37657 

37839 

185 

24 

38021 

38201 

38381 

38560 

38739 

38916 

39093 

39269 

39445 

39619 

177 

25 

39794 

39967 

40140 

40312 

40483 

40654 

40824 

40993 

41162 

41330 

171 

26 

41497 

41664 

41830 

41995 

42160 

42324 

4248  S 

42651 

42813 

42975 

164 

27 

43136 

43296 

43456 

43616 

43775 

43933 

44090 

44248 

44404 

44560 

158 

28 

44716 

44870 

45024 

45178 

45331 

45484 

45636 

45788 

45939 

46089 

153 

29 

46240 

46389 

46538 

46686 

46834 

46982 

47129 

47275 

47421 

47567 

148 

30 

47712 

47856 

48000 

48144 

48287 

48430 

48572 

48713 

48855 

48995 

143 

31 

49136 

49276 

49415 

49554 

49693 

49831 

49968 

50105 

50242 

50379 

138 

32 

50515 

50650 

50785 

50920 

51054 

51188 

51321 

51454 

51587 

51719 

134 

33 

61851 

51982 

52113 

52244 

52374 

52504 

52633 

52763 

52891 

53020 

130 

34 

53148 

53275 

53402 

53529 

53656 

53781 

53907 

54033 

54157 

54282 

126 

35 

54407 

54530 

54654 

54777 

54900 

55022 

55145 

55266 

55388 

55509 

122 

36 

55630 

55750 

55S70 

55990 

56110 

56229 

56348 

56466 

56584 

56702 

119 

37 

56S20 

56937 

57054 

57170 

57287 

57403 

57518 

57634 

57749 

57863 

lie 

38 

57978 

58092 

58206 

58319 

58433 

58546 

58658 

58771 

58883 

58995 

113 

39 

59106 

59217 

59328 

59439 

59549 

59659 

59769 

59879 

59988 

60097 

110 

40 

60206 

60314 

60422 

60530 

60638 

60745 

60852 

60959 

61066 

61172 

107 

41 

61278 

61384 

61489 

61595 

61700 

61804 

61909 

62013 

62118 

62221 

104 

43 

62325 

62428 

62531 

62634 

62736 

62838 

62941 

63042 

63144 

63245 

102 

43 

63347 

63447 

63648 

63648 

63749 

63848 

63948 

64048 

64147 

64246 

99 

44 

64345 

64443 

64542 

64640 

64738 

64836 

64933 

65030 

65127 

65224 

98 

4& 

65321 

65417 

65513 

65609 

65705 

65801 

65896 

65991 

660% 

66181 

96 

46 

66276 

66370 

66t64 

66558 

66651 

66745 

66838 

66931 

670 -.'4 

67117 

94 

47 

67210 

67302 

67394 

67486 

67577 

67669 

67760 

67851 

67942 

68033 

92 

48 

68124 

68214 

68304 

68394 

68484 

68574 

68663 

68752 

68842 

68930 

90 

49 

69020 

69108 

69196 

69284 

69372 

69460 

69548 

69635 

69722 

69810 

88 

50 

69897 

69983 

70070 

70156 

70243 

70329 

70415 

70500 

70586 

70671 

86 

61 

70757 

70842 

70927 

71011 

71096 

71180 

71265 

71349 

71433 

71516 

84 

52 

71600 

71683 

71767 

71850 

71933 

72015 

72098 

72181 

72263 

72345 

82 

53 

72428 

72509 

72591 

72672 

72754 

72835 

72916 

72997 

73078 

73158 

81 

64 

73239 

73319 

73399 

73480 

73559 

73639 

73719 

73798 

73878 

73957 

80 

55 

74036 

74115 

74193 

74272 

74351 

74429 

74507 

74585 

74663 

74741 

78 

66 

74818 

74896 

74973 

750o0 

75127 

75204 

75281 

75358 

75434 

75511 

77 

57 

75587 

75663 

75739 

75815 

75891 

75966 

76042 

76117 

76192 

76267 

75 

58 

76342 

76417 

76492 

76666 

76641 

76715 

76789 

76863 

76937 

77011 

74 

59 

77085 

77158 

77232 

77305 

77378 

77451 

77524 

77597 

77670 

77742 

73 

60 

77815 

77887 

77959 

78031 

78103 

78175 

78247 

78318 

78390 

78461 

72 

61 

78533 

78604 

78675 

78746 

78816 

78887 

78958 

79028 

79098 

79169 

71 

62 

79239 

79309 

79379 

79448 

79518 

79588 

79657 

79726 

79796 

79865 

70 

63 

79934 

80002 

80071 

80140 

80208 

80277 

80345 

80413 

80482 

80550 

69 

54 

80618 

80685 

80753 

80821 

80888' 

80956 

81023 

81090 

81157 

81224 

68 

65 

81291 

81358 

81424 

81491 

81557 

81624 

816VK) 

81756 

81822 

SI  883 

6? 

TABLE    OF    LOGARITHMS. 


79 


ComiKion  or 

Bri^^s  I.og:aritliins 

.  Base  =10. 

No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop. 

66 

81954 

82020 

82085 

82151 

82216 

82282 

82347 

82412 

82477 

82542 

66 

67 

82607 

82672 

82736 

82801 

82866 

82930 

82994 

83068 

83123 

83187 

66 

68 

83250 

83314 

83378 

83442 

83506 

83569 

83632 

83696 

83768 

83821 

64 

69 

83884 

83947 

84010 

84073 

84136 

84198 

84260 

84323 

84385 

84447 

63 

70 

84609 

84671 

84633 

84695 

84757 

84818 

84880 

84941 

85003 

85064 

62 

71 

85125 

85187 

85248 

86309 

85369 

86430 

85491 

85551 

85612 

85672 

61 

72 

86733 

85793 

85853 

85913 

85973 

86033 

86093 

86153 

86213 

86272 

60 

73 

86332 

86391 

86461 

86610 

86569 

86628 

86687 

86746 

86805 

86864 

69 

74 

86923 

86981 

87040 

87098 

87157 

87216 

87273 

87332 

87390 

87448 

58 

75 

87506 

87564 

87621 

87679 

87737 

87794 

87862 

87909 

87966 

88024 

57 

76 

88081 

88138 

88195 

88252 

88309 

88366 

88422 

88479 

88536 

88592 

66 

77 

88649 

88705 

88761 

88818 

88874 

88930 

88986 

89042 

89098 

89153 

56 

78 

89209 

89265 

89320 

89376 

89431 

89487 

89542 

89597 

89652 

89707 

55 

79 

89762 

89817 

89872 

89927 

89982 

90036 

90091 

90145 

90200 

90254 

64 

80 

90309 

90363 

90417 

90471 

90525 

90579 

90633 

90687 

90741 

90794 

54 

81 

90848 

90902 

90955 

91009 

91062 

91115 

91169 

91222 

91275 

91328 

53 

82 

91381 

91434 

91487 

91640 

91692 

91645 

91698 

91750 

91803 

91856 

53 

83 

91907 

91960 

92012 

92064 

92116 

92168 

92220 

92272 

92324 

92376 

52 

84 

92427 

92479 

92531 

92582 

92634 

92685 

92737 

92788 

92839 

92890 

51 

85 

92941 

92993 

93044 

93096 

93146 

93196 

93247 

93298 

93348 

93399 

51 

86 

93449 

93500 

93550 

93601 

93661 

93701 

93761 

93802 

93852 

93902 

50 

87 

93951 

94001 

94051 

94101 

94161 

94200 

94250 

94300 

94349 

94398 

49 

88 

94448 

94497 

94546 

94696 

94fi45 

94694 

94743 

94792 

94841 

94890 

4» 

89 

94939 

94987 

95036 

95085 

95133 

95182 

95230 

95279 

96327 

95376 

48 

90 

95424 

95472 

95520 

96568 

95616 

96664 

95712 

96760 

95808 

95856 

48 

91 

95904 

95951 

95999 

96047 

96094 

96142 

96189 

96236 

96284 

96331 

48 

92 

96378 

96426 

96473 

96520 

96667 

96614 

96661 

96708 

96764 

96801 

47 

93 

96848 

96895 

96941 

96988 

97034 

97081 

97127 

97174 

97220 

97266 

47 

94 

97312 

97359 

97405 

97451 

97497 

97643 

97589 

97635 

97680 

97726 

46 

95 

97772 

97818 

97863 

97909 

97964 

98000 

98046 

98091 

98136 

98181 

46 

96 

98227 

98272 

98317 

98362 

98407 

98462 

98497 

98542 

98587 

98632 

45 

97 

98677 

98721 

98766 

98811 

98855 

98900 

98946 

98989 

99033 

99078 

45 

98 

99122 

99166 

99211 

99265 

99299 

99343 

99387 

99431 

99475 

99519 

44 

99 

99563 

99607 

99661 

99694 

99738 

99782 

99825 

99869 

99913 

99956 

44 

For  extended  table  of  logarithms  see  pages  80-91.  The  table 
above,  being  given  on  two  opposite  pages,  avoids  the  necessity  of  turning  leaves. 
It  contains  no  error  as  great  as  1  in  the  final  figure.  The  proportional  parts,  in 
the  last  column,  give  merely  the  average  difference  for  each  line.  Hence,  when 
dealing  with  small  numbers,  and  using  5-place  logs,  it  is  better  to  find  diflfer- 
encesby  subtraction  :  but  where  a  two-page  table  is  used,  interpolation  is  often 
unnecessary.  Indeed,  the  first  four,  or  even  the  first  three,  places  of  the  man- 
tissas here  given  will  often  be  found  sufiicient.  If  the  first  number  dropped  is 
5  or  more,  increase  by  1  the  last  figure  retained.  Thus,  for  log  660,  mantissa 
=  81954,  or  8195,  or  820. 

Multiplication.    Log  a  6  =  log  a  +  log  h. 

Division.    Log  -  =  log  a  —  log  b. 

Involution   (Powers).    Log  «»»  =  n.  log  a. 

_  l?g_? 
Log^/a-     n 


Evolution  (Roots). 


Characteristics. 


Log  2870  =  3.45788 
"  287  =  2.45788 
"  28.7  =  1.45788 
"    2.87  ==  0.45788 


Log  0.287  =  0.45788  —  1  =  1.45788 
'•  0.0287  =  0.45788  —  2  =  2.45788 
"  0.00287  =  0.45788  —  3  =  3.45788 
♦'.  0.000287  --  0.45788  —  4  =  4.45788 


80 


LOGARITHMS. 


Common  or  Brig^gs  ItO^aritlims.    Base  =  10. 


No. 


Log.  2 


-087,-^ 


— 260:^ 

-303^ 

346 


4321 
475  i 

518 
—561 
—604 
—647 


1000 1 00000 ;, 

01 

02  ' 

03 

04 

05 

06 

07 

08 

09 

loio 

11 

12 
13 
14 
15 
16 
17 
18 
19 

1030 

21 
22 
23 
24' 
25 
26 
27 
28 
29 

1030 

31 
32 
33 
34 
35 
36 
37 
38 


1040 

41 
42 
43 
44 
45 
46 
47 
48 


732^^ 
-7/5  .^ 


860'43 

—903  42 

945|43 

—988 1 42 

01030J- 

072^2 

15742 

-242  43 
42 


-284 

—326 

•368 

410 

452 

494 

—536 

— 57 

620 
—662 

703 

745 
—787 
828 
870 
912 
953 
995 
02036 


49  1—078 


No.      Log.  2 


1050 

51 
52 
53 
54 
55 
56 
57 
58 
59 

1060 

61 
62 
63 
64 
65 
66 
67 
68 
69 

1070 

71 
72 
73 
74 
75 
76 
77 
78 
79 

1080 

81 
82 
83 
84 
85 
86 
87 
88 
89 

1090 

91 
92 
93 
94 
95 
96 
97 


~~ 


02119- 

160 
—202 
—243 

284 

325 

366 

407 
—449 
—490 


41 
42 
41 
41 

;?4i 

41 
41 
42 


—531 

—572 
612 
653 
694 
-735 
—776 
816 
857 


■979 
03019 
—060 

100 
—141 

181 
—222 
—262 

302 

342 

-383 
—423 
—463 
—503 
—543 
—583 
623 
-663 
—703 

-743 
782 
822 
862 

-902 
941 
981 

04021 
060 
100 


No. 


1100 

01 
02 
03 
04 
05 
06 
07 
08 
09 

1110 

11 
12 
13 
14 
15 
16 
•  17 
18 
19 


Log.  2 


04139 
—1797 

218  ': 
-258:^ 
-297 1  ^ 

336  :; 
—376 
—415 
—454 

493 

532 

571 
610 
—650 
-689 
727 
766 
805 
844 


1120 

21 


23 


—922 
■961 
999 

05038 

24  1—077 

25  115 

26  —154 


27 

28 
29 

1130 

31 
32 
33 
34 
35 
36 
37 
38 
39 

1140 

41 

42 

43  j 

44! 

45 

46 

47 

48 


192 
-231 
269 


346 

—385 

—423 

461 

-^00 

—538 

576 

614 

652 

690 

—729 

767 

■805 


918 
956 
994 


49  106032 


No.      Log.  2     No.     Log.  g 


1150  06070 

51  !— 108 

52  !     145 

53  1—183 

54  —221 


55 
56 
57 
58 
59 

1160 

61 
62 
63 
64 
65 
66 
67 
68 
69 

1170 

71 
72 
73 
74 

75 
76 
77 
78 
79 

1180 

81 

82 


258 
—296 

333 
—371 

408 

—446 


—558 


7441' 


1190 

91 

.  92 


-856 
-893 
-930 
-967 
07004 
041 
—078 
115 
151 

188 
■225 

—262 
298 
335 

—37-2 
408 
445 

—482 
518 

— 555 
591 

—628 


93 

664 

94 

700 

95 

—737 

96 

773 

97 

809 

98 

—846 

99 

—882 

1200 

01 
02 
03 
04 
05 
06 
07 
08 
09 

1210 

11 
12 
13 
14 
15 
16 
17 
18 
19 

1220 

21 
22 
23 
24 
25 
26 
27 
28 
29 

1230 

31 
32 
33 
34 
35 
36 
37 


1240 

41 
42 
43 
44 

45 
46 

47 
48 
49 


07918  36 
954:36 
990I37 
080271^ 
-063;^^ 
—0991^^ 
-1351^5 

-171  !^^ 

■243l?2 


—279' 
314*; 
3507 
386' 

42*2; 

^p;c!36 


35 
^36 
.36 


4937 

529  ;: 

—565): 

600^ 

-6361 
—6727 

7077 
—7437 

778 
—814 

849 

884 
—920 

955 

—991 

09026 
061 
096 
132 

—167 
202 

—237 
272 
307 


342 

377 
412 
447 
482 
—517 

-552  r. 

—587 
621 
656  i 


35 
35 
35 

35 
35 
35 
35 
34 
35 
35 


Example: 

To  find  Log.  11826: 
Log.  11830  =  07298 
Dif.  =        10  36 

Log.  11820  =  07262 

11826—  11820  =  6 
Dif.  for  6  under  36 

=  22 
Log.  11826  = 

07262  -h  22  =  07284 


44 

43 

42 

41 

4 

4 

4 

4 

9 

9 

8 

8 

13 

13 

18 

12 

18 

17 

17 

16 

22 

22 

21 

21 

26 

26 

25 

25 

31 

30 

29 

29 

35 

34' 

34 

33 

40 

39 

38 

37 

38 

37 

36 

4 

4 

4 

•  8 

7 

7 

11 

11 

11 

15 

16 

14 

19 

19 

18 

23 

22' 

22 

27 

26 

25 

30 

30 

29 

34 

33 

32 

35 

4 

7 
11 
14 

18 
21 
25 
28 
32 


LOGARITHMS. 


81 


Common  or  Bri^grs  liOgarithms.    Base  =  10. 


No. 


1250 

51 
52 
63 
54 
55 
56 
57 
58 
59 

1260 

61 
62 
63 
64 
65 
66 
67 


1270 

71 
72 
73 
74 
75 
76 
77 
78 
79 

1280 

■  81 
82 
83 
84 
85 
86 
87 


1290 

91 
92 
93 
94 
95 
96 
97 
98 
99 


Log.|§ 


09691 
—726 

760 

795 
—830 

864 
—899 
—934 

968 
10003 


03: 

—072 

—106 

140 

175 

209 

243 

•27 

—312 

346 


—415 

—449 

—483 

—517 

551 

585 

619 

653 

687 

-721 

-755 
—789 

■823 
—857 

890 

924 
—958 

-992 
11025 


■059 
•093 
126 

—160 
193 

-227 
■261 

—294 
327 
361 


34 
33 
34 

34 
34 
33 
33 
34 
33 


No.  I  Log.  2 


1300 

01 
02 
03 
04 
05 
06 
07 
08 
09 

1310 

11 
12 
13 
14 
15 
16 
17 
18 
19 

1320 

21 
22 
23 
24 
25 
26 
27 
28 
29 

1330 

31 
32 
33 
34 
35 
36 
37 
38 
39 


1340 

41 


11394 

—428 
461 
494 

—528 
501 
594 

—628 
661 

—694 

727 
760 
793 
826 

860 : 

—893 
926 

—959 
992 

12024 

057 
090 
123 
156 

-189 
222 
254 
287 

—320 
352 

385 

—418 
450 
483 

—510 
5481 

—581 
613 
646 
678 

710 
743 

775 


42 
43 
44 
45 

46  —905 

47  -937 

48  —969 

49  13001 


•808  ^S, 
840  loo 


No. 


1350 

51 
52 
53 
54 
55 
56 
57 
58 
59 

1360 

61 
62 
63 
64 
65 
66 
67 
68 
69 

1370 

71 
72 
73 
74 
75 
76 
77 
78 
79 


1380 

81 
82 
83 
84 
85 
86 
87 


Log.  § 


13033 

-066 

—098 

—130 

162 

194 

—226 

-258 

290 

—322 

-354 

-386 

—418 

—450 

481 

513 

545 

—577 

—609 

640 

672 
704 
735 
767 

—799 
830 

—862 
893 
-925 
956 


14019 

—051 
082 

—114 
-145 
176 
•208 

—239 
89   270 


1390 

91 
92 
93 
94 
95 
96 
97 
98 
99 


301 
—333 
—364 

395 

426 
457 
—489 
—520 
—551 
—582 


No. 


1400 

01 
02 
03 
04 
05 
06 
07 
08 
09 

1410 

11 
12 
13 
14 
15 
16 
17 
18 
19 

1420 

21 
22 
23 
24 
25 
26 
27 
28 
29 


1430 

31 
32 


Log.l  g 


—675 

—706 

—737 

—768 

—799 

829 

860 

891 

—922 

—953 

983 

15014 

—045 

-076 

106 

—137 

168 

198 

—229 

259 
—290 

320 
—351 

381 
—412 

442 
-473 

503 


534 

—564 

594 

33  —625 

34  —655 


35 


685 

715 

-746 

-776 

806 


1440 

41 
42 
43 
44 

45 

46  16017 

47  —047 

48  —077 

49  —107 


866 
—897 

927 
—957 


No. 


1450 

51 

52 
53 
54 
55 
56 
.  57 
58 


1460 

61 
62 
63 
64 
65 
66 
67 
68 
69 

1470 

71 
72 
73 

74 
75 
76 
77 
78 
79 

1480 

81 
82 


84 


88 
89 

1490 

91 
92 
93 
94 
95 
96 
97 
98 
99 


Log.  § 


16137 

—167 

—197 

—227 

256 

286 

316 

—346 

376 

—406 

435 
465 

—495 
524 
554 

—584 
613 
643 
-673 
702 

-732 
761 

-791 
820 

-850 
879 

-909 
938 
967 


17026 
-056 
—085 
114 
143 
-173 
-202 
231 
260 
289 

—319 
348 
377 

—406 
435 
464 
493 
522 
551 
580 


Example: 

To  find  Log.  12605  : 
Log.  12610  =  10072 
Dif.  =       10  35 

Log.  12600  =  10037 

12605  —  12600  =  5 
Dif.  for  5  under  35 

=  18 
Log.  12605  = 

10037  +  18  =  10055 

6 


35 

34 

33 

32 

31 

30 

29 

1 

4 

3 

3 

3 

3 

3 

3 

1 

2 

7 

7 

7 

6 

6 

6 

6 

2 

3 

11 

10 

10 

10 

9 

9 

9 

3 

4 

14 

14 

13 

13 

12 

12 

12 

4 

5 

18 

17 

17 

16 

16 

15 

15 

5 

6 

21 

20 

20 

19 

19 

18 

17 

6 

7 

25 

24 

23 

22 

22 

21 

20 

7 

8 

28 

27 

26 

26 

25 

24 

23 

8 

9 

32 

31 

30 

29 

28 

27 

26 

9 

A  <lasli  before 
or  after  a  log,  de- 
notes that  its  true 
value  is  less  than 
the  tabular  value 
by  less  than  half  a 
unit  in  the  last 
place.  Thus: 
Log.  1366=1354507 

"  1367=1357685 


82 


LOGARITHMS 


Commoii  or  Brig^^s  liOg^aritlims.    Base  =  10. 


Log.  g     No.     Log.  g 


Log.  2     No.      Log.  g     No.     Log.  g 


No. 


1500 

01 
02 
03 
04 
00 
06 
07 
08 
09 

1510 

11 
12 
13 
14 
15 
16 
17 
18 
19 

1520 

21 
22 
23 
24 
25 
26 
27 
28 
29 

1530 

31 
32 
33 
34 
35 
36 
37 


1540 

41 
42 
43 
44 
45 
46 
47 
48 
49 


17609 

638 

—667 

—696 

—725 

—754 

782 

811 

840 

—869 


926 

955 

—984 

18013 

041 

—070 

-099 

127 

-156 

184 
-213 

241 
-270 

298 
-327 

355 
-384 

412 


469 

—498 

-526 

554 

—583 

—611 

639 

667 

—696 

—724 

752 

780 

808 

—837 

—865 

—893 

—921 

949 

977 

19005 


1550 

51 
52 
53 
54 
55 
'  56 
57 
68 
59 

1560 

61 
62 
63 
64 
65 
66 
67 
68 
69 

1570 

71 
72 
73 
74 
75 
76 
77 
78 
79 

1580 

81 
82 
83 
84 
85 
86 
87 
88 
89 

1590 

91 
92 
93 
94 
95 
96 
97 
98 
99 


19033 
061 
089 
117 
145 
173 
■201 
-229 
-257 

—285 

312 

340 

368 

—396 

—424 

451 

479 

—507 

—535 

562 

■590 
■618 
645 

—673 
700 
728 

—756 
783 

—811 


-866 
893 

-921 
948 

-976 

20003 

080 

—058 

085 

112 


16' 
194 

-222!; 

—249 

276 


.127 
28 
27 
27 
27 
27 

—358 '2^ 


-385; 


No. 


1600 

01 
02 
03 
04 
05 
06 
07 
08 
09 

1610 

11 
12 
13 
14 
15 
16 
17 
18 
19 

1630 

21 
22 
23 
24 
25 
26 
27 
28 
29 

1630 

31 
32 
33 
34 
35 
36 
37 
38 
39 

1640 

41 
42 
43 
44 
45 
46 
47 
48 
49 


20412 

439 

466 

493 

520 

—548 

—575 

—602 

—629 

—656 

—683 
—710 

—737! 
763 
790 
817 
844 
871 
—898 
—925 

—952 

978 

21005 

—032 

—059 

085 

112 

—139 

165 

192 

—219 
245 
272 

—299 
325 

—352 
378! 

—405 
431 

—458 

484 
—511 

537 
— 564i 

590 
—617 
—643 

669: 
—696!; 

722 1  ^ 


1650 

51 
52 
53 
54 
55 
56 
57 
58 
59 

1660 

61 
62 
63 
64 
65 
66 
67 
68 
69 

1670 

71 
72 
73 

74 
75 
76 
77 
78 
79 

1680 

81 
82 
83 
84 
85 
86 


89 

1690 

91 
92 
93 
94 
95 
96 
97 


21748 

—775 
801 
827 

—8541 

—880 
906 
932 
958 

—985 

22011 

—03' 

063 

089 

115 

141 

167 

—194 

—220 

—246 

—272 

—298 

—824 

—350 

—376 

401 

427 

453 

479 

505 

-531 
-557 
—583 

608 

634 
—660 

•686 
—712 

73' 
—763 

—789 

814 

840 

-866 

891 

-917 

-943 

968 


1700 

01 
02 
03 
04 
05 
06 
07 
08 
09 

1710 

11 
12 
13 
14 
15 
16 
17 
18 
19 

1720 

•21 
22 
23 
24 
25 
26 
27 
28 
29 

1730 

31 
32 
33 
34 
35 
36 
37 


1740 

41 
42 
43 
44 
45 
46 
47 
48 
49 


23045 
070 

—096 
121 

—147 
172 

—198 
223 
249 
274 

—300 
325 
350 
-376 
401 
426 

—452 
477 
502 
528 

-553 
578 
603 

—629 
654 

—679 
704 
729 
754 
779 

805 

—830 

■855 


—055 

—080 

—105 

130 

155 

—180 

204 

229 

254 

—279 


Example: 

To  find  Log.  15414: 
Log.  15420  -=  18808 
Dif.     =    10  28 

Log.  15410     =     18780 
15414  —  15410  =  4 
Dif.  for  4   under  28 
=  11 
Log.  15414  = 

18780  4-  11  =  18791 


29 

28 

2*7 

26 

25 

24 

1 

3 

3 

3 

3 

3 

2 

2 

6 

6 

5 

o 

5 

5 

3 

9 

8 

8 

8 

8 

7 

4 

12 

11 

11 

10 

10 

10 

5 

15 

14 

14 

13 

13 

12 

6 

17 

17 

16 

16 

15 

14 

7 

20 

20 

19 

18 

18 

17 

8 

23 

22 

22 

21 

20 

19 

9 

26 

25 

24 

23 

23 

22 

A  dasli  before 
or  after  a  log.  de- 
notes that  its  true 
value  is  less  than 
the  tabular  value 
by  less  than  half  a 
unit  in  the  last 
place.  Thus  : 
Log.  1562=1936810 

"      1563=1939590 


LOGARITHMS. 


83 


Commoii  or  Brigg-s  liogaritbins.    Base  =  10. 


No. 


1750 

52 
54 
56 

58 

1760 

62 
64 


1770 

72 

74 
76 
78 

1780 

82 
84 
86 
88 
1790 
92 
94 
96 
98 

1800 

02 
04 
06 
08 

1810 

12 
14 
16 

18 

1820 

22 
24 
26 
28 
1830 
32 
34 
36 
38 

1840 

42 
44 
46 

48 


Log. 


551 !_„ 
-601^" 
-650:49 

49 
49 
49 
49 
49 
49 


748 
797 
846 
895 
944 
993 

25042 
—091 

139 

188 
—237 

285 
—334 

382 
—431 
—479 

527 
575 

—624 
■672 

—720 

■768 
—816 

•864 
—912 

959 

26007 
-055 

102 

150 
-198 

245 
-293 
-340 

387 
-435 

—482 

—529 

576 

623 

670 


No. 

1850 

52 
54 
56 

58 

1860 

62 
64 
66 
68 
1870 
72 
74 
76 
78 

1880 
82 
84 
86 
88 

1890 
92 
94 
96 
98 

1900 

02 
04 
06 

08 

1910 

12 
14 

16 

18 

1920 

22 
24 
26 
28 
1930 
32 
34 
36 
38 

1940 

42 

■     44 

46 

48 


Log.  1  2 

2671747 

764  47 
-811,47 
—858  47 
—905 1 46 

951 
—998 
27045 

091 
—138 

184 
—231 
—277 

323 
-370 


-416 

—462 

508 

554 

600 

646 

692 

—738 

—784 

■830 

875 

921 

—967 

28012 

—058 

103 
—149 

194 
—240 

■28i 

330 
375 
421 
—466 
—511 
556 
601 
646 
691 
735 

780 

—825 

■870 

914 

—959 


No. 


Log.  T. 


1950  29003 

52 
54 
56 
58 


1960 

62 
64 
66 
68 
1970 
72 
74 
76 
78 


1980 

82 

84 


1990 

92 
94 
96 
98 

2000 

02 
04 
06 
08 

2010 

12 
14 
16 

18 

2020 

22 
24 
26 
28 
2030 
32 
34 
36 
38 

2040 

42 
44 

46 

48 


314!^ 

358-'^ 

-403i 


—447 
—491 
—535 
—579 
—623 


44 
i44 
44 
44 
44 


—667 
710 
754 
—798 
—842 
885 
929 
—973 
30016 
—060 

-103 
146 

-190 
233 
276 

—320 
363 

—406 
449 

492 

535 

578. 
621 
664 

—707 
■750 
792 
835 

—878 
920 

963 
31006 

048 
—091 
—133 


No. 


2050 

52 
54 
56 

58 

2060 

62 
64 
66 
68 
2070 
72 
74 
76 
78 

2080 

82 
84 
86 
88 

2090 

92 
94 
96 
98 

2100 

02 
04 
06 
08 

2110 

12 
14 
16 

18 

2120 

22 
24 
26 
28 
2130 
32 
34 
36^ 
38 

2140 

42 
44 
46 

48 


Log.  2  J^o-  Log.  2 


31176 

—218 

260 

302 

—345 


—387 
—429 
—471 

513 

555 

597 
—639!  42 
-681142 
-723  42 
—765  41 


806 
848 

—890 
931 
973 

32015 
056 

—098 
139 

—181 


—222 
263  42 

—305  41 
■346  41 
387  41 


428 
469 
610 

—552 
—593 

■634 
—675 
715 
756 
797 
838 
■879 
919 
960 
33001 

041 

—082 
122 

—163 
203 


2150 

52 
54 

56 

58 

2160 

62 
64 
66 
68 
2170 
72 
74 
76 
78 

2180 

82 
84 
86 
88 
2190 
92 
94 
96 
98 

2200 

02 
04 
06 


33244 

284 

—325 

—365 

405 

445 

—486 
—526 
—566 
—606 
—646 
—686 
—726 
—766 
—806 

—^46 
885 
925 
965 

34006 
044 
084 

—124 
163 

—203 

242 

—282 
321 
361 

1—400 


12 
14 
16 

18 

2220 

22 
24 
26 
28 
2230 
32 
34 
36 
38 

2240 

42 
44 
46 

48 


439 
—479 
—518 
—557 

596 

635 
674 
713 
—753 
•792 


947 

986 

35025 

—064 

102 

141 

—180 


To  find  Log.  18117 
Log.  18120  =  25816 
Dif.     20  48 

Log.  18100  =  25768 
18117  —  18100  =  17 
Under  48 
Dif.  for  10  =  24 

«    «     7  =  n 

"      "  17 
Log. 18117  = 
25768  +  41  =  25809. 


:    41 


50 

49 

48 

3 

2 

2 

5 

5 

5 

8 

7 

7 

10 

10 

10 

13 

12 

12 

15 

15 

14 

18 

17 

17 

20 

20 

19 

23 

22 

22 

25 

25 

24 

47 

2 


41 

40 

39 

2 

2 

2 

4 

4 

4 

6 

6 

6 

8 

8 

8 

10 

10 

10 

12 

12 

12 

14 

14 

14 

16 

16 

16 

18 

18 

18 

21 

20 

20 

1 

2 
3 
4 

5 
6 

7 

8 

9 

10 


84 


LOGARITHMS. 


Common  or  Brig^g^s  liOgraritbins.    Base  =  10. 


36 

847 

38 

884 

2340 

—922 

42 

—959 

44 

—996 

46 

87033 

48 

—070 

To  find  Log.  23335 
Log.  23340  =  36810 
Dif.       20  37 

Log.    23320    =     36773 
23335  —  23320   =   15 
Under  37 
Dif.  for  10  =  19 
"      "      5  =_9 
"      "  15  =  28 
Log.  23335  = 
36773  +  28  =  36801. 


39 

38 

37 

36 

35 

34 

33 

32 

31 

1 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

4 

4 

4 

4 

4 

3 

3 

3 

3 

3 

6 

6 

6 

5 

5 

5 

5 

5 

5 

4 

8 

8 

7 

7 

7 

7 

7 

6 

6 

5 

10 

10 

9 

9 

9 

9 

8 

8 

8 

6 

12 

11 

11 

11 

11 

10 

10 

10 

9 

7 

14 

13 

13 

13 

12 

12 

12 

11 

11 

8 

16 

15 

15 

14 

14 

14 

13 

13 

12 

9 

18 

17 

17 

16 

16 

15 

15 

14 

14 

10 

20 

19 

19 

18 

18 

17 

17 

16 

16 

1 

3 
3 
4 

5 
6 

7 

8 

9 

10 


LOGARITHMS. 


85 


Common  or  'Briggs  liO^arithms.    Base  =  10. 


No. 


2750 

52 
.54 
56 

58 

2760 

62 
64 
66 
68 
2770 
72 
74 
76 
78 

2780 
82 
84 
86 


2790 

92 
94 
96 

98 


Log.  §' 


-965 

996 

44028 

059 

091 
122 
154 
185  > 

—217 
■248 
279 
311 

—342 
373 

404  .J 
4361 
—4671 
498 
529! 


560 1 . 
— 592p 
—623'^ 
— 654i^ 

6851^ 


32 

^'3 

'31 

2800— 7161  1 

02  —747^1 

31 
31 
31 

31 
30 
31 
31 
31 


04 

—778 

06 

—809] 

08 

—840 

2810 

—871 

12 

-902 

14 

932 

16 

963 

18 

994 

2820 

45025 

22 

—056 

24 

086 

26 

117 

28 

—148 

2830 

-179 

32 

209 

34 

—240 

36 

—271 

38 

301 

2840 

-332 

42 

362 

44 

—393 

46 

423 

48 

—454 

No. 


2850 

52 
54 
56 

58 

2860 

62 
64 
66 
•  68 
2870 
72 
74 
76 
78 

2880 

82 
84 
86  i 
88  I 
2890 
92 
94 
96 
98 

2900 

02 
04 
06 

08 

2910 

12 
14 

16 
18 

2920 

22 
24 
26 
28 
2930 
32 
34 
36 
38 

2940 

42 
44 
46 

48 


Log.  §     No. 


45484 
—515 

545 
—576 

606 

—637 
-667 
697 
■728 

—758 
788 
818 
■849 

—879 
909 

939 

969 
46000 
—030 
—060 
—090 
—120 
—150 
—180 
—210 

—240 
—270 
—300 


419 
—449 
—479 
—509 


568 

—598 

627 

657 

—687 

716 

746 

—776 

805 

—835 
864 

-894  i'^ 
923!; 

—953!" 


2950 

52 
54 
56 

58 

2960 

62 
64 
66 
68 
2970 
72 
74 
76 
78 

2980 

82 
84 
86 
88 
2990 
92 
94 
96 
98 

3000 

02 
04 
06 
08 

3010 

12 
14 
16 

18 

3020 

22 
24 
26 
28 
3030 
32 
34 
36 
38 

3040 

42 
44 
46 

48 


46982 

47012 

041 

070 

—100 

129 
—159 
-188 
217 
246 
—276 
■305 
334 
363 
392 

422 
—451 

■480 
—509 


Log.  s 


567 
596 
625 
654 
683 

712 
741 

-770 
—799 

•828 

•857 
885 
914 
943 
—972 

48001 
029 
058 

—087 
116 
144 
173 

—202 
230 
■259 

287 
316 
344 
—373 
401 


No. 


3050 

52 

54 
56 

58 

3060 

62 
64 
66 
68 
3070 
72 
74 
76 
78 

3080 

82 
84 
86 
88 
3090 
92 
94 
96 
98 

3100 

02 
04 
06 
08 

3110 

12 
14 

16 
18 

3120 

22 
24 
26 
28 
3130 
32 
34 


48430 

458 
—487 

515 
—544 

572 

—601 

■629 

657 

—686 

—714 

742 

770 

—799 

—827 

855 

883 

911 

-940 

-968 

-996 

49024 

052 


3140 

42 
44 

46 

48 


Log.  g     No. 


136 
164 
192 
220 

248 

276 
—304 
—332 

-360 

•388 

415 
443 
471 
—499 
—527 
554 
582 
610 
■638 
665 

—693 
721 

748 
—776 


3150 

52 
54 
56 

58 

3160 

62 
64 
66 
.  68 
3170 
72 
74 
76 
78 

3180 

82 
84 
86 
88 
3190 
92 
94 
96 
98 

3200 

02 
04 
06 
08 

3210 

12 
14 
16 

18 

3220 

22 
24 
26 
28 
3230 
32 
34 
36 
38 

3240 

42 

44 
46 


Log.g 


49831 
—859 

886 
—914 

941 

—969 
996 

50024 
051 

—079 
■106 
133 
161 
188 
215 

—243 

270 

297 

—325 

—352 

379 

406 

433 

—4611 

-488 

-515 
542 
569 
596 
623 

—651 
678 

—705 
■732 
■759 

—786 

—813 

—840 

866 

893 

920 

947 

974 

51001 

—028 

—055 

081 

108 

—135 

—162 


To  find  Log.  29019 : 
Log.  29020  =  46270 
Dif.       20  .30 

Log.   29000  =  46240 
29019  — 29000 -=19 
Under  30 
Dif.  for  10  =  15 
«'      "      9  =J14 
"      "  19  =  29 
Log,  29019  = 
46240  +  29  =  46269. 


32 

31 

30 

29 

28 

1 

2 

2 

2 

1 

1 

2 

3 

3 

3 

3 

3 

3 

5 

5 

5 

4 

4 

4 

6 

6 

6 

6 

6 

5 

8 

8 

8 

7 

7 

6 

10 

9 

9 

9 

8 

7 

11 

11 

11 

10 

10 

8 

13 

12 

12 

12 

11 

9 

14 

14 

14 

13 

13 

10 

16 

16 

15 

15 

14 

A  dash  before 
or  after  a  log.  de- 
notes that  its  true 
value  is  less  than 
the  tabular  value 
by  less  than  half  a 
unit  in  the  last 
place.  Thus : 
Log.  3128=4952667 

**    3130=4955445 


86 


LOGARITHMS. 


Comiiioii  or  Brig-g^s  liOg-arittims.    Base  =  10. 


No.      Log.  §     No.      Log.  ^ 


No. 


3»50 

52 
54 
56 

58 

3^60 

62 

64 


•r  ^ 

Log.lS 
51188  .-,., 

—242  f' 
268  ^^ 
295  f' 


68 
3970 

72 
74 

76 

78 

3^80 

82 
84 
86 
88 
3^90 
92 
94 
96 
98 

3300 

02 
04 
06 
08 

3310 

12 
14 
16 

18 

3320 

22 
24 
26 
28 
3330 
32 
34 


-322! 

348' 
875 
66  —402 

428: 


—455 
481 

—508 
534 

—561 

587 
—614 

640 
—6671 

693; 

— 720! 

—746! 

772! 
—799! 

825 1 

851! 


— 9o7i 

■983! 

52009 

035 

061 


—114 

—140 

166 

192 

218! 

244 1 

270! 

-297! 

—323 

—3491 

— 375! 

—4011 

—427 1 
—453, 
—4791 


3350 

52 
54 
56 
58 

3360 

62 
64 
6t) 
68 
3370 
-  72. 
74 
76 
78 

3380 

82 
84 


3390 

92 
94 


3400 

02 
04 
06 
08 

3410 

12 
14 
16 

18 

3430 

22 
24 
26 
28 
3430 
32 
34 
36 
38 

3440 

42 
44 
46 


52504 
530 
556 

582 
608 

■634 
—660 
686 
711 
737 
■763 
—789 
■815 
840 
■866 

■892 
917 
943 
969 
994 
53020 
046 
071 
097 
122 

—148 
173 

—199 
224 

—260 

275 
■301 
326 
—352 
377 

403 
428 
453 

—479 
504 
529 
555 
580 
605 

-631 

656 
681 
706 
—732 
757 


3450 

52 
54 
56 
58 

3460 

62 
64 
66 

54008 
3470  —033 


53782 
807 
832 
857 
882 

-908 


72 

74 
76 
78 

3480 

82 
84 
86 
88 
3490 
92 
94 
96 
98 

3500 

02 
04 
06 
08 

3510 

12 
14 
16 

18 

3520 

22 
24 
26 
28 
3530 
32 
34 
36 
38 

3540 

42 
44 
46 

48 


—058 
—083 
—108 
—133 

—158 

—183 

—208 

—233 

—258 

—283 

307 

332 

357 

—382 

—407 

—432 

4o6 

481 

—506 

—531 

555 

580 

—605 

—630 

654 

—679 
—704 

728 
—753 

777 

802 
—827 

851 
—876 

900 
—925 

949 
—974 


■^     No. 


3550 

52 
54 
56 
58 

3560 

62 
64 
66 
68 
3570 
72 
74 
76 
78 

3580 

82 
84 
86 
88 
3590 
92 
94 
96 
98 

3600 

02 
04 
06 
08 

3610 

12 
14 
16 
18 

3620 

22 
24 
26 
28 
3630 
32 
34 
36 
38 

3640 

42 
44 
46 

48 


Log.  § 


55023  - 
047  24 


—072 

096 

—121 

—145 

169 
—194 

218 

242 
—267 

291 

315 
—340 

364 

388 
—413 
—437 

461 

485 

509 
—534  25 
—558  24 
-582  i  24 

606  24 
24 

630 

654 

678 
—703 

■727 


—751! 

■775 

—799 

823 

■847 


—871 

—895 

919 

—943 

—967 

—991 

56015 

038  23 

062  24 

086  24 

24 

24 

24 

24 


110  , 

—134 

158 


—182  oo 
205  ^^ 


No.  Log.  § 


3650 

52 
54 
56 
58 

3660 

62 
64 
66 
68 
3670 
72 
74 
76 
78 

3680 

82 
84 
86 
88 
3690 
92 
94 
96 
98 

3700 

02 
04 
06 
08 

3710 

12 
14 
16 
18 

3720 

22 
24 
26 
28 
3730 
32 
34 
36 
38 

3740 

42 

44 
46 

48 


56229 

253 

—277 

—301 

324 

348 
—372 

■396 

419 
—443 
—467 

490 
—514 
—538 

561 

—685 
608 

—632 
656 
679 

—703 
726 

—750 
773 

—797 

820 
-844 

867 
-891 
-914 

937 
—961 

984 
57008 
—031 

054 
■078 

—101 
124 
148 

—171 
194 
217 

—241 
-264 

287 
310 
-334 
-357 
-380 


To  find  Log.  36114: 
Log.  36120  =  55775 
Log.  36100  =  55751 
Dif.         20  24 

36114  —  36100  =  14 
Under  24 
Dif.  for  10  =  12 
''      "      4  =5 
*'      "  14  =  17 
Log.  36114  = 
65751  +  17  =  65768. 


27 

26 

25 

24 

23 

1 

1 

1 

1 

1 

1 

1 

2 

3 

3 

3 

2 

2 

2 

3 

4 

4 

4 

4 

3 

3 

4 

5 

5 

6 

5 

5 

4 

5 

7 

7 

6 

6 

6 

5 

6 

8 

8 

8 

7 

7 

6 

7 

10 

9 

9 

8 

8 

7 

8 

11 

10 

10 

10 

9 

8 

9 

12 

12 

11 

11 

10 

9 

10 

14 

13 

13 

12 

12 

10 

A  dasb  before 
or  after  a  log.  de- 
notes that  its  true 
value  is  less  than 
the  tabular  value 
by  less  than  half  a 
unit  in  the  last 
place.  Thus : 
Log.  3490  =  5428254 
"     3492  =  5430742 


LOGAEITHMS. 


87 


Commoii  or  Brig-g-s  liOg^arithms.    Base  =  10. 


Log. 


57403 
-^61 
519 
576 
634 
692 
749 
—807 
-864 
921 

978 

58035 

092 

149 

206 

263 

■320 

—377 

433 

—490 

546 
602 
659 

—715 
771 
827 
883 
939 
995 

59051 

106 

162 

—218 

273 

-329 

-384 

439 

494 

—550 

—605 

—660 
715 

—770 
824 
879 

—934 
988 

60043 
097 

—152 


No. 


4000 

05 
10 
15 
20 
25 
30 
35 
40 
45 

4050 

55 
60 
65 
70 
75 
80 
85 
90 
95 

4100 

05 
10 
15 
20 
25 
30 
35 
40 
45 

4150 

55 
60 
65 
70 
75 
80 
85 
90 
95 

4300 

05 
10 
15 
20 
25 
30 
35 
40 
45 


hog. 


60206 

260 

314 

—369 

—423 

—477 

■531 

584 

638 

—692 

—746 
799 

—853 
906 
959 

61013 
066 
119 
172 
225 

278 
331 
384 

—437 
490 
542 
595 

—648 
700 
752 

•805 
857 
909 
962 

62014 
066 

—118 
•170 
221 
273 

—325 

—377 
428 

—480 
531 
■583 
634 
685 
■737 

—788 


No. 


4250 

55 
60 
65 
70 
75 
80 
85 
90 
95 

4300 

05 
10 
15 
20 
25 
30 
35 
40 
45 

4350 

55 
60 
65 
70 
75 
80 
85 
90 
95 

4400 

05 
10 
15 
20 
25 
30 
35 
40 
45 

4450 

55 
60 
65 
70 
75 
80 
85 
90 
95 


Log. 


62839 

—890 

—941 

—992 

63043 

—094 

144 

195 

—246 

296 

—347 

397 

—448 

498 

548 

—599 

—649 

—699 

—749 

•799 

—849 
899 

—949 
998 

64048 
•098 
147 

-197 
246 
•296 

345 
—395 
—444 
493 
542 
591 
640 
689 
738 
787 

836 

—885 

933 

982 

65031 

079 

■128 

176 

■225 

—273 


No. 


4500 

05 
10 
15 
20 
25 
30 
35 
40 
45 

4550 

55 
60 
65 
70 
75 
80 
85 
90 
95 

4600 

05 
.  10 
15 
20 
25 
30 
35 
40 
45 

4650 

55 
60 
65 
70 
75 
80 
85 
90 
95 

4700 

05 
10 
15 
20 
25 
30 
35 
40 
45 


Log.  2  No.  Log.  2 


65321 
369 
—418 
—466 
—514 
—562 
—610 
—658 
—706 
753 

801 
—849 

896 

944 
—992 
66039 
—087 
—134 

181 
—229 

—276 
-323 
370 
417 
464 
511 
558 
605 

—652 
•699 

745 

—792 

■839 

885 

■932 

978 

67025 

—071 

117 

—164 

•210 

—256 

302 

348 

394 

440 

486 

—532 

78 

—624 


4750 

55 
60 
65 
70 
75 
80 
85 
90 
95 

4800 

05 
10 
15 
20 
25 
30 
35 
40 
45 

4850 

55 
60 
65 
70 
75 
80 
85 
90 
95 

4900 

05 
10 
15 
20 
25 
30 
35 
40 
45 

4950 

55 
60 
65 
70 
75 
80 
85 
90 
95 


67669 

715 
—761 

806 
—852 

897 
—943 

988 
68034 
—079 

124 

169 

—215 

—260 

—305 

—350 

395 

440 

485 

529 

574 
—619 
—664 

708 
-753 

797 
-842 

886 
-931 

975 

69020 

—064 

108 

152 

—197 

■241 

■285 

—329 

—373 

-417 

—461 
504 
548 

—592 
•636 
679 
723 

—767 
810 
•854 


1 
2 
3 
4 

5 
6 

7 

8 

9 

10 


58 

57 

56 

55 

54 

53 

53 

51 

50 

49 

48 

47 

46 

45 

44  43 

1.2 

1.1 

1.1 

1.1 

1.1 

1.1 

1.0 

1.0 

1.0 

1.0 

1.0 

0.9 

0.9 

0.9 

0.9  0.9 

2.3 

2.3 

2.2 

2.2 

2.2 

2.1 

2.1 

2.0 

2.0 

2.0 

1.9 

1.9 

1.8 

1.8 

1.8  1.7' 

3.0 

3.4 

3.4 

3.3 

3.2 

3.2 

3.1 

3.1 

3.0 

2.9 

2.9 

2.8 

2.8 

2.7 

2.6,  2.6 

4.6 

4.6 

4.5 

4.4 

4.3 

4.2 

4.2 

4:1 

4.0 

3.9 

3.8 

3.8 

3.7 

3.6 

3.5;  3.4 

5.8 

5.7 

5.6 

5.5 

5.4 

5.3 

5.2 

5.1 

5.0 

4.9 

4.8 

4.7 

4.6 

4.5 

4.4!  4.3 

70 

6.8 

6.7 

6.6 

6.5 

6.4 

6.2 

6.1 

6.0 

5.9 

5.8 

5.6 

5.5 

5.4 

5.3  5.2 

8.1 

8.0 

7.8 

7.7 

7.6 

7.4 

7.3 

7.1 

7.0 

6.9 

6.7 

6.6 

6.4 

6.3 

6.2  6.0 

9.3 

9.1 

9.0 

8.8 

8.6 

8.5 

8.3 

8.2 

8.0 

7.8 

7.7 

7.5 

7.4 

7.2 

7.0:  6.9 

10.4 

10.3 

10.1 

9.9 

9.7 

9.5 

9.4 

9.2 

9.0 

8.8 

8.6 

8.5 

8.3 

8.1 

7.9|  7.7 

11.6 

11.4 

11.2 

11.0 

10.8 

10.6 

10.4 

10.2 

10.0 

9.8 

9.6 

9.4 

9.2 

9.0 

8.8 '  8.6 

1 

2 
3 
4 
5 

6 

7 
8 
9 
10 


88 


LOGARITHMS. 


Commoii  or  Brig^g^s  liOg-arithms.    Base  =  10* 


39 

38 

37 

36 

0.8 

0.8 

0.7 

0.7 

1.6 

1.5 

1.5 

1.4 

2.3 

2.3 

2.2 

2.2 

3.1 

3.0 

3.0 

2.9 

3.9  '  3.8 

3.7 

3.6 

4.7  i  4.6 

4.4 

4.3 

5.5    5.3 

5.2 

5.0 

6.2  ,  6.1 

5.9 

5.8 

7.0  ;  6.8 

6.7 

6.5 

7.8 

7.6 

7.4 

7.2 

To  find  Log.  58636 : 
Log.  58650  =  76827 
Log.  58600  =  76790 
Dif.         50  37 

58636  —  58600   =   36 

Under  37 

Dif.  for  10  X  3  =  22.0 

"      ''  6  =    4.4 

"      "  36  =  26.4 

Log.  58636  = 

76790  4-  26  =  76816 


LOGARITHMS. 


89 


Commoii  or  Brig^g^s  liOg^aritliins.    Base  =  10. 


Log.  §     No.    JLog.  g 


No.     Log.  g 


6350 

55 
60 
65 
70 
75 
80 
85 
90 
95 

6300 

05 
10 
15 
20 
25 
30 
35 
40 
45 

6350 

55 
60 
65 
70 
75 
80 
85 
90 
95 

6400 

05 
10 
15 
20 
25 
30 
35 
40 
45 

6450 

55 
60 
65 
70 
75 
80 
85 
90 


79588 
■623 
657 
692 

—727 
761 
■796 

—831 
865 

—900 

934 

—969 

80003 

037 

—072 

106 

140 

175 

—209 

243 

277 

312 

—346 

■380 
—414 

448 
482 
516 
550 
584 

— 618l 

—652 1 
■686 

—720 
■754 
787 
821 

—855 
-889 
922 

—956 
—990 
81023 
—057 

090 
—124 
—158 
—191 

224 


95  —258 


No. 

6500 

05 
10 
15 
20 
25 
30 
35 
40 
45 

6550 

55 
60 
65 

70 
75 
80 
85 
90 
95 

6600 

05 
10 
15 
20 
25 
30 
35 
40 
45 

6650 

55 
60 
65 
70 
75 
80 
85 
90 
95 

6700 

05 
10 
15 
20 
25 
30 
35 
40 
45 


81291 

—325 

358 

391 

—425 

458 

491 

—525 

—558 

-591 

624 

657 

690 

723 

—757 

■790 

—823 

—856 


921 

954 

987 

82020 

—053 

—086 

119 

151 

184 

—217 

249 

282 
■315 
347 
380 

-413 
445 

—478 
510 

—543 
575 

607 

—640 

672 

705 

—737 

769 

■802 

—834 


6750 

55 
60 
65 
.70 
75 
80 
85 
90 
95 

6800 

05 
10 
15 
20 
25 
30 
35 
40 
45 

6850 

55 
60 
65 
70 
75 
80 
85 
90 
95 

6900 

05 
10 
15 
20 
25 
30 
35 
40 
45 

6950 

55 
60 
65 
70 
75 
80 
85 
90 
95 


—963 

—995 

83027 

—059 

—091 

123 

155 

■187 

219 

251 

—283 

315 

—347 

378 

410 

442 

—474 

—506 

537 

569 
—601 

632 

664 
—696 

727 
-759 

790 
-822 

853 

-885 

916 

-948 

979 

84011 

—042 

073 

—105 

136 

167 

198 

—230 

—261 

292 

323 

354 

—386 

—417 

—448 

^479 


No. 


Log.  2  No, 


7000  84510! 

05  1—541 1 
10  — 572i 
15  — 603i 


20 
25 
30 
35 
40 
45 

7050 

55 
60 
65 
70 
75 
80 
85 
90 
95 

7100 

05 
10 
15 
20 
25 
30 
35 
40 
45 

7150 

55 
60 
65 
70 
75 
80 
85 
90 
95 

7200 

05 
10 
15 
20 
25 
30 
35 
40 
45 


■634 
— 665i 
■696! 
726  i 
757 
788 

—819 
-850 
880 
911 

—942 
-973 

85003 
034 

—065 
095 

—126 

156 
—187 

217 
—248 

278 
—309 

339 
—370 

400 

—431 
—461 

4911 
—522 
—5521 

582; 

612 
—643 
—673 

703 

733 

763 
—794 
-824 
—854 
—884 
—914 
—944 
—974 
86004 


7250 

55 
60 
65 
70 
75 
80 
85 
90 
95 

7300 

05 
10 
15 
20 
25 
30 
35 
40 
45 

7350 

55 
60 
65 
70 
75 
80 
85 
90 
95 

7400 

05 
10 
15 
20 
25 
30 
35 
40 
45 

7450 

55 
60 
65 
70 
75 
80 
85 
90 
95 


Log.  g 


86034 

—064 

—094 

•124 

153 

183 

213 

—243 

273 

—303 

332 
362 

—392 
421 
451 

—481 
510 
540 
-570 
.599 

—629 
658 

—688 
717 

—747 
776 

—806 
835 
864 

—894 

923 

—953 

982 

87011 

040 

—070 

—099 

128 

157 

186 

216 

—245 

—274 

—303 

332 

361 

390 

419 

448 

477 


To  find  Log.  63023 : 
Log.  63050  =  79969 
Log.  63000  =  79934 
Dif.         50  35 

63023  —  63000  =  23 
Under  35 
Dif.  for  10  X  2  =  14.0 
"      "  3  =    2J 

"      "  23  =  16.1 

Log.  63023  = 

'9934  -H  16  =  79950. 


32  31 

0.6  0.6 


1.2 
1.9 
2.5 
3.1 
3.8  '  3.7 
4.5  4.3 
5.1  5.0 
5.8  1 5.6 
6.4  6.2 


A  dash  before 
or  after  a  log.  de- 
notes that  its  true 
value  is  less  than 
the  tabular  value 
by  less  than  half  a 
unit  in  the  last 
place.  Thus: 
Log.7400  =  8692317 
"    7405  =  8695251 


90 


LOGARITHMS. 


Common  or  Brigg-s  liOg^arithms.    Base  =  10. 


7500 

05 
10 
15 
20 
25 
30 
35 
40 
45 


87506 

585 

— 564 

—593 

—622 

—651 

679 

708 

737 

—766 

—795 

823 

852 

—881 

—910 

938 

—967 

—996 

88024 

—053 


7650 

55 
60 
65 

70 
75 
80 


7700 

05 
10 
15 
20 
25 
30 
35 
40 
45 


Log.12 


081 
-110 

138 
-167    ^ 

195 1  on 

-224'  ^^ 
252 


649 
677 
705; 

—734 
—762 
—790 1 


I  29 


—423  ; 


—508  28 
536  98 
564  2^ 

—593 

—621 


874] 
902! 


No. 


7750 

55 
60 
65 
70 

75 
80 
85 
90 
95 

7800 

05 
10 
15 
20 
25 
30 
35 
40 
45 

7850 

55 
60 
65 
70 
75 
80 
85 
90 
95 

7900 

05 
10 
15 
20 
25 
30 
35 
40 
45 

7950 

55 
60 
65 
70 
75 
80 
85 
90 
95 


Log. 


958 

986 

89014 

042 

070 

098 

—126 

—154 

—182 

209 

237 

265 

—293 

—321 

348 

376 

—404 

—432 

459 

—487 
—515 

542 
—570 

597 

625 
—653 

680 
—708 

735 

—763 

790 

-818 

845 

—873 

-900 

927 

■955 

982 

90009 

—037 

064 

091 

—119 

—146 

173 

200 

227 

—255 

—282 


No. 

8000 

05 
10 
15 
20 
25 
30 
35 
40 
45 

8050 

55 
60 
65 
70 
75 
80 
85 
90 
95 

8100 

05 
10 
15 
20 
25 
30 
35 
40 
45 

8150 

55 
60 
65 

70 
75 
80 
85 


8^00 
05 
10 
15 
20 
25 
30 
35 
40 
45 


I^og.  s 


90309 

336 

363 

390 

417 

445 

—472 

—499 

—526 

—553 

—580 

—607 

—634 

660 

687 

714 

741 

768 

795 

—822 

—849 

875 

902 

—929 

—956 

982 

91009 

—036 

062 

089 

-116 
142 
169 

—196 
222 

—249 
275 

—302 
328 

—355 

381 
—408 

434 
—461 

487 
— 514j 

540 


566  ;' 

-593  ;, 

6191  o 


No. 


8^50 

55 
60 
65 
70 
75 
80 
85 
90 
95 

830(I^ 

05 
10 
15 
20 
25 
30 
35 
40 
45 

8350 

55 
60 
65 
70 
75 
80 
85 
90 
95 

8400 

05 
10 
15 
20 
25 
30 
35 
40 
45 

8450 

55 
60 
65 
70 
75 
80 
85 
90 
95 


Log. 


91645 

—672 
698 
724 

—751 

-777 
803 
829 
855 

-882 

-908 
-934 

960 

986 
92012 

038 
—065 
—091 
—117 

143 

169 
—195 
221 
—247 
-273 
298 
324 
350 
376 
402 

—428 

—454 

—480 

505 

531 

-557 

-583 

—609 

634 

—660 

•686 
711 
737 
763 
788 
814 

—840 
865 

—891 
916 


No. 


8500 

05 
10 
15 
20 
25 
30 
35 
40 
45 

8550 

55 
60 
65 
70 
75 
80 
85 
90 
95 

8600 

05 
10 
15 
20 
25 
30 
35 
40 
45 

8650 

55 
60 
65 
70 
75 
80 
85 
90 
95 

8700 

05 
10 
15 
20 
25 
30 
35 
40 
45 


92942 

967 

—993 

93018 

—044 

069 

■095 

120 

—146 

171 

—197 
222 
247 

—273 
298 
323 

—349 
374 


—952 

—977 

94002 

—027 

—052 

—077 

101 

126 

151 

—176 


To  find  Log.  83678 : 
Log.  83700  =  92273 
Log.  83650  =  92247 
Dif.         50  36 

83678  —  83650  =  2S 
Under  36 
Dif.  for  10  X  2  =  10.0 
"       "  8=    4.2 

«      "  38  =  14.2 

Log.  83678  = 

92247  -f-  14  =  92261. 


39 

38 

37 

36 

35 

24 

1 

0.6 

0.6 

0.5 

0.5 

0.5 

0.5 

1 

2 

1.2 

1.1 

1.1 

1.0 

1.0 

1.0 

2 

3 

1.7 

1.7 

1.6 

•1.6 

1.5 

1.4 

3 

4 

2.3 

2  2 

2.2 

2.1 

2.0 

1.9 

4 

5 

2.9 

2.8 

2.7 

2.6 

2.5 

2.4 

5 

6 

3.5 

3.4 

3.2 

3.1 

3.0 

2.9 

6 

7 

4.1 

3.9 

3.8 

3.6 

3.5 

3.4 

7 

8 

4.6 

4.5 

4.3 

4.2 

4.0 

3.8 

8 

9 

5.2 

5.0 

4.9 

4.7 

4.5 

4.3 

9 

10 

5.8 

5.6 

6.4 

5.2 

5.0 

4.8 

110 

A  dash  before 
or  after  a  log.  de- 
notes that  its  true 
value  is  less  than 
the  tabular  value 
by  less  than  half 
a  unit  in  the  last 
place.  Thus : 
Log.  8325  =  9203842 
"      8330  =  9206450 


LOGARITHMS. 


91 


Commoii  or  Brig^g^s  liOg^arithms.    Base  =  10. 


Loif. 

94201 

5 

25 

No. 

Log. 

ft 

24 

24 
25 

No. 

Log. 

96614 

24 

No. 
9500 

Log. 

s 

No. 

Log. 

9000 

95424 

9350 

97772 

23 

9750 

98900 

-226 

24 

05 

448 

55 

—638 

23 

05 

795 

23 

55 

-923 

250 

25 

10 

472 

60 

661 

24 

10 

818 

23 

60 

—945 

275 

25 

15 

—497 

24 

24 

65 

—685 

23 

15 

—841 

23 

65 

967 

—300 

25 

20 

—521 

70 

-708 

23 

20 

—864 

22 

70 

989 

-325 

24 

25 

—545 

24 

75 

731 

24 

25 

886 

23 

75 

99012 

349 

25 

30 

—569 

24 
24 

80 

—755 

23 

30 

909 

23 

80 

—034 

374 

25 

35 

—593 

85 

778 

24 

35 

932 

23 

85 

056 

—399 

25 

40 

—617 

24 

90 

—802 

23 

40 

-955 

23 

90 

078 

-424 

24 

45 

—641 

24 

95 

—825 

23 

45 

—978 

22 

95 

100 

448 

25 

9050 

—665 

24 

9300 

848 

24 

9550 

98000 

23 

9800 

—123 

—473 

25 

55 

—689 

24 

05 

—872 

23 

55 

023 

23 

05 

—145 

—498 

24 

60 

—713 

24 

10 

-895 

23 

60 

-046 

22 

10 

—167 

522 

25 

65 

-737 

24 

15 

918 

24 

65 

068 

2o 

15 

189 

-547 

24 

70 

—761 

24 

20 

—942 

23 

70 

091 

23 

20 

211 

571 

25 

75 

—785 

24 

25 

—965 

23 

75 

—114 

23 

25 

233 

696 

26 

80 

—809 

23 

30 

988 

23 

80 

—137 

22 

30 

255 

—621 

24 

85 

832 

24 

35 

97011 

24 

85 

159 

23 

35 

277 

645 

25 

90 

856 

24 

40 

—035 

23 

90 

—182 

22 

40 

—300 

—670 

24 

95 

880 

24 

45 

—058 

23 

95 

204 

23 

45 

—322 

694 

.,- 

9100 

904 

24 

9350 

081 

23 

9600 

227 

23 

9850 

—344 

—719 

24 

05 

—928 

24 

55 

104 

24 

05 

—250 

22 

55 

-366 

743 

25 

10 

—952 

24 
23 

60 

—128 

23 

10 

272 

23 

60 

—388 

-768 

24 

15 

-976 

65 

—151 

23 

15 

—295 

23 

65 

-410 

792 

25 

20 

999 

24 

70 

—174 

23 

20 

—318 

22 

70 

—432 

—817 

24 

25 

96023 

24 

75 

197 

23 

25 

340 

23 

75 

—454 

841 

25 

30 

047 

24 

80 

220 

23 

30 

-363 

22 

80 

—476 

—866 

24 

35 

—071 

24 

85 

243 

24 

35 

385 

23 

85 

—498 

890 

25 

40 

-095 

23 

90 

—267 

23 

40 

—408 

22 

90 

-520 

—915 

24 

45 

118 

24 

95 

—290 

23 

45 

430 

23 

95 

—542 

939 

24 

9150 

142 

94 

9400 

—313 

23 

9650 

—453 

22 

9900 

—564 

963 

25 

55 

—166! ;; 

05 

-336 

23 

55 

475 

23 

05 

585 

—988 

24 

60 

—190 

23 

10 

—359 

23 

60 

—498 

22 

10 

607 

95012 

24 

65 

213 

24 

15 

382 

23 

65 

520 

23 

15 

629 

036 

25 

70 

—237 

24 

20 

405 

23 

70 

—543 

22 

20 

651 

—061 

24 

75 

—261 

23 

25 

428 

23 

75 

565 

23 

25 

673 

085 

24 

80 

284 

24 

30 

451 

23 

80 

—588 

22 

30 

—695 

109 

25 

85 

-308 

24 

35 

474 

23 

85 

—610 

22 

35 

—717 

—134 

24 

90 

—332 

23 

40 

497 

23 

90 

632 

23 

40 

—739 

158 

24 

95 

355 

24 

45 

520 

23 

95 

-655 

22 

45 

760 

182 

25 
24 
24 
24 
24 
25 
24 
24 
24 
24 

9300 

—379 

23 
24 
24 
23 

9450 

543 

23 
23 
23 
23 
23 
23 
23 
23 
22 
23 

9700 

677 

23 
22 
22 
23 
22 
22 
23 
22 
22 
22 

9950 

782 

—207 

05 

402 

55 

566 

05 

-700 

55 

804 

—231 

10 

—426 

60 

689 

10 

-722 

60 

—826 

255 

15 

—450 

65 

612 

15 

744 

65 

—848 

279 

20 

473 

70 

-635 

20 

—767 

70 

—870 

303 

25 

-497,^^ 

520i  11 

-544' 5^ 

75 

—658 

25 

—789 

75 

891 

—328 

30 

80 

—681 

30 

811 

80 

913 

—352 

35 

85 

—704 

35 

—834 

85 

-935 

-376 

40 

90 

—727 

40 

—856 

90 

—957 

400 

45 

—591 

23 

95 

749 

45 

878 

95 

978 

To  tiud  Log.  95544  : 
Log.  95550  =  98023 
Log.  95500  =  98000 
Dif.         50  33 

95544  —  95500  =  44 
Under  23 
Dif.  for  10  X  4  =  18.0 
"      "        4=    1.8 
"      "    44  =  19.8 
Log.  95544  = 
98000  +  20  =  98020. 


25 

34 

23 

23 

21 

1 

0.5 

0.5 

0.5 

0.4 

0.4 

1 

2 

1.0 

1.0 

0.9 

0.9 

0.8 

2 

3 

1.5 

1.4 

1.4 

1.3 

1.3 

3 

4 

2.0 

1.9 

1.8 

1.8 

1.7 

4 

5 

2.5 

2.4 

2.3 

2.2 

2.1 

5 

6 

3.0 

2.9 

2.8 

2.6 

2.5 

6 

7 

8.5 

3.4 

3.2 

3.1 

2.9 

7- 

8 

4.0 

3.8 

3.7 

3.5 

3.4 

8 

9 

4.5 

4.3 

4.1 

4.0 

3.8 

9 

10 

5.0 

4.8 

4.6 

4.4 

4.2 

10 

A  dash  before 
or  after  a  log.  de- 
notes that  its  true 
value  is  less  than 
the  tabular  value 
by  less  than  half  a 
unit  in  the  last 
place.  Thus : 
Log.  9600  =  9822712 
"     9605  =  9824974 


GEOMETRY. 


6E0METM. 


liines,  Fig-nres,  Solids,  defined.    Strictly  speaking  a  geometrical  line 

is  simply  length,  or  distance.  Ttie  lines  we  draw  on  paper  have  not  only  length,  but  breadth  and 
thickness  ;  still  they  are  the  most  convenient  symbol  we  can  employ  for  denoting  a  geometrical  line. 
Sltraisirht  lines  are  also  called  rig-lit  lines.  A  vertical  line  is  one  that  points 
toward  the  center  of  the  earth ;  and  a  horizontal  one  is  at  right  angles  to  a 
vert  one.     A  plane  figure  is  merely  any  flat  surface  or  area  entirely  enclosed 

by  lines  either  straight  or  curved  ;  which  are  called  its  outline,  boundary,  circumf,  or  periphery.  We 
often  confound  the  outline  with  the  fig  itself  as  when  we  speak  of  drawing  circles,  squares,  ic ;  for 
we  actually  draw  only  their  outlines.  Geometrically  speaking,  a  fig  has  length  and  breadth  only ;  no 
thickness.    A  solid  is  any  body ;  it  has  length,  breadth,  and  thickness. 

Geometrically  similar  figs  or  solids,  are  not  necessarily  of  the  same 
sf  2se ;  but  only  of  precisely  the  same  shape.  Thus,  any  two  squares  are,  scien- 
tifically speaking,  similar  to  each  other  ;  so  also  any  two  circles,  cubes,  &c,  no  matter  how  different 
they  niay  be  in  size.     When  they  are  not  only  of  the  same  shape,  but  of  the  same  size,  they  are  said 

to  be  similar,  and  equal. 

The  quantities  of  lines  are  to  each  other  simply  as  their  lengths;  but 

the  quantities,  or  areas,  or  surfaces  of  similar  figures,  are  as,  or  in  proportion 
to,  the  squares  of  any  one  of  the  corresponding  lines  or  sides  which  enclose  the 
figures,  or  which  may  be  drawn  upon  them ;  and  the  quantities,  or  solidities  of 
similar  solids,  are  as  the  cubes  of  any  of  the  corresponding  lines  which  form 

their  edges,  or  the  figures  by  which  thej  are  enclosed. 

Kem. — Simple  aS  the  following  operations  appear,  it  is  only  by  care,  and  good  instruments,  that 
they  are  made  to  give  accurate  results.  SeveraJ  of  them  can  be  much  better  performed  by  means  of  a 
metallic  triangle  having  one  perfectly  accurate  right  angle.  In  the  field,  the  tape-line,  chain,  or  a 
measuring- rod  will  take  the  place  of  the  dividers  and  ruler  used  indoors. 

To  divide  a  given  line,  a  b,  into  two  eqnal  parts. 

From  its  ends  a  and  h  as  centers,  and  with  any  rad  greater  than  one-half  of  a  5, 

ff   describe  the  arcs  c  and  d,  and  join  ef.    If  the'line  a  &  is  very  long,  first  lay  off 

g         equal  dists  a  o  and  b  g,  each  way  from  the  ends,  so  as  to  approach  conveniently 

near  to  each  other  ;  and  then  proceed  as  if  o  fir  were  the  line  to  be  divided.    Or 

measure  a  &  by  a  scale,  and  thus  ascertain  its  center. 

To  divide  a  given  line,  m  n,  into  any^ 
0  given  number  of  equal  parts. 

From  m  and  n  draw  any  two  parallel  lines  m  o  and  n  ee, 
to  an  indefinite  dist ;  and  on  them,  from  m  and  n  step  off  the 
reqd  number  of  equal  parts  of  any  convenient  length :  final- 
ly,join  the  corresponding  points  thus  stepped  off.  Or  only 
one  line,  as  mo,  may  be  drawn  and  stepped  off.  as  to«; 
then  join  sn;  and  draw  the  other  short  lines  parallel  to  it. 

To  divide  a  given  line,  m  n,  into  two  parts  wliicli  sliall  have 
a  given  proportion  to  each  other. 

This  is  done  on  the  same  principle  as  the  last ;  thus,  let  the  proportion  be  as  I  to  3.  First  draw 
any  line  mo;  and  with  any  convenient  opening  of  the  dividers,  make  mx  equal  to  one  step  ;  and  »g 
equal  to  three  steps.     Join  s  n  ;  and  parallel  to  it  draw  x  c.     Then  m  c  is  to  c  «  as  1  is  to  3. 

Angles.  'fVTien  two  straight,  or  right  lines  meet  each  other  at  any  Inclina- 
tion, the  inclination  is  called  an  angle;  and  is  measured  by  the  degrees  con- 
tained in  the  arc  of  a  circle  described  from  the  point  of  meeting  as  a  center.  Since  all  circles,  whether 
large  or  small,  are  supposed  to  be  divided  into  360  degrees,  it  follows  that  any  number  of  degrees  of  a 
small  circle  will  measure  the  same  degree  of  inclination  as  will  the  same  number  of  a  large  one. 

When  two  straight  lines,  as  o  w  and  a  b,  meet  in  such  a  manner  that  the  inclination  o  n  ais  equal 
to  the  inclination  o  nb,  then  the  two  lines  are  said  to  be 
perpendicular  to  each  other;  and  the  angles  o  n  a  and 
0  nb,  are  called  right  angles ;  and  are  each  measd  by,  or 

are  equal  to,  90°,  or  one-fourth  part  of  the  circumf  of  a  circle.  Any  angle, 
asced,  smaller  than  a  right  angle,  is  called  acute  or  sharp ; 
and  one  c  ef,  larger  than  a  right  angle,  is  called  obtuse,  or 

blunt.  When  one  line  meets  another,  as  in  the  first  Fig  on  opposite  page,  the  two  angles  on  the 
same  side  of  either  line  are  called  contiguous,  or  adjacent.  Thus,  vus  and 
vu  w  are  adjacent ;  also  tus  and  tuw ;  sut  and  s  uv ;  wut  and  w  u  v.  The  sum  of  two  adjacent, 
anglys  is  always  equal  to  two  right  angles;  or  to  IbO'^.  Therefore,  if  we  know  the  number  of  de- 
grees contained  in  one  of  them,  and  subtract  it  from  18CP,  we  obtain  the  other. 


4 


GEOMETRY. 


93 


When  two  straight  lines  cross  each  other,  forming  four 
angles,  either  pair  of  those  angles  which  point  in  exactly 
opposite  directions  are  called  opposite,  or  vertical 
ang^les ;  thus,  the  pair  s  u  t  and  v  u  w  are  opposite  an- 
gles ;  also  the  pair  suv  and  t  u  ijy.  The  opposite  angles 
of  any  pair  are  always  equal  to  each  other. 

When  a  straight  line  a  h  crosses  two  parallel  lines  c  d, 
ef,  the  alternate  angles  which  form  a  kind  of  Z  are 
equal  to  each  other.  Thus,  the  angles  don  and  o  nf  are 
equal :  as  are  also  con  and  one.  Also  the  sum  of  the 
two  internal  angles  on  the  same  side  of  a  b,  is  equal  to  two 
right  angles,  or  180°;  thus,  c  o  w  +  o  n/=  180°;  also 
don  -\-  one  =  180°. 

An  interior  ang^le. 

In  any  fig,  is  any  angle  formed  inside  of  that  fig,  by  the  meet- 
ing of  two  of  its  sides,  as  the  angles  c  ab,  ab  c,  b  c  a,  of  tins 
triangle.  All  the  interior  angles  of  any  straight-lined  figure  of 
any  number  of  sides  whatever,  are  together  equal  to  twice  as 
many  right  angles  minus  four,  as  the  figure  has  sides.  Thus,  a 
triangle  has  3  sides ;  twice  that  number  is  6 ;  and  6  right  angles, 
or6  X  90^=540°;  from  which  take  4  right  angles,  or  360°;  and 
there  remain  180^,  which  is  the  number  of  degrees  in  every 
plane,  or  straight-lined  triangle.  This  principle  furnishes  an 
easy  means  of  testing  our  measurements  of  the  angles  of  any 
fig;  for  if  the  sum  of  all  our  measurements  does  not  agree  with 
Ihe  sum  given  by  the  ruie,  it  is  a  proof  that  we  have  committed  some  error. 

An  exterior  ang-le 

Of  any  straight-lined  figure,  is  any  angle,  as  a  b  d,  formed  by  the  meeting  of 
any  side,  as  a  b,  with  the  prolongation  of  an  adjacent  side,  as  c  6;  so  likewise 
the  angles  c  a  s  and  b  c  w.  All  the  exterior  angles  of  any  straight-lined  fig, 
no  matter  how  many  sides  it  may  have,  amount  to  360°;  but,  in  the  case  of 
a  re-entering  angle,  as  y  ij,  the  interior  angle,  g  ij,  exceeds  180°,  and  the 
"exterior"  angle,  g  i  x,  being  =  180°  —  interior  angle,  is  negative.  Thus 
ab  d  -\-  b  cw  -\-  c  a  8  =  360° ;  and  yhj-\-zji—  gix-\-igw  =  360°. 
Angles,  as  a,  b,  c,  g,  h,  arndj,  which  point  outward,  are  called  salient. 


From  any  ^iven  point,  p,  on  a  line  s  t, 
to  draw  a  perp,  p  a. 

'  From  p,  -with  any  convenient  opening  of  the  dividers,  step  off  the 
equals  po,pg.  From  o  and  g  as  centers,  with  any  opening  greater 
than  half  o  g,  describe  the  two  short  arcs  b  and  c;  and  join  a  p. 
Or  still  better,  describe  four  arcs,  and  join  a  y. 

Or  from  p  with  any  convenient  scale  describe  two 
short  arcs  g  and  c  either  one  of  them  with  a  radius  3,  and  the  other 
with  a  rad  4.    Then  from  g  with  rad  5  describe  the  arc  b.    Join  p  a. 


If  the  point  p  is  at  one  end  of  the  line, 
or  very  near  it. 

Extend  the  line,  if  possible,  and  proceed  as  above.  But  if  this 
«annot  be  done,  then  from  any  convenient  point,  w,  open  the  divid- 
ers \,o  p,  and  describe  the  semicircle,  s  p  oj  through  o  w  draw  o  w 
a;  joinp  s. 

Or  use  the  last  foreg^oing;  process  with 

rads  3,  4,  and  5. 


From  a  g-iven  point,  o,  to  let  fall  a 
perp  o  «,  to  a  g^iven  line,  tn  n. 

From  o,  measure  to  the  line  m  n,  any  two  equal  dists,  o  c, 
o  e ;  and  from  c  and  e  as  centers,  with  any  opening  greater 
than  half  of  c  e,  describe  the  two  arcs  a  and  b  ;  join  o  t.  Or 
frorh  any  point,  as  d  on  the  line,  opon  the  dividers  to  o,  and 
describe  the  arc  o  g  ;  make  i  x  equal  to  j"  o ;  and  join  o  x. 


94 


GEOMETRY, 


If  tlie  line,  a  b,  is  on  f  lie  grronnd. 

And  a  perp  is  reqd  to  be  drawn  from  c,  first  measure  oflF  any  two 
equal  dists,  cm,  en.    At  m  aud  n.  hold  the  ends  of  a  piece  of  string, 
tape-line,  or  chain,  man;  then  tighten  out  the  string,  &c,  as  shown 
hymsn;  s  being  its  center.    Then  will  «  c  be  the  reqd  perp.    Or  if 
the  perp  a;  z  is  to  be  drawn  from  the  end  of  the  line  w  x,  first  measure  x  y 
upon  the  line,  and  equal  to  three  feet;  then  holding  the  end  of  a  tape- 
line  at  X,  aud  its  nine  feet  mark  at  y,  hold  the  four  feet  mark  at  z,  keep- 
ing zx&nd  zy  equally  stretched.   Then  z  x  will  be  the  reqd  perp,  because 
8, 4,  and  5,  make  the  sides  of  a  right-angled  triangle.    Instead  of  3, 4,  and 
5,  any  multiples  of  those  numbers  may  be  used,  such  as  6,  8,  and  10 ;  OP 
9,  12,  15,  &o :  also  instead  of  feet,  we  may  use  yards,  chains,  &c. 


Throuf^b  a  g-iven  point,  a,  to  draur  a 
line,  a  c,  parallel  to  another  line, 

e  f. 

"With  the  perp  dist,  a  e,  from  any  point,  n,  in  ef,  describe 
aa  arc,  t;  draw  a  c  just  touching  the  arc. 


"7"^^ 


At  any  point,  a,  in  a  line  a  &, 
to  make  an  an^le  c«  ft^eqnal 
to  a  K'iven  ang-le,  ni  n  o. 

From  n  and  a,  with  any  convenient  rad,  describe 
She  arcs  s  t,  d  e;  measure  s  t,  and  make  e  d  equal 
k>  it ;  through  a  d  draw  a  c. 


To  bisect,  or  divide  any  ang-le,  taocy,  into 
two  equal  parts. 

From X  set  oCf  any  two  equal  dists,  xr,  x  a.  From  r  and  a  with  any  rad 
describe  two  arcs  intersecting,  as  at  o ;  and  join  o  x.  If  the  two  sides  of 
the  angle  do  not  meet,  as  c  /  and  g  h,  either  first  extend  them  until  they 
do  meet ;  or  else  draw  lines  x  w,  and  x  y,  parallel  to  them,  and  at  eqoiJ 
dista  from  them,  so  as  to  meet ;  then  proceed  as  before. 


All  angles,  as  n  a  m,  n  o  m.  at  the  ciroumf  of  a  semicircle,  and  stand 
ing  on  its  diam  n  m,  are  right  angles ;  or,  as  it  is  usually  expressed, 

all  ansfles  in  a  semicircle  are  right  angles. 

An  angle  n  s  x  at  the  centre  of  a  circle,  is  twice  as  great  as  an  angle  h 
m  X  at  the  circumf,  when  both  stand  upon  the  same  arc  n  x. 


All  angles,  as  y  d  p,  y  e  p,  y  g  p,  &t  the  circumf  of  a  circle,  and  otanding 
upon  the  same  arc,  as  y  p,  are  equal  to  each  other ;  or,  as  usually  expressed, 

all  angles  in  the  same  segment  of  a  circle  are 
equal. 


The  complement  of  an  angle  is  what  it  lacks  of  90°.     Thus,  the  com< 

piemen t  of  80°  is  90°  —  80°  =  10° ;  and  that  of  210°  is  90°  —  210°  —  — 120°. 
The  snpplement  of  an  angle  is  what  it  lacks  of  180°.  Thus,  the  supple, 
ment  of  80°  is  180°  —  80°  =  100° ;  and  that  of  210°  is  180°  —  210°  =  —  30°. 
But  ordinarily  we  may  neglect  the  signs  -J-  and  — ,  before  complements  and 
supplements,  and  call  the  complement  of  an  angle  its  diff  from  90°*  aa4 
the  supplement  its  diff  from  180°. 


ANGLIB. 


95 


Angeles  in  a  Parallelogram. 

A  parallelogram  is  any  four-sided  straight-lined  fig-< 
ure  whose  opposite  sides  are  equal,  as  ah  c  d;  or  a 
square,  &c.  Any  line  drawn  across  a  parallelogram 
between  2  opposite  angles,  is  called  a  diagonal,  as  a  c, 
OT  b  d.  A  diag  divides  a  parallelogram  into  two  equal 
parts  ;  as  does  also  any  line  m  n  drawn  through  the 
center  of  either  diag ;  and  moreover,  the  line  m  n 
itself  is  div  into  two  equal  parts  by  the  diag.  Two 
diags  bisect  each  other ;  they  also  divide  the  parallel- 
ogram into  four  triangles  of  equjtl  areas.  The  sum 
of  the  two  angles  at  the  ends  of  any  one  side  is  =  180° ;  thus,  dab  +  abc  —  abc-j- 
b  c  d  —  180° ;    and  the  sum  of  the  four  angles,  d  a  b,  a  b  c,  b  c  d,  c  d  a  =  360°. 

The  sum  of  the  squares  of  the  four  sides,  is  equal  to  the  sum  of  the  squares  of  the 
two  diags. 

To  reduce  Minutes  and  SSeconds  to  Degrees  and  decimals 
of  a  Deg-ree,  etc. 

In  any  given  angle — 
Humber  of  degrees  =  Number  of  minutes  -f-  60. 

=  Number  of  seconds  -j-  3600." 

Number  of  minutes  =  Number  of  degrees  X  60. 
=  Number  of  seconds  -^  60. 

Number  of  seconds  =  Number  of  degrees  X  3600. 
=  Number  of  minutes  X  60. 

Table  of  Minutes  antl  Seconds  in  Decimals  of  a  Degree, 
and  of  i^ieconds  in  Decimals  of  a  Minute. 

(The  columns  of  Mins  and  Degs  answer  equally  for  Sees  and  Mins.) 

Mins.  Deg.     Mins.  Deg.      Mins.  Deg.      i  Sees.     Deg.      Sees.    Deg.  |  Sees.     Deg. 


In  each  equivalent,  the  last  < 

digit  repeats  inde 

fini 

itely. 

See 

*  below : 

1 

0.016 

21    0.350 

41 

0.683 

1 

0.00027 

21 

0.00583    41 

0.01138 

2 

0.033 

22    0.366 

42 

0.700 

2 

0.00055 

22 

0.00611     42 

0.01166 

3 

0.050 

23    0.383 

43 

0.716 

3 

0.00083 

23 

0.00638    43 

0.01194 

4 

0.066 

24    0.400 

44 

0.733 

4 

0.00111 

24 

0.0U666  1  44 

0.01222 

5 

0.083 

25    0.416 

45 

0.750 

5 

0.00138 

25 

0.00694  1  45 

0.01250 

6 

0.100 

26    0.433 

46 

0.766 

6 

0.00166 

26 

0.00722    46 

0.01277 

7 

0.116 

27    0.450 

47 

0.783 

7 

0.00194 

27 

0.00750    47 

0.01305 

8 

(U33 

28    0.466 

48 

0.800 

8 

0.00222 

28 

O.U0777  '  48 

0.01333 

9 

0.150 

29    0.483 

49 

0.816 

9 

0.00250 

29 

0.00805    49 

0.01361 

10 

0.166 

30  ■  0.500 

50 

0.833 

10 

0.00277 

30 

0.00833  1  60 

0.01388 

11 

0.183 

31     0.516 

51 

0.850 

11 

0.00305 

31 

0.00861  1  51 

0.01416 

12 

0.200 

32    0.533 

52 

0.866 

12 

0.00333 

32 

0.00888  1  52 

0.01444 

13 

0.216 

33    0.550 

53 

0.883 

13 

0.00361 

33 

0.00916  1  53 

0.01472 

14 

0.233 

34    0.566 

54 

0.900 

14 

0.00388 

34 

0.00944 

54 

0.01500 

15 

0.250 

35    0.583 

55 

0.916 

15 

0.00416 

35 

0.00972 

55 

0.01527 

16 

0.266 

36    0.600 

56 

0.933 

16 

0.00444 

36 

0.01000 

56 

0.01555 

17 

0.283 

37    0.616 

57 

0.950 

17 

0.00472 

37 

0.01027 

57 

0.01583 

18 

0.300 

38    0.633 

58 

0.966 

18 

0.005('0 

38 

0.01055 

58 

0.01611 

19 

0.316 

39    0.650 

59 

0.983 

19 

0.00527 

39 

0.01083 

59 

0.01638 

20 

0.333 

40    0.666 

60 

1.000 

20 

0.00555 

40 

0.01111 

60 

0.01666 

Sees 

Min. 

Sees.  Min. 

Sees 

Min. 

Sees 

.     Deg. 

Sees.  Deg. 

Sees.   Deg. 

*  Each  equivalent  is  a  repeating  decimal,  thus : 
2  minutes  ==  0.0333333  ....  degree        12  seconds  =  0.2000000 
7        "        -=  0.1166666  ....        "  1  second    =  0.0002777 

12        "        =  0.2000000  ....        "  50  seconds  =  0.0138888 


.  minute 
.  degree 


96 


ANGLES. 


Approximate  Measurement  of  Angles. 

(1)  The  four  fingrers  of  the  hand,  held  at  right  angles  to  the  arm  and 

at  arm's  length  from  the  eye,  cover  about  7  degrees.  And  an  angle  of  7°  corre- 
sponds to  about  12.2  feet  in  100  feet ;  or  to  36.6  feet  in  100  yards ;  or  to  645  feet  in  a 
mile. 

(2)  By  means  of  a  two-foot  rnle,  either  on  a  drawing  or  between  dis- 
tant objects  in  the  field.  If  the  inner  edges  of  a  common  two-foot  rule  be  opened 
to  the  extent  shown  in  the  column  of  inches,  they  will  be  inclined  to  each  other 
at  the  angles  shown  in  the  column  of  angles.  Since  an  opening  of  ]/^  inch  (up 
to  19  inches  or  about  105°)  corresponds  to  from  about  3^°  to  1°,  no  great  accuracy 
is  to  be  expected,  and  beyond  105°  still  less;  for  the  liability  to  error  then  in- 
creases very  rapidly  as  the  opening  becomes  greater.  Thus,  the  last  }/^  inch  cor- 
responds to  about  12° 

Angles  for  openings  intermediate  of  those  given  may  be  calculated  to  the 
nearest  minute  or  two,  by  simple  proportion,  up  to  23  inches  of  opening,  or 
about  147°. 

Table  of  Ang-les  corresponding  to  openin^^s  of  a  2-foot  rule. 

(Original). 

Correct. 


Ins. 

Deg.  min. 

Ins. 

Deg.  minJ 

Tns. 

Deg.  min. 

Ins. 

Deg.min. 

Ins. 

Deg.min. 

Ins. 

Deg.  min 

K 

1 

12 

4M 

20 

24 

8M 

40 

IS 

12>i 

61 

23 

16>i 

85 

14 

20>i 

115  5 

1 

48 

21 

40 

51 

62 

5 

86 

3 

116  12 

H 

2 

24 

H 

21 

37 

]4 

41 

29 

J4 

62 

47 

14 

86 

52 

H 

117  20 

3 

GO 

22 

13 

42 

7 

63 

28 

87 

41 

118  30 

H 

3 

36 

H 

22 

50 

H 

42 

46 

H 

64 

11 

H 

88 

31 

H 

119  40 

4 

11 

23 

27 

43 

24 

64 

53 

89 

21 

120  52 

1 

4 

47 

5 

24 

3 

9 

44 

3 

13 

65 

35 

17 

90 

12 

21 

122  6 

5 

23 

24 

39 

44 

42 

66 

18 

91 

3 

123  20 

H 

5 

58 

H 

25 

16 

H 

45 

21 

U. 

67 

1 

H 

91 

54 

H 

124  S6 

6 

34 

25 

53 

45 

59 

67 

44 

92 

46 

125  54 

K 

7 

10 

^ 

26 

30 

H 

46 

38 

H 

6a 

28 

H 

93 

38 

H 

127  14 

7 

46 

27 

7 

47 

17 

69 

12 

94 

31 

128  35 

H 

8 

22 

% 

27 

44 

H 

47 

56 

H 

69 

55 

H 

95 

24 

H 

129  59 

8 

58 

28 

21 

48 

35 

70 

38 

96 

17 

131  25 

2 

9 

34 

6 

28 

58 

10 

49 

15 

14 

71 

22 

18 

97 

11 

22 

132  53 

10 

10 

29 

35 

49 

54 

72 

6 

98 

5 

134  24 

H 

10 

46 

J^ 

30 

11 

^ 

50 

34 

H 

72 

51 

H 

99 

00 

34 

135  58 

11 

22 

30 

49 

51 

13 

73 

86 

99 

55 

137  35 

H 

11 

58 

yi 

31 

26 

H 

51 

53 

H 

74 

21 

M 

100 

51 

H 

139  16 

12 

34 

32 

3 

52 

33 

75 

6 

101 

48 

Ul  1 

H 

IS 

10 

H 

32 

40 

H 

53 

13 

H 

75 

51 

H 

102 

45 

H 

142'  51 

13 

46 

33 

17 

53 

53 

76 

36 

103 

43 

144  46 

3 

14 

22 

7 

33 

54 

11 

54 

34 

15 

77 

22 

19 

104 

41 

23 

146  48 

14 

58 

34 

33 

55 

14 

78 

8 

105 

40* 

148  56 

M 

15 

34 

H 

35 

10 

H 

55 

55 

H 

78 

54 

H 

106 

39 

H 

151  If 

16 

10 

35 

47 

56 

35 

79 

40 

107 

40 

153  41 

^ 

16 

46 

14 

36 

25 

H 

57 

16 

H 

80 

27 

H 

108 

41 

H 

156  3fc 

17 

22 

37 

3 

57 

57 

81 

14 

109 

43 

159  48 

H 

17 

59 

H 

37 

41 

H 

5b 

38 

H 

82 

2 

H 

110 

46 

H 

163  2T 

18 

35 

38 

19 

59 

19 

82 

49 

111 

49 

168  18 

4 

19 

12 

8 

38 

57 

12 

60 

00 

16 

83 

37 

20 

112 

53 

24 

180  00 

19 

48 

39 

35 

60 

41 

84 

26 

113 

58 

(3)  With  the  same  table,  usin^  feet  instead  of  inches.  From 
the  given  point  measure  12  feet  toward*  each  object,  and  place  marks.  Measure 
the  distance  in  feet  between  these  marks.  Suppose  the  first  column  in  the  table  to 
be  feet  instead  of  inches.     Then  opposite  the  distance  in  feet  will  be  the  angle. 

"%  foot  =•  1.5  inches. 

1  in.    =  .083  ft.     I  4  ins.  =  .333  ft.  I  7  ins.  =  .583  ft.  I  10  ins.  =    .833  ft. 

2  ins.  =  .167  ft.       5  ins.  =  .416  ft.      8  ins.  =  .607  ft.      11  ins.  =    .917  ft. 

3  ins.  =  .25  ft.      1  6  ins.  -=  .5  ft.       |  9  ins.  =  .75  ft,    I  12  ins.  =  1.0  ft. 

(4)  Or,  measure  toward  *  each  object  100  or  any  other  number  of 
feet,  and  place  marks.    Measure  the  distance  in  feet  between  the  marks.    Then 

Sine  of  half  _  half  the  distance  between  the  marks 

the  angle     ~  the  distance  measured  toward  one  of  the  objects* 

Find  this  sine  in  the  table  pp.  98,  etc. ;  take  out  the  corresponding  angle  and 
multiply  it  by  2 

(5)  See  last  paragraph  of  foot-note,  pp  152  and  153. 


*  If  it  is  inconvenient  to  measure  toward  the  objects,  measure  directly  /rom  thenu 


SIKES,  TANGENTS,  ETC. 


97 


Sines,  Tangents,  Ac. 

$iine,  a  8,  of  any  angle,  a  ch,  or  -which  is  the  same  thins,  the  sine  of  any  circular  arc,  a  6, 
which  subtends  or  measures  the  angle,  is  a  straight  line  drawn  from  one  end,  as  a,  of  the  arc,  at  right 
fciidjles  to,  and  terminating  at,  the  rad  c  b,  drawn  to  the  other  end  b  of  the  arc.  It  is,  therefore,  equal 
to  half  the  chord  a  n,  of  the  arc  abn,  which  is  equal  to  twice  the  arc  a6  ;  or.  the  sine  of  an  angle  ia 
always  squal  to  half  the  chord  of  twice  that  angle ;  and  vice  versa,  the  chord  of  an  angle  is  alwayj 
equal  to  twice  the  sine  of  half  the  angle. 

f  he  sine  <  c  of  an  angle  tcb,  or  of  an  arc  in 

tab,  of  90°,  is  equal  to  the  rad  of  the  arc 
or  of  the  circle  ;  and  this  sine  of  90°  is 
greater  than  that  of  any  other  angle. 

Cosine  c  *  of  an  angle  a  cb, 

l8  that  part  of  the  rad  which  lies  between 
the  sine  and  the  center  of  the  circle.  It 
is  always  equal  to  the  sine  y  a  of  the 
complement  t  c  aof  a  c  b;  or  of  what  a 
c  b  wants  of  being  90°.  The  prefix  co  be- 
fore sines,  &c,  meaus  complement ;  thus, 
cosine  means  sine  of  the  complement. 

Versed  sine  5  &  of  any  angle 

o  c  6,  is  that  part  of  the  diam  which  lies 
between  the  sine,  and  the  outer  end  6. 
It  is  very  common,  but  erroneous,  when 
speaking  of  bridges,  &c,  to  call  the  rise 
or  height  s  6  of  a  circular  arch  abn,  its 
rersed  sine;  while  it  is  actually  the  versed 
sineofonly  half  lAie  arch.  Thisabsurdity 
should  cease ;  for  the  word  rise  or  height 
is  not  only  more  expressive,but  is  correct. 
Tan8-eiit6z<;orarf,ofanyangle 
a  c  2),  is  a  line  drawn  from,  and  at  right 
angles  to,  the  end  6  or  a  of  either  rad  c  b, 
or  c  a,  which  forms  one  of  the  legs  of  the 
angle ;  and  terminating  as  at  w,  or  d,  in 
the  prolongation  of  the  rad  which  forms 
the  other  leg.  This  last  rad  thus  pro- 
longed, that  is.  c  w,  or  c  d,  as  the  case  may 

lie,  is  the  secant  of  the  angle 
«  c  6.  The  angle  tcb  being  supposed 
to  be  equal  to  90°,  the  angle  tea  becomes  the  complement  of  the  angle  a  e  &,  or  what  aeb  want! 
of  being  90° ;  and  the  sine  y  a  of  this  complement ;  its  versed  sine  f  y  ;  its  tangent  t  o ;  and  its  seoant 
c  0,  are  respectively  the  co-sine,  co-versed  sine;  co-tangent;  and  co-secant,  of  the  angle  a  c  b.  Or, 
vice  versa,  the  sine,  &c,  of  a  c  b,  are  the  cosine,  &c,  of  tea;  because  the  angle  a  c  6  is  the  comple- 
ment of  the  angle  t  c  a.  When  the  rad  c  6,  c  a.  or  c  t,  is  assumed  to  be  equal  to  unity,  or  1,  the  cor- 
responding sines,  tangents,  &c,  are  called  natwrai  ones;  and  their  several  lengths"for  ditT  angles, 
for  said  rad  of  unity,  have  been  calculated:  constituting  the  well-known  tables  of  nat  sines,  &c.  In 
any  circle  whose  rad  is  either  larger  or  smaller  than  1,  the  sines,  <fcc,  of  the  angles  will  be  in  the 
same  proportion  larger  or  smaller  than  those  in  the  tables,  and  are  consequently  found  by  mult  the 
sine,  &c,  of  the  table,  by  said  larger  or  smaller  rad. 

The  folloTFing:  table  of  natural  sines.  Ac.  does  not  contain  nat 
versed  sines,  co- versed  sines,  secants,  nor  cosecants,  but  these  may  be  found  thus; 
for  any  angle  not  exceeding  90  degrees. 

Versed  Sine.     Prom  1  take  the  nat  cosine. 

Co-versed  Sitie.     From  I  take  the  nat  sine. 
.   Secant.     Divide  1  by  the  nat  cosine. 

^"'-""'nnt.  O'vide  ]  by  the  nat  sine. 
F«r  an^gles  exceeding  90^^  ;  to  find  the  sine,  cosine,  tangent,  cotang,  seoant,  or  cosec,  (but  not 
the  versed  sine  or  co-versed  sine),  take  the  angle  from  180° ;  if  between  180°  and  270°  take  180°  from 
the  angle ;  if  bet  270°  and  360°,  take  the  angle  from  360°.  Then  in  each  case  take  from  the  table  the 
Bine,  cosine,  tang,  or  cotang  of  the  remainder,  find  its  secant  or  cosec  as  directed  above.  For  tAe 
versed  sine;  if  between  90°  and  270°,  add  cosine  to  I ;  if  bet  270°  and  360°,  take  cosine  from  1.  CThe 
engineer  seldom  needs  sines,  &c,  exceeding  180°. 

To  find  the  nat  sine,  cosine,  tan^,  secant,  versed  sine,  Ae^ 
of  an  an;s-Ie  containing^  seconds.     First  find  that  due  to  the  given  deg 

and  min  ;  then  the  next  greater  one.  Take  their  diff.  Then  as  60  sec  are  to  this  difif,  so  are  the  sec 
only  of  the  given  angle  to  a  dec  quantity  to  be  a<lded  to  the  one  first  taken  out 
if  it  is  a  sine,  tang,  secant,  &c  ;  or  to  be  subtracted  from  it  if  it  is  a  cosine, 
cotang,  cosecant,  &c. 

The  tangents  in  the  table  are  strict  trigonometrical  ones ;  that  is, 
tangents  to  given  angles;  and  which  must  extend  to  meet  the  secants  of  the  angles 
to  which  they  belong.  Ordinary,  or  geometrical  tangents,  as  those  on 
p  162,  may  extend  as  far  as  we  please.  In  the  field  practice  of  railroad 
curves,  two  trigonometrical  tangents  terminate  where  they  meet  each  other. 
Each  of  these  tangs  is  tlie  tang  of  half  the  curve.  It  is  usually,  but  improperly, 
called  "the  tang  of  the  curve."^  "Apex  dist  of  the  curve,"  as  suggested  by  Mr 
Shunk,  would  be  better. 


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TABLE  OF  CHORDS. 


143 


The  table  of  chords,  below,  furnishes  the  means  of  laying  down  angles  on 
paper  more  accurately  than  by  an  ordinary  protx'actor.  To  do  this,  after  having  drawn 
and  measured  the  first  side  (say  ac)  of  the  figure  that  is 
to  be  plotted;  from  its  end  c  as  a  center,  describe  an  arc 
ny  of  a  circle  of  suflacient  extent  to  subtend  the  angle  at 
that  point.  The  rad  en  with  which  the  arc  is  described 
should  be  as  great  as  convenience  will  permit ;  and  it  is  to 
be  assumed  as  unity  or  1 ;  and  must  be  decimally  divided, 
and  subdivided,  to  be  used  as  a  scale  for  laying  down  the 
chords  taken  from  the  table,  in  which  their  lengths  are 
given  in  parts  of  said  rad  1.  Having  described  the  arc,  find 
in  the  table  the  length  of  the  chord  n  t  corresponding  to 
the  angle  act.  Let  us  suppose  this  angle  to  be  45°;  then 
we  find  that  the  tabular  chord  is  .7654  of  our  rad  1.  There- 
fore from  n  we  lay  off  the  chord  nt,  equal  to  .7654  of  our  radius-scale ;  and  the  lint 
cs  drawn  through  the  point  t  will  form  the  reqd  angle  act  oi  45^^.  And  so  at  each 
angle.  The  degree  of  accuracy  attained  will  evidently  depend  on  the  length  of  the 
rad,  and  the  neatness  of  the  drafting.  The  method  becomes  preferable  to  the  com- 
mon protractor  in  proportion  as  the  lengths  of  the  sides  of  the  angles  exceed  the  rad 
of  the  protractor.  With  a  protractor  of  4  to  6  ins  rad,  and  with  sides  of  angles  not 
much  exceeding  the  same  limits,  the  protractor  will  usually  be  preferable.  The  di- 
viders in  boxes  of  instruments  are  rarely  fit  for  accurate  arcs  of  more  than  about  6 
ins  diam.  In  practice  it  is  not  necessary  to  actually  describe  the  whole  arc,  but 
merely  the  portion  near  t,  as  well  as  can  be  judged  by  eye.  We  thus  avoid  much  use 
of  the  India-rubber,  and  dulling  of  the  pencil-point.  For  larger  radii  we  may  dis- 
pense with  the  dividers,  and  use  a  straight  strip  of  paper  with  the  length  of  the  rad 
marked  on  one  edge ;  and  by  laying  it  from  c  toward  s,  and  at  the  same  time  placing 
another  strip  (with  one  edge  divided  to  a  radius-scale)  from  n  toward  t,  we  can 
by  trial  find  their  exact  point  of  intersection  at  the  required  point  t.  In  such  mat- 
ters, practice  and  some  iu<^enuity  are  very  essential  to  satisfactory  results.  We  can- 
not devote  more  space  to  the  subject. 


CHORDS  TO  A  EADIUS  1. 


M. 

0° 

1° 

,o 

3° 

40 

5° 

6° 

7° 

8° 

9° 

10° 

M. 

0' 

.0000 

.0175 

.0349 

.0524 

.0698 

.0872 

.1047 

.1221 

.1395 

.1569 

.1743 

0' 

2 

.0006 

.0180 

.0355 

.0529 

.0704 

.0878 

.1053 

.1227 

.1401 

.1575 

.1749 

2 

4 

.0012 

.0186 

.0361 

.0535 

.0710 

.0884 

.1058 

.1233 

.1407 

.1581 

.1755 

4 

(> 

.0017 

.0192 

.0366 

.0541 

.0715 

.0890 

.1064 

.1238 

.1413 

.1587 

.1761 

6 

8 

.0023 

.0198 

.0372 

.0547 

.0721 

.0896 

.1070 

.1244 

.1418 

.1592 

.1766 

8 

10 

.0029 

.0204 

.0378 

.0553 

.0727 

.0901 

.1076 

.1250 

.1424 

.1598 

.1772 

10 

12 

.0035 

.0209 

.0384 

.0558 

.0733 

.0907 

.1082 

.1256 

.1430 

.1604 

.1778 

12 

14 

.0041 

.0215 

.0390 

.0564 

.0739 

.0913 

.1087 

.1262 

.1436 

.1610 

.1784 

14 

16 

.0047 

.0221 

.0396 

.0570 

..0745 

.0919 

.1093 

.1267 

.1442 

.1616 

.1789 

16 

18 

.0052 

.0227 

.0401 

.0576 

.0750 

.0925 

.1099 

.1273 

.1447 

.1621 

.1795 

18 

20 

.0058 

.0233 

.0407 

.0582 

.0756 

.0931 

.1105 

.1279 

.1453 

.1627 

.1801 

20 

22 

.0064 

.0239 

.0413 

.0588 

.0762 

.0;-»36 

.1111 

.1285 

.1459 

.1633 

.1807 

22 

24 

.0070 

.0244 

.0419 

.0593 

.0768 

.0942 

.1116 

.1291 

.1465 

.1639 

.1813 

24 

26 

.0076 

.0250 

.0425 

.0599 

.0774 

.0948 

.1122 

.1296 

.1471 

.1645 

.1818 

26 

28 

.0081 

.0256 

.0430 

.0605 

.0779 

.0954 

.1128 

.1302 

.1476 

.1650 

.1824 

28 

30 

.0087 

.0262 

.0268 

.0436 

.0611 

.0785 

.0960 

.1134 

.1.308 

.1482 

.1656 

.1830 

30 

32 

.0093 

.0442 

.0617 

.0791 

.0965 

.1140 

.1314 

.1488 

.1662 

.1836 

32 

.0448 

.0622 

.0797 

.0971 

.1145 

.1320 

.1494 

.1668 

.1842 

34 

.0454 

.0628 

.0803 

.0977 

.1151 

.1.325 

,1,500 

.1674 

.1847 

36 

.0460 

.0634 

.0808 

.0983 

.11.57 

.1.331 

,1.505 

.1679 

.1853 

38 

40 

.0116 

.0291 
.0297 

.0465 

.0640 

.0814 

.0989 

.1163 

.1.337 

.1511 

.1685 

.1859 

40 

42 

.0122 

.0471 

.0646 

.0820 

.0994 

.1169 

.1343 

.1517 

.1691 

.1865 

42 

44 

.0128 

.0303 

.0477 

.0651 

.0826 

.1000 

.1175 

.1349 

.1.523 

,1697 

.1871 

44 

46 

.0134 

.0308 

.0483 

.0657 

.0832 

.1006 

.1180 

.1.355 

.1.529 

,1703 

.1876 

46 

48 
50 

.0140 

.0314 

.0489 

.0663 

.0838 

.1012 

.1186 

.1.360 

.1534 

,1708 

.1882 

48 

.0145 

.0320 

.0494 

.0669 

.0843 

.1018 

.1192 

.1366 

.1540 

,1714 

.1888 

50 

52 

.0151 

.0326 

.0,500 

.0675 

.0849 

.1023 

.1198 

.1.372 

,1546 

.1720 

.1894 

52 

.0332 

.0506 

.0681 

.0855 

.1029 

.1204 

.1378 

,15.52 

.1726 

.1900 

54 

.0.337 

.051 2 

.0686 

.0861 

.10.35 

.1209 

.1384 

.1558 

,1732 

.1905 
.1911 

56 

58 

.0169 

.0343 

.0518 

.0692 

.0867 

.1041 

.1215 

.1389 

.1563 

,1737 

58 

SO 

.0175 

.0349 

.0524 

.0698 

.0872 

.1047 

.1221 

.1395 

.1569 

.1743 

.1917 

60 

144 


TABLE    OF   CHORDS. 


Table  of  Cbords,  io  parts  of  a  rad  1;  for  protraeting*— Continued.! 


M. 

11° 

12° 

13° 

14° 

15° 

16° 

17° 

18° 

19° 

20° 

M. 

0' 

.1917 

.2091 

.2264 

.2437 

.2611 

.2783 

.2956 

.3129 

.3301 

.3473 

0' 

2 

.1923 

.2096 

.2270 

.2443 

.2616 

.2789 

.2962 

.3134 

.3307 

.3479 

2 

4 

.1928 

.2102 

.2276 

.2449 

.2622 

.2795 

.2968 

.3140 

.3312 

.3484 

4 

6 

.1934 

.2108 

.2281 

.2455 

.2628 

.2801 

.2973 

.3146 

.3318 

.3490 

6 

8 

.1940 

.2114 

.2287 

.2460 

.2634 

.2807 

.2979 

.3152 

.3324 

.3496 

8 

10 

.1946 

.2119 

.2293 

.2466 

.2639 

.2812 

.2985 

.3157 

.3330 

.3502 

10 

J2 

.1952 

.2125 

.2299 

.2472 

.2645 

.2818 

.2991 

.3163 

.3335 

.3507 

12 

U 

.1957 

.2131 

.2305 

.2478 

.2651 

.2824 

.2996 

.3169 

.3341 

.3513 

14 

16 

.1963 

.2137 

.2310 

.2484 

.2657 

.2830 

.3002 

.3175 

.3347 

.3519 

16 

18 

.1969 

.2143 

.2316 

.2489 

.2662 

.2835 

.3008 

.3180 

.3353 

.3525 

18 

20 

.1975 

.2148 

.2322 

.2495 

.2668 

.2841 

.3014 

.3186 

.3358 

.3530 

20 

22 

.1981 

.2154 

.2328 

.2501 

.2674 

.2847 

.3019 

.3192 

.8364 

.3536 

22 

M 

.1986 

.2160 

.2333 

.2507 

.2680 

.2853 

.3025 

.3198 

.3370 

.3542 

24 

26 

.1992 

.2166 

.2339 

.2512 

.2686 

.2858 

.3031 

.3203 

.3376 

.3547 

26 

28 

.1998 

.2172 

.2345 

.2518 

.2691 

.2864 

.3037 

.3209 

.3381 

.3553 

28 

30 

.2004 

.2177 

.2.351 

.2524 

.2697 

.2870 

.3042 

.3215 

.3387 

.3559 

30 

32 

.2010 

.2183 

.2357 

.2530 

.2703 

.2876 

.3048 

.3221 

.3393 

.3565 

32 

U 

.2015 

.2189 

.2362 

.2536 

.2709 

.2881 

.3054 

.3226 

.3398 

.3570 

34 

36 

.2021 

.2195 

.2368 

.2541 

.2714 

.2887 

.3060 

.3232 

.3404 

.3576 

36 

38 

.2027 

.2200 

.2374 

.2547 

.2720 

.2893 

.3065 

.3238 

.3410 

.3582 

38 

40 

.2033 

.2206 

.2380 

.2553 

.2726 

.2899 

.3071 

.3244 

.3416 

.3587 

40 

42 

.2038 

.2212 

.2385 

.2559 

.2732 

.2904 

.3077 

.3249 

.3421 

.3593 

42 

44 

.2044 

.2218 

.2391 

.2564 

.2737 

.2910 

.3083 

.3255 

.3427 

.3599 

44 

46 

.2050 

.2224 

.2397 

.2570 

.2743 

.2916 

.3088 

.3261 

.3433 

.3605 

46 

48 

.2056 

.2229 

.2403 

.2576 

.2749 

.2922 

.3094 

.3267 

.3439 

.3610 

48 

50 

.2062 

.2235 

.2409 

.2582 

.2755 

.2927 

.3100 

.3272 

.3444 

.3616 

60 

52 

.2067 

.2241 

.2414 

.2587 

.2760 

.2933 

.3106 

.3278 

.3450 

.3622 

52 

54 

.2073 

.2247 

.2420 

.2593 

.2766 

.2939 

.3111 

.3284 

.3456 

.3628 

54 

56 

.2079 

.2253 

.2426 

.2599 

.2772 

.2945 

.3117 

.3289 

.3462 

.3633 

.56 

58 

.2085 

.2258 

.2432 

.2605 

.2778 

.2950 

.3123 

.3295 

.3467 

..3639 

58 

60 

.2091 

.2264 

.2437 

.2611 

.2783 

.2956 

.3129 

.3301 

.3473 

.3645 

60 

M. 

21° 

22° 

23° 

24° 

25° 

26° 

27° 

28° 

29° 

30° 

M. 

0' 

.3645 

.3816 

.3987 

.4158 

.4329 

.4499 

.4669 

.4838 

.5008 

.5176 

0' 

2 

.3650 

.3822 

.3993 

.4164 

.4334 

.4505 

.4675 

.4844 

.5013 

.5182 

4 

.3656 

.3828 

.3999 

.4170 

.4340 

.4510 

.4680 

.4850 

.5019 

.5188 

6 

.3662 

.3833 

.4004 

.4175 

.4346 

.4516 

.4686 

.4855 

.5024 

.5198 

8 

.3668 

.3839 

.4010 

.4181 

.4352 

.4522 

.4692 

.4861 

.5030 

.5199 

10 

.3673 

.3845 

.4016 

.4187 

.4357 

.4527 

.4697 

.4867 

.5036 

.5204 

12 

.3679 

.3850 

.4022 

.4192 

.4363 

.4533 

.4703 

.4872 

.5041 

.5210 

14 

.3685 

.3856 

.4027 

.4198 

.4369 

.4539 

.4708 

.4878 

.5047 

.5216 

16 

.3690 

.3862 

.4033 

.4204 

.4374 

.4544 

.4714 

.4884 

.5053 

.5221 

18 

.3696 

.3868 

.4039 

.4209 

.4380 

.4550 

.4720 

.4889 

.5058 

.5227 

20 

.3702 

.3873 

.4044 

.4215 

.4386 

.4556 

.4725 

.4895 

.5064 

.5233 

20 

22 

.3708 

.3879 

.4050 

.4221 

.4391 

.4561 

.4731 

.4901 

.5070 

.5238 

22 

24 

.3713 

.3885 

.4056 

.4226 

.4397 

.4567 

.4737 

.4906 

.5075 

.5244 

24 

26 

.3719 

.3890 

.4061 

.4232 

.4403 

.4573 

.4742 

.4912 

.5081 

.5249 

26 

28 

.3725 

.3896 

.4067 

.4238 

.4408 

.4578 

.4748 

.4917 

.5086 

.5255 

28 

30 

.3730 

.3902 

.4073 

.4244 

.4414 

.4584 

.4754 

.4923 

.5092 

.5261 

30 

32 

.3736 

.3908 

.407« 

.4249 

.4420 

.4590 

.4759 

.4929 

.5098 

.5266 

82 

34 

.3742 

..3913 

.4084 

.4255 

.4425 

.4595 

.4765 

.4934 

.5103 

.5272 

34 

36 

.3748 

.3919 

.4090 

.4261 

.4431 

.4601 

.4771 

.4940 

.5109 

.5277 

36 

88 

.3753 

..3925 

.4096 

.4266 

.4437 

.4607 

.4776 

.4946 

.5115 

.5283 

38 

40 

.3759 

.3930 

.4101 

.4272 

.4442 

.4612 

.4782 

.4951 

.5120 

.5289 

40 

42 

.3765 

.3936 

.4107 

.4278 

.4448 

.4618 

.4788 

.4957 

.5126 

.5294 

42 

44 

.8770 

.3942 

.4113 

.4283 

.4454 

.4624 

.4793 

.4963 

.5131 

.5300 

44 

46 

.3776 

..3947 

.4118 

.4289 

.44.S9 

.4629 

.4799 

.4968 

.5137 

.5306 

46 

48 

.3782 

.3953 

.4124 

.4295 

.4465 

.4635 

.4805 

.4974 

.5143 

.5311 

48 

50 

.3788 

.3959 

.4130 

.4300 

.4471 

.4641 

.4810 

.4979 

.5148 

.5317 

50 

52 

.3793 

.3965 

.4135 

.4.306 

.4476 

.4646 

.4816 

.4985 

..5154 

.5322 

52 

54 

.3799 

..3970 

.4141 

.4312 

.4482 

.4652 

.4822 

.4991 

..5160 

.5328 

54 

56 

.3805 

..3976 

.4147 

.4317 

.4488 

.4658 

.4827 

.4996 

.5165 

.5334 

bS 

58 

.3810 

..3982 

.4153 

.4323 

.4493 

.4663 

.4833 

.5002 

.5171 

.53.39 

58 

eo 

.3816 

.3987 

.4158 

.4329 

.4499 

.4669 

.4838 

.5008 

.5176 

.5345 

«0 

TABLE   OF    CHORDS. 


145 


Table  of  cliords,iii  parts  of  a  rad  1;  for  protracting— Continued 


M.      31° 


.5345 
.5350 
.5356 


.5378 
.5384 
.5390 
.5395 
.5401 

.5406 
.5412 
.5418 
.1423 
.5429 


32° 


.5513 
.5518 
.5524 
.5530 
.5535 
.5541 


.5546 
.5552 
.5557 
.5563 
.5569 


.5574 
.5580 
.5585 
.5591 
.5597 


.5691 
.5697 
.5703 
.5708 


.5714 
.5719 
.5725 
.5730 
.5736 


.5742 
.5747 
.5753 

.5758 
.5764 


34° 


.5847 
.5853 
.5859 
.5864 
.5870 
.5875 


.5881 
.5886 


.5897 
.5903 


35° 


.6014 
.6020 
.6025 
.6031 
.6036 
.6042 


.5909 
.5914 
.5920 
.5925 
.5931 


.6075 
.6081 
.(i086 
.6032 
.6097 


37° 


.6180 
.6186 
.6191 
.6197 
.6202 
.6208 


.6214 
.6219 
.6225 
.6230 
.6236 


.6346 
,6352 
.6357 
.6363 
.6368 
.6374 


.6379 
.6385 
.6390 
.6396 
.6401 


.6241    .6407 


.6247 
.6252 
.6258 
.6263 


,6412 
.6118 
.6423 
.6429 


38° 


.6511 
.6517 
.6522 
.6528 
.6533 
.6539 


.6544 
.6550 
.6555 
.6561 


.6572 
.6577 
.6583 
.6588 
.6594 


39° 


.66«2 
.6687 
.6693 
.6698 
.6704 


.6709 
.6715 
.6720 
.6725 
.6731 


.6736 
.6742 
.6747 
.6753 
.6758 


40°  M. 


.6840 
.6846 
.6851 
.6857 
.6862 
.6868 


.6873 
.6879 

.6884 


.6906 
.6911 
.6917 
.6922 


.5434 
.5440 
.5446 
.5451 
.5457 


.5602 
.5608 
.5613 
.5619 
.5625 


.5769 
.5775 
.5781 
.5786 
.5792 


.5936 
.5942 
.5947 
.5953 
.5959 


.6103 
.6108 
.6114 
.6119 
.6125 


.6274 
.6280 
.6285 
.6291 


,6434 
,6440 
.6445 
.6451 
.6156 


.6599 
.6605 
6610 
.6616 
.6621 


.6764 
.6769 

.6775 
.6780 
.6786 


.6928 
.6933 
.6939 
.o944 
.6950 


.5462 
.5468 
.5474 
.5479 
.5485 

.5490 
.5496 
.5502 
.5507 
.5513 


.5630 
.5636 
.5641 
.5647 
.5652 


.5797 
.5803 
.5808 
.5814 
.5820 


.5964 
.5970 
.5975 
.5981 
.5986 


.6130 
.6136 
.6142 
.6147 
.6153 


.0236 
.6302 
.6307 
.6313 
.6318 


.6462 
I  .6467 
I  .6473 

I  .6478 
!  .6484 


.6627 
.6632 
.6638 
.6643 
.6649 


.6791 
,6797 
.6802 
.6808 
.6813 


.6955 
.6961 


.,5664 
.5669 
.5675 


.5825 
.5831 
.5836 
.5842 
.5847 


.5992 
.5997 
.6003 


.6158 
.6164 


.6175 
.6180 


.6324    .6189 
.6+95 


.6335 
.6341 
.6346 


.6500 
.6506 
.6511 


.6654 
.6660 
.6665 


.6819 
.6824 
.6829 
.6835 
,6840 


.6988 
.6993 
.6999 
.70t4 


M, 

41° 

.7004 

42° 

43° 

44° 

45° 

46° 

47° 

48° 

49° 

50° 

M. 

0' 

.7167 

.7330 

.7492 

.7654 

.7815 

.7975 

.8135 

.8294 

.8452 

0' 

2 

.7010 

.7173 

.7335 

.7498 

.7659 

.7820 

.7980 

.8140 

.8299 

,8458 

2 

4 

.7015 

.7178 

.7341 

,7503 

.7664 

.7825 

.7986 

.8145 

.8304 

.8463 

4 

6 

.7020 

.7184 

.7346 

,7508 

.7670 

.78;ii 

.7991 

.8151 

.8;^10 

.8468 

6 

8 

.7026 

.7189 

.7352 

.7514 

.7675 

.7836 

.7996 

.8156 

.8315 

,8473 

8 

10 

.7031 

.7195 

.7.357 

.7519 

.7681 

.7841 

,8002 

.8161 

.8320 

,8479 

10 

12 

.7037 

.7200 

.7362 

.7524 

.7686 

.7847 

.8007 

,8167 

.8326 

.8484 

12 

14 

.7042 

.7205 

.7368 

.7530 

.7691 

.7852 

.8012 

.8172 

.8331 

,8489 

14 

16 

.7048 

.7211 

.7373 

.7535 

.7697 

.7857 

.8018 

,8177 

.8336 

,8495 

16 

18 

.7053 

.7216 

.7379 

.7541 

.7702 

.7863 

.8023 

.8183 

.8341 

,8500 

18 

20 

.7059 

.7222 

.7384 

.7546 

.7707 

.7868 

.8028 

.8188 

.8347 

,8505 

20 

22 

.7064 

.7227 

.7390 

.7551 

.7713 

.7873 

.8034 

.8193 

.8352 

,8510 

22 

24 

.7069 

.7232 

.7395 

.7557 

.7718 

.7879 

.8039 

8198 

.8357 

.8516 

24 

26 

.7075 

,7238 

.7400 

.7562 

.7723 

.7884 

.8044 

.8204 

.8363 

.8521 

26 

28 

.7080 

.7243 

.7406 

.7.568 

.7729 

.7890 

.8050 

.8209 

.8368 

.8526 

28 

IK) 

.7086 

.7249 

.7411 

.7573 

,7734 

.7895 

.8055 
.8060 

.8214 

.8373 

.8531 

30 

32 

.7091 

.7254 

.7417 

.7578 

.7740 

.7900 

.8220 

.8378 

.8537 

32 

34 

.7097 

.7260 

.7422 

.7.584 

.7745 

.7906 

.8066 

.8225 

.8384 

.8542 

54 

36 

.7102 

.7265 

.7427 

.7589 

.7750 

.7911 

.8071 

.8230 

.8389 

.85+7 

36 

38 

.7108 

.7270 

.7433 

.7595 

.7756 

.7916 

.8076 

,82.36 

.8394 

.8552 

3d 

40 

,7113 

.7276 

.7438 

.7600 

.7761 

.7922 

.8082 

.8241 

.8400 

.8558 

40 

42 

.7118 

.7281 

.7443 

.7605 

.7766 

.7927 

.8087 

.8246 

.8405 

.8.563 

42 

44 

.7124 

.7287 

.7449 

.7611 

.7772 

.7932 

.8092 

.8251 

.8410 

.85«? 

44 

46 

.7129 

.7292 

.7454 

.7616 

.7777 

.7938 

.8098 

.8257 

.8415 

.8.573 

46 

48 

.7135 

.7298 

.7460 

.7621 

.7782 

.7943 

.8103 

.8262 

.8421 

.8579 

48 

50 

.7140 

.7303 

.7465 

.7627 

.7788 

.7948 

.8108 

.8267 

.8426 

.8584 

50 

52 

.7146 

.7308 

.7471 

.7632 

.7793 

.7954 

.8113 

,8273 

.8431 

.8589 

52 

54 

.7151 

.7314 

.7476 

.7638 

.7799 

..7959 

.8119 

.8278 

.8437 

.8594 

54 

56 

.7156 

.7319 

.7481 

.7643 

.7804 

.7964 

.8124 

.8283 

.8442 

.8600 

56 

58 

,7162 

.7325 

.7487 

.7648 

.7809 

.7970 

.8129 

.8289 

.8447 

.8605 

,5« 

30 

.7167 

.7330 

.7492 

,7654 

,7815 

.7975 

,8135 

.829A 

.8452 

.8610 

6C« 

lu 


146 


TABLE    OF   CHORDS. 


Table  of  ebor€ls,iii  parts  of  a  rad  1 ;  for  protracting:  — C 


M. 

51° 

52° 

53° 

54° 

55° 

56° 

57° 

58° 

59° 

60° 

M. 

0' 

.8610 

.8767 

.8924 

.9080 

.9235 

.9389 

.9543 

.9696 

.9848 

1.0000 

0* 

2 

.8615 

.8773 

.8929 

.9085 

.9240 

.9.395 

.9548 

.9701 

.9854 

1.0005 

t 

4 

.8621 

.8778 

.8934 

.9090 

.9245 

.9400 

.9553 

.9706 

.9859 

1.0010 

4 

6 

.8626 

.8783 

.8940 

.9095 

.9250 

.9405 

.9559 

.9711 

.9864 

1.0015 

6 

8 

.8631 

.8788 

.8945 

.9101 

.9256 

.9410 

.9564 

.9717 

.9869 

1.0020 

8 

10 

.8636 

.8794 

.8950 

.9106 

.9261 

.9415 

.9569 

.9722 

.9874 

1.0025 

10 

12 

.8642 

8799 

.8955 

.9111 

.9266 

.9420 

.9574 

.9727 

.9879 

1.0030 

12 

U 

.8647 

.8804 

.8960 

.9116 

.9271 

.9425 

.9579 

.9732 

.9884 

1.0035 

14 

16 

16652 

.8809 

.8966 

.9121 

.9276 

.9430 

.9584 

.9737 

.9889 

1.0040 

16 

18 

.8657 

.8814 

.8971 

.9126 

.9281 

.9436 

.9589 

.9742 

.9894 

1.0045 

18 

20 

.8663 

.8820 

.8976 

.9132 

.9287 

.9441 

.9594 

.9747 

.9899 

1.0050 

20 

22 

.8668 

.8825 

.8981 

.9137 

.9292 

.9446 

.9599 

.9752 

.9904 

1.0055 

22 

24 

.8673 

.8830 

.8986 

.9142 

.9297 

.9451 

.9604 

.9757 

.9909 

1.0060 

24 

26 

.8678 

.8835 

.8992 

.9147 

.9302 

.9456 

.9610 

.9762 

.9914 

1.0065 

26 

28 

.8684 

.8841 

.8997 

.9152 

.9307 

.9461 

.9615 

.9767 

.9919 

1.0070 

28 

S9 

.8689 

.8846 

.9002 

.9157 

.9312 

.9466 

.9620 

.9772 

.9924 

1.0075 

30 

32 

.8694 

.8851 

.9007 

.9163 

.9317 

.9472 

.9625 

.9778 

.9929 

1.0080 

32 

34 

.8699 

.8856 

.9012 

.9168 

.9323 

.9477 

.9630 

.9783 

.99.34 

1.0086 

34 

36 

.8705 

.8861 

.9018 

.9173 

.9328 

.9482 

.9635 

.9788 

.9939 

1.0091 

36 

38 

.8710 

.8867 

.9023 

.9178 

.9333 

.9487 

.9640 

.9793 

.9945 

1.0096 

38 

40 

.8715 

.8872 

.9028 

.9183 

.9338 

.9492 

.9645 

.9798 

.9950 

1.0101 

40 

42 

.8720 

.8877 

.9033 

.9188 

.9343 

.9497 

.9650 

.9803 

.9955 

1.0106 

42 

44 

.8726 

.8882 

.9038 

.9194 

.9348 

.9502 

.9655 

.9808 

.9960 

1.0111 

44 

46 

.8731 

.8887 

.9044 

.9199 

.9353 

.9507 

.9661 

.9813 

.9965 

1.0116 

46 

48 

.8736 

.8893 

.9049 

.9204 

.9359 

.9512 

.9666 

.9818 

.9970 

1.0121 

48 

50 

.8741 

.8898 

.9054 

.9209 

.9364 

.9518 

.9671  - 

.9823 

.9975 

1.0126 

50 

52 

.8747 

.8903 

.9059 

.9214 

.9369 

.9523 

.9676 

.9828 

.9980 

1.0131 

52 

54 

.8752 

.8908 

.9064 

.9219 

.9374 

.9528 

.9681 

.9833 

.9985 

1.0136 

54 

56 

.8757 

.8914 

.9069 

.9225 

.9379 

.9533 

.9686 

.9838 

.9990 

1.0141 

56 

58 

.8762 

.8919 

.9075 

.9230 

.9384 

.9538 

.9691 

.9843 

.9995 

1.0146 

58 

60 

.8767 

.8924 

.9080 

.9235 

.9389 

.9543 

.9696 

.9848 

1.0000 

1.0151 

60 

M. 

6J° 

62° 

63° 

64° 

65° 

66° 

67° 

68° 

69° 

70° 

M. 

0' 

1.0151 

1.0301 

1.0450 

1.0598 

1.0746 

1.0893 

1.1039 

1.1184 

1.1328 

1.1472 

0- 

2 

1.0156 

1.0306 

1.0455 

1.0603 

1.0751 

1.0898 

1.1044 

1.1189 

1.1333 

1.1476 

2 

4 

1.0161 

1.0311 

1.0460 

1.0608 

1.0756 

1.0903 

1.1048 

1.1194 

1.1338 

1.1481 

4 

6 

1.0166 

1.0316 

1.0465 

1.0613 

1.0761 

1.0907 

1.1053 

1.1198 

1.1342 

1.1486 

6 

8 

1.0171 

1.0321 

1.0470 

1.0618 

1.0766 

1.0912 

1.1058 

1.1203 

1.1347 

1.1491 

H 

10 

1.0176 

1.0326 

1.0475 

1.0623 

1.0771 

1.0917 

1.1063 

1.1208 

1.1352 

1.1495 

10 

12 

.  0181 

1.0331 

1.0480 

1.0628 

1.0775 

1.0922 

1.1068 

1.1213 

1.1357 

1.1500 

12 

14 

1.0186 

1.0336 

1.0485 

1.0633 

1.0780 

1.0927 

1.1073 

1.1218 

1.1362 

1.1505 

14 

16 

1.0191 

1.0341 

1.0490 

1.0638 

1.0785 

1.0932 

1.1078 

1.1222 

1.1366 

1.1510 

16 

18 

1.0196 

1.0346 

1.0495 

1.0643 

1.0790 

1.0937 

1.1082 

1.1227 

1.1371 

1.1514 

la 

20 

1.0201 

1.0351 

1.0500 

1.0648 

1.0795 

1.0942 

1.1087 

1.1232 

1.1376 

1.1519 

20 

22 

1.0206 

1.0356 

1.0504 

1.0653 

1.0800 

1.0946 

1.1092 

1.1237 

1.1381 

1.1524 

22 

24 

1.0211 

1.0361 

1.0509 

1.0658 

1 .0805 

1.0951 

1.1097 

1.1242 

1.1386 

1.1529 

24 

26 

1.0216 

1.0366 

1.0514 

1.0662 

1.0810 

1.0956 

1.1102 

1.1246 

1.1390 

1.1533 

26 

28 

1.0221 

1.0370 

1.0519 

1.0667 

1.0815 

1.0961 

1.1107 

1.1251 

1.1395 

1.1538 

28 

30 

1.0226 

1.0375 

1.0524 

1.0672 

1.0820 

1.0966 

1.1111 

1.1256 

1.1400 

1.1543 

30 

32 

1 .0231 

1.0380 

1.0529 

1.0677 

1.0824 

1.0971 

1.1116 

1.1261 

1.1405 

1.1548 

32 

34 

1.0236 

1.0385 

1.0534 

1.0682 

1.0829 

1.0976 

1.1121 

1.1266 

1.1409 

1.1552 

34 

.% 

1.0241 

1.0390 

1.0539 

1.0687 

1.0834 

1.0980 

1.1126 

1.1271 

1.1414 

1.1557 

36 

88 

1.0246 

1 .0395 

1.0544 

1.0692 

1.0839 

1.0985 

1.1131 

1.1275 

1.1419 

1.1562 

3S 

40 

1.0251 

1.0400 

1.0549 

1.0697 

1.0844 

1.0990 

1.1136 

1.1280 

1.1424 

1.1567 

40 

42 

1.0256 

1 .0405 

1.0554 

1.0702 

1 .0849 

1.0995 

1.1140 

1.1285 

1.1429 

1.1571 

42 

44 

1.0261 

1.0410 

1.0559 

1.0707 

1.0854 

1.1000 

1.1145 

1.1?<'0 

1.1433 

1.1576 

44 

46 

1.0266 

1.0415 

10564 

1.0712 

1.0859 

1.1005 

1.1150 

1.12^.. 

1.1438 

1.1581 

46 

4fi 

1.0271 

1.0420 

1.0569 

1.0717 

1.0863 

1.1010 

1.11.55 

1.1299 

1.1443 

1.1586 

48 

50 

1.0276 

1.0425 

1.0574 

1 .0721 

1.0868 

1.1014 

1.1160 

1.1304 

1.1448 

1.1590 

50 

52 

1.0281 

1.0430 

1.0579 

1.0726 

1.0873 

1.1019 

1.1165 

1.1309 

1.1452 

1.1595 

52 

54 

1.0286 

1.0435 

1.0584 

1.0731 

1.0878 

1.1024 

1.1169 

1.1314 

1.1457 

1.1600 

54 

56 

1.0291 

1.0440 

1.0589 

1.0736 

1.0883 

1.1029 

1.1174 

1.1319 

1.1462 

1.1605 

56 

58 

,  1.0296 

1.0445 

1.0593 

1.0741 

1.0888 

1.1034 

1.1179 

1.1.323 

1.1467 

1.1609 

58 

60 

1.0301 

1.0460 

1.0698 

1.0746 

1.0893 

1.1039 

1.1184 

1.1S2S 

1.1472 

1.1j614 

«U 

TABLE   OF   CHORDS. 


147 


Table  of  Cliovds,  in  parto  of  a  rad  1 ;  for  protracting:  — 

-Contiuued 

M. 

71° 

72° 

73° 

74° 

75° 

76° 

77° 

78° 

79° 

80° 

M, 

0' 

1.1614 

1.1756 

1.1896 

1.2036 

1.2175 

1.2313 

1.2450 

1.2586 

1.2722 

1.2856 

0 

2 

1.1619 

1.1760 

1.1901 

1.2041 

1.2180 

1.2318 

1.2455 

1.2591 

1.2726 

1.2860 

2 

4 

1.1624 

1.1765 

1.1906 

1.2046 

1.2184' 

1.2322 

1.2459 

1.2595 

1.2731 

1.2865 

♦ 

fi 

1.1623 

1.177e 

1.1910 

1.2050 

1.2189 

1.2327 

1.2464 

1.2600 

1.2735 

1.2869 

6 

8 

1.1633 

1.1775 

1.1915 

1.2055 

1.2194 

1.2332 

1.2468 

1.2604 

1.2740 

1.2874 

ft 

10 

1.1638 

1.1779 

1.1920 

1.2060 

1.2198 

1.2336 

1.2473 

1.2609 

1.2744 

1.2^78 

10 

Vf. 

1.1642 

1.1784 

1.1924 

1.2064 

1.2203 

1.2341 

1.2478 

1.2614 

1.2748 

1.2882 

12 

14 

1.1647 

1.1789 

1.1929 

1.-2069 

1.2208 

1.2345 

1.2482 

1.2618 

1.2753 

1.2887 

U 

1« 

1.1652 

1.1793 

1.1934 

1.2073 

1.2212 

1.2350 

1.2487 

1.2623 

1.2757 

1.2891 

IB 

1« 

1.1657 

1.1798 

1.1938 

1.2078 

1.2217 

1.2354 

1.2491 

1.2627 

1.2762 

1.2896 

1« 

20 

1.1661 

1.1803 

1.1943 

1.2083 

1.2221 

1.2359 

1.2496 

1.2632 

1.2766 

1.2900 

20 

22 

1.1666 

1.1807 

1.1948 

1.2087 

1.2226 

1.2364 

1.2500 

1.2636 

1.2771 

1.2305 

22 

24 

1.1671 

l.lPi2 

1.1952 

1.2092 

1.2231 

1.2368 

1.2505 

1.2641 

1.2775 

1.2909 

24 

26 

1.1676 

1.1817 

1.1957 

1.2097 

1.2235 

1.2373 

1.2509 

1.2645 

12780 

1.2914 

26 

28 

1.1680 

J-1821 

11962 

1.2101 

1.2240 

1.2377 

1.2514 

1.2650 

1.2784 

1.2918 

28 

30 

1.1685 

1.1826 

1.1966 

1.2108 

1.2244 

1.2382 

1.2518 

1.2654 

1.2789 

1.2922 

30 

32 

1.1690 

1.1831 

1.1971 

1.2111 

1.2249 

1.2386 

1.2523 

1.2659 

1.2793 

1.2927 

32 

34 

1.1694 

1.1836 

1.1976 

1.2115 

1.2254 

1.2391 

1.2528 

1.2663 

1.2798 

1.2931 

34 

36 

1.1699 

1.1840 

1.1980 

1.2120 

1.2258 

1.2;i96 

1.2532 

1.2668 

1.2802 

1.2936 

3« 

38 

1.1704 

1.1845 

1.1985 

1.2124 

1.2263 

1.2400 

1.2537 

1.2672 

1.2807 

1.2940 

38 

40 

1.170!* 

1.1850 

1.1990 

1.2129 

1.2267 

1.2405 

1.2541 

1.2677 

1.2811 

1.2945 

40 

42 

1.171S 

1.1854 

1.1994 

1.2134 

1.2272 

1.2409 

1.2546 

1.2681 

1.2816 

1.2949 

42 

44 

i.nis 

1.1859 

1.1999 

1.2138 

1.2277 

1.2114 

1.2550 

1.2686 

1.2820 

1.2954 

44 

46 

1.1723 

1.1864 

1.2004 

1.2143 

1.2281 

1.2418 

1.2555 

1.2690 

1.2825 

1.2958 

46 

48 

1.1727 

1.1868 

1.2008 

1.2148 

1.22K6 

1.2423 

1.2559 

1.2695 

1.2829 

1.2962 

48 

50 

1.1732 

1.1873 

1.2013 

1.2152 

1.2-290 

1.2428 

1.2564 

1.2699 

1.2833 

1.2967 

50 

52 

1.17.37 

1.1878 

1.2018 

1.2157 

1.2295 

1.2432 

1.2568 

1.2704 

1.2838 

1.2971 

52 

54 

1.1742 

1.1882 

1.2022 

1.2161 

1.2299 

1.2437 

1.2573 

1.2708 

1.2842 

1.2976 

54 

56 

1.1746 

1.1887 

1.2027 

1.2166 

1.2304 

1.2441 

1.2577 

1.2713 

1.2847 

1.2980 

56 

58 

1.1751 

1. 1892 

1.2032 

1.2171 

1.2309 

1.2446 

1.2.582 

1.2717 

1.2851 

1.2985 

58 

60 

1.1756 

1.1896 

1.2036 

1.2175 

1.2313 

1.2450 

1.2586 

1.2722 

1.2856 

1.2989 

60 

K. 

81° 

82° 

83° 

84° 

85° 

86° 

87° 

88° 

89° 

M. 

0' 

1.2;)89 

1.3121 

1.3252 

1.3.383 

1.3512 

1.3640 

1.3767 

1.3893 

1.4018 

0' 

2 

1.2993 

1.3126 

1.3257 

1.3387 

1.3516 

1.3614 

1.3771 

1.3897 

1.40-22 

2 

4 

1.2998 

1.3130 

1.3261 

1.3391 

1.35-20 

1.3648 

1.3776 

1.3902 

1.4026 

4 

6 

1.3002 

1.3134 

1.3265 

1-3396 

1.35-25 

1.3653 

1.3780 

1.3906 

1.4031 

• 

8 

1.3007 

1.3139 

1.3270 

1.3400 

1.352;) 

1.3657 

1.3784 

1.3910 

14035 

8 

10 

1.3011 

1.3143 

1.3274 

1.3404 

1.3533 
1.3538 

1.3661 
1.3665 

1.3788 

1.3914 

1.4039 

10 

12 

1.3015 

1.3147 

1.3279 

1.3409 

1.3792 

1.3918 

1.4043 

12 

14 

1.3020 

1.3152 

1.3283 

1.3413 

1.3542 

1.3670 

1.3797 

1.3922 

1.4047 

14 

16 

1.3024 

1.3156 

1.3287 

1.3417 

1.3546 

1.3674 

1.3501 

1.3927 

1.4051 

16 

18 

1.3029 

1.3161 

1.3292 

1.3421 

1.3550 

1.3678 

1.3805 

1.3931 

1.4055 

18 

20 

1.3033 

1.3165 

1.3296 

1.3426 

1.3555 

1.3682 

1.3809 

1.3935 

1.4060 

SO 

22 

1.3038 

1.3169 

1.3300 

1.3430 

1.3559 

1.3687 

1.3813 

1.3939 

1.4064 

22 

24 

1.3042 

1.3174 

1.3.305 

1.31.34 

1.3563 

1.3691 

1.3818 

1.3943 

1.4068 

24 

26 

1.3016 

1.3178 

1.3309 

1.3439 

1.3567 

1.3695 

1.3822 

1 .3947 

1.4072 

26 

28 

1.3051 

1.3183 

1.3313 

1.3443 

1.3572 

1.3699 

1.3826 

1.3952 

1.4076 

28 

30 

1.3055 

1.3187 

1.3318 

1.3447 

1.3576 

1.3704 

1.3830 

1.3956 

1.4080 

30 

32 

1.3060 

1.3191 

1.3322 

1.3452 

1.3580 

I. .3708 

1.3834 

1.3960 

1.4084 

32 

34 

1  3064 

1.3196 

1.3323 

1.3456 

1.3585 

1.3712 

1.3839 

1.3964 

1.4089 

34 

S6 

1.3068 

1.3200 

1.3331 

1.3460 

1.3589 

1  3716 

1.3843 

1.3968 

1.4093 

36 

38 

1.3073 

1.3204 

1.3335 

1.3465 

1.3593 

1.3721 

1 .3847 

1.3972 

1.4097 

38 

40 

1.3077 

1.3209 

1.3339 

1.3169 

1.3597 

1.3725 

1.3851 

1.3977 

1.4101 

40 

42 

1.3082 

1.3213 

1.3344 

1.3473 

1.3602 

1.3729 

1.3855 

1.3981 

1.4105 

42 

44 

1.3086 

1.3218 

1.3348 

1.3477 

1.3606 

1.3733 

1.3860 

1.3985 

1.4109 

44 

46 

1.3090 

1.3222 

1.3352 

1.3482 

1.3610 

1.3738 

1.3864 

1.3989 

1,4113 

46 

48 

1.3095 

1..3226 

1.3357 

1..3486 

1.3614 

1.3742 

1.3868 

1.3993 

1.4117 

48 

50 

1.3099 

1.3231 

1.3361 

1.3490 

1.3619 

1.3746 

1.3872 

1.3997 

1.4122 

50 

52 

1.3104 

1.3235 

1.3365 

1.3495 

1.3623 

1.3750 

1.3876 

1.4002 

1.4126 

52 

64 

1.3108 

1 .3239 

1.3.370 

1.3499 

1.3627 

1.3754 

1.3881 

i.40on 

1.4130 

64 

66 

1.S112 

1.3244 

1.3374 

1.3503 

1.3631 

1.3759 

1.3885 

1.4010 

1.4134 

66 

58 

1.3117 

1.3248 

1.3378 

1.3508 

1..3636 

1.3763 

1..3ft89 

1.4014 

1.4138 

58 

60 

1.3121 

1.3252 

1.3383 

1.3512 

1.3640 

1.3767 

1.3893 

1.4018 

1.4142 

60 

148 


POLYGOKS. 


POIiYOONS. 


Pentagon. 

5  siUtA. 

Any  straight-sided  fig  is  called  a  polygon.    If  all  tne  sides  and  angles  are  equal,  it  is  a  resHlat 
polygon  ;  if  not,  it  is  irregular.     Of  course  the  number  of  polygons  is  infinite. 


Table  of  Regular  Polyg'ons. 


Number 

Name 

Area  = 

Radius  of  cir- 

Interior  angle 

Angle  at  cen* 

of 

of 

(square  of  one 

subtended 

Sides. 

Polygon. 

side)  mult  by 

mult  by 

sides. 

by  a  side. 

M 

Equilateral 
triangle. 

1     .433013 

.577350 

60° 

120° 

4 

Square. 

1.000000 

.707107 

90° 

90° 

5 

Pentagon. 

1.720477 

.850651 

108° 

72° 

6 

Hexagon. 

2.598076 

1.000000 

120° 

60° 

7 

Heptagon. 

3.638912 

1.152382 

128°  34.2857' 

51°  25.7143' 

8 

Octagon. 

4.828427 

1.306563 

135° 

45° 

9 

Nonagon. 

fi.l81S24 

1.461902 

140° 

40° 

10 

Decagon. 

7.694209 

1.618034 

144° 

36° 

11 

Undecagon. 

9.365640 

1774733 

147°  16.3636' 

32°  43.6364' 

12 

Dodecagon. 

11.196152 

1.931854 

150° 

30° 

Area  of  any  regular  polygon  =  length  of  one  side,  ab  X  perp  p  drawn  from  cen  of  fig  t« 
cen  of  side  X  hulf  the  number  of  sides. 

Sum  of  interior  angles,  a  b  c,  etc,  of  any  polygon,  regular  or  irregular  =  180°  X 
(number  of  side.s  —  2). 

Angle  at  cen  subtended  by  a  side,  in  any  regular  polygon  =  360°  -^  number  of  aides* 


TRIA?rOI.ES. 


We  speak  here  of  plane  triangles  only ;  or  those  having  straight  sides. 

A  triangle  is  equilateral  when  all  its  sides  aie  equal,  as  a  ;  isosceles  when  only  two  sides 
are  equal,  as  B;  scalene  when  all  the  sides  are  unequal,  as  C.  D.  and  E  ;  ucutc-angled  when 
all  its  angles  are  acute,  or  each  less  than  90°,  as  A,  B,  and  C  ;  right-angled  when  it  contains  a 
right  angle,  as  D  ;  obtuse-angled  when  it  contains  an  obtuse  augle.  ur  one  gi  eater  than  90^.  as  E. 

All  the  three  angles  of  any  triangle  are  equal  to  two  right 
angles,  or  180 -' ;  tlieretore,  it  we  know  two  or  them,  we  can  tind  the  third  by  ^ 

subtracting  their  sum  from  180°.   All  triangles  which  liavc  equal  bases,  ^___„^ J? 

and  equal  perp  heights,  have  also  equal  iirciis;  tliusthe  areas  ot  awe,  aw  d.and 
a  w  c,  are  equal  to  each  other.  The  area  of  any  triangle  is  equal  to  half 
that  of  any  parallelogram  which  has  an  equal  base, and  unequal  [..erp  height.  The 
areas  of  triangles  which  have  equal  bases,  but  diff  perp  heights,  are  to 
each  other  as,  or  in  proportion  to.  their  perp  heights;  thus  the  triangle  awn, 
•with  a  perp  height  s  n,  equal  to  but  one-half  that  (s  e)  of  the  three  other  trian- 
gles,  but  with  the  same  base  a  w,  has  also  but  half  the  area  of  either  of  those  "" 
others. 

Area  of  any  triangle.  Figs  A,  B,  C,  D,  E,  =  half  the  base,  S.  X  the  height,  or  perp  dist  p  to 
the  opposite  angle.  Any  side  may  be  talsen  as  the  base  of  a  triangle ;  but  the  perp  height  must  always 
be  measured  from  thoside  so  assumed;  to  do  which,  the  side  must  sometimes  be  prolonged,  as  in 
Fig  E  ;  but  the  prolongation  is  not  to  be  considered  as  a  part  of  the  base. 

Area  of  any  equilateral  triangle  =  .433013  X  square  of  one  side. 

To  find  area,  having  the  three  sides. 

Add  them  opether  ;  div  the  sum  by  2 ;  from  the  -lalf  sum,  subtract  each  side  separately;  mult  the 
half  sum  and  the  three  remainders  continuously  together ;  take  the  sq  rt  of  the  prod. 

Ex.— The  three  Bides  =  20,  30,  40  ft.  Here  20 +  30  +  40  =90;  and —  =  45.  And  45  — 20  =  25; 
45  —  30  =  15;  and  45  —  40  =  5.  And  45  X  25  X  15x5  =  84375'  and  the  sq  rt  of  84375  is  290.47  sq  ft, 
«rea  reqd. 


TRIANGLES. 


149 


To  find  area,  liaTing:  one  side  and  tbe  2  angrles  at  its  ends. 

Add  the  2  angles  together:  take  the  sum  from  180°;  the  rem  will  be  the  angle  opp  the  given  side. 
Find  the  nat  sine  of  this  angle  ;  also  find  the  nat  sines  of  the  other  angles,  and  mult  them  together. 
Then  as  the  nat  sine  of  the  single  angle,  is  to  the  prod  of  the  nat  sines  of  the  other  2  angles,  so  is  the 
square  of  the  given  side  to  double  the  reqd  area. 

To  find  area,  having:  two  sides,  and  the  included  ang^le. 

Mult  together  the  two  sides,  and  the  nat  sine  of  the  included  angle ;  div  by  2. 

Ex.— Sides  650  ft  and  980  ft ;  included  angle  69"^  20'.    By  the  table  we  find  the  nat  sine  .93561 
^^      ,         650  X  980  X  .9356        „„„„„, 
therefore, =:  297988.6  square  ft  area. 

To  find  area,  having-  the  three  ang^les  and  the 
perp  heig-ht,  a  b. 

Find  the  nat  sines  of  the  three  angles  ;  mult  together  the  sines  of  the  angles 
d  and  o  ;  div  the  sine  of  the  angle  b  by  the  prod  ;  mult  the  quot  by  the  square 
of  the  perp  height  a  b;  div  by  2. 

To  find  any  side,  as  a  o,  having  the  three 
angles,  d,  b  and  o,  and  the  area. 

(Sinp  of  rf  X  sine  of  o)  j  sine  of  ft  t  »  twice  the  area  :  square  of  d  o. 

The  perp  height  of  an  equilateral  triangle  is  e(|ual  to  one  side  X  .866025.  Hence  one  of 
its  sides  is  equal  to  the  perp  height  div  by  . 866025  or  to  perp  height  X  M547.  Or,  to  find  a  Klde« 
mult  the  sq  rt  of  its  area  by  1.51967.  The  side  of  an  equilateral  triangle,  mult  by  .658037  —  side  of  $ 
square  of  the  same  area ;  or  mult  by  .742517  it  gives  the  diam  of  a  circle  oif  the  same  area. 

The  following  apply  to  any  plane  triangle,  whether  oblique  or  right-angled : 

1,  The  three  angles  amount  to  180^,  or  two  right  angles. 

8.  Any  exterior  angle,  as  A  C  7i,  is  equal  to  the  two  interior  and  opposite 
•nes,  A  and  B. 

8.  The  greater  side  is  opposite  the  greater  angle, 

4.  The  sides  are  as  the  sines  of  the, opposite  angles.  Thus,  the  side  a  is  to 
the  side  b  as  the  sine  of  A  is  to  the  sine  of  B. 

5.  If  any  angle  as  s  be  bisected  by  a  line  s  o,  the  two  parts  mo,  on  of 
the  opposite  side  m  n  will  be  to  each  other  as  the  other  two  sides  am,  sn ; 
or,  m  0 :  o  n ::  s  m :  8  n. 

6.  If  lines  be  drawn  from  each  angle  r  s  t  to  the 
center  of  the  opposite  side,  they  will  cross  each 
other  at  one  point,  a,  and  the  short  part  of  each 
of  the  lines  will  be  the  third-part  of  the  whole  Hue. 
Also,  a  is  the  een  of  grav  of  the  triangle. 

7.  If  lines  be  drawn  bisecting  the  three  angles,  they  will  meet  at  a  point 
^V"~.^^^\^          perpendicularly  equidistant  from  each  side,  and  consequently  the  centev 

fl^^        \   ^^^-<:^>.  1  of  the  greatest  circle  that  can  be  drawn  in  the  tria/nglo. 

^      8.  If  a  line  s  n  be  drawn  parallel  to  any  side  c  a, 
the  two  triangles  ran^rca,  will  be  similar. 

9.  To  divide  any  triangle  a  c  r  into  two  equal  parts  by  a  line  a  n  parallel  to 
any  one  of  its  sides  c  o.  On  either  one  of  the  other  sides,  as  o  r,  as  a  diam, 
iescribe  a  semicircle  a  o  r;  and  find  its  middle  o.    From  r  (opposite  c  a),  with 

radius  r  o,  describe  the  arc  o  n.     Prom  n  draw  n  s.  par- 
allel to  c  a. 

10.  To  find  the  greatest  parallelogram  that  can  be 
drawn  in  any  given  triangle  o  nb.  Bisect  the  three  sides  at  a  c  e,  and  join 
a  c,  a  e,  e  c.  Then  either  a  e  b  c,  a  e  c  o,  or  a  c  en,  each  equal  to  half  the 
triangle,  will  be  the  reqd  parallelogram.  Any  of  these  parallelograms  can 
plainly  be  converted  into  a  rectangle  of  equal  area,  and  the  greatest  that  can  be 
drawn  in  the  triangle. 
„  ,  10^4.  If  a  line  a  c  bisects  any  two  sides  o  5,  o  n,  of  a  triangle,  it  will  be  par- 

allel to  the  third  side  n  b.  and  half  as  long  as  it. 

11.  To  find  the  greatest  square  that  can  be  drawn  in  any  triangle  axr.    From 
an  angle  as  a  draw  a  perp  a  n  to  the  opposite  side  xr,  and  find  its  length.   Then 

0  n,  or  a  side  v  t  oi  the  square  will  — ; 

xr-{-  an 

Rem.— If  the  triangle  is  such  that  two  or  three  such  perps  can  be  drawn,  then 
two  or  three  equal  squares  may  be  found. 


mo  n 


150 


PLANE  TRIGONOMETRY. 


Rig:Iit-aii$>:led  Triaiig^les. 

A.11  the  foregoing  apply  also  to  right-angled  triangles;  but  what  follow  apply  to  them  only. 

Call  the  right  angle  A,  and  the  others  B  and  C ;  and  call  the  sides  respaotivelj 
opposite  to  them  a,  b,  and  c.    Then  is 

B  n  =  -~'—^  ~        "  '^ 

Sine  0 


IV  **  -  "Sin( 

Xa  6=aX 


=  c  X  Sec  E 


p^  =  6X  See  0=1/524.  A 


Cosine  ( 
Sine  B  =  a  X  Cos  0  =  c  X  Cot  C  =  c  X  Tang  B, 
Sine  C  =  aXCosB-6X  Tang  C. 

e  &  e 

Also  Sine  of  C  =  -  ;  Cos  C  =  -  /  Tang  C  =  j" . 


And  Sine  of  B  = 


r  Cos  B  =■  ~  ;   Tang  B  =  -. 


And  Sine  of  A  or  90°  =  1.     Cos  A  =  0.    Tang  A  =:  infinity.    Sec  A  =  Infinity. 
I  If  from  the  right  angle  o  a  line  0  w  be  drawn  perp  to  the  hypothenuse  or  long  side  A  g,  then  th€ 
two  small  triangles   0  to  h,  o  tv  g,  and   the  large  one  ohg,  will  be  similar. 
Or  g  w:w  o',:w  o:  IV  h;  and  gw  X  w  h  =  w  0^. 

5.  A  line  drawn  from  the  right  angle  to  the  center  of  the  long  sida  will 
be  half  as  long  as  sa'd  side. 

3.  If  on  the  three  sides  oh,  o  g,  gh  we  draw  three  squares  t,  u,  v,  or 
three  circles,  or  triangles,  or  any  other  three  figs  that  are  similar,  then  the 
area  of  the  largest  one  is  equal  to  the  sum  of  the  areas  of  the  two  others. 

4«  In  a  triangle  whose  sides  are  as  .3,  4,  and  5  fas  are  those  of  the  tri> 
angle  A  B  C),  the  angles  are  very  approximately  90";  53"  7'  48.38";  and 
360  52'  11,62'/,  Their  Sines,  1. ;  .8;  and  .6.  Their  Tangs,  infinity  ;  1.3333 ; 
and  .75. 

6,  One  whose  sides  are  as  7,  7,  and  9.9,  has  very  approv.  one  angle  of  9(fi 
and  two  of  45°  each,  near  enough  for  all  practical  purposes. 


PLANE  TEIGONOMETEY. 


fzAtra  trigonometry  teaches  how  to  find  certain  unknown  parts  of  plane,  or  straight  -  sided'  irt* 
angles,  by  means  of  other  parts  which  are  known ;  and  thus  enables  us  to  measure  inaccessible  dis- 
tances, &c.  A  triangle  consists  of  six  parts,  namely,  three  sides,  and  three  angles;  and  if  we  know 
any  three  of  these,  (except  the  three  angles,  and  in  the  ambiguous  case  under  "  Case  2,")  we  can  find 
the  other  three.   The  following  four  cases  include  the  whole  subject;  the  student  shotLJ  commit  them 


to  memory, 

Case  1.  Having  any  two  ang^les,  and  one  side, 
to  find  tlie  otlier  sides  and  an$?Ie. 

Add  the  two  angles  together  ;  and  subtract  their  sum  from  ISO©;  the  rem 
♦ill  be  the  third  ang'.e.     And  for  the  sides,  as 
Sine  of  the  angle      ,     Sine  of  the  angle 
opp  the  given  side     •    opp  the  reqd  side 
Use  the  side  thus  found,  as  the  given  one ;  and  in  the  same  manner 
the  third  side. 


Fig-.  W 


given  side  :  reqd  side. 


Case  2. 

b 


a 
Figr.  X. 


Having  two  sides,  6«,  ac.  Fig  X,  and  the  angle  fihe^ 
opposite  to  one  of  them,  to  find  the  other  side  and  anglei* 


Side  a  c  opp        The  other 
the  given  an-    *    given  side  ; 
gle  ab  c  b  a 


Sine  of  the 
given  angle 
ab  c 


Sine  of  angle  6  d  a  or 
bca  opposite  the  other 
given  side  b  a. 


Having  found  the  sine,  take  out  the  corresponding  angle  from  the  table  of 
nat  sines,  but,  in  doing  so,  if  the  side  o  c  opp  the  given  angle  iff 

shorter  than  the  other  given  side  b  a,  bear  in  mind  that  an  angle  and  its  sup- 
plement have  the  same  sine.  Thus,  in  Fig  X,  the  sine,  as  found  above,  is 
opp  the  angle  6  c  a  in  the  table.  But  a  c,  if  shorter  than  b  a,  can  evidently  b» 
laid  ofiF  in  the  opp  direction,  a  d,  in  which  case  b  da  ia  the  supplement  of  6  c  a. 
If  a  c  is  as  long  as,  or  longer  than,  b  a,  there  can  be  no  doubt ;  for  in  that  case 
it  cannot  be  drawn  toward  b,  but  only  toward  n.  and  the  angle  ic  a  will  be 
found  «t  once  iu  the  table,  oop  the  sine  as  found  above. 


PLAKE  TRIGONOMETR'X. 


151 


When  the  two  angles,  ah  c,  b  e  a,  have  been  found,  find  the  remaintng  side  by  Case  I. 
Vor  the  remaining  angle,  bac,  add  together  the  angle  ab  c  first  given,  and  the  one,  6  c  a,  foam 
aa  above.     Deduct  their  sum  from  180°. 

Case  3.  Having:  two  sidesi,  and  tbe  ang^le  included 
between  tliem. 

Take  the  angle  rrom  180°;  the  rem  will  be  the  sum  of  the  two  unknown  angles.  Div  this  sum  bj 
S;  and  find  tbe  nat  tang  of  the  quot.    Theu  as 

The  sum  of  the     ,    ipu-j-  Ji»    .    .    Tang  of  half  the  sum  of    ,    Tang  of  half 
two  given  sides    •     ^neirain    ,    ,    the  two  unknown  angles    •     their  diflF. 

Take  from  the  table  of  nat  tang,  the  angle  opposite  this  last  tang.  Add  this  angle  to  the  half  sum 
•f  the  two  unknown  angles,  and  it  will  give  the  angle  opp  the  longest  given  side  ;  and  subtract  it 
from  the  same  half  sum,  for  the  angle  opp  the  shortest  given  side.  Having  thus  found  the  angles, 
Bad  the  third  side  bj  Case  1. 

As  a  practical  example  of  the  use  of  Case  3,  we  can  ascertain  the  dist  n  m,  across  a  deep  pond,  by 
measuring  two  lines  n  o  and  m  n;  and  the  angle  n  o  m.  Prom  these  data  we  may  calculate  nm  ;  or 
by  drawing  the  two  sides,  and  the  angle  on  paper,  by  a  scale,  we  can  afterward  measure  n  m  on 
the  drawing. 


The  base  ! 


Case  4.  Having:  the  three  sides, 

T»  find  the  three  anglea ;  upon  one  side  a  &  as  a  base,  draw  (or  suppose  to  be  drawn)  a  perp  c  g  troru 

fche  opposite  angle  c.   Find  the  diff  between  the  other  two  sides,  a  c  and  c  b  ;  also  their  sum.   Then,  as 
Sum  of  the  ,  .  Difif  of  other  .   Diff  of  the  two 

other  two  sides  •  •  two  aides  •  parts  ag  and  bg,  of  the  base. 
Add  half  this  diff  of  the  parts,  to  half  the  base  a  6;  the  sum  will  be  the  longest  part  ag;  which 
taken  from  the  whole  base,  gives  the  shortest  part  g  b.  By  this  means  we  get  in  each  of  the  small  tri- 
angles a  c.  g  and  cgb,  two  sides,  (namely,  a  e  and  a  g;  and  c  b  and  g  b;)  and  an  augie  (namely,  the 
right  angle  c  g  a,  or  c  g  b)  opposite  to  one  of  the  given  aides.  Therefore,  use  Case  2  for  finding  the 
aagles  a  and  6.    When  that  is  done,  take  their  sum  from  180<^.  for  the  angle  a  c  b. 

Or«  Sd  mode  t  call  half  the  sum  of  the  three  sides,  s;  and  call  the 
two  sides   which  form  either  angle,  m  and  n.    Then  the  nat  sine  of 

half  that  angle  will  be  equal  to  \  /(«  —  "*)  XJ^«r^>. 
V  mXn 

Ex.  1.    To  find  the  dist  from  n  to  an  inac- 
cessible object  c. 

Measure  a  line  ab  ;  and  from  its  ends  measure  the  angles  cab  and 
cb  a.  Thus  having  found  one  side  and  two  angles  of  the  triangle  a  be, 
calculate  a  c  by  means  of  Case  1.  Or  if  extreme  accuracy  is  not  reqd, 
draw  the  line  a  6  on  paper  to  any  convenient  scale  ;  then  by  means  of  a 
protractor  lay  off  the  angles  c  ab,  cb  a;  and  draw  a  c  and  c  b  ;  then 
measure'  a  c  by  the  same  scale. 

Ex.  2.    To  find  the  heigrht  of  a  vertieta 
object,  n  a. 

Place  the  instrument  for  measuring  angles,  at  any  conve. 
nient  spot  o ;  also  meas  Che  dist  oa\  or  if  o  a  cannot  be  actually 
measd  in  consequence  of  some  obstacle,  calculate  it  by  the 
same  process  as  a  c  in  Pig  1.  Then,  first  directing  the  instru- 
ment horizontally,*  as  o  8,  measure  the  angle  of  depression, 
s  o  a,  say  12^ ;  also  the  angle  son,  say  30^^.  These  two  angles 
added  together,  give  the  angle  a  o  n,  42^.  Now.  in  the  small 
triangle  o  s  a  we  have  the  angle  os  a  equal  to  90°.  because  on 
is  vert,  and  o  s  hor ;  and  since  the  three  angles  of  any  triangle 
are  equal  to  180°,  if  we  subtract  the  angles  osa  (90°).  and  soa 
(12°)  from  180°,  the  rem  (78^)  will  be  the  angle  o  a  s  or  o  a  n. 
Therefore,  in  the  triangle  on  a,  we  have  one  side  o  a;  and  twa 
angles  a  on,  and  o  a  n,  to  calculate  the  side  a  n  by  Case  1. 

"  Angeles  and  dists  on  sloping:  ground  must  be  measured  hor- 
izon tally.    The  graduated  hor 

circle  of  the  instrument  evidently  meaa< 
t'"" "-."."' "V"-V----'--'^0    \  »res  the  angle  between  two  objects  bori 

'       ■---"  zontally,  no  matter  how  much  higher  one 

-iQ      \  of  them  may  be  than  the  other  ;  onepei- 

\  haps  requiring  the  telescope  of  the  instru- 

ment  to  be   directed  upward  toward  it; 
and  the  other  downward.     If,  therefore, 
the  sides  of  triangles  lying  upon  sloping 
C  \   ground,  are  not  also  measd  hor,  there  can 

be  no  aocordance  between  the  two.    Tbaa 


152 


PLANE   TRIGONOMETRY. 


Rbm.  If,  as  in  Fig  3,  it  should  be  necessary  to  ascertain  the  vert  height  an  from  a  point  o,  entirely 
Bbovo  it,  then  both  the  angles  measd  at  o,  namelv,  son,  and  a  o  a,  will  be  angles  of  depression,  or 
below  the  hor  line  o  s  assumed  to  measure  them  from.  In  this  case  we  have  the  side  o  a  as  before: 
the  angle  noa  =  8oa  —  8on;  and  the  angle  o  an  =  l&)°—  (oaa  (90°),  and  $  o  a;^  to  calcolatt 
«n  by  Case  1. 


Fig.  3. 


Figr.4. 


Orlf,  as  in  Fig 4,  the  observations  are  to  be  taken  from  a  point  o,  entirely  below  the  object  an,  thep 
botk  the  angles  g  o  a,  s  o  n,  will  be  angles  of  elevation,  or  above  the  assumed  hor  line  ot.  Here  wd 
have  in  the  triangle  o  n  a,  the  given  side  o  a  as 
before ;  the  angle  a  o  n~  a  on  — «oo;  and  the 
angle ona^lSOC-—  (o an (90°), and  no  «,  ^  to 
calculate  a  n  by  Case  1, 

If  the  object  a  n,  as  in  Fig  5,  instead  of  being 
vert,  is  inclined;  and  instead  of  its  vert  height, 
we  wish  to  tind  its  length  a  n,  we  must  first  as- 
certain its  angle  y  ti  of  inclination  to  the  hori 
ton  ;  to  which  angle  each  of  the  angles  oan  will 
be  equal.  To  tind  this  angle  yti,  suspend  a  plumb- 
line  t  y,  of  any  convenieut  kuown  length,  from  the 
object  an;  and  measure  also  y  t  horizontally. 
Then  say  as 

y  t  :  iji  '.  :  I  '.  nat  tang  of  angle  y  t  i. 

From  the  table  of  nat  tangs  take  out  the  angle 
yti  found  opposite  this  nat  tang;  and  use  it  for 
the  angles  o  «  n  or  o«o;  instead  of  the  90°  of  Figs 
S  and  4.  Also  when  the  object  inclines,  the  side 
a  o  of  the  triangle  must  be  measd  in  line,  or  in 
range  with  the  inclination.  If  the  object,  as  the 
rock  a  n,  Fig  6,  is  curved  or  irregular,  a  pole  o  a 
may  be  planted  sloping  in  the  direction  a  n ;  and 


Fig.  5. 


in  the  triangle  ab  c,  upon  sloping  ground,  the  instrument  at  o,  measures  the  hor  angle  ton;  and  not 
the  angle  bac.  Therefore,  the  side  which  corresponds  with  this  hor  angle  i  o  n.  is  the  hor  dist  i  n ; 
and  not  the  sloping  dist  6  c.  In  other  words,  when  sides  and  angles  are  on  sloping  ground,  we  do 
not  seek  their  actual  measures;  but  their  ftor  ones.  This  remark  applies  to  all  surveying  for  farms, 
railroads,  triangulations  of  countries,  &c,  &c  ;  and  the  want  of  a  strict  attention  to  it,  is  one  cause 
of  the  small  errors,  almost  unavoidable,  (and  fortunately,  of  but  triding  consequence  in  practice), 
which  oocur  in  all  ordinary  field  operations. 

When  a  sextant  is  nsed*  angles  between  objects  at difF altitudes,  asp  and 

q,  may  be  measd  hor,  by  first  planting  two  vert  rods 
0  and  8.  in  range  with  the  objects;  and  then  taking 
the  hor  angle  o  n  «,  subtended  by  the  rods. 

Ang-les  may  be  measd  without 
any  inst,  thus:  Measure  100  ft  toward 

each  object,  and  drive  stakes  j  measure  the  dist  across 
from  one  stake  to  the  other.  Half  this  dist  will  be 
thesineof  Tioi/theangletoaradof  100:  and  if  we  move 
the  decimal  pdint  two  places  to  the  left,  we  get  the  nat 
aineoT  this  one  half  of  the  angle  to  arad  of  1.  as  in  the 
tables.  Thus,  suppose  the  dist  to  be  80.64  feet ;  then 
40.32  is  the  sine  of  half  the  angle  ;  and  ,4032  will  be 
the  nat  sine,  opposite  to  which  in  the  table  of  nat 
■ines  we  find  the  angle  23°  47'  ;  which  mult  bv  2  gives 
il°  34',  the  reqd  angle.  If  obstacles  prevent  measuring  toward  the  objects,  we  may  measure  directly 
from  them  ;  because,  when  two  lines  intersect,  the  opposite  angles  are  equal.  A  rough  measurement 
may  be  made  bv  sticking  three  pins  vert,  and  a  few  ins  apart,  into  a  small  piece  of  board,  nailed  hor 
to  the  top  of  a  post.  The  pins  would  occupy  the  positions  n  o  «,  of  the  last  figure.  Pencil-linof=  may 
then  be  drawn,  connecting  the  pin-holes  ;  and  the  angle  be  measd  with  a  protractor.  By  nailing  a 
piece  of  board  vert  to  a  tree,  and  then  drawing  upon  it  a  short  hor  line,  by  means  of  a  pocket  car- 
penters' spirit-level,  vert  angles  of  elevation  and  depression  mav  be  taken  roughly  in  the  same  way. 
In  this  way  the  writer  has  at  times  availed  himself  of  the  outer  door  of  a  house,  by  opening  it  until  i» 
pointed  toward  some  mountain-peak,  the  dist  of  which  he  knew  approximately  ;  but  of  the  height  of 


PLANE   TRIGONOMETRY. 


153 


its  angle  yti  of  inclination  with  the  horizon  found  as  before; 
in  which  case  the  dist  aw  is  calculated.  Or  if  the  vert  height  en 
is  sought,  the  point  c  may  first  be  found  by  sighting  upward 
along  a  plumb-line  held  above  the  head. 

Ex.  3.    To  find  tbe  approximate  tieight, 
«  a?^  of  a  iiiountaiii. 

Of  which,  perhaps,  only  the  very  summit,  x,  is  visible  above 
interposing  forests,  or  other  obstacles ;  but  the  dist,  mi,  of  which 
is  known.  In  this  case,  first  direct  tbe  instrument  bor,  as  m  h', 
and  then  measure  the  anglb  i  m  x. 
Then  in  the  triangle  imx  we  have 
one  side  mi;  the  measd  angle  imx, 
and  the  angle  mix  (90°),  to  find  im 
by  Case  1.  But  to  this  i  x  we  must 
add  i  o,  equal  to  the  height  ym  of  the 
instrument  above  the  ground;  and 
also  o  8.  Now,  o  s  is  apparently  due 
•  entirely  to  the  curvature  of  the  earth, 
which  is  equal  to  very  nearly  8  ins,  or 
.667  ft  in  one  mile  ;  and  increases  as 
the  squares  of  the  dists ;  being  4 
times  8  ins  in  2  miles  ;  9  times  8  ins 
in  3  miles,  &c.  But  this  is  somewhat  diminished  by  the  refraction  of  the  atmosphere  ;  which  varies 
with   temperature,  moisture,  &c ;  but  alwayi  tends  to  make  the  object   x   appear  higher  than  it 

actually  is.       At  an  average,  this  deceptive  elevation  amounts  to  about-— th  part  of  the  curvature  of 

the  earth ;  and  like  the  latter,  it  varies  with  the  aquares  of  the  dists.  Consequently  if  we  subtract  — - 

part  from  8  ins,  or  .667  ft,  we  have  at  once  the  combined  eflFect  of  curvature  and  refraction  for  one 
mile,  equal  to  6.857  ins,  or  .6714  ft ;  and  for  other  dists,  as  shown  in  the  following  table,  bj  the  use 
of  which  we  avoid  the  necessity  of  making  separate  allowances  for  curvature  and  refraction. 


¥ig,7. 


Table  of  allowances  to  be  added  for 

curvature  of  the  eartb: 

and  for  refraction ;  combined. 

Dist. 

Allow. 

Dist. 

Allow. 

Dist. 

Allow. 

Dist. 

Allow. 

in  yards. 

feet. 

in  miles. 

feet. 

in  miles. 

feet. 

in  miles. 

feet. 

100 

.002 

K 

.036 

6 

20.6 

20 

229 

150 

.004 

x| 

.143 

7 

28.0 

22 

277 

200 

.007 

/x 

,321 

8 

36.6 

25 

357 

300 

.017 

1 

.572 

9 

46.3 

30 

614 

400 

.030 

IK 

.893 

10 

57.2 

35 

700 

500 

.046 

W2, 

1.29 

11 

69.2 

40 

915 

600 

.066 

m 

1.75 

12 

82.3 

45 

1158 

700 

.090 

2 

2.29 

13 

96.6 

50 

1429 

800 

.118 

2^ 

3.57 

14 

112 

55 

1729 

900 

.149 

3^ 

5.14 

15 

129 

60 

2058    . 

1000 

.185 

33^ 

•      7.00 

16 

146 

70 

2801 

1200 

.266 

4 

9.15 

17 

165 

80 

3659 

1500 

.415 

4^ 

11.6 

18 

185 

90 

4631 

2000 

.738 

/^ 

14.3 

19 

206 

100 

5717 

Hence,  if  a  person  whose  eye  is  5.14  ft,  or  112  ft  above  the  sea,  sees  an  object  just  at  the  sea's 
horizon,  that  object  will  be  about  3  miles,  or  14  miles  distant  from  him. 

A  horizontal  line  is  not  a  level  one,  for  a  straight  line  cannot  be  a 

level  one.  The  curve  of  the  earth,  as  exemplified  in  an  expanse  of  quiet  water,  is  level.  In  Fig  7, 
if  we  suppose  the  curved  line  ty  a  g  to  represent  the  surface  of  the  sea,  then  the  points  ty$  and  g  are 
on  a  level  with  each  other.  They  need  not  be  equidistant  from  the  center  of  the  earth,  for  the  sea  at 
the  poles  is  about  13  miles  nearer  it  than  at  the  equator;  yet  its  surface  is  everywhere  on  a  level. 

Up,  and  down,  refer  to  sea  level,  l^evel  means  parallel  to  the  curvature 
of  the  sea;  and  horizontal  means  tangential  to  a  level. 

Ex.  4.    If  the  inaccessible  vert  height  c  d,  Fig^  8, 

Is  80  situated  that  we  cannot  reach  it  at  all,  then  place  the  instrument  for  measuring  angles,  at  any 
convenient  spot  n ;  and  in  range  betwe'en  n  and  d,  plant  two  staffs,  whose  tops  o  and  i  shall  range 
precisely  with  n,  though  they  need  not  be  on  the  same  level  or  hor  line  with  it.  Measure  n  o:  also 
from  n  measure  the  angles  on  d  and  o  n  c.     Then  move  the  instrument  to  the  precise  spot  previously 

which  he  had  no  idea.  For  allowance  for  curvature  and  refraction  see  above  Table, 
A  triang-le  whose  sides  are  as  3,  4,  and  5,  is  right  angled;  and  one 

whose  sides  are  as  7 ;  7  ;  and  9.  9;  contains  1  right  angle;  and  2  angles  of  45°  each.  As  it  is  fre- 
quently necessary  to  lay  down  angles  of  45"  and  90°  on  the  ground,  these  proportions  may  be  used  for 
the  purpose,  by  shaping  a  portion  of  a  tape-line  or  chain  into  such  a  triangle,  and  driving  a  stake  at 
each  angle. 


154 


PLANE   TRIGONOMETKY. 


•ooupied  by  the  top  o  of  the  staff;  and  from  o  measure  the  angles  iod  and  doc.  This  being  doiie,  sub 

tract  the  augle  i  o  c  from 

180° ,  the  rem  will  be  the 

angle  c  o  n.     Consequent- 
ly in  the  triangle  noc,  we 

have  one  side  n  o,  and  two 

angles,  en  o  and  con,  to 
5*  find  by  Case  1  the  side  t 

Again,  take  the  angle  iod  - 

from  180°;  the  remainder 

"will  be  the  angle  n  o  d,  so 

that  in  the  triangle  dn  o 

we  have  one  side  n  o,  and 

the  two  angles  dn  o   and 

nod,  to  And  by  Case  1 

the  side  od.     Finally,  in 
•  the  triangle  cod,  we  have 

two  sides  c  o  and  o  d,  and 

their  included  angle  cod, 

to  find  c  d,  the  reqd  vert 

height. 

Rkm.  If  c  d  were  in  a  valley,  or  on  a  hill,  and  the  observations  reqd  to  be  made  from  either  higher 
or  lower  ground,  the  operation  would  be  precisely  the  same. 

£x.  5.     See  Ex  10. 
To  find  the  dist  ao,  ¥!§  9,  between  two  entirely  inaccessible 
objects. 


Fiff.  8. 


Fig-.  9. 


m  measure  the  angles  o 
Fig  9,  and  the  angles  a 


Tig.  10. 


Measure  a  side  n  m:  at  w  measure  the  angles  anm  and  onm;  also  at 
amn.  This  being  done,  we  have  in  the  triangle  a n  m,  one  side  n  m, 
nma;  hence,  by  Case  1,  we  can  calculate  the  side  a 7i. 
Again,  in  the  triangle  o  m  n  we  have  one  side  n  m,  and 
the  two  angles  c  mn,  and  mno;  hence,  by  Case  1,  we  can 
calculate  the  side  n  o.  This  being  done,  we  have  in  the 
triangle  ano,  two  sides  an,  and  no;  and  their  included 
angle  ano;  hence,  by  Case  3,  we  can  calculate  the  side 
ao,  which  is  the  reqd  dist.  It  is  plain  that  in  this  manner 
we  may  obtain  also  the  position  or  direction  of  the  inacces- 
sible line  ao ;  for  we  can  calculate, the  angle  nao  ;  and  can 
therefrom  deduce  that  of  ao;  and  thus  be  enabled  to  run 
a  line  parallel  to  it,  if  required.  By  drawing  n  m  on  pa- 
per by  a  scale,  and  laying  down  the  four  measd  angles, 
the  dist  a  o  may  be  measd  upon  the  drawing  by  the  same  scale. 

If  the  position  of  the  inaccessible  dist  c  n.  Fig  10,  be  such  that 
^e  can  place  a  stakes  in  line  with  it, we  may  proceed  thus  :  Place 
the  instrument  at  any  suitable  point  s,  and  take  the  angles  p  s  c 
andc«n.  Also  find  the  angle  ops,  and  measure  the  distj's.  Then 
in  the  triangle  ^  s  c  fi:;d  s  c  by  Case  1 ;  agnin,  the  exterior  angle 
n  c  «,  being  equal  to  the  two  interior  and  opposite  angles  c  p  s, 
and  p  s  c,  we  have  in  the  triangle  c  s  n,  one  side  and  two  angles 
to  fiad  c  n  by  Case  1. 

Ex.6.   To  find  a  dist  a  &^Fi^  11.  of  which 
the  ends  only  are   accessible. 

Prom  a  and  h,  measure  any  two  lines  ache,  meeting  at  c ;  also 
measure  the  angle  ach.  Then  in  the  triangle  a  ft  c  we  have  two 
sides,  and  the  included  angle,  to  find  the  third  side  a  h  by  Case  3. 

Ex.  7.  To  find  the  vert  hei$£ht  o  m,  of  a 
hill,  above  a  ^iven  point  i. 

Place  the  instrument  at  t;  measure  a  m.  Directing 
the  instrument  hor,  as  an,  take  the  angle  nam.  Then, 
since  a  n  m  is  90°  Fig  12,  we  have  one  side  a  m,  and 
two  angles,  nam  and  anm,  to  find  n  m  by  Case  I. 
Add  no,  equal  to  at,  the  height  of  the  instrument. 
Also,  if  the  hill  is  a  long  one,  add  for  curvature  of  the 
earth,  and  for  refraction,  as  explained  in  Example  3,  ^i  --' 

Pig  7.  Or  the  instrument  may  be  placed  at  the  top  of 
the  hill ;  and  an  angle  of  depression  measured  j  instead 
of  the  angle  of  elevation  nam. 

Rem.  1.  It  is  plain,  that  if  the  height  om  be  previously 
known,  and  we  wish  to  ascertain  the  dist  from  its  sum- 
mit m  to  any  point  i,  the  same  measurement  as  before, 
of  the  angle  nam,  will  enable  us  to  calculate  a  m  by 
Case  1.  So  in  Ex.  2,  if  the  height  na  be  known,  the  angles  measd  in  that  examplfe,  will  enable  as. 
to  oompate  the  dist  a  o ;  so  also  in  Figs  3,  4,  5,  and  7  j  in  all  of  which  the  procen  is  so  plain  as  te 
require  no  furtker  explanation. 

Rem.  2.  The  height  of  a  vert  object  by  mcanS  Of  itS  ShadoW.  Plant  one  end  of 
a  straight  stick  vert  in  the  ground  ;  and  measure  ts  shadow ;  also  measure  the  length  of  the  shadow 
of  the  object.      Then,  as  the  length  of  the  shadow  of  the  stick  is  to  the  length  of  the  stick  above 


Fig.  11. 


¥1^.12. 


J 


PLANE   TRIGONOMETRY. 


155 


Fig.  12^^ 


round,  so  is  the  length  of  the  shadow  of  the  object,  to  its  height.    If  the  object  is  inclined,  the  stick 
lUst  be  equally  inclined. 

Rem.  3.  Or  the  lieig^ht  of  a  vert  object  mw, 

Fig  12^^,  whose  distance  r  m  is  known,  may  be  found  by 
its  reflection  in  a  vessel  of  water,  or  in  a  piece  of 

looking  glass  placed  perfectly  horizontal  at  r;  for  as  r  a  is  to  the  height 

-        a  i  of  the  eye  above  the  reflector  r,  so  is  r  m  to^  xji*  ^    I  Ql/ 

-'CI  the  height  mn  of  the  object  above  r.  /t/j-^^  X  ic*  -*-*'Z3 

Rem.  4.    Or  let  o  c,  Fig  12}4,  be 

I  planted  pole,  or  a  rod  held  vf  rt  by  an  assistant.    Then 

taud  at  a  proper  dist  back  from  it,  and  keeping  the  eyes  steady,  let  marks  be 
lade  at  o  and  c,  where  the  lines  of  sight  i  n  and  t  m  strike  the  rod.     Then 
c  is  to  c  o,  so  18  t  m  to  m  n. 

The  following  examples  may  be  regarded  as  substitutes  for  strict  trigonom*. 

try  :  and  will  at  times  be  useful,  in  case  a  table  of  sines,  &c,  is  not  at  hand  for 

making  trigonometrical  calculations. 

Ex.  8.  To  find  the  dist  ab,  of  which  one  end  only 
is  accessible. 

Drive  a  stake  at  any  convenient  point  a ;  from  a  lay  off  any  angle  b  a  e.  In 
the  line  a  c,  at  any  convenient  point  c,  drive  a  stake ;  and  from  c  lay  off  an  angle 
acd,  equal  to  the  angle  6  a  c.  In  the  line  c  d,  at  any  convenient  point,  as  cJ, 
drive  a  stake.  Then,  standing  at  d,  and  looking  at  b,  place  a  stake  o  in  range 
withdh;  and  at  the  same  time  in  the  line  a  c.  Measure  a  o,  o  c,  and  erf  j  than, 
from  the  principle  of  similar  triangles,  as 

oe:ed:ino:ab. 


<S 


Fig.  13. 


Fig.  15.       X 


Or  tlins: 

Fig  14,  n  h  being  the  dist,  place  a  stake  at  n  ;  and  lay  off  the  angle  h  n  m  90°. 
At  any  convenient  dist  n  m,  place  a  stake  m.  Msike  the  angle  h  m  y  =  90°  ;  and 
place  a  stake  at  y,  in  range  with  Ii  n.  Measure  n  y  and  n  m ;  then,  from  the 
principle  of  similar  triangles,  as 

n  y  m  m  '.  I  rt  m  '.  ti  h. 

Or  thus.  Fig  14.    Lay  off  the  angle  hnm  =  90°,  placing  a  stake 

m,  at  any  convenient  dist  n  m.  Measure  n  m.  Also  measure  the  angle  nmh. 
Find  nat  tang  of  n  m  A  by  Table  Mult  this  nat  tang  by  n  m.     The  prod 

will  be  n  h. 
Or  thus.     Lay  off  angle  hnm  =  90°.    From  m  measure  the 

angle  n  m  h,  and  lay  off  angle  nmy  equal  to  it,  placing  &  stake  at  y  in  range 

with  h  n.    Then  is  /»  y  =  n  h. 

Or  thus,  without  measuring 
any  ang'le; 

t  u  being  the  dist.  Make  m  r  of  any  convenienfe 
length,  in  range  with  t  u.  Measure  any  v  o ;  and 
o  X  equal  to  it,  in  range.  Measure  u  o ;  and  o  y 
equal  to  it  in  range.  Place  a  stake  z  in  range  with 
both  X  y,  and  t  o.  Then  will  y  z  he  both  equal  to 
t  u,  and  parallel  to  it. 


Fig.  16, 


Or  thus,  without  measuring  any  angle. 

Drive  two  stakes  t  and  u,  in  range  with  the  object  s.     From  t  lay  off  any 
convenient  dist  t  x,  in  any  direction.     From  u  lay  ofif  w  w  parallel  to  «_x, 
placing  w  in  range  with  i  s.     "  '         " 
X  t.     Then,  as 


Make  u  v  equal  to  t  x.    Measure  u; «,  t>  *,  and 


Or  thus.    At  a  lay  off  angle  o  a c  =  5°  43'.    Lay     ^ 
off  0  0  at  right  angles  to  ao.     Measure  oc.     Then 
flo  =  10  0  c,  too  long  on^y  1  part  in  935.6,  or  5.643  feet 
in  a  mile,  or  .1069  foot  (full  U  inches)  in  100  feet. 


156 


PLANE   TRIGONOMETRY. 


Fig.  17. 


Ex.  9.    To  find  the  dist  a  &,  of  wliicb  th« 
ends  only  are  accessible. 

Prom  a  lay  off  the  angle  h  ac;  and  from  5,  the  angle  ah  d,  each 
90°.  Make  a  c  and  h  d  equal  to  each  other.  Then,  cd  =  afe.  Or 
a  h  may  be  considered  as  the  dist  across  the  river  in  Figs  15,  13,  or 
14 ;  and  be  ascertained  in  the  same  way.  Or  measure  any  dist,  Fig 
17,  a  o;  and  make  on  in  line  and  equal  to  it.  Also  treasure  ho\ 
and  piake  om  in  line  and  equal  to  it.  Then  will  mn  be  both  paral. 
lei  to  a  5,  and  equal  to  it. 


Fig.  18. 


Ex.  10.    See  Ex.  4.    To  find  tlie  entirely 

inaccessible  dist  y  z,  and  also 

its  direction. 

At  any  two  convenient  points  a  and  6,  from  each  of  which 
y  and  z  can  be  seen,  drive  stakes.  Then  we  have  the  four 
corners  of  a  four-sided  figure,  in  which  are  given  the  directions 
of  three  of  its  sides,  and  of  its  two  diags.  These  data  enable  us 
to  lay  out  on  the  ground,  the  small  four-sided  fig  a  c  o  t,  exactly 
similar  to  the  large  one.  Thus,  in  the  line  a  b  place  a  stake 
C.:  and  make  co  parallel  to  62:;  o  being  at  the  same  time  in 
range  of  the  diag  a  z.  Also,  from  c  make  c  i  parallel  to  6  y; 
i  being  at  the  same  time  in  range  of  a  y.  Then  will  i  o  be  in 
the  same  direction  as  y  z,  or  parallel  to  it.  Measure  ac,  ab, 
and  io;  then  evidently,  from  the  principle  of  similar  figures,  as 
a  c  :  ab  :  '.  i  o  '.  y  z. 

If  y  z  were  a  visible  line,  such  as  a  fence  or  road,  we  could 
from  a  divide  it  into  any  required  portions.  Thus,  if  we  wish 
to  place  a  stake  halfway  between  y  and  z,  first  place  one  half- 
way between  %  and  o ;  then  standing  at  a,  by  means  of  signals, 
place  a  person  in  range  on  y  z.  Or,  to  find  along  ab,  a.  point  t 
perp  to  y  «  at  y,  first  make  oi  s'=  90° ;  and  measure  a  8.  Then, 
as  - 

oiiasiiyziat. 

Ex.  11.  To  find  the  position  of  a  point,  n.  Tig  19, 

By  meana  of  two  angles  a  n  b  and  b  n  c.  taken  from  it        tlie  three  objects  a  b  c,  whose  positions 
and  dists  apart  are  known. 

The  use  of  this  problem  is  more  frequent  in  marine  than  in  land  surveying.  It  is  chiefly  employed 
for  determining  the  position  n  of  a 
boat  from  which  soundings  are  being 
taken  along  a  coast.  As  the  boat 
moves  from  point  to  point  to  take 
fresh  soundings,  it  becomes  necessary 
to  make  a  fresh  observation  at  each 
point,  in  order  to  define  its  position 
on  the  chart.  An  observation  consists 
in  the  measurement  by  a  sextant  of 
the  two  angles  anb,  bn  c.  to  the  sig- 
nals ab  c,  previously  arranged  on  the 
shore.  When  practicable,  this  method 
shoald  be,  rejected ;  and  the  observa- 
tions taken  to  the  boat  at  the  same 
instant,  by  two  observers  on  shore,  at 
'  two  of  the  stations.  The  boat  to  show 
a  signal  at  the  proper  moment.  The 
most  expeditious  mode  of  fixing  the 
point  n  upon  the  map,  is  to  draw  three 
lines,  forminfit  the  two  angles,  and  ex- 
tended indefinitely,  on  a  piece  of  trans- 
parent paper.  Place  the  paper  upon  the  map,  and  move  it  about  until  the  three  lines  pass  through 
the  three  stations  ;  then  prick  through  the  point  n  wherever  it  happens  to  come. 

Instead  of  the  transparent  paper,  an  instrument  called  a  station  pointer  may  be  used  when  there 
are  many  points  to  be  fixed. 

But  the  position  of  the  point  n  can  be  found  more  correctly  by  describing  two  circles,  as  in  Fig  19, 
each  of  which  shall  pass  through  n  and  two  of  the  station  points.  The  question  is  to  find  the  centers 
o  and  »  of  two  such  circle?.  This  is  very  simple.  We  know  that  the  angle  a  o  6  at  the  center  of  a  circle  is 
twice  as  great  as  any  angle  «  n  ft  at  the  circumf  of  the  same  circle,  when  both  are  subtended  by  the 
same  chord  a  b.  Consequently,  if  the  angle  anb,  observed  from  the  boat,  is  say,  50°,  the  angle  aob 
must  be  100°.  And,  since  the  three  angles  of  every  plane  triangle  are  equal  to"  180°,  the  two  angles 
o  ab  and  o  b  a  are  together  equal  to  180°  —  100°  =  80°.  And,  since  the  two  sides  a  o  and  b  o  are 
equal  (being  radii  of  the  same  circle),  therefore,  the  angles  oab  and  oba  are  equal ;  and  each  equal  to 
80° 
—-  =  40°.     Consequently,  on  the  map  we  have  only  to  lay  down  at  a  and  b,  two  angles  of  40°;  the 

r>int  0  of  intersection  will  be  the  center  of  the  circle  a  b  n.   Proceed  in  the  same  way  with  the  angle 
f»  c,  to  find  the  center  at.    Then  the  intersection  of  the  two  eircles  at  n  will  be  the  point  sought. 


Fig.  19. 


PARALLELOGRAMS. 


157 


PARAIil^EI^OORAMS. 

Square.  Rectangle.         Rhombus.  Rhomboid. 

a        


P 


or 


P\ 


"fm 


P\  --4 


A  PARALLELOGRAM  is  any  figure  of  four  straight  sides,  the  opposite  ones  of  which 
ire  paral?el.  There  are  but  four,  as  in  the  above  figs.  The  rhombus,  like  the  rhom- 
}ohedron,  Fig  3,  p  195,  is  sometimes  called  "rhomb."  In  the  square  and  rhombus 
ill  the  four  sides  are  equal ;  in  the  rectangle  and  rhomboid  only  the  opposite  ones 
ire  equal.  In  any  parallelogram  the  four  angles  amount  to  four  right  angles,  or 
J60° ;  and  any  two  diagonally  opposite  angles  are  equal  to  each  other ;  hence,  having 
me  angle  given,  the  other  three  can  readily  be  found.  In  a  square,  or  a  rhombus,  a 
liag  divides  each  of  two  angles  into  two  equal  parts  ;  but  in  the  two  other  parallel- 
)grams  it  does  not. 

To  find  the  area  of  any  parallelogram. 

Multiply  any  side,  as  S,  by  the  perp  height,  or  dist  p  to  the  opposite  side.   Or,  multiply  togethtf 
wo  sides  and  nat  sine  of  their  included  angle. 
The  dlas  a  b  of  any  eauare  is  equal  to  one  side  mult  by  1.41421 ;  and  a  side  is  equal  to 

^■^^^^ ;  or,  to  diag  mult  by  .707107. 

The  side  of  a  square  equal  in  area  to  a  gtvex  circle*  is  equal  to  diam  X  .886227. 

The  side  of  the  greatest  square,  t/iat  can  be  inscribed  in 
I  criven  circle,  is  equal  to  diam  X  .707107. 

The  side  of  a  square  mult  by  1.51967  gives  the  side  of  an  equl- 
Bteral  triangle  of  the  same  area.  All  parallelograms  as  A 
nd  C,  which  have  equal  bases,  «  c,  and  equal  perp  heights  n 
,  have  aiso  equal  areas ;  aud  the  area  of  each  is  twice  that  of  a  tri- 
ngle  having  the  same  base,  and  perp  height.  The  area  of  a 
quare  Inscribed  in  a  circle  is  equal  to  twice  the  square  of  the 
ad. 

In  every  parallelogram,  the  4  squares  drawn  on  its  sides  have  a  united  area  equal  to  that  of 
lie  two  squares  drawn  on  its  2  diags.  If  a  larger  square  be  drawn  on  the  diag  a  i  of  a  smaller 
ijuare.  its  area  will  be  twice  that  of  said  smaller  square.  Either  diag  of  any  parallelogram 
ivides  it  into  two  equal  triangles,  and  the  2  diags  div  it  into  4  triangles  of  equal  areas.  Tlie  two 
Hags  of  any  parallelogram  divide  each  other  into  two  equal  parts.  Any  line  drawn  through 
he  center  of  a  diag  divides  the  parallelogram  into  two  equal  parts. 

Remark  1.— The  area  of  any  fig  whatever  as  B  that  is  enclosed  by  four  straight 
[nes,  may  be  found  thus  :  Mult  together  the  two  diags  am,  n  b;  and  the  nat  sine  of  the  least  angle 

o  6 ;  orn  o  wi,  formed  by  their  intersection.  Div  the  product  by  2.  This  is  useful  in  land  surveying, 
'hen  obstacles,  as  is  often  the  case,  make  it  difficult  to  measure  the  sides  of  the  6g  or  held ;  while  it 
lay  be  easy  to  measure  the  diags ;  and  after  finding  their  point  of  intersection  o,  to  measure  the  re- 
Hired  angle.  Sut  if  the  fig  is  to  be  drawn,  the  parts  o  a,  o  &,  o  n,  o  m  of  the  diags  must  also 
e  measd. 

Rem.  3.  —  The  sides  of  a  parallelogram,  triangle,  and  many  other  figs  may  be 
bund,  when  only  the  area  and  angles  are  given,  thus  :  Assume  some  particular  one  of  its 
ides  to  be  ot  the  length  1  ;  and  calculate  what  its  area  would  be  if  that  were  the  case.  Then  as  the 
q  rt  of  the  area  thus  found  is  to  this  side  1,  so  is  the  sq  rt  of  the  actual  given  area,  to  the  corre* 
ponding  actual  side  of  the  fig. 


On  a  ^tven  line  wx^ta  draiF  a  square* 

From  w  and  x,  with  rad  w  x,  describe  the  arcs  xry  and  wre. 
Prom  their  intersection  r,and  with  rad  equal  to  3^  of  w  x,  describe 
8  8  s.  From  to  and  x  draw  w  n  and  xm  tangential  to  a  s  «,  and 
ending  at  the  other  arcs ;  join  nm. 


158 


TRAPEZOIDS  AND  TRAPEZIUMS. 


TRAPEZOIDS. 

n  t .       rn      n      m 


is  any  fig  with  four  straight  sides,  of  which  no  two  are  parallel. 


c        a  8     c  a      8    c 

A  trapezoid  a  c  n  m,  is  any  figure  with  four  straight  aides,  only  two  of  which,  as  ac  and  n  i)t,  ar€ 
parallel. 

To  find  tlie  area  of  any  trapezoid. 

Add  together  the  two  parallel  sides,  a  c  and  m  n ;  mult  the  sum  by  the  perp  dist  s  t  b«f  fen. 
them ;  div  the  prod  by  2.  See  the  following  rules  for  trapeziums,  which  are  all  equally  appL  *)ie 
to  trapezoids  ;  also  see  Remarks  after  Parallelograms. 

TRAPEZIUMS. 


A  trapezium 

To  find  the  area  of  any  trapezinm,  having"  g-iven  the  diag* 
bOf  or  a  c,  between  either  pair  of  opposite  ang-les;  and  also 
the  two  perps,  n.,  n,  from  the  other  two  an^rles. 

Add  together  these  two  perps  ;  mult  the  sum  by  the  diag;  div  the  prod  by  2. 

HaTing;  the  four  sides ;  and  either  pair  of  opposite  ang^les, 

as  (the,  <t  o  c;  or  b  a  o,  and  b  c  o. 

Consider  the  trapezium  as  divided  into  two  triangles,  in  each  of  which  are  given  two  sides  and  the 
included  angle.  Find  the  area  of  each  of  these  triangles  as  directed  under  the  preceding  head  "  Tri- 
angles," and  add  them  together. 

Having-  the  four  angles,  and  either  pair  of  opposite  sides. 

Begin  with  one  of  the  sides,  and  the  two  angles  at  its  ends.  If  the  sum  of  these  two  angles  exceeds 
180°,  subtract  each  of  them  from  180°,  and  make  use  of  the  rems  instead  of  the  angles  themselves. 
Then  consider  this  aide  and  its  two  adjacent  angles  (or  the  two  rems,  as  the  case  may  be)  as  those 
at  a  triangle;  and  find  its  area  as  directed  for  that  otMt  under  the  preceding  head  "  Triangle."  I>o 
the  same  with  the  other  given  side,  and  its  two  adjacent  angles,  (or  their  rems,  as  the  case  may  be.) 
Subtract  the  least  of  the  areas  thus  found,  from  the  greatest;  theYem  will  be  the  reqd  area. 

Having-  three  sides;  and  the  tivo  included  angles. 

Mult  together  the  middle  side,  and  one  of  the  adjacent  sides  ;  mult  the  prod  by  the  i^at  sine  of  their 
included  angle ;  call  the  result  a.  Do  the  same  with  the  middle  side  and  its  other  adjacent  side, 
and  the  nat  sine  of  the  other  included  angle ;  call  the  result  b.  Add  the  two  angles  together ;  find 
the  diff  between  their  sum  and  180°,  whether  greater  or  less  ;  find  the  nat  sine  of  this  diff;  molt 
together  the  two  given  sides  which  are  opposite  one  another ;  mult  the  prod  by  the  nat  sine  just  fomnd ; 
call  the  result  c.  Add  together  the  results  a  and  b  ;  then,  if  the  sum  of  the  two  given  angles  is  leas 
than  180°,  subtract  c  from  the  sum  of  a  and  b  ;  half  the  rem  will  be  the  area  of  the  trapezium.  But 
if  the  sum  of  the  two  given  angles  be  greater  than  180°,  add  together  the  three  results  a,  6,  and  c; 
half  their  sum  will  be  the  area. 

Having^  the  two  diagonals,  and  either  angle  formed  by  their 
intersection. 

See  Remarks  after  Parallelograms. 

In  railroad  measurements 

Of  excavation  and  embankment,  the  trapezium 
tm  n  0  frequently  occurs  ;  as  well  as  the  two  5-sided 
figures  im  n  0  t  and  I  mn  o  s;  in  all  of  which  m  n 
represents  the  roadway  ;  rs.rc,  and  r  t  the  center- 
depths  or  heights ;  I  u  and  o  v  the  side-depths  er 
heights,  as  given  bv  the  level;  Im  and  n o  the  side- 
slopes. 

The  same  general  rule  for  area  applies  to  all  three 
of  these  figs ;  namely,  mult  the  extreme  hor  width 
uvhj  half  the  center  depth  r  s,  re.  or  r  t,  as  the 
case  may  be.  Also  mult  one  fourth  of  the  width  of 
roadway  m  n,  by  the  sum,  of  the  two  side-depths  I  u 
and  o  V.  Add  the  two  prods  together ;  the  sum  is  the 
reqd  area.  This  rule  applies  whether  the  two  side- 
slopes  m  I  and  n  o  have  the  same  asgle  of  inclination  or  not.  In  railroad  work,  etc.,  the  mid- 
way hor  width,  center  depth,  and  side  depths  of  a  prismoid  are  respectively  =  The  half  sams  of 
toe  corresponding  end  ones,  a'nd  thus  can  be  found  without  actual  measurement. 


POLYGONS. 


159 


To  draw  a  hexagron,  each  side  of  which  shall 
be  equal  to  a  g^iven  line,  a  b. 

Prom  a  and  h,  with  rad  a  h,  describe  the  two  arcs;  from  their  intersection, 
»,  with  the  same  rad,  describe  a  circle;  around  the  circumf  of  which  step  off 
the  same  rad.  ' 

Side  of  a  hexag^on  =  nn  X  .57735. 


To  draTT  an  octag'on,  w^ith  each  side 
equal  to  a  g^iven  line,  c  e. 

From  cand  e  draw  two  perps,  cp,  ep.  Also  prolong  cc  toward 
/  and  g;  and  from  c  and  c,  with  rad  equal  c  c,  draw  the  two 
quadrants;  and  find  their  centers  ft  A;  join  c  A,  and  e  ft;  ^'^^ 
h  s  and  ft  < parallel  to  cp;  and  make  each  of  tliem  equal  to 
xuake  c  o,  and  e  o,  each  equal  to  ft  ft ;  join  o  0,0  a,  and  o  t. 


draw 
to  ce; 


Side  of  an  octag^on  =  w  n  X  .41421354. 


To  draw  an  ocfag^on  in  a  g^iven  square. 

From  each  corner  of  the  square,  and  with  a  rad  equal  to  half  its  diag, 
describe  the  four  arcs;  and  join  the  points  at  which  they  cut  the  sides  of  the 
square.  , 

To  draw  any  regrular  polygon,  with  each  side 
equal  to  m  n. 

Div  360  degrees  by  the  number  of  sides;  take  the  quot  from  180°;  div  the 

rem  by  2.   Thi»  will  give  the  angle  cmn,  or  cum.    htm  and  n  lay  down  these 

angles  by  a  protractor:  the  sides  of  these  augles  will  meet  at«i  point,  c.  from 

,  which  describe  the  circle  mny;  and  around  its  circumf  step  oft  dists  equal  to 

mn. 

In  any  circle,  m  n  y^  to  draw  any  regular 
polygon. 

DivSGO^  by  the  number  of  sides;  thequot  will  be  the  angle  men.  at  the  center. 
Lay  off  this  angle  by  a  protractor ;  and  its  chord  m  n  will  be  one  side ;  which 
step  off  Around  tl*  circumf. 

To  reduce  any  polygon,  as  abed efa,  to  a  triangle  of  the 
same  area. 


If-  we  produce  the  side /a  toward  ro;  and  draw  6  g  parallel  to  a  c,  and  join  g  c.  we  get  equal  trl. 
angles  a  oft,  and  a  c  g,  both  on  the  same  base  a  c ;  and  both  of  ibe  same  perp  height,  inasmuch  aa 
they  are  between  the  two  parallels  a  c  and  g  b.  But  the  part  a  c  i  forms  a  portion  of  both  these  trl* 
angles,  or  in  other  words,  is  common  to  both.  Therefore,  if  it  be  taken  away  from  both  triangles, 
the  remaining  parts,  t  c  6  of  one  of  them,  and  i  g  a  o(  the  other,  are  also  equal.  Therefore,  if  the 
part  icbbe  left  off  from  the  polygon,  and  the  part  ig  ab&  taken  into  it,  the  polygon  g  fed  cig  will 
have  the  same  area  as  afe  d  c  b  a;  but  it  will  have  but  five  sides,  while  the  other  has  six.  Again, 
if  e  s  be  drawn  parallel  to  d /,  and  ds  joined,  we  have  upon  the  same  base  e«,  and  between  the  same 
parallels  e  a  and  df,  the  two  equal  triangles  e  s  d,  and  e  s/,  with  the  part  eos  common  to  both  ;  and 
consequently  the  remaining  part  e  o  d  ot  one.  and  o  sf  of  the  other,  are  equal.  Therefore,  if  o  s/  be 
left  off  from  the  polygon,  and  e  o  oJ  be  taken  into  it,  the  new  polygon  gad  eg,  Fig  2,  will  have  the  same 
area  as  gfe  d  c  g;  but  it  has  but  four  sides,  while  the  other  has  five.  Finally,  if  g  s,  Fig  2,  be 
extended  toward  n ;  and  d  n  drawn  parallel  to  c  s  ;  and  c  n  joined,  we  have  on  the  same  base  c  s,  and 
between  the  same  parallels  c  s  and  d  n,  the  two  equal  triangles  e  a  n,  and  cad,  with  the  part  cat 
common  to  both.  Therefore,  if  we  leave  out  cdt,  and  take  itx.a  t  n,  we  have  the  triangle  gnc  equal 
to  tliepolygon  </s  rf  cw,  Fig2;  or  to  a/fl  rfc  b  a,  Fig  1. 

This  simple  method  is  applicable  to  polygons  of  any  number  of  eidea. 


160 


POLYGONS. 


To  reduce  a  larg^e  fig:,  abed  efg,  to  a  smaller 
Niinilar  one. 

From  any  interior  point  o,  whicli  had  better  be  near  the  center,  draw  lines 
to  all  the  angles  a,  b,  c,  <fec.  Join  these  lines  by  others  parallel  to  the  sides 
of  the  fig.  If  it  should  be  reqd  to  enlarge  a  small  fig,  draw,  from  any  point 
o  within  it,  lines  extending  beyond  its  angles  ;  and  join  these  lines  by  others 
parallel  to  the  sides  of  the  small  Mg. 


To  reduce  a  map  to  one  on  a  smaller  scale. 

The  best  method  is  by  dividing  the  large  map  into  squares  by  faint  lines,  with  a  very  soft  leai 
pencil;  and  then  drawing  the  reduced  map   upon   a  sheet  of 
smaller  squares.     A  pair  of  proportional  dividers   will  assist   ^  \C 

much  in  fixing  points  intermediate  uf  the  sides  of  the  squares.     \    n     \y,\  h 

If  the  larg«  map  would  be  injured  by   drawing    and  rubbing     \  ^      ^M  j (^ 

•at  the  •qaares,  threads  may  be  stretched  across  it  to  form  tlie       \  \   >^  | 


In  a  rectang-ular  fig",  g  h  s  d, 

Representing  an  open  panel,  to  find  the  points  o  o  o  o  in  its 
sides ;  and  at  equal  dists  from  the  angles  .9,  and  s ;  for  inserting 
a  diag  piece  o  o  o  o,  of  a  given  width  I  I,  measured  at  right 
angles  to  its  length.  From  g  and  s  as  centers,  describe  several 
concentric  arcs,  as  in  the  Fig>.  Draw  upon  transparent  paper, 
two  parallel  lines  a  a,  c  c,  at  a  distance  apart  equal  toll:  and 
placing  these  line.s  on  top  of  the  panel,  move  them  about  until  it 
is  shown  by  the  arcs  that  the  four  dists  g  o,  go,  s  o,  s  o.  are 
equal.  Instead  of  the  transparent  paper,  a  strip  of  common 
paper,  of  the  width  1 1  may  be  used. 

Rem.  Many  problems  which  would  otherwise  be  very  diflBcult, 
may  be  thus  solved  with  an  accuracy  sufficient  for  practical 

purposes,  by  means  of  transparent  paper. 

To  find  the  area  of  any  irreg^ular  poly* 
gpon,  anb  c  m. 

Div  it  into  triangles,  as  anh,  am  c,  and  06  c;  in  each  of 
which  find  the  perp  dist  0,  between  its  base  a  6,  a  c,  or  b  c;  and 
the  opposite  angle  n,  m,  or  a;  mult  each  base  by  its  perp  dist; 
add  all  the  prods  together ;  div  by  2, 


To  find  approx  tlie  area  of  a  long*  ir« 

reg^ular  fig,  ».Habcd.    Between  its  ends  ah,cA, 


space  off  equal  dists,  (the  shorter  they  are  the  more  accurate  will  be  the  result,)  through  which 
draw  the  intermediate  parallel  lines  1,  2,  3,  &c,  across  the  breadth  of  the  fig.  Measure  the  lengths 
of  these  intermediate  lines  ;  add  them  together ;  to  the  sum  add  half  the  sum  of  the  two  end  breadths 
a  6  and  c  d.  Mult  the  entire  sum  by  one  of  the  equal  spaces  between  the  parallel  lines.  The  prod 
will  be  the  area  This  rule  answers  as  well  if  either  one  or  both  the  ends  terminate  in  points,  as  at  m 
and  n.  In  tne  last  of  these  cases,  both  a  b  and  c  d  will  be  included  in  tne  iniermediate  lines;  and 
half  the  two  end  breadths  will  be  0,  or  nothing. 

To  find  the  area  of  any  irregular  figure. 

Draw  around  it  lines  which  shall  enclose  within  them  (as  nearly  as 
can  be  judged  by  the  eye)  as  much  space  not  belonging  to  the  figure  as 
they  exclude  space  belonging  to  it.  The  area  of  the  simplified  figure 
thus  formed,  being  in  this  manner  rendered  equal  to  that  of  the  com- 
plicated one,  may  be  calculated  by  dividing  it  into  triangles,  Ac.  By 
using  a  piece  of  fine  thread,  the  proper  position  for  the  new  boundary 
lines  may  be  found,  before  drawing  them  in. 

Areas  of  irregular  figures  may  be  found  from  a  drawing,  by  laying 
upon  it  a  piece  of  transparent  paper  carefully  ruled  into  small  squares,  each  of  a  given  area,  say  10 
20,  or  100  sq.  ft.  each  ;  and  by  first  counting  the  whole  squares,  and  then  adding  the  fractions  of 
squares. 


^=^^-s^ 


CIRCLES. 


161 


A  circle  is  th*  area  Included  within  a  curved  line  of  such  a  character  that  every  point  In  it  Is 
•qually  distant  from  a  certain  point  within  it,  called  its  center.  The  curved  line  itself  is  called  the 
•iroumference,  or  periphery  of  the  circle ;  or  very  commonly  it  is  called  the  circle. 

To  find  tlie  circumtereiice. 

If  ult  diam  by  S.1416,  which  gives  too  much  by  only  .148  of  an  inch  in  a  mile.  Or»  as  US  is  to  366 
«o  is  diam  to  circumf ;  too  great  1  inch  in  186  miles.  Or,  mult  diam  by  3-^  ;  too  great  by  about  1 
part  in  2485.     Or,  mult  area  by  12.566,  and  take  sq  root  of  prod. 


To  find  the  diam. 

Div  the  oircumf  by  3.1416  ;  or,  as  355  is  to  113,  so  is  circumf  to  diam  ;  or,  mult  the  ciroumf.  by  7  ; 
and  div  the  prod  by  22,  which  gives  the  diam  too  small  by  oojy  about  one  part  in  2485 ;  or,  mult  the 
area  by  1.2732;  and  take  the  sq  rt  of  the  prod. 

The  diam  is  to  the  circumf  more  exactly  as  1  to  3.14159265. 

To  find  the  area  of  a  circle. 

Square  the  diam;  mult  this  square  by  .7854;  or  more  accurately  by  .78589816;  or  square  the  cir- 
cumf ;  mult  this  square  by  .07958 :  or  more  accurately  by  .07957747  ;  or  mult  half  the  diam  by  half  the 
oircumf;  or  refer  to  the  following  table  of  areas  of  circles.     Also  area  =  sq  of  rad  X  3.1416. 

The  area  of  a  circle  is  to  the  area  of  any  circumscribed  straight-sided  fig,  as  the  circumf  of  the 
circle  is  to  the  circumf  or  periphery  of  the  fig.  The  area  of  a  square  inscribed  in  a  circle,  is  equal  to 
twice  the  square  of  the  rad.    Of  a  circle  in  a  square,  —  square  X  .7854. 

It  is  convenient  to  remember,  in  rounding  off  a  square  corner  a  &  c,  by  a  quarter  of     ^  ^ 

a  circle    that   the    shaded   area  «'>''>■  omml    tn   »hniit.    1    nnrr.  CnnrrpnMv  -914.fi'i  rtf  t.hi^       **|- 

whole  square  abed. 


To  find  the  diam  of  a  circle  equal  in  area  to  a  ^iven  sqnare. 

Mult  one  side  of  the  square  by  1.12838. 

To  find  the  rad  of  a  circle  to  circnmscribe  a  g^iven  square. 

Mult  one  side  by  .7071 ;  or  take  ^  the  diag. 

To  find  the  side  of  a  square  equal  in  area  to  a  g^iven  circle. 

Mult  the  diam  by  .88623. 

To  find  the  side  of  the  grreatest  square  in  a  §:iven  circle. 

Mult  diam  by  .7071.     The  area  of  the  greatest  square  that  can  be  inscribed  in  a  circle  is  equal  t* 
twrice  the  square  of  the  rad.   The  diam  X  by  1.3468  gives  the  side  of  an  equilateral  triangle  of  equal  aroa. 


To  find  the  center  c,  of  a  given  circle. 

Draw  any  chord  a  b ;  and  from  the  middle  of  it  o,  draw  at  rght  angles  !• 
it,  a  diam  d  g ;  find  the  center  c  of  this  diam. 


To  describe  a  circle  througrh  any  three 
points,  ah  Cf  not  in  a  straig-ht  line. 

Join  the  points  by  the  lines  ab,  he;  from  the  centers  of  these  lines  dravf 
the  dotted  perps  meeting,  as  at  o,  which  will  be  the  center  of  the  circle. 
Or  from  b,  with  any  convenient  rad,  draw  the  arc  m  n;  and  from  a  and  c, 
with  the  same  rad,  draw  arcs  y  and  z;  then  two  lines  drawn  through  the 
intersections  of  these  arcs,  will  meet  at  the  center  o. 

To   describe  a   circle   to    touch    the   three 
ang^les  of  a  triang^le  is  plainly  the  same  as  this. 
To  inscribe  a  circle  in  a  triang-ledraw  two  lines 

bisecting  any  two  of  the  angles.     Where  these  lines  meet  is  the  center  of 
the  circle. 


162 


CIRCLES. 


To  ibrwL-w  a  tang-ent,  iei,  to  a  circle,  from  an^ 
driven  point,  e^  in  its  circumf. 

Through  the  center  n,  and  the  giren  point  e,  draw  n  o;  make  e  o  equal  i« 
e  71 ;  from  n  and  o,  with  any  rad  greater  than  half  of  o  n,  describe  the  tw« 
pairs  of  arc  i  i;  join  their  intersections  i  i. 

Here,  and  in  the  following  three  figs,  the  tangents  are  ordinary  or  geo- 
metrical  ones ;  and  may  end  where  we  please.  But  the  trigonometricai 
*"     tangent  of  a  given  angle,  must  end  in  a  secant. 

Or  from  e  lay  off  two  equal  distances  e  c,  e  /;  and  draw  i  i 

parallel  to  c  t. 

To  draiir  a  tan^,  ash,  to  a  circle,  from  a  point, 
a,  iFlilcli  is  ontside  of  tlie  circle. 

Draw  a  c,  and  on  it  describe  a  semicircle ;  through  the  intersection,  $,  drsp«9 
asb.    Here  c  is  the  center  of  the  circle. 


To  drai¥  a  tangr,  g  Ji*  from  a  circular  arc,  </  <»  c. 

Of  which  n  a  is  the  rise.    With  rad  g  a,  describe  an  arc,  8  a  o.     Make  (  ^ 
equal  to  s  a.    Through  t  draw  g  h. 


To  draw  a  tang:  to  two  circles. 

First  draw  the  line  m  n,  just  touching  the  two 
circles ;  this  gives  the  direction  of  the  tang.  Then 
from  the  centers  of  the  circles  draw  the  radii,  o  o,  perp 
to  mn.  The  points  1 1  are  the  tang  points.  If  the 
tang  is  in  the  position  of  the  dotted  line,  s  y,  the  ope- 
ration is  the  same. 


•^y 


If  any  two  cbords,  as  a  b,  o  c,  cross  eacb  otber, 

then  as  0  w  :  w  &  : :  a  w  :  n  c.  Hence,  nhy^an  =  ony^nc.  That 
is,  the  product  of  the  two  parts  of  one  of  the  lines,  is  =  tbe  pro* 
lluct  of  the  two  parts  of  the  other  line. 


CIRCLES. 


163 


TABIiE  1  OF  CIRCL.es. 
Diameters  in  units  and  eightlis,  &>c. 


Diam. '  Circumf. 

Area. 

Diam. 

Circumf. 

Area. 

Diam. 

Circumf. 

Area. 

Diam. 

Circumf. 

ArMi. 

1-6 1     .omsi 

.00019 

3.~^ 

10.9956 

9.621l|  103^ 

31.8086 

80.516 

1934 

60.4757 

291.04 

1-32      .0:*,S175 

.00077 

9-16 

11.1919 

9.9678J       U 

32,2013 

82.516 

% 

60.8684 

294.8S 

3-641      ,U72fi2 

.00173 

% 

11.3883 

10.321 

% 

32.5940 

84.541 

Yz 

61.2611 

298.6J 

1-16 

.196350 

.00307 

11-16 

11.5846 

10.680 

H 

32.9867 

86.590 

% 

61.6538 

302.49 

3-32 

.294524 

.00690 

H 

11.7810 

11.045 

% 

33.3794 

88.664 

H 

62.0435 

306.35 

H 

.392699 

.01227 

13-16 

11.9773 

11.416 

% 

33.7721 

90.763 

y» 

62.4392 

310.24 

5-32 

.490874 

.01917 

% 

12.1737 

11.793 

% 

34.1648 

92.886 

20 

62.8319 

314.1S 

3-16 

589049 

.02761 

15-16 

12.3700 

12.177 

11. 

34.5575 

95.033 

% 

63.2246 

318.10 

7-32 

.687223 

.03758 

4. 

12.5664 

12.566 

H 

34.9502 

97.205 

Yi 

63.6173 

322.0S 

H 

.78.3398 

.04909 

1-16 

12.7627 

12.962 

35.3429 

99.402 

% 

64.0100 

326.06 

9-32 

.88.^573 

.06213 

% 

12  9591 

13.364 

% 

35.7356 

101.62 

H 

64.4026 

330.06 

5-16 

.981748 

.07670 

3-16 

13.1554 

13.772 

}4 

36.1283 

103.87 

% 

64.7953 

334.10 

11-32 

1.07;)92 

.09281 

34 

13.3518 

14.186 

Yt 

36.5210 

106.14 

% 

65.1880 

338.16 

Vs 

1.17810 

.11045 

5-16 

13.5481 

14.607 

% 

36.9137 

108.43 

Ys 

65.5807 

342.25 

13-32 

1.27627 

.12962 

% 

13.7445 

15.033 

% 

37.3064 

110.75 

21 

05.9734 

346.38 

7-16 

1.37445 

.15033 

7-16 

13.9408 

15.466 

12. 

37.6991 

113.10 

H 

66.3661 

350.50 

15-32 

1.47262 

.17257 

}4 

14.1372 

15.904 

% 

38.0918 

115.47 

H 

66.7588 

354.66 

H 

1.57080 

.19635 

9-16 

14.3335 

16.349 

Yi 

38.4845 

117.86 

% 

67.1515 

358.84 

17-32 

1.66897 

.22166 

% 

14.5299 

16.800 

% 

38.8772 

120.28 

3^ 

67.5442 

363.05 

"9-16 

1.76716 

.24850 

11-16 

14.7262 

17.257 

Ji 

39.2699 

122.72 

% 

67.9369 

367.28 

19-32 

1.86532 

.27688 

H 

14.9226 

17.721 

% 

39.6626 

125.19 

% 

68.3296 

371.54 

% 

1.96350 

.30680 

13-16 

15.1189 

18.190 

% 

40.0553 

127.68 

Y» 

68.7223 

375.88 

21-32 

2.06167 

.33824 

Vs 

15.3153 

18.666 

% 

40.4480 

130.19 

22. 

69.1150 

380.13 

11-16 

2.15984 

.37122 

15-16 

15.5116 

19.147 

13. 

40.8407 

132.73 

% 

69.5077 

384.46 

23-32 

2.25802 

.40574 

5. 

15.7080 

19.635 

H 

41.2334 

135.30 

69.9004 

388.82 

H 

2.35619 

.44179 

i-16 

15.9043 

J0.129 

H 

41.6261 

137.89 

% 

70.2931 

393.20 

25-32 

2.46437 

.47937 

H 

16.1007 

20.629 

% 

42.0188 

140.50 

70.6858 

397.61 

13-16 

2.55254 

.51849 

3-16 

16.2970 

ai.l35 

14 

42.4115 

143.14 

Y% 

71.0785 

402.04 

27-32 

2.65072 

.55914 

}4 

16.4934 

21.648 

42.8042 

145.80 

71.4712 

406.49 

% 

2.74889 

.60132 

5-16 

16.6897 

22.166 

H 

43.1969 

148.49 

Y» 

71.8639 

410.97 

29-32 

2.84707 

.64504 

% 

16.8861 

22.691 

y% 

43.5896 

151.20 

23. 

72.2566 

415.48 

1516 

2.94524 

.69029 

7-16 

17.0824 

23.221 

14. 

43.9823 

153.94 

H 

72.6493 

420.00 

31-32 

3.04342 

.73708 

^ 

17.2788 

23.758 

% 

44.3750 

156.70 

34 

73.0420 

424.56 

1. 

3.14159 

.78540 

9-16 

17.4751 

24.301 

y< 

44.7677 

159.48 

Ys 

73.4347 

429.13 

1-16 

3.33794 

.88664 

% 

17.6715 

24.850 

% 

45.1604 

162.30 

}4 

73.8274 

433.74 

14     3.53429 

.99402 

11-16 

17.8678 

25.406 

H 

45.5531 

165.13 

% 

74.2201 

438.36 

3-16!   3.73064 

1.1075 

H 

18.06  i2 

25.967 

% 

45.9458 

167.99 

74.6128 

443.01 

34  1   3.92699 

1.2272 

13-16 

18.2605 

26.535 

46.3385 

170.87 

Yt 

75.0055 

447.69 

5-16  i   4.12334 

1.3530 

% 

18.4569 

27.109 

Yi 

46.7312 

173.78 

24. 

75.3982 

452.39 

^1   4  31969 

1.4849 

15-16 

18.6532 

27.688 

15. 

47.1239 

176.71 

H 

75.7909 

457.11 

7-16    4.51604 

1.6230 

6. 

18.8496 

28.274 

^ 

47.5166 

179.67 

34 

76.1836 

461.86 

H     4.71239 

1.7671 

H 

19.2423 

29.465 

Y 

47.9093 

182.65 

f» 

76.5763 

466.64 

9-16     4.90874 

1.9175 

y< 

19.6350 

80.680 

% 

48.3020 

185.66 

76.9690 

471.44 

%\   5.10509 

2.0739 

% 

20.0277 

31.919 

Yi. 

48.6947 

188.69 

% 

77.3617 

476.26 

11-16     5.30144 

2.2365 

H 

20.4204 

83.183 

% 

49.0874 

191.75 

% 

77.7544 

481.11 

H'   5.49779 

2.4053 

% 

20.8131 

84.472 

% 

49.4801 

194.83 

Yi 

78.1471 

485.91 

1»-16    5.69414 

2.5802 

% 

21.2058 

35.785 

y» 

49.8728 

197.93 

2b 

78.5398 

490.8T 

y,     5.89049 

2.7612 

y» 

21.5984 

37.122 

16. 

50.2655 

201.06 

H 

78.9325 

495.79 

15-16    6.08684 

2.9483 

7. 

21.9911 

38.485 

H 

50.6582 

204.22 

Q, 

79.3252 

500.74 

2.        i   6.28319 

3.1416 

H 

22.3838 

39.871 

H 

51.0509 

207.39 

% 

79.7179 

505.71 

1-16     6.47953 

3.3410 

Va. 

22.7765 

41.282 

H 

51.4436 

210.60 

H 

80.1106 

510.71 

H     6.67588 

3.5466 

Va 

23.1692 

42.718 

H 

51.8363 

213.82 

Y% 

80.5033 

515.72 

3-16;   6.87223 

3.7583 

y% 

23.5619 

44.179 

H 

52.2290 

217.08 

Vi 

80.8960 

520.77 

J4!   7.06858 

3.9761 

% 

23.9546 

45.664 

H 

52.6217 

220.35 

Yt 

81.2887 

525.84 

5  16     7.26493 

4.2000 

% 

24.3473 

47.173 

% 

53.0144 

223.65 

26.' 

81.6814 

530.93 

%'    7.46128 

4.4301 

% 

24.7400 

48.707 

17. 

53.4071 

226.98 

H 

82.0741 

536.05 

7-16     7.65763* 

4.6664 

8. 

25.1327 

50.265 

H 

53.7998 

230.33 

% 

82.4668 

541.19 

}4\    7.85398 

4.9087 

H 

25.5254 

51.849 

.54.1925 

233.71 

% 

82.8595 

546.35 

9-16     8.05033 

5.1572 

yi 

25  9181 

53.45G 

% 

54.5852 

237.10 

^ 

83.2522 

551.55 

%\   8.24668 

5.4119 

% 

26.3108 

55.088 

M 

54.9779 

240.53 

Yt 

83.6449 

556.76 

11-16     8.44;W3 

5.6727 

}4 

26.7035 

56.745 

% 

55.3706 

243.98 

Y< 

84.0376 

562.00 

^l   8.63938 
IS-lS    8.83573 

5.9396 

27.0962 

58.4?« 

H 

55.7633 

247.45 

Yt 

84.4303 

567.27 

6.2126 

H 

27.4889 

60.132 

y% 

56.1560 

250.95 

27. 

84.8230 

572.56 

%'   9.03208 

6.4918 

% 

27.8816 

61.862 

18. 

56.5487 

254.47 

■  H 

85.2157 

577.87 

15-16    9.22843 

6.7771 

9. 

28.2743 

63.617 

^ 

66.9414 

258.02 

Yi 

85.6084 

583.21 

g.        i   9.42478 

7.0686 

H 

28.6670 

65.397 

34 

57.3341 

261.59 

% 

86.0011 

588.57 

1-16     9.62113 

7.3662 

y* 

29.0597 

67.201 

% 

57.7268 

265.18 

u, 

86  39.38 

593.98 

%\   9.81748 

7.6699 

% 

29.4524 

69.029 

3t 

58.1195 

268.80 

% 

86.7865 

599..37 

3-16  10.0138 

7.9798 

34 

29.8451 

70.882 

% 

58.5122 

272.45 

% 

87.1792 

'504.81 

J4  110.2102 

8.2958 

30.2378 

72.760 

H 

58.9049 

276.12 

% 

87.5719 

610.27 

5-16!  10.4065 

8.6179 

H 

30.6305 

74.662 

% 

59.2976 

279.81 

28 

87.9646 

615.75 

%i  10.6029 

8.9462 

H 

31.0232 

76.589 

19. 

59.6903 

283.53 

% 

88.3573 

621.26 

M6 

10.7992 

9.2806 

10. 

31.4159 

78.540 

H 

60.0830 

287.27 

Y4, 

88.7500 

626.86 

164 


CIRCLES. 


TABIiE  1  OF  CIRCLES— (Continued). 
Diameters  in  units  and  ei^litlis,  Ac. 


Diam. 

Circumf. 

Area. 

Diam. 

Circumf. 

Area. 

Diam. 

Circumf. 

Area. 

Diam. 

Circumf. 

Aretu 

asrs 

89.1427 

632.36 

38. 

119.381 

1134.1 

47^ 

149.618 

1781.4 

57^ 

179.856 

2574.1 

H 

89.5354 

637.94 

H 

119.773 

1141.6 

H 

150.011 

1790.8 

% 

180.249 

2585.4 

% 

89.9281 

643.55 

H 

120.166 

1149:1 

Vb 

150.404 

1800.1 

}i 

180.642 

2596.7 

H 

90.3208 

649.18 

% 

120.559 

1156.6 

48. 

150.796 

1809.6 

.H 

181.034 

3608.0 

% 

90.7135 

654.84 

H 

120.951 

1164.2 

H 

151.189 

1819.0 

% 

181.427 

2619.4 

29. 

91.1062 

660.52 

% 

121.344 

1171.7 

151.582 

1828.5 

y% 

181.820 

2630.7 

H 

91.4989 

666.23 

% 

121.737 

1179.3 

% 

151.975 

1837.9 

58. 

182.212 

2642.1 

91.8916 

671.96 

% 

122.129 

1186.9 

Ml 

152.367 

1847.5 

H 

182.605 

2653.5 

y» 

92.2843 

677.71 

39. 

122.522 

1194.6 

H 

152.760 

1857.0 

Ya 

182.998 

2664.9 

14 

92.6770 

683.49 

H 

122.915 

1202.3 

% 

153.153 

1866.5 

% 

183.390 

2676.4 

% 

93.0697 

689.30 

123.308 

1210.0 

y% 

153.545 

1876.1 

M 

183.783 

2687.8 

H 

93.4624 

695.13 

% 

123.700 

1217.7 

49. 

153.938 

1885.7 

V% 

184.176 

2699.3 

% 

93.8551 

700.98 

)4 

124.093 

1225.4 

H 

154.331 

1895.4 

H 

184.569 

2710.9 

SO. 

94.2478 

706.86 

% 

124.486 

1233.2 

H 

154.723 

1905.0 

% 

184.961 

2722.4 

H 

94.6405 

712.76 

% 

124.878 

1241.0 

% 

155.116 

1914.7 

59 

185.354 

2734.0 

H 

95.0332 

718.69 

y» 

125.271 

1248.8 

H 

155.509 

1924.4 

H 

185.747 

2745.6 

% 

95.4259 

724.64 

40. 

125.664 

1256.6 

H 

155.902 

1934.2 

h 

186.139 

2757.2 

95.8186 

730.62 

H 

126.056 

1264.5 

H 

156.294 

1943.9 

% 

186.532 

2768.8 

% 

96.2113 

736.62 

126.449 

1272.4 

% 

156.687 

1953.7 

14 

186.925 

2780.5 

% 

96.6040, 

742.64 

% 

126.842 

1280.3 

50." 

157.080 

1963.5 

% 

187.317 

2792.2 

^"^ 

96.9967 

748.69 

H 

127.235 

1288.2 

H 

157.472 

1973.3 

% 

187.710 

2803.9 

M. 

97.3894 

754.77 

% 

127.627 

1296.2 

157.865 

1983.2 

% 

188.103 

2815.7 

% 

97.7821 

760.87 

% 

128.020 

1304.2 

% 

158.258 

1993.1 

60. 

188.496 

2827.4 

98.1748 

766.99 

% 

128.413 

1312.2 

14 

158.650 

2003.0 

% 

188.888 

2839.2 

% 

98.5675 

773.14 

41. 

128.805 

'  1320.3 

% 

159.043 

2012.9 

Ya 

189.281 

2851.0 

14 

98.9602 

779.31 

H 

129.198 

1328.3 

% 

159.436 

2022.8 

% 

189.674 

2862.9 

% 

99.3529 

785.51 

129.591 

1336.4 

K 

•  159.829 

2032.8 

y. 

190.066 

2874.8 

% 

99.7456 

791.73 

% 

129.983 

1344.5 

51. 

160.221 

2042.8 

% 

190.459 

2886.6 

% 

100.138 

797.98 

^ 

130.376 

1352.7 

H 

160.614 

2052.8 

% 

190.852 

2898.6 

B2 

100.531 

804.25 

% 

130.769 

1360.8 

M 

161.007 

2062.9 

K 

191.244 

2910.5 

H 

100.924 

810.54 

% 

131.161 

1369.0 

% 

161.399 

2078.0 

61. 

191.637 

2922.5 

101.316 

816.86 

% 

131.554 

1377.2 

}4 

161.792 

2083.1 

% 

192.0.50 

2934.5 

% 

101.709 

823.21 

42 

131.947 

1385.4 

% 

162.185 

2093.2 

H 

192.423 

2946.5 

H 

102.102 

829.58 

M 

132.340 

1393.7 

H 

162.577 

2103.3 

Va 

192.815 

2958.5 

102.494 

835.97 

132.732 

1402.0 

% 

162.970 

2113.5 

}4 

193.208 

2970.6 

8/ 

102.887 

842.39 

y 

133.125 

1410.3 

52. 

163.363 

2123.7 

% 

193.601 

2982.7 

yi 

103.280 

848.83 

14 

133.518 

1418.6 

^ 

163.756 

2133.9 

% 

193.993 

2994.8 

18. 

103.673 

855.30 

133.910 

1427.0 

164.148 

2144.2 

% 

194.386 

3006.9 

H 

104.065 

861.79 

H 

134.303 

1435.4 

% 

164.541 

2154.5 

62. 

194.779 

3019.1 

H 

104.458 

868.31 

% 

134.696 

1443.8 

% 

164.934 

2164.8 

"A 

195.171 

3031.3 

% 

104.851 

874.85 

43. 

135.088 

1452.2 

% 

165.326 

2175.1 

% 

195.564 

3043.5 

14 

105.243 

881.41 

H 

135.481 

1460.7 

H 

165.719 

2185.4 

% 

195.957 

3055.7 

% 

105.636 

888.00 

135.874 

1469,1 

% 

166.112 

2195.8 

% 

196.350 

3068.0 

Vi 

106.029 

894.62 

y 

136.267 

1477.6 

53. 

166.504 

2206.2 

% 

196.742 

3080.8 

y» 

106.421 

901.26 

rz 

136.659 

1486.2 

H 

166.897 

2216.8 

% 

197.135 

3092.6 

14. 

106.814 

907.92 

% 

137.052 
1.37.445 

1494.7 

y* 

167.290 

2227.0 

% 

197.528 

3104.9 

H 

107.207 

914.61 

% 

1503.3 

% 

167.683 

2237.5 

63. 

197.920 

3117.2 

107.600 

921.32 

y% 

137.837 

1511.9 

}4 

168.075 

2248.0 

% 

198.313 

312«.« 

% 

107.992 

928.06 

44. 

1.38.230 

1520.5 

% 

168.468 

2258.5 

Ya 

198.706 

3142.0 

/^ 

108.385 

934.82 

M 

138.623 

1529.2 

% 

168.861 

2269.1 

% 

199.098 

3154.fi 

% 

108.778 

941.61 

M 

139.015 

1537.9 

% 

169.253 

2279.6 

M 

199.491 

3166.t 

% 

109.170 

948.42 

% 

139.4C8 

1546.6 

54. 

169.646 

2290.2 

% 

199.884 

3179.4* 

H 

109.563 

955.25 

3^ 

139.801 

1555.3 

% 

170.039 

230<3.8 

% 

200.277 

3191.9 

ii. 

109.956 

962.11 

^ 

140.194 

1564.0 

170.431 

2311.5 

% 

200.669 

3204.4 

H 

110,348 

969.00 

% 

140.586 

1572.8 

% 

170.824 

2322.1 

64. 

201.062 

3217.0 

110.741 

975.91 

% 

140.979 

1581.6 

H 

171.217 

2332.8 

H 

201.455 

3229.6 

% 

111.134 

982.84 

45. 

141.372 

1590.4 

% 

171.609 

2343.5 

Ya 

201.847 

3242.2 

K 

111.527 

989.80 

M 

141.764 

1599.3 

% 

172.002 

2354.3 

% 

202.240 

3254.8 

H 

111.919 

996.78 

142.157 

1608.2 

y% 

172..395 

2365.0 

K 

202.633 

3267.5 

a/ 

112.312 

1003.8 

y 

142.550 

1617.0 

65. 

172.788 

2375.8 

% 

203.025 

3280.1 

r/ 

112.705 

1010.8 

14 

142.942 

1626.0 

H 

173.180 

2386.6 

% 

203.418 

3202.8 

S6. 

113.097 

1017.9 

% 

143.335 

1634.9 

M 

173.573 

2397.5 

% 

203.811 

3305.6 

H 

113.490 

1025.0 

H 

143.728 

1643.9 

% 

173.966 

2408.3 

65. 

204.204 

3318.3 

H 

113.883 

1032.1 

% 

144.121 

1652.9 

H 

174.358 

2419.2 

H 

204.596 

3331.1 

% 

114.275 

1039.2 

46.' 

144.513 

1661.9 

% 

174.751 

2430.1 

204.989 

3343.9 

H 

114.668 

1046.3 

H 

144.906 

1670.9 

% 

175.144 

2441.1 

y» 

205.382 

3356.7 

H 

115.061 

1053.6 

H 

145.299 

1680.0 

K 

175.536 

2452.0 

Y2 

205.774 

3369.6 

S 

115.454 

1060.7 

% 

145.691 

1689.1 

56. 

175.929 

2463.0 

% 

206.167 

3882.4 

% 

115.846 

1068.0 

A 

146.084 

1698.2 

H 

176.322 

2474.0 

% 

206.560 

3396.8 

n. 

116.239 

1075.2 

% 

146.477 

1707.4 

H 

176.715 

2485.0 

% 

206.952 

3406.2 

H 

116.632 

1082.5 

% 

146.869 

1716.5 

% 

177.107 

2496.1 

66. 

207.345 

3421.2 

^ 

117.024 

1089.8 

% 

147.262 

1725.7 

% 

177.500 

2507.2 

% 

207.738 

3484.2 

% 

117.417 

1097.1 

47 

147.655 

1734.9 

% 

177.893 

2518.3 

u 

208.131 

.3447.2 

H 

117.810 

1104.5 

% 

148.048 

1744.2 

H 

178.285 

2529.4 

% 

208.523 

3460.2 

118.202 

1111.8 

148.440 

1753.5 

% 

178.678 

2540.6 

^ 

208.916 

3473.2 

9i 

118.596 

1119.2 

% 

lis.ass 

1762.7 

57. 

179.071 

2551.8 

% 

209.309 

3486.3 

H 

118.988 

U26.7 

H 

149.226 

1772.1 

H 

179.463 

2568.0 

H 

209.701 

3499.4 

CIRCIiBS. 


165 


TABIiE  1  OF  CIRCIiES— (Continued). 
Diameters  in  units  and  eigbtlis,  Ae, 


Mam. 

Ciraamf. 

Area. 

Diam. 

Circumf. 

Area. 

Olam. 

Circamf. 

Area. 

Diam. 

Gireumf. 

ArM. 

.,.'* 

210.094 

3512.5 

7534 

236.405 

4447.4 

83)^ 

262.716 

5492.4 

92. 

289.027 

6647.« 

210.487 

3525.7 

H 

236.798 

4462.2 

% 

268.108 

5508.8 

yi 

289.419 

6666.7 

H 

210.879 

3538.8 

4 

237.190 

4477.0 

Y» 

263.501 

5525.3 

yi 

289.812 

6683.8 

}4 

211.272 

3552.0 

% 

237.583 

4491.8 

84. 

263.894 

5541.8 

% 

290.205 

6701.9 

211.665 

3565.2 

H 

237.976 

4506.7 

H 

264.286 

5558.3 

yi 

290.597 

6720.1 

212.058 

3578.5 

% 

238.368 

4521.5 

Y* 

264.679 

5574.8 

Ys 

290.990 

6738.2 

g 

212.450 

3591.7 

76. 

238.761 

4536.5 

Yt 

265.072 

5591.4 

Yi 

291.383 

6756.4 

212.843 

3605.0 

yi 

239.154 

4551.4 

yi 

265.465 

5607.9 

Ys 

291.775 

6774.7 

K 

213.236 

3618.3 

yi 

239.546 

4566.4 

% 

265.857 

5624.5 

93. 

292.168 

6792.9 

68. 

213.628 

3631.7 

% 

239.939 

4581.3 

% 

266.250 

5641.2 

H 

292.561 

6811.1 

H 

214.021 

3645.0 

yi 

240.332 

4596.3 

% 

266.643 

5657.8 

H 

292.954 

6829.5 

% 

214.414 

3658.4 

% 

240.725 

4611.4 

85. 

267.035 

5674.5 

Ys 

293.346 

6847.8 

214.806 

3671.8 

H 

241.117 

4626.4 

yi 

267.428 

5691.2 

yi 

293.739 

6866.1 

IZ 

215.199 

3685.3 

% 

241.510 

4641.5 

yi 

267.821 

5707.9 

Ys 

294.132 

6884.5 

\i 

215.592 

3698.7 

77. 

241.903 

4656.6 

% 

268.213 

5724.7 

Yi 

294.524 

6902.9 

H 

215.984 

3712.2 

yi 

242.295 

4671.8 

^ 

268.606 

5741.5 

Ys 

294.917 

6921.3 

•u 

216.377 

3725.7 

yi 

242.688 

4686.9 

% 

268.999 

5758.3 

94. 

295.310 

6939.8 

69. 

216.770 

37.S9.3 

•% 

243.081 

4702.1 

H 

269.392 

5775.1 

yi 

295.702 

6958.2 

% 

217.163 

3752.8 

1^1    243.473 

4717.3 

% 

269.784 

5791.9 

Yi 

296.095 

6976.7 

yi 

217.555 

3766.4 

%\    243.866 

4732.5 

86. 

270.177 

5808.8 

Ys 

296.488 

6995.3 

% 

217.948 

3780.0 

«^l    244.259 

4747.8 

H 

270.570 

5825«7 

Yl 

296.881 

7013.8 

yi 

218.341 

3793.7 

%     244.652 

4763.1 

yi 

270.962 

5842.6 

Ys 

297.273 

7032.4 

218.733 

3807.3 

78.          245.044 

4778.4 

Ys 

271.355 

5859.6 

% 

297.666 

7051.0 

H 

219.126 

3821.0 

yi     245.437 

4793.7 

yi 

271.748 

5876.5 

Ys 

298.059 

7069.6 

% 

219.519 

3834.7 

}4\    245.830 

4809.0 

% 

.272.140 

5893.5 

95. 

298.451 

7088.2 

70. 

219.911 

3848.5 

%     246.222 

4824.4 

H 

272.533 

5910.6 

yi 

298.844 

7106.9 

yi 

220.304 

3862.2 

y^ 

246.615 

4839.8 

Y» 

272.926 

5927.6 

Yi 

299.237 

7125.6 

220.697 

3876.0 

% 

247.008 

4855.2 

87. 

273.319 

5&44.7 

■      Ys 

299.629 

7144.? 

% 

22-1.090 

3888.8 

247.400 

4870.7 

yi 

273.711 

5961.8 

yi 

300.022 

7163.0 

yi 

221.482 

3903.6 

•u 

247.793 

4886.2 

274.104 

5978.9 

Ys 

300.415 

7181.e 

% 

221.875 

3917.5 

79. 

248.186 

4901.7 

% 

274.497 

5996.0 

Yi 

300.807 

7200.6 

% 

222.268 

3931.4 

H 

248.579 

4917.2 

y^ 

274.889 

6013.2 

Ys 

301.200 

7219.4 

% 

222.660 

3945.3 

yi 

248.971 

4932.7 

H 

275.282 

6030.4 

96. 

301.593 

7238.2 

n. 

223.053 

3959.2 

% 

249.364 

4948.3 

r/ 

275.675 

6047.6 

M 

301.986 

7257.1 

H 

223.446 

3973.1 

yi 

249.757 

4963.9 

% 

276.067 

6064.9 

Yi 

302.378 

7276  0 

223.838 

3987.1 

% 

250.149 

4979.5 

88. 

276.460 

6082.1 

Ys 

302.771 

7294.9 

y 

224.231 

4001.1 

H 

250.542 

4995.2 

yi 

276.853 

6099.4 

¥> 

303.164 

7313.8 

yi 

224.624 

4015.2 

y% 

250.935 

5010.9 

yi 

277.246 

6116.7 

% 

303.556 

7332.8 

% 

225.017 

4029.2 

80. 

251.ar27 

5026.5 

Ys 

277.638 

6134.1 

H 

303.949 

7351.8 

% 

225.409 

4043.3 

%\    251.720 

5042.3 

yi 

278.031 

6151.4 

Ys 

304.342 

7370.8 

% 

225.802 

4057.4 

.%\    252.113 

5058.0 

Ys 

278.424 

6168.8 

97. 

304.734 

7389.8 

t2. 

226.195 

4071.5 

Yi'    252.506 

5073.8 

H 

278.816 

6186.2 

yi 

305.127 

7408.9 

yi 

226.587 

4085.7 

yi\    252.898 

5089.6 

Ys 

279.209 

6203.7 

Yi 

305.520 

7428. C 

226.980 

4099.8 

%\    253.291 

5105.4 

89. 

■   279.602 

6221.1 

Ys 

305.913 

7447.1 

^ 

227.373 

4114.0 

%■    253.684 

5121.2 

yi 

279.994 

6238.6 

y^ 

306.305 

7466.2 

227.765 

4128.2 

%!    254.076 

5137.1 

yi 

280.387 

6256.1 

Ys 

306.698 

7485.3 

^ 

228.158 

4142.5 

81.          254.469 

5153.0 

Ys 

280.780 

6273.7 

Yi 

307.091 

7504.8 

^ 

228.551 

4156.8 

1^1    254.862 

5168.9 

yi 

281.173 

6291.2 

Ys 

307.483 

7523.7 

% 

228.944 

4171.1 

M:    255.204 

5184.9 

Ys 

281.565 

6308.8 

98. 

307.876 

7543.C 

78. 

229.336 

4185.4 

Yi     255.647 

5200.8 

H 

281.958 

6326.4 

yi 

308.269 

7562.2 

yi 

229.729 

4199.7 

y^\    256.040 

5216.8 

Ys 

282.351 

6344.1 

Yi 

308.661 

7581  .S 

230.122 

42141 

%^    256.438 

5232.8 

90. 

282.743 

6361.7 

Ys 

309.054 

7600.8 

% 

230.514 

4228.5 

%.    256.825 

5248.9 

yi 

283.136 

6379.4 

yi 

309.447 

7620.1 

H 

230.907 

4242.9 

%     257.218 

5264.9 

yi 

283.529 

6397.1 

Ys 

309.840 

7639.5 

% 

231.300 

4257.4 

82.          257.611 

5281.0 

% 

283.921 

6414.9 

Yi 

310.232 

7658.S 

% 

231.692 

4271.8 

yi     258.003 

5297.1 

y^ 

284.314 

6432.6 

Ys 

310.625 

7678.3 

% 

232.085 

4286.3 

H     258.396 

5313.3 

284.707 

6450.4 

99. 

311.018 

7697.7 

74. 

232.478 

4300.8 

J/g!    258.789 

5329.4 

%/ 

285.100 

6468.2 

yi 

311.410 

7717.1 

yi 

232.871 

4315.4 

yi 

259.181 

5345.6 

yi 

285.492 

6486.0 

Yi 

311.803 

7736.6 

23S263 

4329.9 

% 

259.574 

5361.8 

91. 

285.885 

6503.9 

% 

312.196 

7756.1 

% 

233.656 

4344.5 

Va 

259.967 

5378.1 

yi 

286.278 

16521.8 

yi 

312.588 

7775.6 

H 

•«4.049 

4359.2 

Yt 

260.359 

5394.3 

M 

286.679 

6539.7 

Ys 

312.981 

7795.2 

% 

234.441 

4373.8 

83. 

260.752 

5410.6 

Ys 

287.063 

6557.6 

Yi 

313.374 

7814.8 

% 

234.834 

4388.5 

•  H 

261.145 

5426.9 

yi 

287.456 

6575.5 

Ys 

313.767 

7834.4 

X 

235.227 

4403.1 

Yi 

261.538 

5443.3 

% 

287.848 

6593.5 

100. 

314.159 

785*.(i 

76. 

235.619 

4417.9 

yt 

261.930 

5459.6 

H 

288.241 

6611.5 

H 

236.012 

4432.6 

yi 

262.323 

5476.0 

}s 

288.634 

6629.6 

166 


CIRCLES. 


TABIii:  3  OF  CIRCIiHS. 
Diameters  in  units  and  tenths. 


Dia. 

Circumf. 

Area. 

Dia. 

Circumf. 

Are*. 

Dia. 

Circumf. 

Area. 

0.1 

.314159 

.007854 

6.3 

19.79203 

31.17245 

12.5 

39.26991 

122.7185 

.2 

.628319 

.031416 

.4 

20.10619 

32.16991 

.6 

39.58407 

124.6898 

.3 

.942478 

.070686 

.5 

20.42035 

33.18307 

.7 

39.89823 

126.6769 

.4 

1.256637 

.125664 

.6 

20.73451 

34.21194 

.8 

40.21239 

128.6796 

.5 

1.570796 

.196350 

.7 

21.04867 

35.25652 

.9 

40.52655 

130.6981 

.6 

1.884956 

.282743 

.8 

21.36283 

36.31681 

13.0 

40.84070 

132.7323 

.7 

2.199115 

.384845 

.9 

21.67699 

37.39281 

.1 

41.15486 

134.7822 

.8 

2.513274 

.502655 

7.0 

21.99115 

38.48451 

o 

41.46902 

136.8478 

.9 

2.827433 

.636173 

.1 

22.30531 

39.59192 

'.3 

41.78318 

138.9291 

1.0 

3.141593 

.785398 

.2 

22.61947 

40.71504 

A 

42.09734 

141.0261 

.1 

3.455752 

.950332 

.3 

22.93363 

41.85387 

.5 

42.41150 

143.1388 

.2 

3.769911 

1.13097 

.4 

23.24779 

43.00840 

.6 

42.72566 

145.2672 

^ 

4.084070 

1.32732 

.5 

23.56194 

44.17865 

.7 

43.03982 

147.4114 

.4 

4.398230 

1.53938 

.6 

23.87610 

45.36460 

.8 

43.35398 

149.5712 

.5 

4.712389 

1.76715 

.7 

24.19026 

46.56626 

.9 

43.66814 

151.7468 

.6 

5.026548 

2.01062 

.8 

24.50442 

47.78362 

14.0 

43.98230 

153.9380 

.7 

5.340708 

2.26980 
2.5^69 

.9 

24.81858 

49.01670 

.1 

44.29646 

156.1450 

.8 

5.654867 

8.0 

25.13274 

50.26548 

.2 

44.61062 

158.3677 

.9 

5.969026 

2.83529 

.1 

25.44690 

51.52997 

.3 

44.92477 

160.6061 

2.0 

6.283185 

3.14159 

.2 

25.76106 

52.81017 

.4 

45.23893 

162.8602 

.1 

6.597345 

3.46361 

.3 

•26.07522 

64.10608 

.5 

45.55309 

165.1300 

^ 

6.911504 

3.80133 

.4 

26.38938 

55.41769 

.6 

45.86725 

167.4155 

.3 

7.225663 

4.15476 

.5 

26.70354 

56.74502 

.7 

46.18141 

169.7167 

.4 

7.539822 

4.52389 

.6 

27.01770 

58.08805 

.8 

46.49557 

172.0336 

.5 

7.853982 

4.90874 

.7 

27.33186 

59.44679 

.9 

46.80973 

174.3662 

.6 

8.168141 

5.30929 

.8 

27.64602 

60.82123 

15.0 

47.12389 

176.7146 

.7 

8.482300 

5.72555 

.9 

27.96017 

62.21139 

.1 

47.43805 

179.0786 

.8 

8.796159 

6.15752 

9.0 

28.27433 

63.61725 

.2 

47.75221 

181.4584 

.9 

9.110619 

6.60520 

.1 

28.58849 

65.03882 

.3 

48.06637 

183.8539 

3.0 

9.424778 

7.06858 

.2 

28.90265 

66.47610 

.4 

48.38053 

186.2650 

.1 

9.738937 

7.54768  • 

.3 

29.21681 

.5 

48.69469 

188.6919 

.2 

10.05310 

8.04248 

.4 

29.53097 

69^39^78 

.6 

49.00885 

191.1345 

.3 

10.36726 

8.55299 

.5 

29.81513 

70.88218 

.7 

49.32300 

193.5928 

.4 

10.68142 

9.07920 

.6 

30.15929 

72.38229 

.8 

49.63716 

196.0668 

.5 

10.99557 

9.62113 

.7 

30.47345 

73.89811 

.9 

49.95132 

198.5565 

,6 

11.30973 

10.17876 

.8 

30.78761 

75.42964 

16.0 

50.26548 

201.0619 

.7 

11.62389 

10.75210 

.9 

31.10177 

76.97687 

.1 

50.57964 

203.5831 

.8 

11.93805 

11.34115 

10.0 

31.41593 

78.53982 

.2 

50.89380 

206.1199 

.•9 

12.25221 

11.94591 

.1 

31.73009 

80.11847 

.3 

51.20796 

208.6724 

4.0 

12.56637 

12.56637 

.2 

32.04425 

81.71282 

.4 

51.52212 

211.2407 

.1 

12.88053 

13.20254 

.3 

32.35840 

83.32289 

.5 

51.83628 

213.8246 

.2 

13.19469 

13.85442 

.4 

32.67256 

84.94867 

.6 

52.15044 

216.4243 

.3 

13.50885 

14.52201 

.5 

32.98672 

86.59015 

.7 

52.46460 

219.0397 

.4 

13.82301 

15.20531 

.6 

33.30088 

88.24734 

.8 

52.77876 

221.6708 

.5 

14.13717 

15.90431 

.7 

33.61504 

89.92024 

.9 

53.09292 

224.3176 

.6 

14.45133 

16.61903 

.8 

33.92920 

91.60884 

17.0 

53.40708 

226.9801 

.7 

14.76549 

17.34945 

.9 

34.24336 

93.31316 

.1 

53.72123 

229.6583 

.8 

15.07964 

18.09557 

11.0 

34.55752 

95.03318 

.2 

54.03539 

232.3522 

.9 

15.39380 

18.85741 

.1 

34.87168 

96.76891 

.3 

54.34955 

235.0618 

5.0 

15.70796 

19.63495 

.2 

35.18584 

98.52035 

.4 

54.66371 

237.7871 

.1 

16.02212 

20.42821 

.3 

35.50000 

100.2875 

.5 

54.97787 

240.5282 

.2 

16.33628 

21.23717 

.4 

35.81416 

102.0703 

.6 

55.29203 

243.2849 

.3 

16.65044 

22.06183 

.5 

36.12832 

103.8689 

.7 

55.60619 

246.0574 

.4 

16.96460 

22.90221 

.6 

36.44247 

105.6832 

.8 

55.92035 

248.8456 

.5 

17.27876 

23.75829 

.7 

36.75663 

107.5132 

.9 

56.23451 

251.6494 

.6 

17.59292 

24.63009 

.8 

37.07079 

109.3588 

18.0 

56.54867 

254.4690 

.7 

17.90708 

25.51759 

.9 

37.38495 

111.2202 

.1 

56.86283 

257.3043 

.8 

18.22124 

26.42079 

12.0 

37.69911 

113.0973 

.2 

57.17699 

260.1553 

.9 

18.53540 

27.33971 

.1 

38.01327 

114.9901 

.3 

57.49115 

263.0220 

6.0 

18.84956 

28.27433 

.2 

38.32743 

116.8987 

.4 

57.80530 

265.9044 

.1 

19.16372 

29.22467 

.3 

38.64159 

118.8229 

.5 

58.11946 

268.8025 

.2 

19.47787 

30.19071 

.4 

38.95575 

120.7628 

.6 

58.43362 

271.7163 

CIRCLES. 


167 


TABIii:  3  OF  CIRCIiES—CContinaed). 
]>iaineters  in  units  and  tenths. 


Dia. 

Circumf. 

Area. 

Dia. 

Circumf. 

Area. 

Dia. 

Circumf. 

Area. 

18.7 

68.74778 

274.6459 

24.9 

78.22566 

486.9547 

31.1 

97.70353 

759.6450 

.8 

59.06194 

277.5911 

25.0 

78.53982 

490.8739 

2 

98.01769 

764.5380 

.9 

59.37610 

280.5521 

.1 

78.85398 

494.8087 

!3 

98.331&3 

769.4467 

19.0 

59.69026 

283.5287 

.2 

79.16813 

498.7592 

.4 

98.64601 

774.3712 

.1 

60.00442 

286.5211 

.3 

79.48229 

502.7255 

.5 

98.96017 

779.3113 

2 

60.31858 

289.5292 

.4 

79.79645 

506.7075 

.6 

99.27433 

784.2672 

!3 

60.^3274 

292.5530 

.5 

80.11061 

510.7052 

.7 

99.58849 

789.2388 

.4 

60.94690 

295.5925 

.6 

80.42477 

514.7185 

.8 

99.90265 

794.2260 

.5 

61.26106 

298.6477 

.7 

80.73893 

518.7476 

.9 

100.2168 

799.2290 

.6 

61.57522 

301.7186 

.8 

81.05309 

522.7924- 

32.0 

100.5310 

804.2477 

.7 

61.88938 

304.8052 

.9 

81.36725 

526.8529 

.1 

100.8451 

809.2821 

.8 

62.20353 

307.9075 

26.0 

81.68141 

530.9292 

.2 

101.1593 

814.3322 

.9 

62.51769 

311.0255 

.1 

81.99557 

535.0211 

.3 

101.4734 

819.3980 

20.0 

62.83185 

314.1593 

.2 

82.30973 

539.1287 

.4 

101.7876 

824.4796 

.1 

63.14601 

317.3087 

.3 

82.62389 

543.2521 

.5 

102.1018 

829.5768 

.2 

63.46017 

320.4739 

.4 

82.93805 

547.3911 

.6 

102.4159 

834.6898 

.3 

63.77433 

323.6547 

.5 

83.25221 

551.5459 

.7 

102.7301 

839.8184 

.4 

64.08849 

326.8513 

.6 

83.56636 

555.7163 

.8 

103.0442 

844.9628 

.5 

64.40265 

330.0636 

.7 

83.88052 

559.9025 

.9 

103.3584 

850.1228 

.6 

64.71681 

333.2916 

.8 

84.19468 

564.1044 

33.0 

103.6726 

855.2986 

.7 

65.03097 

336.5353 

.9 

84.50884 

568.3220 

.1 

103.9867 

860.4901 

.8 

65.34513 

339.7947 

27.0 

84.82300 

572.5553 

.2 

104.3009 

865.6973 

.9 

65.65929 

343.0698 

.1 

85.13716 

576.8043 

.3 

104.6150 

870.9202 

21.0 

65.97345 

346.3606 

.2 

85.45132 

581.0690 

.4 

104.9292 

876.1588 

.1 

66.28760 

349.6671 

.3 

85.76548 

585.3494 

.5 

105.2434 

881.4131 

.2 

66.60176 

352.9894 

.4 

86.07964 

589.6455 

.6 

105.5575 

886.6831 

.3 

66.91592 

356.3273 

.5 

86.39380 

593.9574 

.7 

105.8717 

891.9688 

.4 

67.23008 

359.6809 

.6 

86.70796 

598.2849 

.8 

106.1858 

897.2703 

.5 

67.54424 

363.0503 

.7 

87.02212 

602.6282 

.9 

106.5000 

902.5874 

.6 

67.85810 

366.4354 

.8 

87.33628 

606.9871 

34.0 

106.8142 

907.9203 

.7 

68.17256 

369.8361 

.9 

87.65044 

611.3618 

.1 

107.1283 

913.2688 

.8 

68.48672 

373.2526 

28.0 

87.96459 

615.7522 

.2 

107.4425 

918.6331 

.9 

68.80088 

376.6848 

.1 

88.27875 

620.1582 

.3 

107.7566 

924.0131 

22.0 

69.11504 

380.1327 

.2 

88.59291 

624.5800 

.4 

108.0708 

929.4088 

.1 

69.42920 

383.5963 

.3 

88.90707 

629.0175 

.5 

108.3849 

934.8202 

.2 

69.74336 

387.0756 

.4 

89.22123 

633.4707 

.6 

108.6991 

940.2473 

.3 

70.05752 

390.5707 

.5 

89.53539 

637.9397 

.7 

109.0133 

945.6901 

.4 

70.37168 

394.0814 

.6 

89.84955 

642.4243 

.8 

109.3274 

951.1486 

.5 

70.68583 

397.6078 

.7 

90.16371 

646.9246 

.9 

109.6416 

956.6228 

.6 

70.99999 

401.1500 

.8 

90.47787 

651.4407 

36.0 

109.9557 

962.1128 

.7 

71.31415 

404.7078 

.9 

90.79203 

655.9724 

-.1 

110.2699 

967.6184 

.8 

71.62831 

408.2814 

29.0 

91.10619 

660.5199 

.2 

110.5841 

973.1397 

.9 

71.94247 

411.8707 

.1 

91.42035 

665.0830 

.3 

110.8982 

978.6768 

28.0 

72.25663 

415.4756 

.2 

91.73451 

669.6619 

.4 

111.2124 

984.2296 

.1 

72.57079 

419.0963 

.3 

92.04866 

674.2565 

.5 

111.5265 

989.7980 

.2 

72.8&495 
73.19911 

422.7327 

.4 

92.36282 

678.8668 

.6 

111.8407 

995.3822 

.3 

426..3848 

.5 

92.67698 

683.4928 

.7 

112.1549 

1000.9821 

.4 

73.51327 

430.0526 

.6 

92.99114 

688.1345 

.8 

112.4690 

1006.5977 

.5 

73.82743 

433.7361 

.7 

93.30530 

692.7919 

.9 

112.7832 

1012.2290 

.6 

74.14159 

437.4354 

.8 

93.61946 

697.4650 

36.0 

113.0973 

1017.8760 

.7 

74.45575 

441.1503 

.9 

93.93362 

702.1538 

.1 

113.4115 

1023.5387 

.8 

74.76991 

444.8809 

30.0 

94.24778 

706.8583 

.2 

113.7257 

1029.2172 

.9 

75.08406 

448.6273 

.1 

94.56194 

711.5786 

.3 

114.0398 

1034.9113 

24.0 

75.89822 

452.3893 

.2 

94.87610 

716.3145 

.4 

114.3540 

1040.6212 

.1 

75.71238 

456.1671 

.3 

95.19026 

721.0662 

.5 

114.6681 

1046.3467 

.2 

76.02654 

459.9606 

.4 

95.50442 

725.8336 

.6 

114.9823 

1052.0880 

.3 

76.34070 

463.7698 

.5 

95.81858 

730.6166 

.7 

115.2965 

1057.8449 

.4 

76.65486 

467.5947 

.6 

96.13274 

735.4154 

.8 

115.6106 

1063.6176 

.5 

76.96902 

471.4352 

.7 

96.44689 

740.2299 

.9 

115.9248 

1069.4060 

.6 

77.28318 

475.2916 

.8 

96.76105 

745.0601 

37.0 

116.2389 

1075.2101 

,7 

77.59734 

479.1636 

.9 

97.07521 

749.9060 

.1 

116.5531 

1081.0299 

.8 

77.91150 

483.0513 

31.0 

97.38937 

754.7676 

.2 

116.8672 

1086.8654 

168 


CIRCLES. 


TABIiE  3  OF  CIRCI.es— (Continued). 
I>iaiiieters  in  units  and  tenths. 


Oia. 

Circunif. 

Area. 

Dia. 

Circumf. 

Area. 

Dia. 

Circumf. 

Area. 

37.3 

117.1814 

1092.7166 

43.5 

136.6593 

1486.1697 

49.7 

156.1372 

1940.0041 

.4 

117.4956 

1098.5835 

.6 

136.9734 

1493.0105 

.8 

156.4513 

1947.8189 

.5 

117.8097 

1104.4662 

.7 

137.-2876 

1499.8670 

.9 

156.7655 

1955.6493 

.6 

118.1239 

1110.3645 

.8 

137.6018 

1506.7393 

50.0 

157.0796 

1963.4954 

.7 

118.4380 

1116.2786 

.9 

137.9159 

1513.6272 

.1 

157.3938 

1971.3572 

.8 

118.7522 

1122.2083 

44.0 

138.2301 

1520.5308 

.2 

157.7080 

1979.2348 

.9 

119.0664 

1128.1538 

.1 

138.5442 

1527.4502 

.3 

158.0221 

1987.1280 

38.0 

119.3805 

1134.1149 

.2 

138.8584 

1534.3853 

.4 

158.3363 

1995.0370 

.1 

119.6947 

1140.0918 

.3 

139.1726 

1541.3360 

.5 

158.6504 

2002.9617 

.2 

120.0088 

1146.0844 

.4 

139.4867 

1548.3025 

.6 

158.9646 

2010.9020 

.3 

120.3230 

1152.0927 

.5 

139.8009 

1555.2847 

.7 

159.2787 

2018.8581 

.4 

120.6372 

1158.1167 

.6 

140.1150 

1562.2826 

.8 

159.5929 

2026.8299 

.5 

120.9513 

1164.1564 

.7 

140.4292 

1569.2962 

.9 

159.9071 

2034.8174 

.6 

121.2655 

1170.2118 

.8 

140.7434 

1576.3255 

61.0 

160.2212 

2042.8206 

.7 

121.5796 

1176.2&30 

.9 

141.0575 

1583.3706 

.1 

160.5354 

2050.8395 

.8 

121.8938 

1182.3698 

45.0 

141.3717 

1590.4313 

.2 

160.8495 

2058.8742 

.9 

122.2080 

1188.4724 

.1 

141.6858 

1597.5077 

.3 

161.1637 

2066.9245 

39.0 

122.5221 

1194.5906 

.2 

142.0000 

1604.5999 

A 

161.4779 

2074.9905 

.1 

122.8363 

1200.7246 

.3 

142.3141 

1611.7077 

.5 

161.7920 

2083.0723 

.2 

123.1504 

1206.8742 

.4 

142.6283 

1618.8313 

.6 

162.1062 

2091.1697 

.3 

123.4646 

1213.0396 

.5 

142.9425 

1625.9705 

.7 

162.4203 

2099.2829 

.4 

123.7788 

1219.2207 

.6 

143.2566 

1633.1255 

.8 

162.7345 

2107.4118 

.5 

124.0929 

1225.4175 

.7 

143.5708 

1640.2962 

.9 

163.0487 

2115.5563 

.6 

124.4071 

1231.6300 

.8 

143.8849 

1647.4826 

52.0 

163.3628 

2123.7166 

.7 

124.7212 

1237.8582 

.9 

144.1991 

1654.6847 

.1 

163.6770 

2131.8926 

.8 

125.0354 

1244.1021 

46.0 

144.5133 

1661.9025 

.2 

163.9911 

2140.0843 

.9 

125.3495 

1250.3617 

.1 

144.8274 

1669.1360 

.3 

164.3053 

2148.2917 

40.0 

125.6637 

1256.6371 

.2 

145.1416 

1676.3853 

.4 

164.6195 

2156.5149 

.1 

125.9779 

1262.9281 

.3 

145.4557 

1683.6502 

.5 

164.9336 

2164.7537 

.2 

126.2920 

1269.2348 

.4 

145.7699 

1690.9308 

.6 

165.2478 

2173.0082 

.3 

126.6062 

1275.5573 

.5 

146.0841 

1698.2272 

.7 

165.5619 

2181.2785 

.4 

126.9203 

1281.8955 

.6 

146.3982 

1705.5392 

.8 

165.8761 

2189.5644 

.5 

127.2345 

1288.2493 

.7 

146.7124 

1712.8670 

.9 

166.1903 

2197.8661 

.6 

127.5487 

1294.6189 

.8 

147.0265 

1720.2105 

53.0 

166.5044 

2206.1834 

.7 

127.8628 

1301.0042 

.9 

147.3407 

1727.5697 

.1 

166.8186 

2214.5165 

.8 

128.1770 

1307.4052 

47.0 

147.6549 

1734.9445 

.2 

167.1327 

2222.8653 

.9 

128.4911 

1313.8219 

.1 

147.9690 

1742.3351 

.3 

167.4469 

2231.2298 

41.0 

128.8053 

1320.2543 

.2 

148.2832 

1749.7414 

.4 

167.7610 

2239.6100 

.1 

129.1195 

1326.7024 

.3 

148.5973 

1757.1635 

.5 

168.0752 

2248.0059 

.2 

129.4336 

1333.1663 

.4 

148.9115 

1764.6012 

.6 

168.3894 

2256.4175 

.3 

129.7478 

1339.6458 

.5 

149.2257 

1772.0546 

.7 

168.7035 

2264.8448 

A 

130.0619 

1346.1410 

.6 

149.5398 

1779.5237 

.8 

169.0177 

2273.2873 

.5 

130.3761 

1352.6520 

.7 

149.8540 

1787.0086 

.9 

169.3318 

2281.7466 

„6 

130.6903 

13.59.1786 

.8 

150.1681 

1794.5091 

54.0 

169.6460 

2290.2210 

.7 

131.0044 

1365.7210 

•  .9 

150.4823 

1802.0254 

.1 

169.9602 

2298.7112 

.8 

131.3186 

1372.2791 

48.0 

150.7964 

1809.5574 

.2 

170.2743 

2307.2171 

.9 

131.6327 

1378.8529 

.1 

151.1106 

1817.1050 

.3 

170.5885 

2315.7386 

42.0 

131.9469 

1385.4424 

.2 

151.4248 

1824.6684 

.4 

170.9026 

2324.2759 

.1 

132.2611 

1392.0476 

.3 

151.7389 

1832.2475 

.5 

171.2168 

2332.8289 

.2 

132.5752 

1398.6685 

.4 

152.0531 

1839.8423 

.6 

171.5310 

2341.3976 

.3 

132.8894 

1405.3051 

.5 

152.3672 

1847.4528 

.7 

171.8451 

2349.9820 

.4 

133.2035 

1411.9574 

.6 

152.6814 

1855.0790 

.8 

172.1593 

2358.5821 

.5 

133.5177 

1418.6254 

.7 

152.9956 

1862.7210 

.9 

172.4734 

2367.1979 

.6 

133.8318 

1425.3092 

.8 

153.3097 

1870.3786 

55.0 

172.7876 

2375.8294 

.7 

134.1460 

1432.0086 

.9 

153.6239 

1878.0519 

.1 

173.1018 

2384.4767 

.8 

134.4602 

1438.7238 

49.0 

153.9380 

1885.7410 

.2 

173.4159 

2393.1396 

.9 

134.7743 

1445.4546 

.1 

154.2522 

1893.4457 

.3 

173.7301 

2401.8183 

48.0 

135.0885 

1452.2012 

.2 

154.5664 

1901.1662 

.4 

174.0442 

2410.5126 

.1 

135.4026 

1458.9635 

.3 

154.8805 

1908.9024 

.5 

174.3584 

2419,2227 

.2 

135.7168 

1465.7415 

.4 

155.1947 

1916.6543 

.6 

174.6726 

2427.9485 

.3 

1:36.0310 

1472.5352 

.5 

155.5088 

1924.4218 

.7 

174.9867 

24:^.6899 

.4 

136.3451 

1479.3446 

.6 

155.8230 

1932.2051 

.8 

175.3009 

2445.4471 

CIRCLES. 


169 


TABIiE  2  OF  CIRCXES— (Continued). 
JDiameters  in  units  and  tentlis. 


Dia. 

Circumf. 

Area. 

Dia. 

Circumf. 

Area. 

Dia. 

Circumf. 

Area. 

o5.9 

175.6150 

2454.2200 

62.1 

195.0929 

3028.8173 

68.3 

214.5708 

3663.7960 

66.0 

175.9292 

2463.0086 

.2 

195.4071 

3038.5798 

.4 

214.8849 

3674.5324 

.1 

176.2433 

2471.8130 

.3 

195.7212 

3048.3580 

.5 

215.1991 

3685.2845 

.2 

176.5575 

2180.6330 

.4 

196.0354 

3058.1520 

.6 

215.5133 

3696.0523 

.3 

176.8717 

2489.4687 

.5 

196.13495 

3067.9616 

.7 

215.8274 

3706.8359 

.4 

177.1858 

2498.3201 

.6 

196.6637 

3077.7869 

.8 

216.1416 

3717.6351 

.5 

177.5000 

2507.1873 

.7 

196.9779 

3087.6279 

.9 

216.4557 

3728.4500 

.6 

177.8141 

2516.0701 

.8 

197.2920 

3097.4847 

69.0 

216.7699 

3739.2807 

.7 

178.1283 

2524.9687 

.9 

197.6062 

3107.3571 

.1 

217.0841 

3750.1270 

.8 

178.4425 

2533.8830 

63.0 

197.9203 

3117.2453 

.2 

217.3982 

3760.9891 

.9 

178.7566 

2542.8129 

.1 

198.2345 

3127.1492 

.3 

217.7124 

3771.8668 

57.0 

179.0708 

2551.7586 

.2 

198.5487 

3137.0688 

.4 

218.0265 

3782.7603 

.1 

179.3849 

2560.7200 

.3 

198.8628 

3147.0040 

.5 

218.3407 

3793.6695 

.2 

179.6991 

2569.6971 

.4 

199.1770 

3156.9550 

.6 

218.6548 

3804.5944 

.3 

180.0133 

2578.6899 

.5 

199.4911 

3166.9217 

.7 

218.9690 

3815.5350 

.4 

180.3274 

2587.6985 

.6 

199.8053 

3176.9042 

.8 

219.2832 

3826.4913 

.5 

180.6416 

2596.7227 

•  .7 

200:1195 

3186.9023 

.9 

219.5973 

3837.4633 

.6 

180.9557 

2605.7626 

.8 

200.4336 

3196.9161 

70.0 

219.9115 

3848.4510 

.7 

181.2699 

2614.8183 

.9 

200.7478 

3206.9456 

.1 

220.2256 

3859.4544 

.8 

181.5841 

2623.8896 

64.0 

201.0619 

3216.9909 

.2 

220.5398 

3870.4736 

.9 

181.8982 

2«32.9767 

.1 

201.37(a 

3227.0518 

.3 

220.8540 

3881.5084 

68.0 

182.2124 

2642.0794 

.2 

201.6902 

3237.1285 

.4 

221.1681 

3892.5590 

.1 

182.5265 

2651.1979 

.3 

202.(X)44 

3247.2209 

•  .5 

221.4823 

3903.6252 

.2 

182.8407 

2660.3321 

.4 

202.3186 

3257.3289 

.6 

221.7964 

3914.7072 

.3 

183.1549 

2669.4820 

.5 

202.6327 

3267.4527 

.7 

222.1106 

3925.8049 

.4 

183.4690 

2678.6476 

.6 

2(J2.9469 

3277.5922 

.8 

222.4248 

3936.9182 

.5 

183.7832 

2687.8289 

.7 

203.2610 

3287.7474 

.9 

222.7389 

3948.0473 

.6 

184.0973 

2697.0259 

.8 

203.5752 

3297.9183 

71.0 

223.0531 

3959.1921 

.7 

181.4115 

2706.2386 

.9 

203.8894 

3308.1049 

.1 

223.3672 

3970.3526 

.8 

184.7256 

2715.4670 

65.0 

204.2035 

3318.3072 

.2 

223.6814 

3981.5289 

.9 

185.0398 

2724.7112 

.1 

204.5177 

3328.5253 

.3 

223.9956 

3992.7208 

59.0 

185.3540 

2733.9710 

.2 

204.8318 

3338.7590 

.4 

224.3097 

4003.9284 

.1 

185.6681 

2743.2466 

.3 

205.1460 

3349.0085 

.5 

224.6239 

4015.1518 

.2 

185.9823 

2752.5378 

.4 

205.4602 

3359.2736 

.6 

224.9380 

4026.3908 

.3 

186.2964 

2761.8448 

.5 

205.7743 

3369.5545 

.7 

225.2522 

4037.6456 

.4 

186.6106 

2771.1675 

.6 

206.0885 

3379.8510 

.8 

225.5664 

4048.9160 

.5 

186.9248 

2780.5058 

.7 

206.4026 

3390.1633 

.9 

225.8805 

4060.2022 

.6 

187.2389 

2789.8599 

.8 

206.7168 

3400.4913 

72.0 

226.1947 

4071.5U41 

.7 

187.5531 

2799.2297 

.9 

207.0310 

3410.8350 

.1 

226.5088 

4082.8217 

.8 

187.8672 

2808.6152 

66.0. 

207.3451 

3421.1944 

.2 

226.8230 

4094.1550 

.9 

188.1814 

2818.0165 

.1 

207.659S 

3431.5695 

.3 

227.1371 

4105.5040 

•0.0 

188.4956 

2827.4334 

.2 

207.9734 

3441.9603 

.4 

227.4513 

4116.8687 

.1 

188.8097 

2836.8660 

.3 

208.2876 

3452.3669 

.5 

227.7655 

4128.2491 

.2 

189.1239 

2846.3144' 

.4 

208.6018 

3462.7891 

.6 

228.0796 

4139.6452 

.3 

189.4380 

2855.7784 

.5 

208.9159 

3473.2270 

.7 

228.3938 

4151.0571 

.4 

189.7522 

2865.2582 

.6 

209.2301 

3483.6807 

.8 

228.7079 

4162.4846 

.5 

190.0664 

2874.7536 

.7 

209.5442 

3494.1500 

.9 

229.0221 

4173.9279 

.6 

190.3805 

2884.2648 

.8 

209.8584 

3504.6351 

73.0 

229.3363 

4185.3868 

.7 

190.6947 

2893.7917 

.9 

210.1725 

3515.1359 

.1 

229.6504 

4196.8615 

.8 

191.0088 

2903.3343 

67.0 

210.4867 

3525.6524 

*.2 

229.9646 

4208.3519 

.9 

191.3230 

2912.8926 

.1 

210.8009 

3536.1845 

.3 

230.2787 

4219.8579 

61.0 

191.6372 

2922.4666 

.2 

211.1150 

3546.7324 

.4 

230.5929 

4231.3797 

.1 

191.9513 

2932.0563 

.3 

211.4292 

3557.2960 

.5 

230.9071 

4242.9172 

.2 

192.2655 

2941.6617 

.4 

211.7433 

3567.8754 

.6 

231.2212 

4254.4704 

,3 

192.5796 

2951.2828 

.5 

212.0575 

3578.4704 

.7 

231.5354 

4266.0394 

.4 

192.8938 

2960.9197 

.6 

212..3717 

3589.0811 

.8 

231.8495 

4277.6240 

.5 

193.2079 

2970.5722 

.7 

212.6858 

3^599.7075 

.9 

232.1637 

4289.2243 

.6 

193.5221 

2980.2405 

.8 

213.0000 

3610.3497 

74.0 

232.4779 

4300.8403 

.7 

193.8363 

2989.9244 

.9 

213.3141 

3621.0075 

.1 

2.32.7920 

4312.4721 

.8 

194.1504 

2999.6241 

68.0 

213.6283 

3631.6811 

.2 

233.1062 

4324.1195 

.9 

194.4646 

3009.3395 

.1 

213.9425 

3642.3704 

.3 

233.4203 

4335.7827 

62.0 

194.7787 

3019.0705 

.2 

214.2566 

3653.0754 

.4 

233.7345 

4347.4616 

170 


CIRCLK8. 


TABIiE  2  OF  ClRCIiES— (Continued). 
]>iaineters  in  units  and  tenttis> 


Bia. 

Circumf. 

Area. 

Dia. 

Circumf. 

Area. 

Dia. 

Circumf. 

Area. 

74.5 

234.0487 

4359.1562 

80.7 

253.5265 

5114.8977 

86.9 

273.0044 

5931.0206 

.6 

234.3628 

4370.8664 

.8 

■  253.8407 

5127.5819 

87.0 

273.3186 

5944.6787 

.7 

234.6770 

4382.5924 

.9 

254.1548 

5140.2818 

.1 

273.6327 

5958.3525 

.8 

2S4.9911 

4394.3341 

81.0 

254.4690 

5152.9974 

-2 

273.9469 

5972.0420 

.9 

235.3053 

4406.0916 

.1 

254.7832 

5165.7287 

.3 

274.2610 

5985.7472 

75.0 

235.6194 

4417.8647 

.2 

255.0973 

5178.4757 

.4 

274.5752 

5999.4681 

.1 

235.9336 

4429.6535 

.3 

255.4115 

5191.2384 

.5 

274.8894 

6013.2047 

.2 

236.2478 

4441.4580 

.4 

255.7256 

5204.0168 

.6 

275.2035 

6026.9570 

.3 

236.5619 

4453.2783 

.5 

256.0398 

5216.8110 

.7 

275.5177 

6040.7250 

.4 

236.8761 

4465.1142 

.6 

256.3540 

5229.6208 

.8 

275.8318 

6054.5088 

.5 

237.1902 

4476.9659 

.7 

256.6681 

5242.4463 

.9 

276.1460 

6068.3082 

.6 

237.5044 

4488.8332 

.8 

256.9823 

5255.2876 

88.0 

276.4602 

6082.1234 

.7 

237.8186 

4500.7163 

.9 

257.2964 

5268.1446 

.1 

276.7743 

6095.9542 

.8 

238.1327 

4512.6151 

82.0 

257.6106 

5281.0173 

.2 

277.0885 

6109.8008 

.9 

238.4469 

4524.5296 

.1 

257.9248 

5293.9056 

.3 

277.4026 

6123.6631 

16.0 

238.7610 

4536.4598 

.2 

258.2389 

5306.8097 

.4 

277.7168 

6137.5411 

.1 

239.0752 

4548.4057 

.3 

258.5531 

5319.7295 

.5 

278.0309 

6151.4348 

.2 

239.3894 

4560.3673 

.4 

258.8672 

5332.6650 

.6 

278.3451 

6165.3442 

.3 

239.7035 

4572.3446 

.5 

259.1814 

5345.6162 

.7 

278.6593 

6179.2693 

.4 

240.0177 

4584.3377 

.6 

259.4956 

5358.5832 

.8 

278.9734 

6193.2101 

.5 

240.3318 

4596.3464 

.7 

259.8097 

5371.5658 

.9 

^79.2876 

6207.1666 

.6 

240.6460 

4608.3708 

.8 

260.1239 

5384.5641 

89.0 

279.6017 

6221.1389 

.7 

240.9602 

4620.4110 

.9 

260.4380 

5397.5782 

.1 

279.9159 

6235.1268 

.8 

241.2743 

4632.4669 

88.0 

260.7522 

5410.6079 

.2 

280.2301 

6249.1304 

.9 

241.5885 

4644.5384 

.1 

261.0663 

5423.6534 

.3 

280.5442 

6263.1498 

77.0 

241.9026 

4656.6257 

.2 

261.3805 

5436.7146 

.4 

280.8584 

6277.1849 

.1 

242.2168 

4668.7287 

.3 

261.6^7 

5449.7915 

.5 

281.1725 

6291.2356 

.2 

242.531C 

4680.8474 

.4 

262.0088 

5462.8840 

.6 

281.4867 

6305.3021 

.3 

242.8451 

4692.9818 

.5 

262.3230 

5475.9923 

.7 

281.8009 

6319.3843 

.4 

243.1593 

4705.1319 

.6 

262.6371 

5489.1163 

.8 

282.1150 

6333.4822 

.5 

243.4734 

4717.2977 

.7 

262.9513 

5502.2561 

.9 

282.4292 

6347.5958 

.6 

243.7876 

4729.4792 

.8 

263.2655 

5515.4115 

90.0 

282.7433 

6361.7251 

.7 

244.1017 

4741.6765 

.9 

263.5796 

5528.5826 

.1 

283.0575 

6375.8701 

.8 

244.4159 

4753.8894 

84.0 

263.8938 

5541.7694 

.2 

283.3717 

6390.0309 

.9 

244.7301 

4766.1181 

.1 

264.2079 

5554.9720 

.3 

283.6858 

6404.2073 

18.0 

245.0442 

4778.3624 

.2 

264.5221 

5568.1902 

.4 

284.0000 

6418.3995 

.1 

245.3584 

4790.6225 

.8 

264.8363 

5581.4242 

.5 

284.3141 

6432.6073 

.2 

245.6725 

4802.8983 

.4 

265.1504 

5594.6739 

.6 

284.6283 

6446.8309 

.3 

245.9867 

4815.1897 

.5 

265.4646 

5607.9392 

.7 

284.9425 

6461.0701 

.4 

246.3009 

4827.4969 

.6 

265.7787 

5621.2203 

.8 

285.2566 

6475.3251 

.5 

246.6150 

4839.8198 

.7 

266.0929 

5634.5171 

.9 

285.5708 

6489.5958 

.6 

246.9292 

4852.1584 

.8 

266.4071 

5647.8296 

91.0 

285.8849 

6503.8822 

.7 

247.243? 

4864.5128 

.9 

266.7212 

5661.1578 

.1 

286.1991 

6518.1843 

.8 

247.5575 

4876.8828 

85.0 

267  0354 

5674.5017 

.2 

286.5133 

6532.5021 

Jd 

247.8717 

4889.2685 

.1 

267  3495 

5687.8614 

.3 

286.8274 

6546.8356 

79.0 

248.1858 

4901.6699 

.2 

267  6637 

5701.2367 

.4 

287.1416 

6561.1848 

.1 

248.5000 

4914.0871 

.3 

267.9779 

5714.6277 

.5 

287.4557 

6575.5498 

.2 

248.8141 

4926.5199 

.4 

268.2920 

5728.0345 

.6 

287.7699 

6589.9304 

.3 

249.1283 

4938.9685 

.5 

268.6062 

5741.4569 

.7 

288.0840 

6604.3268 

.4 

249.4425 

4951.43^8 

.6 

268.9203 

5754.8951 

.8 

288.3982 

6618.7388 

.5 

249.7566 

4963.9127 

.7 

269.2345 

5768.3490 

.9 

288.7124 

6633.1666 

.6 

250.0708 

4976.4084 

.8 

269.5486 

5781.8185 

92.0 

289.0265 

6647.6101 

.7 

250.3849 

4988.9198 

.9 

269:8628 

5795.3038 

.1 

289.3407 

6662.0692 

.8 

250.6991 

5001.4469 

86.0 

270.1770 

5808.8048 

.2 

289.6548 

6676.5441 

.9 

251.0133 

5013.9897 

.1 

270.4911 

5822.3215 

.3 

289.9690 

6691.0347 

8«.0 

251.3274 

5026.5482 

.2 

270.8053 

5835.8539 

.4 

290.2832 

6705.5410 

.1 

251.6416 

5039  1??5 

.3 

271.1194 

5849.4020 

.5 

290.5973 

6720.0630 

.2 

251.9557 

5051.7124 

.4 

271.4386 

5862.9659 

.6 

290.9115 

6734.6008 

.3 

252.2699 

5064.3180 

.5 

271.7478 

5876.5454 

.7 

291.2256 

6749.1542 

.4 

252.5840 

5076.9394 

.6 

272.0619 

5890.1407 

.8 

291.5398 

6763.7233 

.5 

252.8982 

5089.5764 

.7 

272.3761 

5903.7516 

.9 

291.8540 

6778.3082 

.6 

253.2124 

5102.2292 

.8 

272.6902 

5917.3783 

93.0 

292.1681 

6792.9087 

CIRCLES. 


171 


TABIiE  2  OF  CIRCIiES— (Continued). 
Diameters  in  units  and  tentbs. 


Ma. 

Circumf. 

Area. 

Dia. 

Circumf. 

Area. 

Dia. 

Circumf. 

Area. 

9B.1 

292.4828 

6807.5250 

95.5 

300.0221 

7163.0276 

97.8 

307.2478 

7512.2078 

.2 

292.7964 

6822.1569 

.6 

300.:^63 

7178.0366 

.9 

307.5619 

7527.5780 

.3 

293.1106 

6836.8046 

.7 

300.6504 

7193.0612 

98.0 

307.8761 

7542.9640 

.4 

293.4248 

6851.4680 

.8 

300.9646 

7208.1016 

.1 

308.1902 

7558.8656 

.5 

293.7389 

6866.1471 

.9 

301.2787 

7223.1577 

.2 

308.5044 

7573.7830 

.6 

294.0531 

6880.8419 

9ft.0 

301.5929 

7238.2295 

.3 

308.8186 

7589.2161 

.7 

294.3672 

6895.5524 

.1 

301.9071 

7253.3J70 

.4 

309.1327 

7604.6648 

.8 

294.6814 

6910.2786 

.2 

302.2212 

7208.4202 

.o 

309.4469 

7620.1293 

.9 

294.9956 

6925.0205 

.3 

302.5354 

7283.5391 

.6 

309.7610 

7635.6095 

94.0 

295.3097 

6939.7782 

.4 

302.8495 

7298.6737 

.7 

310.0752 

7651.1054 

.1 

295.6239 

6954.5515 

.0 

303.1637 

7313.8240 

.8 

310.3894 

7666.6170 

.2 

295.9380 

6969.3406 

.6 

303.4779 

7328.9901 

.9 

310.7035 

7682.1444 

.3 

296.2522 

6984.1453 

.7 

303.7920 

7344.1718 

99.0 

311.0177 

7697.6874 

.4 

296.5663 

6998.9658 

.8 

304.1062 

7359.3693 

.1 

314.3318 

7713.2461 

.5 

296.8805 

7013.8019 

.9 

304.4203 

7374.5824 

.2 

311.6460 

7728.8206 

.6 

297.1947 

7028.6538 

97.0 

304.7345 

7389.8113 

.3 

311.9602 

7744.4107 

.7 

297.5088 

7043.5214 

.1 

305.0486 

7405.0559 

.4 

312.2743 

7760.0166 

.8 

297.8230 

7058.4047 

.2 

305.3628 

7420.3162 

.5 

312.5885 

7775.6382 

.9 

298.1371 

7073.3037 

.3 

305.6770 

7435.5922 

.6 

312.9026 

7791.2754 

#6.0 

298.4513 

7088.2184 

.4 

305.9911 

7450.8839 

.7 

313.2168 

7806.9284 

.1 

298.7655 

7103.1488 

.5 

306.3053 

7466.1913 

.8 

313.5309 

7822.5971 

.2 

299.0796 

7118.0950 

.6 

306.6194 

7481.5144 

.9 

313.8451 

7838.2815 

.3 

299.3938 

7133.0568 

.7 

306.9336 

7496.a532 

100.0 

314.1593 

7853.9816 

A 

299.7079 

7148.0343 

Circumferences  when  the  diameter  has  more  than  one 
place  of  decimals. 


Diam. 

Circ. 

Diam. 

Circ. 

1 
Diam. 

Circ. 

Diam. 

Circ. 

Diam. 

Circ. 

.1 

.314159 

.01 

.031416 

.001 

.003142 

.0001 

.000314 

.00001 

.000031 

.2 

.628319 

.02 

.062832 

.002 

.006283 

.0002 

.000628 

.00U02 

.000063 

.3 

.942478 

.03 

.094248 

.003 

.009425 

.0003 

.000942 

.00003 

.000094 

.1 

1.256637 

.04 

.125664 

.004 

.012566 

.0004 

.001257 

.00004 

.000126 

.5 

1.570796 

.05 

.157080 

.005 

.015708 

.0005 

.001571 

.00005 

.000157 

.6 

1.884956 

.06 

.188496 

..006 

.018850 

.0006 

.001885 

.00006 

.000188 

.7 

2.199115 

.07 

.219911 

.007 

.021991 

.0007 

.002199 

.00007 

.000220 

.8 

2.513274 

.08 

.251327 

.008 

.025133 

.0008 

.002513 

.00008 

.000251 

.9 

2.827433 

.09 

.282743 

.009 

.028274 

.0009 

.002827 

.00009 

.000283 

Diameter  =  3.12699 
Circumference  = 

Circ  for  dia  of        3.1    • 
•*  .02 

"  .006 

***  .0009 

«  .00009 


Examples. 


Sum  of 
■■  9.738937 
:  .062832 
:  .018850 
:  .002827 
:     .000283 


9.823729 


Circumfce  =» 
Diameter  = 

Dia  for  circ  of 


9.823729 
9.738937  ■■ 


Sum  oi 
3.1 


.062832  =     .02 

.021960" 

.018850  =,     .006 

.003110 

.002827  =     .0009 


.000283 

.000283  »     .00009 


172 


CIRCLES. 


TABI.E  3  OF   CIRCIiES. 
Diams  in  units  and  twelfths;  as  in  feet  and  inebes. 


Dia. 

Circumf. 

Area, 

Dia.  Circumf. 

1 

Area. 

Dia. 

Circumf. 

Area. 

Pt.In. 

Feet. 

Sq.  ft. 

Ft.In. 

Feet. 

Sq.  ft. 

Ft.In. 

Feet. 

Sq.ft. 

5  0 

15.70796 

19.63495 

10  0 

31.41593 

78.53982 

0  1 

.261799 

.005454 

1 

15.96976 

20.29491 

1 

31.bV//3 

79.85427 

2 

.523599 

.021817 

2 

16.23156 

20.96577 

2 

31.93953 

81.17968 

3 

.785398 

.049087 

3  1  16.49336 

21.64754 

3 

32.20132 

82.51589 

4 

1.047198 

.087266 

4 

16.75516 

22.34021 

4 

32.46312 

83.86307 

5 

1.308997 

.136354 

5 

17.01696 

23.04380 

5 

32.72492 

85.22115 

6 

1.570796 

.196350 

6 

17.27876 

23.75829 

6 

32.98672 

86.59015 

7 

1.832596 

.267254 

7 

17.54056 

24.48370 

7 

33.24852 

87.97005 

8 

2.094395 

.349066 

8 

17.80236 

25.22001 

8 

33.51032 

89.36086 

9 

2.356195 

•441786 

9 

18.06416 

25.96723 

9 

33.77212 

90.76258 

10 

2.617994 

.545415 

10 

18.32596 

26.72535 

10 

34.03392 

92.17520 

11 

2.879793 

.659953 

11 

18.58776 

27.49439 

11 

34.29572 

93.59874 

1  0 

3.14159 

.785398 

6  0 

18.84956 

28.27433 

11  0 

34.55752 

95.03318 

1 

3.40339 

.921752 

1 

19.11136 

29.06519 

1 

34.81932 

96.47858 

2 

3.66519 

1.06901 

2 

19.37315 

29.86695 

2 

35.08112 

97.93479 

3 

3.92699 

1.22718 

3 

19.63495 

30.67962 

3 

35.34292 

99.40196 

4 

4.18879 

1.39626 

4 

19.89675 

31.50319 

4 

35.60472 

100.8800 

5 

4.45059 

1.57625 

5 

20.15855 

32.33768 

5 

35.86652 

102.3690 

6 

4.71239 

1.76715 

6 

20.42035 

33.18307 

6 

36.12832 

103.8689 

7 

4.97419 

1.96895 

7 

20.68215 

34.03937 

7 

36.39011 

105.3797 

8 

5.23599 

2.18166 

8 

20.94395 

34.90659 

8 

36.65191 

106.9014 

9 

5.49779 

2.40528 

9 

21.20575 

35.78470 

9 

36.91371 

108.4340 

10 

6.76959 

2.63981 

10 

21.46755 

36.67373 

10 

37.17551 

109.9776 

11 

6.02139 

2.88525 

11 

21.72935 

37.57367 

11 

37.43731 

111.5320 

S  0 

6.28319 

3.14159 

7  0 

21.99115 

38.48451 

12  0 

37.69911 

113.0973 

1 

6.54496 

3.40885 

1 

22.25295 

39.40626 

1 

37.96091 

114.6736 

2 

6.80678 

3.68701 

2 

22.51475 

*  40.33892 

2 

38.22271 

116.2607 

3 

7.06858 

3.97608 

3 

22.77655 

41.28249 

3 

38.48451 

117.8588 

4 

7.33038 

4.27606 

4 

23.03835 

42.23697 

4 

38.74631 

119.4678 

5 

7.59218 

4.58694 

5 

2S.30015 

43.20235 

5 

39.00811 

121.0877 

6 

7.85398 

4.90874 

6 

23.56194 

44.17865 

6 

39.26991 

122.7185 

7 

8.11578 

5.24144 

7 

23.82374 

45.16585 

7 

39.53171 

124.3602 

8 

8.37758 

5.58505 

8 

24.08554 

46.16396 

8 

39.79351 

126.0128 

9 

8.63938 

5.93957 

9 

24.34734 

47.17298 

9 

40.05531 

127.6763 

10 

8.90118 

6.30500 

10 

24.60914 

48.19290 

10 

40.31711 

129.3507 

11 

9.16298 

6.68134 

11 

24.87094 

49.22374 

11 

40.57891 

131.0360 

8  0 

9.42478 

7.06858 

8  0 

25.13274 

50.26548 

13  0 

40.84070 

132.7323 

1 

9.68658 

7.46674 

1 

25.39454 

51.31813 

1 

41.10250 

134.4394 

2 

9.94838 

7.87580 

2 

25.65634 

52.38169 

2 

41.36430 

136.1575 

3 

10.21018 

8.29577 

3 

25.91814 

53.45616 

3 

41.62610 

137.8865 

4 

10.47198 

8.72665 

4 

26.17994 

54.54154 

4 

41.88790 

139.6263 

5 

10.73377 

9.16843 

5 

26.44174 

55.63782 

5 

42.14970 

141.3771 

6 

10.99557 

9.62113 

6 

26.70354 

56.74502 

6 

42.41150 

143.1388 

7 

11.25737 

10.08473 

7 

26.96534 

57.86312 

7 

42.67330 

144.9114 

8 

11.51917 

10.55924 

8 

27.22714 

58.99213 

8 

42.93510 

146.6949 

9 

11.78097 

11.04466 

9 

27.48894 

60.13205 

9 

43.19690 

148.4893 

10 

12.04277 

11.54099 

10 

27.75074 

61.28287 

10 

43.45870 

150.2947 

11 

12.30457 

12.04823 

11 

28.01253 

62.44461 

11 

43.72050 

152.1109 

4  0 

12.56637 

12.56637 

9  0 

28.27433 

63.61725 

14  0 

43.98230 

153.938a 

1 

12.82817 

13.09542 

1 

28.53613 

64.80080 

1 

44.24410 

155.7761 

2 

13.08997 

13.63538 

2 

28.79793 

65.99526 

2 

44.50590 

157.6250 

3 

13.35177 

14.18625 

3 

29.05973 

67.20063 

3 

44.76770 

159.4849 

4 

13.61357 

14.74803 

4 

29.32153 

68.41691 

4 

45.02949 

161.3557 

5 

13.87537 

15.32072 

5 

29.58333 

69.64409 

5 

45.29129 

163.2374 

6 

14.13717 

15.90431 

6 

29.84513 

70.88218 

6 

45.55309 

165.1301 

7 

14.39897 

16.49882 

7 

30.10693 

72.13119 

7 

45.81489 

167.0335 

8 

14.66077 

17.10423 

8 

30.36873 

73.39110 

8 

46.07669 

168.9479 

9 

14.92257 

17.72055 

9 

30.63053 

74.66191 

9 

46.33849 

170.8732 

10 

15.18436 

18.34777 

10 

30.89233 

75.94364 

10 

46.60029 

172.8094 

U 

15.44616 

18.98591 

11 

31.15413 

77.23627 

11 

46.86209 

174.7565 

CIRCLES. 


173 


Bia 


TABIii:  3  OF  CIRCI.es— (Continued). 
\  in  units  and  twelfths;  as  in  feet  and  incbes. 


m. 

Circumf. 

Area. 

Dia. 

Circumf. 

Area. 

Dia. 

Circumf. 

Area. 

Pt.In. 

Feet. 

Sq.  ft. 

Ft.In. 

Feet. 

Sq.  ft. 

Ft.Iu. 

Feet. 

Sq.  ft. 

[6  0 

47.12389 

176.7146 

20  0 

62.83185 

314.1593 

25  0 

78.53982 

490.8739 

1 

47.38569 

178.6835 

1 

63.09365 

316.7827 

1 

78.80162 

494.1518 

2 

47.64749 

180.6634 

2 

63.35545 

319.4171 

2 

79.06342 

497.4407 

3 

47.90929 

182.6542 

3 

63.61725 

322.0623 

3 

79.32521 

500.7404 

4 

48.17109 

184.6558 

4 

63.87905 

324.7185 

4 

79.58701 

504.0511 

5 

48.43289 

186.6684 

5 

64.14085 

327.3856 

5 

79.84881 

507.3727 

6 

48.69469 

188.6919 

6 

64.40265 

330.0636 

6 

80.11061 

510.7052 

7 

48.95649 

190.7263 

7 

64.66445 

332.7525 

7 

80.37241 

514.0486 

S 

49.21828 

192.7716 

8 

64.92625 

335.4523 

8 

80.63421 

517.4029 

9 

49.48008 

194.8278 

9 

65.18805 

338.1630 

9 

80.89601 

520.7681 

10 

49.74188 

196.8950 

10 

65.44985 

340.8846 

10 

81.15781 

524.1442 

11 

50.00368 

198.9730 

11 

65.71165 

343.6172 

11 

81.41961 

527.5312 

L«  0 

50.26548 

201.0619 

21  0 

65.97345 

346.3606 

26  0 

81.68141 

530.9292 

1 

50.52728 

203.1618 

1 

66.23525 

349.1149 

1 

81.94321 

534.3380 

2 

50.78908 

205.2725 

2 

66.49704 

351.8802 

2 

82.20501 

537.7578 

3 

51.05088 

207.3942 

3 

66.75884 

354.6564 

3 

82.46681 

541.1884 

4 

51.31268 

209.5268 

4 

67.02064 

357.4434 

4  1  82.72861 

544.6300 

5 

51.57448 

211.6703 

5 

67.28244 

360.2414 

5 

82.99041 

548.0825 

6 

51.83628 

213.8246 

6 

67.54424 

363.0503 

6 

83.25221 

551.5459 

7 

52.09808 

215.9899 

7 

67.80604 

365.8701 

7 

83.51400 

555.0202 

8 

52.35988 

218.1662 

8 

68.06784 

368.7008 

8 

83.77580 

558.5054 

9 

52.62168 

220.3533 

9 

68.32964 

371.5424 

9 

84.03760 

562.0015 

10 

52.88348 

222.5513 

10 

68.59144 

374.3949 

10 

84.29940 

565.5085 

11 

53.14528 

224.7602 

11 

68.85324 

377.2584 

11 

84.56120 

569.0264 

17  0 

53.40708 

226.9801 

22  0 

69.11504 

380.1327 

27  0 

84.82300 

572.5553 

1 

53.66887 

229.2108 

1 

69.37684 

383.0180 

1 

85.08480 

576.0950 

2 

53.93067 

231.4525 

2 

69.63864 

385.9141 

2 

85.34060 

579.6457 

3 

54.19247 

233.7050 

3 

69.90044 

388.8212 

3 

85.60840 

583.2072 

4 

54.45427 

235.9085 

4 

70.16224 

391.7392 

4 

85.87020 

586.7797 

5 

54.71607 

238.2429 

5 

70.42404 

394.6680 

5 

86.13200 

590.3631 

6 

54.97787 

240.5282 

6 

70.68583 

397.6078 

6 

86.39380 

593.9574 

7 

55.23967 

242.8244 

7 

70.94763 

400.5585 

7 

86.65560 

597.5626 

8 

55.50147 

245.1315 

8 

71.20943 

408.5201 

8 

86.91740 

601.1787 

9 

55.76327 

247.4495 

9 

71.47123 

406.4926 

9 

87.17920 

604.8057 

10 

56.02507 

249.7784 

10 

71.73303 

409.4761 

10 

87.44100 

608.4436 

11 

56.28687 

252.1183 

11 

71.99483 

412.4704 

11 

87.70279 

612.0924 

L8  0 

56.54867 

254.4690 

28  0 

72.25663 

415.4756 

28  0 

87.96459 

615.7522 

1 

56.81047 

256.8307 

1 

72.51843 

418.4918 

1 

88.22639 

619.4228 

2 

57.07227 

'259.2032 

2- 

72.78023 

421.5188 

2 

88.48819 

623.1044 

8 

57.33407 

261.5867 

3 

73.04203 

424.5568 

3 

88.74999 

626.7968 

4 

57.59587 

263.9810 

4 

73.30383 

427.6057 

4 

89.01179 

630.5002 

5 

57.85766 

266.3863 

5 

73.56563 

430.6654 

5 

89.27359 

634.2145 

6 

58.11946 

268.8025 

6 

73.82743 

433.7361 

6 

89.53539 

637.9397 

7 

58.38126 

271.2296 

7 

74.08923 

436.8177 

7 

89.79719 

641.6758 

8 

58.64306 

273.6676 

8 

74.35103 

439.9102 

8 

90.05899 

645.4228 

9 

58.90486 

276.1165 

9 

74.61283 

443.0137 

9 

90.32079 

649.1807 

10 

59.16666 

278.5764 

10 

74.87462 

446.1280 

10 

90.58259 

652.9495 

11 

59.42846 

281.0471 

11 

75.13642 

449.2532 

11 

90.84439 

656.7292 

L»  0 

59.69026 

283.5287 

24  0 

75.39822 

452.3893 

29  0 

91.10619 

660.5199 

1 

59.95206 

286.0213 

1 

75.66002 

455.5364 

1 

91.36799 

664.3214 

2 

60.21386 

288.5247 

2 

75.92182 

458.6943 

2 

91.62979 

668.1339 

3 

60.47566 

291.0391 

3 

76.18362 

461.8632 

3 

91.89159 

671.9572 

4 

60.73746 

293.5644 

4 

76.44542 

465.0430 

4 

92.15338 

675.7915 

5 

60.99926 

296.1006 

5 

76.70722 

468.2337 

5 

92.41518 

679.6367 

6 

61.26106 

298.6477 

6 

76.96902 

471.4352 

6 

92.67698 

683.4928 

7 

61.52286 

301.2056 

7 

77.23082 

474.6477 

7 

92.93878 

687.3597 

8 

61.78466 

303.7746 

8 

77.49262 

477.8711 

8 

93.20058 

691.2377 

9 

62.04645 

306.3544 

9 

77.75442 

481.1055 

9 

93.46238 

695.1265 

10 

62.30825 

308.9451 

10 

78.01622 

484.3507 

10 

93.72418 

699.0262 

11 

62.57005 

311.5467 

11 

78.27802 

487.6068 

11 

93.98598 

702.9368 

174 


CIRCLES. 


TABIiE  3  OF  CIRCIiES— (Continued). 
Diams   in  units  and  twelfths;  as  in  feet  and  incbes. 


Dia. 

Circumf. 

Area* 

Dia. 

Circumf. 

Area. 

Dia. 

Circumf. 

Area. 

ft.  In. 

Feet. 

Sq.  ft. 

Ft.In. 

Feet. 

Sq.  ft. 

Ft.In. 

Feet. 

Sq.  ft. 

SO  0 

94.24778 

706.8583 

85  0 

109.9557 

962.1128 

40  0 

125.6637 

1256.6371 

1 

94.50958 

710.7908 

1 

110.2175 

966.6997 

1 

125.9255 

1261.8785 

2 

94.77138 

714.7341 

2 

110.4793 

971.2975 

2 

126.1873 

1267.1309 

3 

95.03318 

718.6884 

8 

110.7411 

975.9063 

3 

126.4491 

1272.3941 

4 

95.29498 

722.6535 

4 

111.0029 

980.5260 

4 

126.7109 

1277.6683 

5 

95.55678 

726.629';; 

5 

111.2647 

985.1566 

5 

126.9727 

1282.9534 

6 

95.81858 

730.6166 

6 

111.5265 

989.7980 

6 

127.2345 

1288.2493 

7 

96.08088 

734.6145 

7 

111.7883 

994.4504 

7 

127.4963 

1293.5562 

8 

96:34217 

738.6233 

8 

112.0501 

999.1137 

8 

127.7581 

1298.8740 

9 

96.60397 

742.6431 

9 

112.3119 

1003.7879 

9 

128.0199 

1304.2027 

10 

96.86577 

746:6737 

10 

112.5737 

1008.4731 

10 

128.2817 

1309.5424 

11 

97.12757 

750.7152 

11 

112.8355 

1013.1691 

11 

128.5435 

1314.8929 

81  0 

97.38937 

754.7676 

36  0 

113.0973 

1017.8760 

41  0 

128.8053 

1320.2543 

1 

97.65117 

758.8310 

1 

113.3591 

1022.5939 

1 

129.0671 

1325.6267 

2 

97.91297 

762.9052 

2 

113.6209 

1027.3226 

2 

129.3289 

1331.0099 

3 

98.17477 

766.9904 

3 

113.8827 

1032.0623 

3 

129.5907 

1336.4041 

4 

98.48657 

771.0865 

4 

114.1445 

1036.8128 

4 

129.8525 

1341.8091 

5 

98.69837 

775.1934 

5 

114.4063 

1041.5743 

5 

130.1143 

1347.225^ 

6 

98.96017 

779.3113 

6 

114.6681 

1046.3467 

6 

130.3761 

1352.652ft 

7 

99.22197 

783.4401 

7 

114.9299 

1051.1300 

7 

130.6379 

1358.0898 

8 

99.48377 

787.5798 

8 

115.1917 

1055.9242 

8 

130.8997 

1363.5385 

9 

99.74557 

791.7304 

9 

115.4535 

1060.7293 

9 

131.1615 

1368.9981 

10 

100.0074 

795.8920 

10 

115.7153 

1065.5453 

10 

131.4233 

1374.4686 

11 

100.2692 

800.0644 

11 

115.9771 

1070.3723 

11 

131.6851 

1379.9500 

82  0 

100.5310 

804.2477 

37  0 

116.2389 

1075.2101 

42  0 

131.9469 

1385.4424 

1 

100.7928 

808.4420 

1 

116.5007 

1080.0588 

1 

132.2087 

1390.9456 

2 

101.0546 

812.6471 

2 

116.7625 

1084.9185 

2 

132.4705 

1396.4598 

3 

101.3164 

816.8632 

3 

117.0243 

1089.7890 

3 

132.7323 

1401.9848 

4 

101.5782 

821.0901 

4 

117.2861 

1094.6705 

4 

132.9941 

1407.5208 

5 

101.8400 

825.3280 

5 

117.5479 

1099.5629 

5 

133.2559 

1413.0676 

6 

102.1018 

829.5768 

6 

117.8097 

1104.4662 

6 

133.5177 

1418.6254 

7 

102.3636 

833.8365 

7 

118.0715 

1109.3804 

7 

133.7795 

1424.1941 

8 

102.6254 

838.1071 

8 

118.3333 

1114.3055 

8 

134.0413 

1429.7737 

9 

102.8872 

842.3886 

9 

118.5951 

1119.2415 

9 

134.3031 

1435.3642 

10 

103.1490 

846.6810 

10 

118.8569 

1124.1884 

10 

134.5649 

1440.9656 

11 

103.4108 

850.9844 

11 

119.1187 

1129.1462 

11 

134.8267 

1446.5780 

83  0 

103.6726 

855.2986 

38  0 

119.3805 

1134.1149 

43  0 

135.0885 

1452.2012 

1 

103.9344 

859.6237 

1 

119.6423 

1139.0946 

1 

135.3503 

1457.8353 

2 

104.1962 

863.9598 

2 

119.9041 

1144.0851 

2 

135.6121 

1463.4804 

3 

104.4580 

868.3068 

3 

120.1659 

1149.08C6 

3 

135.8739 

1469.1364 

4 

104.7198 

872.6646 

4 

120.4277 

1154.0990 

4 

136.1357 

1474.8032 

5 

104.9816 

877.0334 

5 

120.6895 

1159.1222 

5 

136.3975 

1480.4810 

6 

105.2434 

881.4131 

6 

120.9513 

1164.1564 

6 

136.6593 

1486.1697 

7 

105.5052 

885.8037 

7 

121.213J 

1169.2015 

7 

136.9211 

1491.8693 

8 

105.7670 

890.2052 

8 

121.4749 

1174.2575 

8 

137.1829 

1497.579S 

9 

106.0288 

894.6176 

9 

121.7367 

1179.3244 

9 

137.4447 

1503.3012 

10 

106.2906 

899.0409 

10 

121.9985 

1184.4022 

10 

137.7065 

1509.0335 

11 

106.5524 

903.4751 

11 

122.2603 

1189.4910 

11 

137.9683 

1514.7767 

84  0 

106.8142 

907.9203 

39  0 

122.5221 

1194.5906 

44  0 

138.2301 

1520.5308 

1 

107.0759 

912.3763 

1 

122.7839 

1199.7011 

1 

138.4919 

1526.2959 

2 

107.3377 

916.8433 

2 

123.0457 

1204.8226 

2 

138.7537 

1532.0718 

3 

107.5995 

921.3211 

3 

123.3075 

1209.9550 

3 

139.0155 

1537.8587 

4 

107.8613 

925.8099 

4 

123.5693 

1215.0982 

4 

139.2773 

1543.6565 

5 

108.1231 

930.3096 

5 

123.8311 

1220.2524 

5 

139.5391 

1549.4651 

6 

108.3849 

934.8202 

6 

124.0929 

1225.4175 

6 

139.8009 

1555.2847 

7 

108.6467 

939.3417 

7 

124.3547 

1230.5935 

7 

140.0627 

1561.1152 

8 

108.9085 

943.8741 

8 

124.6165 

1235.7804 

8 

140.3245 

1566.9566 

9 

103.1703 

948.4174 

9 

124.8783 

1240.9782 

9 

140.5863 

1572.8089 

10 

109.4321 

952.9716 

10 

125.1401 

1246.1869 

10 

140.8481 

1578.6721 

11 

109.6939 

957.5367 

11 

125.4019 

1251.4065 

11 

141.1099 

1584.5462 

CIRCLES. 


175 


TABIiE  3  OF  CIRCIiES— (Continued). 
Diams  in  units  and  tivelfths;  as  in  feet  and  inches. 


Di;i. 

Circamf. 

Area. 

Dia. 

Circumf. 

Area. 

Dia. 

Circumf. 

Area. 

Ft.In. 

Feet. 

Sq.  ft. 

Ft.In. 

Feet. 

Sq.  ft. 

Ft.In. 

Feet. 

Sq.  ft. 

45  0 

141.3717 

1590.4313 

50  0 

157.0796 

1963.4954 

55  0 

172.7876 

2375.8294 

1 

141.6335 

1596.3272 

1 

157.3414 

1970.0458 

1 

173.0494 

2383.0344 

2 

141.8953 

1602.2341 

2 

157.6032 

1976.6072 

2 

173.3112 

2390.2502 

3 

142.1571 

1608.1518 

3 

157.8650 

1983.1794 

3 

173.5730 

2397.4770 

4 

142.4189 

1614.0805 

4 

158.1268 

1989.7626 

4 

173.8348 

2404.7146 

5 

142.6807" 

1620.0201 

5 

158.3886 

1996.3567 

5 

174.0966 

2411.9632 

6 

142.9425 

1625.9705 

6 

158.6504 

2002.9617 

6 

174.3584 

2419.2227 

7 

143.2043 

1631.9319 

7 

158.9122 

2009.5776 

7 

174.6202 

2426.4931 

8 

143.4661 

1637.9042 

8 

159.1740 

2016.2044 

8 

174.8820 

2433.7744 

9 

143.7279 

1643.8874 

9 

159.4358 

2022.8421 

9 

175.1438 

2441.0666 

10 

143.9897 

1649.8816 

10 

159.6976 

2029.4907 

10 

175.4056 

2448.3697 

11 

144.2515 

1655.8866 

11 

159.9594 

2036.1502 

11 

175.6674 

2455.6837 

46  0 

144.5133 

1661.9025 

51  0 

160.2212 

2042.8206 

56  0 

175.9292 

2463.0086 

1 

144.7751 

1667.9294 

1 

160.4830 

2049.5020 

1 

176.1910 

2470.3445 

2 

145.0369 

1673.9671 

2 

160.7448 

2056.1942 

2 

176.4528 

2477.6912 

3 

145.2987 

1680.0158 

3 

161.0066 

2062.8974 

3 

176.7146 

2485.0489 

4 

145.5605 

1686.0753 

4 

161.2684 

2069.6114 

4 

176.9764 

2492.4174 

5 

145.8223 

1692.1458 

5 

161.5302 

2076.3364 

5 

177.2382 

2499.7969 

6 

146.0841 

1698.2272 

6 

161.7920 

2083.0723 

6 

177.5000 

2507. 187» 

7 

146.3459 

1704.3195 

7 

162.0538 

2089.8191 

7 

177.7618 

2514.5886 

8 

146.6077 

1710.4227 

8 

162.3156 

2096.5768 

8 

178.0236 

2522.0008 

9 

146.8695 

1716.5368 

9 

162.5774 

2103.3454 

9 

178.2854 

2529.4239 

10 

147.1313 

1722.6618 

10 

162.8392 

2110.1249 

10 

178.5472 

2536.8579 

11 

147.3931 

1728.7977 

11 

163.1010 

2116.9153 

11 

178.8090 

2544.3028 

47  0 

147.6549 

1734.9445 

52  0 

163.3628 

2123.7166 

57  0 

179.0708 

2551.7586 

1 

147.9167 

1741.1023 

1 

163.6246 

2130.5289 

1 

179.3326 

2559.2254 

2 

148.1785 

1747.2709 

2 

163.8864 

2137.3520 

2 

179.5944 

2566.7030 

8 

148.4403 

1753.4505 

3 

164.1482 

2144.1861 

3 

179.8562 

2574.1916 

4 

148.7021 

1759.6410 

4 

164.4100 

2151.0310 

4 

180.1180 

2581.6910 

5 

148.9639 

1765.8423 

5 

164.6718 

2157.8869 

5 

180.3798 

2589.2014 

6 

149.2257 

1772.0546 

6 

164.9336 

2164.7537 

6 

180.6416 

2596.7227 

7 

149.4875 

1778.2778 

7 

165.1954 

2171.6314 

7 

180.9034 

2604.2549 

8 

149.7492 

1784.5119 

8 

165.4572 

2178.5200 

8 

181.1652 

2611.7980 

9 

150.0110 

1790.7569 

9 

165.7190 

2185.4195 

9 

181.4270 

2619.3520 

10 

150.2728 

1797.0128 

10 

165.9808 

2192.3299 

10 

181.6888 

2626.9169 

11 

150.5346 

1803.2796 

11 

166.2426 

2199.2512 

11 

181.9506 

2634.4927 

4S  0 

150.7964 

1809.5574 

53  0 

166.5044 

2206.1834 

58  0 

182.2124 

2642.0794 

1 

151.0582 

1815.8460 

1 

166.7662 

2213.1266 

1 

182.4742 

2649.6771 

2 

151.3200 

1822.1456 

2 

167.0280 

2220.0806 

2 

182.7360 

2657.2856 

3 

151.5818 

1828.4560 

3 

167.2898 

2227.0456 

3 

182.9978 

2664.9051 

4 

151.8436 

1834.7774 

4 

167.5516 

2234.0214 

4 

183.2596 

2672.5354 

5 

152.1054 

1841.1096 

5 

167.8134 

2241.0082 

5 

183.5214 

2680.1767 

6 

152.3672 

1847.4528 

6 

168.0752 

2248.0059 

6 

183.7832 

2687.8289 

7 

152.6290 

1853.8069 

7 

168.3370 

2255.0145 

7 

184.0450 

2695.4920 

8 

152.8908 

1860.1719 

8 

168.5988 

2262.0340 

8 

184.3068 

2703.1659 

9 

153.1526 

1866.5478 

9 

168.8606 

2269.0644 

9 

184.5686 

2710.8508 

10 

153.4144 

1872.9346 

10 

169.1224 

2276.1057 

10 

184-8304 

2718.5467 

11 

153.6762 

1879.3324 

11 

169.3842 

2283.1579 

11 

185.0922 

2726.2534 

49  0 

153.9380 

1885.7410 

54  0 

169.6460 

2290.2210 

59  0 

185.3540 

2733.9710 

1 

154.1998 

1892.1605 

1 

169.9078 

2297.2951 

1 

185.6158 

2741.6995 

2 

154.4616 

1898.5910 

2 

170.1696 

2304.3800 

2 

185.8776 

2749.4390 

3 

154.7234 

1905.0323 

3 

170.4314 

2311.4759 

3 

186.1394 

2757.1893 

4 

154.9852 

1911.4846 

4 

170.6932 

2318.5826 

4 

186.4012 

2764.9506 

5 

•155.2470 

1917.9478 

5 

170.9550 

2325.7003 

5 

186.6630 

2772.7228 

6 

155.5088 

1924.4218 

6 

171.2168 

2332.8289 

6 

186.9248 

2780.5058 

7 

155.7706 

1930.9068 

7 

171.4786 

2339.9684 

7 

187.1866 

2788.2998 

8 

156.0324 

1937.4027 

8 

171.7404 

2347.1188 

8 

187.4484 

2796.1047 

9 

156.2942 

1943.9095 

9 

172.0022 

2354.2801 

9 

187.7102 

2803.9205 

10 

156.5560 

1950.4273 

10 

172.2640 

2361.4523 

10 

187.9720 

2811.7472 

11 

156.8178 

1956.9559 

11 

172.5258 

2368.6354 

11 

188.2338 

2819. 584i 

176 


CIRCLES. 


TABIiE  3  OF  CIRCIiES— (Continued). 
Biams  in  units  and  twelftlis;  as  in  feet  and  inebes. 


Dia. 

Circumf. 

Area. 

Dia. 

Circumf. 

Area. 

Dia. 

Circumf. 

Area. 

Ft.Iu. 

Feet. 

Sq.  ft. 

Ft.In. 

Feet. 

Sq.  ft. 

Ft.In. 

Feet. 

Sq.  ft. 

60  0 

188.4956 

2827.4334 

66  0 

204.2035 

3318.3072 

70  0 

219.9115 

3848.4510 

1 

188.7574 

2835.2928 

1 

204.4653 

3326.8212 

1 

220.1733 

3857.6194 

2 

189.0192 

2843.1632 

2 

204.7271 

3335.3460 

2 

220.4351 

3866.7988 

3 

189.2810 

2851.0444 

3 

204.9889 

3343.8818 

3 

220.6969 

3875.9890 

4 

189.5428 

2858.9366 

4 

205.2507 

3352.4284 

4 

220.9587 

3885.1902 

5 

189.8046 

2866.8397 

5 

205.5125 

3360.9860 

5 

221.2205 

3894.4022 

6 

190.0664 

2874.7536 

6 

205.7743 

3369.5545 

6 

221.4823 

3903.6252 

7 

190.3282 

2882.6785 

7 

206.0361 

3378.1339 

7 

221.7441 

3912.8591 

8 

190.5900 

2890.6143 

8 

206.2979 

3386.7241 

8 

222.0059 

3922.1039 

9 

190.8518 

2898.5610 

9 

206.5597 

3395.3253 

9 

222  2677 

3931.3596 

10 

191.1136 

2906.5186 

10 

206.8215 

3403.9375 

10 

222.5295 

3940.6262 

11 

191.3754 

2914.4871 

11 

207.0833 

3412.5605 

11 

222.7913 

3949.9037 

61  0 

191.6372 

2922.4666 

66  0 

207.3451 

3421.1944 

71  0 

223.0531 

3959.1921 

1 

191.8990 

2930.4569 

1 

207.6069 

3429.8392 

1 

223.3149 

3968.4915 

2 

192.1608 

2938.4581 

2 

207.8687 

3438.4950 

2 

223.5767 

3977.8017 

3 

192.4226 

2946.4703 

3 

208.1305 

3447.1616 

3 

223.8385 

3987.1229 

4 

192.6843 

2954.4934 

4 

208.3923 

3455.8392 

4 

224.1003 

3996.4549 

6 

192.9461 

2962.5273 

5 

208.6541 

3464.5277 

5 

224.3621 

4005.7979 

C 

193.2079 

2970.5722 

6 

208.9159 

3473.2270 

6 

224.6239 

4015.1518 

7 

193.4697 

2978.6280 

7 

209.1777 

3481.9373 

7 

224.8867 

4024.5165 

8 

193.7315 

2986.6947 

8 

209.4395' 

3490.6585 

8 

225.1475 

4033.8922 

9 

193.9933 

2994.7723 

9 

209.7013 

3499.3906 

9 

225.4093 

4043.2788 

10 

194.2551 

3002.8608 

10 

209.9631 

3508.1336 

10 

225.6711 

4052.6763 

11 

'  194.5169 

3010.9602 

11 

210.2249 

3516.8875 

11 

225.9329 

4062.0848 

62  0 

194.7787 

3019.0705 

67  0 

210.4867 

3525.6524 

72  0 

226.1947 

4071.5041 

1 

195.0405 

3027.1918 

1 

210.7485 

3534.4281 

1 

226.4565 

4080.9343 

2 

195.3023 

3035.3239 

2 

211.0103 

3543.2147 

2 

226.7183 

4090.3756 

3 

195.5641 

3043.4670 

3 

211.2721 

3552.0123 

3 

226.9801 

4099.8275 

4 

195.8259 

3051.6209 

4 

211.5339 

3560.8207 

4 

227.2419 

4109.2905 

5 

196.0877 

3059.7858 

5 

211.7957 

3569.6401 

5 

227.5037 

4118.7643 

6 

196.3495 

3067.9616 

6 

212.0575 

3578.4704 

6 

227.7655 

4128.2491 

7 

196.6113 

3076.1483 

7 

212.3193 

3587.3116 

7 

228.0273 

4137.7448 

8 

196.8731 

3084.3459 

8 

212.5811 

3596.1637 

8 

228.2891 

4147.2514 

9 

197.1349 

8092.5544 

9 

212.8429 

3605.0267 

9 

228.5509 

4156.7689 

10 

197.3967 

3100.7738 

10 

213.1047 

3613.9006 

10 

228.8127 

4166.2973 

n 

197.6585 

3109.0041 

11 

213.3665 

3622.7854 

11 

229.0745 

4175.8866 

«3  0 

197.9203 

3117.2453 

68  0 

213.6283 

3631.6811 

73  0 

229.3363 

4185.3868 

1 

198.1821 

3125.4974 

1 

213.8901 

3640.5877 

1 

229.5981 

4194.9479 

2 

198.4439 

3133.7605 

2 

214.1519 

3649.505S 

2 

229.8599 

4204.5200 

3 

198.7057 

3142.0344 

3 

214.4137 

3058.4337 

3 

230.1217 

4214.1029 

4 

198.9675 

3150.3193 

4 

214.6755 

3667.3731 

4 

230.3835 

4223.6968 

5 

199.2293 

3158.6151 

5 

214.9373 

3676.3234 

5 

230.6453 

4233.3016 

6 

199.4911 

3166.9217 

6 

215.1991 

8685.2845 

6 

230.9071 

4242.9172 

7 

199.7529 

3175.2393 

7 

215.4609 

3694.2566 

7 

231.1689 

4252.5438 

8 

200.0147 

3183.5678 

8 

215.7227 

3703.2396 

8 

231.4307 

4262.1813 

9 

200.2765 

3191.9072 

9 

215.9845 

3712.2335 

9 

231.6925 

4271.8297 

10 

200.5383 

3200.2575 

10 

216.2463 

3721.2383 

10 

231.9543 

4281.4890 

11 

200.8001 

3208.6188 

11 

216.5081 

3730.2540 

11 

232.2161 

4291.1592 

64  0 

201.0619 

3216.9909 

69  0 

216.7699 

3739.2807 

74  0 

232.4779 

4300.8408 

1 

201.3237 

3225.3739 

1 

217.0317 

3748.3182 

1 

232.7397 

4310.5324 

2 

201.5855 

3233.7679 

2 

217.2935 

3757.3666 

2 

233.0015 

4320.2353 

3 

201.8473 

3242.1727 

3 

217.5553 

3766.4260 

8 

233.2633 

4329.9492 

4 

202.1091 

3250.5885 

4 

217.8171 

3775.4962 

•  4 

233.5251 

4339.6739 

6 

202.3709 

3259.0151 

5 

218.0789 

3784.5774 

5 

233.7869 

4349.4096 

6 

202.6327 

3267.4527 

6 

218.3407 

3793.6695 

6 

234.0487 

4359.1562 

7 

202.8945 

3275.9012 

7 

218.6025 

3802.7725 

7 

234.3105 

4368.9136 

8 

203.1563 

3284.3606 

8 

218.8643 

3811.8864 

8 

234.5723 

4378.6820 

9 

203.4181 

3292.8309 

9 

219.1261 

3821.0112 

9 

234.8341 

4388.4613 

10 

203.6799 

3301.3121 

10 

219.3879 

3830.1469 

10 

235.0959 

4398.2515 

11 

203.9417 

3309.8042 

11 

219.6497 

3839.2935 

U 

235.3576 

4408.0526 

CIRCLES. 


177 


TABIiB  3  OF  CIRCIiES— (Continued). 
Diams  in  units  and  twelfths;  as  in  feet  and  Incbes. 


Dia. 

Circttinf. 

Area. 

Dia. 

Circumf. 

Area. 

Dia. 

Circumf. 

Area. 

rUa. 

Feet. 

Sq.  ft. 

Ft.Ia. 

Feet. 

Sq.  ft. 

Ft.In. 

Feet. 

Sq.  ft. 

75  0 

235.6194 

4417.8647 

80  0 

251.3274 

5026.5482 

85  0 

267.0354 

5674.5017 

1 

235.8812 

4427.6S76 

1 

251.5892 

5037.0257 

1 

267.2972 

5685.6337 

2 

236.1430 

4437.5214 

2 

251.8510 

5047.5140 

2 

267.5590 

5696.7765 

8 

236.4048 

4447.3662 

3 

252.1128 

5058.0133 

3- 

267.8208 

5707.9302 

4 

236.6666 

4457.2218 

4 

252.3746 

5068.5234 

4 

268.0826 

5719.0949 

5 

236.9284 

4467.0884 

6 

252.6364 

5079.0445 

5 

268.3444 

5730.2705 

6 

237.1902 

4476.9659 

6 

252.8982 

5089.5764 

6 

268.6062 

5741.4569 

7 

237.4520 

4486.8543 

7 

253.1600 

5100.1193 

7 

268.8680 

5752.6543 

8 

237.7138 

4496.7536 

8 

253.4218 

5110.6731 

8 

269.1298 

5763.8626 

9 

237.9756 

4506.6637 

9 

253.6836 

5121.2378 

9 

269.3916 

5775.0818 

10 

238.2374 

4:)16.5849 

10 

253.9454 

5131.8134 

10 

269.6534 

5786.3119 

11 

238.4992 

4526.5169 

11 

254.2072 

5142.3999 

11 

269.9152 

5797.5529 

76  0 

238.7610 

4536.4598 

81  0 

254.4690 

5152.9974 

86  0 

270.1770 

5808.8048 

1 

239.0228 

4546.4136 

1 

254.7308 

5163.6057 

1 

270.4388 

5820.0676 

2 

239.2846 

4556.3784 

2 

254.9926 

5174.2249 

.  2 

270.7006 

5831.3414 

3 

239.5464 

4566.3540 

3 

255.2544 

5184.8551 

3 

270.9624 

5842.6260 

4 

239.8082 

4576.3406 

4 

255.5162 

5195.4961 

4 

271.2242 

5853.9216 

5 

240.0700 

4586.3380 

5 

255.7780 

5206.1481 

5 

271.4860 

5865.2280 

6 

240.3318 

4596.3164 

6 

2»6.0398 

5216.8110 

6 

271.7478 

5876.5454 

7 

240.5936 

4606.3657 

7 

256.3016 

5227.4847 

7 

272.0096 

5887.8737 

8 

240.8554 

4616.3959 

8 

256.5634 

5238.1694 

8 

272.2714 

5899.2129 

9 

241.1172 

4626.4370 

9 

256.8252 

5248.8650 

9 

272.5332 

5910.5630 

10 

241.3790 

4636.4890 

10 

257.0870 

5259.5715 

10 

272.7950 

5921.9240 

11 

241.6408 

4646.5519 

11 

257.3488 

5270.2889 

11 

273.0568 

5933.2959 

t!   0 

241.9026 

4656.6257 

82  0 

257.6106 

5281.0173 

87  0 

273.3186 

5944.6787 

1 

242.1644 

4666.7104 

1 

257.8724 

5291.7565 

1 

273.5804 

5956.0724 

2 

2i2.4262 

4676.8061 

2 

258.1342 

5302.5066 

2 

273.8422 

5967.4771 

8 

242.6880 

4686.9126 

3 

258.3960 

5313.2677 

3 

274.1040 

5978.8926 

4 

242.9i98 

4697.0301 

4 

258.6578 

5324.0396 

4 

274.3658 

5990.3191 

5 

243.2116 

4707.1584 

5 

258.9196 

5334.8225 

5 

274.6276 

6001.7564 

6 

243.4734 

4717.2977 

6 

259.1814 

5345.6162 

6 

274.8894 

6013.2047 

7 

213.7352 

4727.4479 

7 

259.4432 

5356.4209 

7 

275.1512 

6024.6639 

8 

243.9970 

4737.6090 

8 

259.7050 

5367.2365 

8 

275.4130 

6036.1340 

9 

244.2588 

4747.7810 

9 

259.9668 

5378.0630 

9 

275.6748 

6047.6149 

10 

244.5206 

4757.9639 

10 

260.2286 

5388.9004 

10 

275.9366 

6059.1068 

11 

244.7824 

4768.1577 

11 

260.4904 

5399.7487 

11 

276.1984 

6070.6097 

SS  0 

245.0442 

4778.3624 

83  0 

260.7522 

5410.6079 

88  0  276.4602 

6082.1234 

1 

245.3060 

4788.5781 

1 

261.0140 

5421.4781 

1 

276.7220 

6093.6480 

2 

245.5678 

4798.8046 

2 

261.2758 

5432.3591 

2 

276.9838 

6105.1835 

3 

^5.8296 

4809.0420 

3 

261.5376 

5443.2511 

3 

277.2456 

6116.7300 

4 

246.0914 

4819.2904 

4 

261.7994 

5454.1539 

4 

277.5074 

6128.2873 

5 

246.3532 

4829.5497 

5 

262.0612 

5465.0677 

5 

277.7692 

6139.8656 

6 

246.6150 

4839.8198 

6 

262.3230 

5475.9923 

6 

278.0309 

6151.4348 

7 

246.8768 

4850.1009 

7 

262.5848 

5486.9279 

7 

278.2927 

6163.0248 

8 

247.1386 

4860.3929 

8 

262.8466 

5497.8744 

8 

278.5545 

6174.6258 

9 

247.4004 

4870.6958 

9 

263.1084 

5508.8318 

9 

278.8163 

6186.2377 

10 

247.6622 

4881.0096 

10 

263.3702 

5519.8001 

10 

279.0781 

6197.8605 

11 

247.9240 

4891.3343 

11 

263.6320 

5530.7793 

11 

279.3399 

6209.4942 

99  0 

248.1858 

4901.6699 

84  0 

263.8938 

5541.7694 

89  0 

279.6017 

6221.1389 

1 

248.4476 

4912.0165 

1 

264.1556 

5552.7705 

1 

279.8635 

6232.7944 

2 

248.7094 

4922.3739 

2 

264.4174 

5563.7824 

2 

280.1253 

6244.4608 

3 

248.9712 

4932.7423 

3 

264.6792 

5574.8053 

3 

280.3871 

6256.1382 

4 

249.2330 

4943.1215 

4 

264.9410 

5585.8390 

4 

280.6489 

6267.8264 

5 

249.4948 

4953.5117 

5 

265.2028 

5596.8837 

6 

280.9107 

6279.5256 

6 

249.7566 

4963.9127 

6 

265.4646 

5607.9392 

6 

281.1725 

6291.2356 

7 

250.0184 

4974.3247 

7 

265.7264 

5619.0057 

7 

281.4343 

6302.9566 

8 

250.2802 

4984.7476 

8 

265.9882  5630.0831 

8 

281.6961 

6314.6885 

9 

250.5420 

4995.1814 

9 

266.2500  5641.1714 

9 

281.9579 

6326.4313 

10 

250.8038 

5005.6261 

10  266.5118  '  5652.2706 

10 

282.2197 

6338.1850 

11 

251.0656 

5016.0817 

11  266.7736  1  5663.3807 

11 

282.4815 

6849.9496 

12 


178 


CIRCLES. 


TABIiE  3  OF  CIRCL.es— (Continued). 
Dlams  in  units  and  twelfths;  as  in  feet  and  inebes. 


Dia. 

Circumf. 

Area. 

Dia. 

Circumf. 

Area. 

Dia. 

Circumf. 

Area.  ' 

Ft.In. 

Feet. 

Sq.  ft. 

Ft.In. 

Feet. 

Sq.  ft. 

Ft.In. 

Feet. 

Sq.  ft. 

90  0 

282.7433 

6361.7251 

93  5 

293.4771 

6853.9134 

96  9 

303.9491 

7351.768ft 

1 

283.0051 

6373.5116 

6 

293.7389 

6866.1471 

10 

304.2109 

7364.4386 

2 

283.2669 

6385.3089 

7 

294.0007 

6878.3917 

11 

304.4727 

7377.1195 

3 

•283.5287 

6397.1171 

8 

294.2625 

6890.6472 

97  0 

301.7345 

7389.8113 

4 

283.7905 

6408.9363 

9 

294.5243 

6902.9135 

1 

304.9963 

7402.5140 

5 

284.0523 

6420.7663 

10 

294.7861 

6915.1908 

2 

305.2581 

7415.2277 

6 

284.3141 

6432.6073 

11 

295.0479 

6927.4791 

3 

305.5199 

7427.9522 

7 

284.5759 

6444.4592 

94  0 

295.3097 

6939.7782 

4 

305.7817 

7440.6877 

8 

284.8377 

6456.3220 

1 

295.5715 

6952.0882 

5 

306.0435 

7453.4340 

9 

285.0995 

6468.1957 

2 

295.8333 

6964.4091 

6 

306.3053 

7466.1913 

10 

285.3613 

6480.0803 

3 

296.0951 

6976.7410 

7 

306.5671 

7478.9595 

11 

285.6231 

6491.9758 

4 

296.3569 

6989.0837 

8 

306.8289 

7491.7385 

•1  0 

285.8849 

6503.8822 

5 

296.6187 

7001.4374 

9 

307.0907 

7504.5285 

1 

286.1467 

6515.7995 

6 

296.8805 

7013.8019 

10 

307.3525 

7517.3294 

2 

286.4085 

6527.7278 

7 

297.1423 

7026.1774 

11 

307.6143 

7530.1412 

3 

286.6703 

6539.6669 

8 

297.4041 

7038.5638 

98  0 

307.8761 

7542.9640 

4 

286.9321 

6551.6169 

9 

297.6659 

7050.9611 

1 

308.1379 

7555.7976 

5 

287.1939 

6563.5779 

10 

297.9277 

7063.3693 

2 

308.3997 

7568.6421 

6 

287.4557 

6575.5498 

11 

298.1895 

7075.7884 

3 

308.6615 

7581.4976 

7 

287.7175 

6587.5325 

95  0 

298.4513 

7088.2184 

4 

308.9233 

7594.3639 

8 

287.9793 

6599.5262 

1 

298.7131 

7100.6593 

6 

309.1851 

7607.2412 

9 

288.2411 

6611.5308 

2 

298.9749 

7113.1112 

6 

309.4469 

7620.1293 

10 

288.5029 

6623.5463 

3 

299.2367 

7125.5739 

7 

309.7087 

7633.0284 

11 

288.7647 

6635.5727 

4 

299.4985 

7138.0476 

8 

309.9705 

7645.9384 

•2  0 

289.0265 

6647.6101 

5 

299.7603 

7150.5321 

9 

310.2323 

7658.8593 

1 

289.2883 

6659.6583 

6 

300.0221 

7163.0276 

10 

310.4941 

7671.7911 

2 

289.5501 

6671.7174 

7 

300.2839 

7175.5340 

11 

310.7559 

7684.7338 

3 

289.8119 

6683.7875 

8 

300.5457 

7188.0513 

99  0 

311.0177 

7697.6874 

4 

290.0737 

6695.8684 

9 

300.8075 

7200.5794 

1 

311.2795 

7710.6510 

5 

290.3355 

6707.9603 

10 

301.0693 

7213.1185 

2 

311.5413 

7723.6274 

6 

290.5973 

6720.0630 

11 

301.3311 

7225.6686 

3 

311.8031 

7736.6137 

7 

290.8591 

6732.1767 

96  0 

301.5929 

7238.2295 

4 

312.0649 

7749.6100 

8 

291.1209 

6744.3013 

1  i  301.8547 

7250.8013 

5 

312.3267 

7762.6191 

9 

291.3827 

6756.4368 

2  302.1165 

7263.3840 

6 

312.5885 

7775.6382 

10 

291.6445 

6768.5832 

3 

302.3783 

7275.9777 

7 

312.8503 

7788.6681 

11 

291.9063 

6780.7405 

4 

302.6401 

7288.5822 

8 

313.1121 

7801.7090 

93  0 

292.1681 

6792.9087 

5 

302.9019 

7301.1977 

9 

313.3739 

7814.7608 

1 

292.4299 

6805.0878 

6 

303.1637 

7313.8240 

10 

313.6357 

7827.8235 

2 

292.6917 

6817.2779 

7  303.4255 

73L'6.4618 

n 

313.8975 

7840.8971 

S 

292.9535 

6829.4788 

8  !  303.6873 

7339.1095 

100  0 

314.1593 

7858.9816 

4 

293.2153 

6&41.6907 

1 

Diam, 

1-64" 
1-32 
3-64 
1-16 
5-64 
3-32 
7-64 

ru 

6-32 
11-64 

3-16 
13^ 


Circumf, 

Dlam, 

Circumf, 

Diam, 

foot 

Inch. 

foot 

Inch 

.004091 

7-32 

.057269 

27-64 

.008181 

15-64 

.061359 

7-16 

.012272 

H 

.065450 

29-64 

.016362 

17-64 

.069540 

15-32 

.020453 

9-32 

.0736ol 

31-64 

.024544 

19-64 

.077722 

.^ 

.028634 

5-16 

.081812 

.032725 

21-64 

.086903 

17-32 

.036816 

11-32 

.089994 

36-64 

.040906 

23-64 

.094084 

9-16 

.044997 

% 

.098175 

37-64 

.049087 

25-64 

.102265 

19-32 

.063178 

13-32 

.106356 

39-64 

Circumf, 

Diam, 

Circumf, 

Diam, 

foot. 

Inch. 

5-8 

foot. 

Inch. 

.110447 

.163625 

53-64 

.114537 

41-64 

.167716 

27-32 

.118628 

21-32 

.171806 

56-64 

.122718 

43-64 

.176896 

7-8 

.126809 

11-16 

.179987 

57-64 

.130900 

46-64 

.184078 

29-32 

.134990 

23-32 

.188168 

69-64 

.139081 

47-64 

.192259 

15-16 

.143172 

% 

.196360 

61-64 

.147262 

49-64 

.200440 

31-32 

.151353 

26-32 

.204631 

63-64 

.165443 

61-64 

.208621 

1 

.169534 

13-16 

.212712 

Circumf, 
foot. 

.216803 
.220893 
.224984 
.229074 
.233161 
.237256 
.241346 
.245437 
.249628 
.263618 
.257709 
.261799 


CIRCULAR  ARCS. 
CIRCUIiAB  ARCS. 


179 


e 
Fig:.  2. 

Kules  for  Fig.  1  apply  to  all  arcs  equal  to,  or  less  than,  a  semi-circle. 

Fig,  2      "  "  "  or  greater  than,  a  semi-circle. 


Cliorcl,  a  bf  of  irliole  arC)  a  db, 

--  2  X  \/radiu82  —  (radius  —  rise)2.    Fig.  1. 
=  2  X  V  radius^  —  (rise  —  radiu8)2.    Fig.  2. 
=  2  X  N/rise  X  (2  X  radius  —  rise).     Figs.  1  and  2. 
■■  2  X  radius  X  sine  of  }4  a  cb.    Figs.  1  and  2. 
rise 


2  X 


Figs.  1  and  2. 


tangent  of  ahd.* 
-  2  X  db^  X  cosine  of  abd*    Figs.  1  and  2. 
=  2  X  \/d62  _  risez.     Figs.  1  and  2.§ 
=  approximately  8  X  dbi^  —  3X  length  of  arc  a  d  6  ^.    Fig.  1. 


,.      ^  ^  »rc  a  d  6  in  degrees      _,.       ,       ,_ 
«=  2  TT  radius  X  ^^ — •   ^igs-  1  and  2 ; 

•^  .01745  X  radius  X  arc  adb  in  degrees.     Figs.  1  and  2.    Ife-^^ _ 

=  circumference  of  circle  —  length  of  small  arc  subtending  angle  ach.     Fig.  2. 
S  X^db^  —  chord  a&.** 


=  approximately  =^ ^ 

J^ig.  1. 

*  a  6  d  is  =  J^  of  lihe  angle  a  c  6,  subtended  by  the  arc.    In  Fig.  2  the  latter  angle 

exceeds  180". 

§ d 6  =  chord  of  d i 6,  or  of  half  adb  = 

\/ri8e2  +  (^  a  6)2.     Figs.  1  and  2. 

f  If  rise  = 
.5     chord, 
.4         « 
.333     « 
.3 

multiply  the  result  by 
1.036 
1.0196 
1.0114 
1.0083 

If  rise  =- 
.25    chord, 
.2         " 
.125      « 
.1          « 

multiply  the  result  by 
1.0044 
1.0021 
1.00036 
1.00016 

••If  rise  = 
.5     chord 
A 

.333      « 
.3         «* 

multiply  the  result  by 
1.012 
1.0065 
1.0038 
1.0028 

If  rise  = 
.25   chord 
.2         « 
.125     « 
.1 

multiply  the  result  ^w 

1.0015 
1.0007 
1.00012 
1.00006 

180 


CIRCULAR  ARCS. 


E^ig.l 


Continued  from  p.  179. 


Fig.   2. 


Knits  for  Fig.  1  apply  to  all  arcs  equal  to  or  less  than  a  semi-circle. 
"        "    Fig.  2    "  «  •   «♦  Qf  greater  than  a  semi-circle. 

Radius,  c  a,  ed,  ci  or  cb, 

^  Q4ab)2  +  riseg  ^  j^igs.  1  and  2. 
2  X  rise 

^      ^^^^      ,  Figs.  1  and  2. 
2  X  rise 

%_ab___  ^  YisB.  1  and  2.  _  K  ^'^ 


sine  of  y^acb 
rise  d  e 
1  —  cosine  of  ^acb 


,  Figs.  1  and  2. 
,  Fig.  1. 


Bine  of  }4bcdl 

rise  de 

1  4-  cosine  of  ^  a  c  6  f 


,  Rgs.  1  and  2. 


,  Fig.  i. 


Rise,  or  middle  ordinate,  d9p 

=-  radius  —  v  radiusS  —  (J^  a  fc)^,  Fig.  1. 
"=  radius  +  \/radiu82  —  (i^  a  6)2,  Fig.  2. 
=  radius  X  (1  —  cosine  of  bed ||),  Fig.  1. 
=  radius  X  (1  +  cosine  of  6cd||),f  Fig.  2. 


db^^ 


.  ,  Figs,  1  and  2. 


2  X  radius 
-^  }4ab  X  tangent  of  a 6 d *  Figs.  1  and  2. 

^  approximately  /'^  °  ^      .  Fig.  1. 
2  X  radius 

When  radius  =  chord  a  6,  the  result  is  6.7  parts  in  100  too  shoft. 
«  **       =-  3  X  chord  a  b,  the  result  is  0.7  parts  in  100  too  short 


Side  ordinate,  as  n  t, 

:  ]/ radius*  —  e  n*  +  rise  —  radius,  Figs.  1  and  2. 
anX  nb 


=  approximately  ^  ^  ^^^^^.  Fig.  l.H 


*  a  6  d  is  =  34  of  the  angle  acb,  subtended  by  the  arc. 

f  Strtetly,  this  should  read  1  mimis  cosine ;  but  the  cosines  of  angles  between  90* 
and  270°  must  then  be  regarded  as  miwxs  or  negative.  Our  rule,  therefore,  amounti 
to  the  same  thing. 

§  d  6  =  chord  of  d  i  &,  or  of  half  a  d  6,  =  v  rise* 


(^a6)2. 


Figs.  1  and  2. 

\bcd  =-  half  the  angle  acb  subtended  by*  the  arc.     In  Fig.  2,  the  latter  angle 
exceeds  180°. 
f  When  radius  =  chord  a  b,  this  makes  d  e  6.7  parts  in  100  too  short. 

"  "    =  3  X  chord  a  b,  this  makes  d  e  0.7  parts  in  100  too  short. 

The  proportionate  error  is  greater  with  the  side  ordinates. 


CIRCULAE  ARCS. 


181 


Angle,  ach,  subtended  by  arc,  adb. 

An  angle  and  its  supplement  (as  hce  and  bed,  Fig.  2)  have  the  same  sine,  the 
ime  cosine  and  the  same  tangent. 

Caution.    The  following  sines,  etc.,  are  those  of  only  half  acb. 


Sine  of  1^  ac 6  =  -Mj?L^  ,  Figs.  1  and  2. 
radius 


Cosine  of  3^  a  c  6  = 
Tangent  of  }^  acb 


radius  —  rise 

radius        * 
__        Hah 
radius  —  rise 


,  rise  —  radius     „.     ^ 

.  1;  =  • r-. ,  Fig.  2. 


.  Fig.l;  = 


radius 
rise  —  radius 


,  Fig.  2 


To  describe  tbe  arc  of  a  circle  too  large  for  tbe  dividers. 
Ist  Metliod.    Let  a  c  be  the  chord,  and  o  b  the  height,  of  the  required  arc,  as 

b 


laid  down  on  the  drawing.  On  a  separate  strip  of  paper,  semn,  draw  a  c.  o  b.  and  a& 
also  b  e,  parallel  to  the  chord  a  c.  It  is  well  to  make  6  s,  and  b  e,  each  a  little  longer 
than  a  b.  Then  cut  off  the  paper  carefully  along  the  lines  s  b  and  b  e,  so  as  to  leave 
remaining  only  the  strip  s  ab  emn.  Now,  if  the  straight  sides  s  b  and  &  e  be  applied 
to  the  drawing,  so  that  any  parts  of  them  shall  touch  at  the  same  time  the  points  a 
and  6,  or  b  and  c,  the  point  &  on  the  strip  will  b©  in  the  circumference  of  the  arc, 
and  may  be  pticked  oflf.  Thus,  any  number  of  points  in  the  arc  may  be  found,  and 
afterward  united  to  form  the  curve. 
/dd  Metl&od*    Draw  the  span  ab;  the  rise  re;  and  ac,b  c    From  c  with  radius 


c  r  describe  a  circle.  Make  each  of  the  arcs  o  t  and  1 1  equal  to  ro  or  r  i;  and  draw 
c  t.  cl.  Divide  ct,  cl,  cr,  each  into  half  as  many  equal  parts  as  the  curve  is  to  be  divided 
into.  Draw  the  lines  61,  62,  63;  and  a4,  a5,  tt6,  extended  to  meet  the  first  ones  at 
e,  «,  h.  Then  c,  *.  h,  are  points  in  one  half  the  curve.  Then  for  the  other  half,  draw 
similar  lines  from  a  to  7,  8,  9;  and  others  from  b  to  meet  them,  as  before.  Trace 
the  curve  by  hand. 


182  CIRCULAR    ARCS. 

Remark.  —It  may  frequently  be  of  use  to  remember,  that  ia  any  arc  &  o  «,  not 


exceeding  29°,  or  in  o\  her  words,  whose  chord  b  s  is  at  least  sixteen  times  its  ris«,  th« 
middle  ordinate  a  o,  will  be  one-halt  of  a  c,  quite  n«ar  enough  tor  many  pur- 
poses; b  c  and  s  c  being  langents  to  the  arc.f  And  vice  ver«a,  if  in  such  an  arc  we 
make  o  c  equal  a  o,  then  will  c  be,  very  nearly,  the  point  at  which  tangents  from  th© 
ends  of  the  arc  will  meet.  Also  the  middle  ordinate  n,  of  tlie  lialf  arc  o  b,  or 
o*,  will  be  approximately  34  of  a  o,  the  middle  ordinate  of  th©  whole  arc.  Indeed, 
this  last  observation  will  apply  near  enough  for  many  approximate  uses  even  if  the 
arc  be  as  great  as  45° ;  for  if  in  that  case  we  take  }^ofo  a  for  the  ordinate  n,  n  will 
then  be  but  1  part  in  103  too  small;  and  therefore  the  principle  may  often  be  used 
in  drawings,  for  finding  points  in  a  curve  of  too  great  radius  to  be  drawn  by  the 
dividers;  for  in  the  same  manner,  %  of  n  will  be  the  middle  ordinate  for  the  arc  n  b 
or  n  o;  and  so  on  to  any  extent.  Below  will  be  found  a  table  by  'wKlcli  tlie 
rise  or  middle  ordinate  of  a  lialf  arc  can  be  obtained  with  greater 
accuracy  when  required  for  more  exact  drawings. 


CIRCUIiAR  ARCS  IN  FREaUEMT  USE. 

The  fifth  column  is  of  use  for  finding  points  for  drawing  arcs  too  large  for  the 
beam-compass,  on  the  principle  given  above.  In  even  the  largest  oflBce  drawings  it 
will  not  be  necessary  to  use  more  than  the  first  three  decimals  of  the  fifth  column; 
and  after  the  arc  is  subdivided  into  parts  smaller  than  about  35°  each,  the  first  two 
decimals  .25  will  generally  suflBce.    Original. 


Rise 

For 

For  rise 

Rise 

1      For 

For 

in       Degrees 

For  rad 

length  of 

of  half 

in 

Degrees 

For  rad  j  length  of 

rise  of 

parts    in  whole 

mult  rise 

aro  mult 

arc 

parts 

in  whole 

mult  rise  jarc  mult 

halfare 

of          arc. 

by 

chord 

mult  rise 

of 

arc. 

by       1   chord 

mult 

chord. 

by 

by 

chord. 

by 

riseby 

1-50 

9     9.75 

313. 

1.00107 

.2501 

Xh 

o         / 
56     8.70 

8.5 

1.04116 

.2538 

1-45 

10   10.75 

253.625 

1.00132 

.2501 

63   46.90 

6.625 

1.05356 

.2549 

1-40 

11   26.98 

200.5 

1.00167 

.2502 

.155 

68   53.63 

6.70291 

1.06288 

.2557 

1-35 

13     4.92 

153.625 

1.00219 

,2502 

1-6 

73  44.39 

6. 

1.07250 

.25«6 

1-30 

15   15.38 

113. 

1.00296 

.2503 

.18 

79   11.73 

4.35803 

1.0842S 

.2576 

1-25 

18   17.74 

78.625 

1.00426 

.2504 

1-5 

87   12.34 

3.625 

1.10347 

.2593 

1-20 

22  50.54 

50.5 

1.00665 

.2506 

.207107 

90 

3.41422 

1.11072 

.2599 

1-19 

24     2.16 

45.625 

1.00737 

.2507 

.225 

96  54.67 

2.96913 

1.12997 

.2615 

1-18 

25  21.65 

41. 

1.00821 

.2508 

M 

106  15.61 

2.5 

1.15912 

.2639 

1-17 

26   50.36 

36.625 

1.00920 

.2509 

.275 

115   14.59 

2.15289 

1.19082 

.2665 

1-16 

28   30.00 

32.5 

1.01038 

.2510 

.3 

123   5130 

1.88889 

1.22495 

.2692 

1-15 

30   22.71 

28.625 

1.01181 

.2511 

yk 

134  45.62 

1.625 

1.27401 

.2729 

1-14 

32  31.22 

25. 

1.01355 

.2513 

.365 

144  30.98 

1.43827 

1.32413 

.2766 

1-13 

34  59.08 

21.625 

1.01571 

.2515 

.4 

154  3835 

1.28125 

1.38322 

.2808 

1-12 

37  50.96 

18.5 

L01842 

.2517 

.425 

161   27.52 

1.19204 

1.42764 

.2838 

1-11 

41   13.16 

15.625 

1.02189 

.2520 

.45 

167   56.S3 

1.11728 

1.47377 

.2868 

1-10 

45   14.38 

13. 

1.02646 

.2525 

.475 

174     7.49 

1.054U2 

1.52152 

.2899 

1-9 

50     6.91 

10.625 

1.03260 

.2530 

.5 

180 

1. 

1.57080 

.2929 

^▲t  29°  o  c  thus  found  will  be  but  about  3  parts  too  ihort  in  lOO. 


MENSURATION. 


liengttas  of  circular  arcs.    If  arc  exceeds  a  semicircle,  seep  184 

Knowing  its  chord  and  height,  divide  the  height  by  the  chord.  Find  in  the  column  of  heights  the 
number  equal  to  this  quotient.  Take  out  the  corresponding  number  from  the  column  of  lengths, 
Multiplj  this  last  number  by  the  length  of  the  given  chord. 


TABI.E   OF   CIRCUI.AR   ARCS. 

No  error* 

H'ghts. 

Lengths. 

H'ghts. 

Lengths. 

H'ghts, 

Lengths. 

H'ghts. 

Lengths. 

H'ghts. 

Lengths. 

.00^ 

1.00002 

.076 

1.01535 

.151 

1.05973 

.226 

1.13108 

.301 

1,22636 

002 

1.00002 

,077 

1.01573 

.152 

1.06051 

.227 

1.13219 

.302 

1,22778 

-003 

1.00003 

.078 

1.01614 

.153 

1.06130 

.228 

1,13331 

.303 

1.22920 

,004 

1.00004 

.079 

1.01656 

.154 

1.06209 

.229 

1,13444 

.304 

1.23063 

.005 

1.00007 

.080 

1.01698 

.155 

1.06288 

.230 

1.13557 

.305 

1.23206 

,006 

1.00010 

,081 

1.01741 

.156 

1.06368 

.231 

1.13671 

.306 

1,23349 

.007 

1.00013 

.082 

1.01784 

.157 

1.06449 

.232 

1,13785 

.307 

1.23492 

,008 

1.00017 

.083 

1.01828 

.158 

1.06530 

.233 

1,13900 

.308 

1.23636 

.009 

1.00022 

.084 

1.01872 

.159 

1.06611 

.234 

1,14015 

.309 

1.23781 

.010 

1.00027 

,085 

1.01916 

.160 

1.06693 

.235 

1.14131 

.310 

1.23926 

.011 

1.00032 

,086 

1.01961 

.161 

1.06775 

.236 

1.14247 

.311 

1.24070 

.012 

1.00038 

.087 

1,02006 

.162 

1.06858 

.2.37 

1.14363 

.312 

1.24216 

.013 

1.00045 

,088 

1,02052 

.163 

1.06941 

.238 

1.14480 

,313 

1.24361 

.OH 

1.00053 

,089 

1,02098 

.164 

1.07025 

.239 

1.14597 

.314 

1.24507 

.01ft 

1.00061 

,090 

1.02146 

.165 

1.07109 

.240 

1.14714 

.315 

1.24654 

.016 

1,00069 

.091 

1.02192 

.166 

1.07194 

.241 

1.14832 

.316 

1.24801 

017 

1,00078 

.092 

1,02240 

.167 

1.07279 

.242 

1.14951 

.317 

1. '24948 

.018 

1.00087 

,093 

1,02289 

.168 

1.07365 

.243 

1.15070 

.318 

1.25095 

.019 

1.00097 

,094 

1,02339 

,169 

1.07451 

.244 

1.15189 

.319 

1.25243 

.020 

1.00107 

.095 

1.02389 

,170 

1.07537 

.245 

1.15308 

.320 

1.25391 

.021 

1.00117 

,096 

1.02440 

.171 

1,07624 

.246 

1.15428 

.321 

L  25540 

,022 

1.00128 

087 

1.02491 

.172 

1.07711 

.247 

1.15549 

.322 

1.2566* 

.023 

1.00140 

,098 

1,02542 

,173 

1.07799 

.248 

1.15670 

.323 

1.25838 

.024 

1.00153 

,099 

1.02593 

,174 

1.07888 

.249 

1.15791 

.324 

1.25988 

.025 

1  00167 

.100 

1.02646 

.175 

1.07977 

.250 

1.15912 

.325 

1.26138 

.026 

1.C0182 

.101 

1.02698 

.176 

1.08066 

.251 

1.16034 

.326 

1.26288 

.027 

1.00196 

.102 

1.02752 

.177 

1.08156 

.252 

1.16156 

.327 

1.26487 

.028 

1,00210 

.103 

1,02806 

.178 

1.08246 

.258 

1.16279 

.328 

1.26568 

.029 

1.00225 

,104 

1.02860 

,179 

1.08337 

.254 

1.16402 

.329 

1.26740 

,030 

]  .00240 

,105 

1.02914 

,180 

1.08428 

.255 

1,16526 

.330 

1,26892 

.031 

1.00256 

,106 

1.02970 

,181 

1.08519 

.256 

1.16650 

,331 

1.27044 

i)32 

1.00272 

,107 

1.03026 

,182 

1.08611 

.257 

1.16774 

,332 

1.271«« 

.033 

1.00289 

,108 

1.03082 

.183 

1.08704 

.258 

1,16899 

.333 

1.27.S4S 

.034 

1.00307 

,109 

1.03139 

,184 

1.08797 

.259 

1,17024 

.334 

1. '27502 

.035 

1.00327 

,110 

1.03196 

,185 

1.08890 

.260 

1.17150 

,335 

1.27656 

.036 

1.00345 

.111 

1.03254 

,186 

1.08984 

,261 

1.17276 

.336 

1.27810 

.037 

1,00364 

.112 

1.03312 

,187 

1.09079 

.262 

1.17403 

.337 

1.27964 

.038 

1.00384 

.113 

1.03371 

.188 

1.09174 

.263 

1.17530 

,338 

1.28118 

.039 

1.00405 

.114 

1.03430 

.189 

1.09269 

.264 

1.17657 

.339 

1.2827S 

.040 

1.00426 

.115 

1.03490 

.190 

1.09365 

.265 

1.17784 

.340 

1.28428 

.041 

1.00447 

.116 

1.03551 

.191 

1.09461 

.266 

1.17912 

.341 

1.28583 

.042 

1.00469 

.117 

1,03611 

,192 

1.09557 

.267 

1.18040 

.342 

1. '28739 

.043 

1.00492 

.118 

1.03672 

.193 

1.09654 

.268 

1.18169 

.343 

1.28895 

.044 

1.00515 

.119 

1.03734 

.194 

1.09752 

.269 

1.18299 

.344 

1.29052 

.045 

1.00539 

.120 

1,03797 

.195 

1.09850 

.270 

1.18429 

.345 

1.29209 

.046 

1.00563 

.121 

1.03860 

.196 

1.09949 

.271 

1.18559 

.346 

1. -29366 

.047 

1 .00587 

.122 

.  1.03923 

.197 

1.10048 

.272    . 

1.18689 

.347 

1.29523 

.048 

1.00612 

.123 

1.03987 

,198 

1.10147 

.273 

1.18820 

.348 

1.29681 

.049 

1.00638 

,124 

1.04051 

.199 

1.10247 

.274 

1.18951 

,349 

1,29839 

.050 

1.00665 

.125 

104116 

.200 

1.10347 

.275 

1.19082 

.350 

1,29997 

.051 

1 .00692 

.126 

1.04181 

.201 

1.10447 

.276 

1.19214 

,351 

1,30156 

.052 

1  00720 

.127 

1.04247 

.202 

1.10548 

.277 

1.19346 

.352 

1.30315 

053 

1  00748 

.128 

1.04313 

.203 

1.10650 

.278 

1.19479 

.353 

1.30474 

.054 

1.00776 

,129 

1.04380 

.204 

1.10752 

.279 

1.19612 

,354 

1.30634 

.055 

1.00805 

.130 

1.04447 

.205 

1,10855 

.280 

1.19746 

.355 

1.30794 

,056 

1.00834 

.131 

1.04515 

.206 

1,10958 

,281 

1.19880 

.356 

1.30954 

,OR7 

I     1.00864 

.132 

1.04584 

.207 

1.11062 

.282 

1.20014 

,357 

1.3111* 

,058 

1.00895 

.133 

1.04652 

.208 

1.11165 

.283 

1.20149 

.358 

1.31276 

-059 

1.00926 

,134 

1.04722 

.209 

1.11269 

.284 

1.20284 

,359 

1.31437 

.060 

1,00957 

.135 

1.04792 

.210 

1,11374 

.285 

1.20419 

.360 

1.31599 

.061 

1.00989 

.136 

1.04862 

.211 

1.11479 

,286 

1.20555 

.361 

1.31761 

.062 

1.01021 

.137 

1 .04932 

.212 

1.11584 

,287 

1 .20691 

.362 

1.3192S 

,063 

1.01054 

.138 

1 .05003 

.213 

1.11690 

,288 

1.20827 

.363 

1.32086 

,064 

1.01088 

,139 

1.05075 

.214 

1.11796 

,289 

1.20964 

.364 

1.32249 

,065 

1.01123 

.140 

1.05147 

.215 

1.11904 

,290 

L21102 

,365 

1.32413 

.066 

1.01158 

.141 

1.05220 

,216 

1.12011 

.291 

1.21239 

,366 

1.32577 

.067 

1,01193 

.142 

1 .05293 

.217 

1.12118 

,292 

1.21377 

.367 

1.32741 

.068 

1.01228 

.143 

1.05367 

,218 

1.12225 

,293 

1.21515 

.368 

1.32905 

.069 

1.01264 

.144 

1.05441 

.219 

1.12334 

,294 

1.21654 

,369 

1.33069 

.070 

1,01302 

.145 

1.05516 

,220 

1.12444 

,295 

1.21794 

.370 

1.. 33*234 

.071 

1,01338 

.146 

1.05591 

,221 

1.12554 

.296 

1.21933 

,371 

1.33399 

.072 
.073 
.074 
.075 

1,01376 

.147 

1.05667 

.222 

1.12664 

.297 

1.22073 

.372 

1.33564 

l!oi414 

.148 

1.05743 

,223 

1.1'2774 

,-.i98 

1.22213 

373 

1.33730 

1  01453 

.149 

1,05819 

.224 

1.12885 

.299 

1.22354 

,374 

1.33896 

1.01493 

.150 

1,05896 

.225 

1  12997 

.300 

1.22495 

.375 

1.34063 

184 


MENSURATION. 


TABIi£ 

OF  €IR€UIiAR 

ARCS  —  (CONTINUKD.) 

H'ghts. 

Lengths. 

H'ghts. 

Lengths. 

H'ghts. 

Lengths. 

H'ghts. 

Lengths. 

H'ghts. 

Lengths. 

.376 

1.34229 

.401 

1.38496 

.426 

1.42945 

.451 

1.47565 

.476 

1.52346 

.377 

1.34396 

.402 

1.38671 

.427 

1.43127 

.452 

1.47753 

.477 

1.52541 

.378 

1.34563 

.403 

1.38846 

.428 

1.43309 

.453 

1.47942 

.478 

1.5273e 

.379 

1.34731 

.404 

1.39021 

.429 

i.  43491 

.454 

1.48131 

.479 

1.52931 

.380 

1.34899 

.405 

1.39196 

.430 

1.43673 

.455 

1.48320 

.480 

1.53126 

.381 

1.35068 

.406 

1.39372 

.431 

1.43856 

.456 

1.48509 

.481 

1.53322 

.382 

1.35237 

.407 

1.39548 

.432 

1.44039 

.457 

1.48699 

.482 

1.53618 

.383 

1.35406 

.408 

1.39724 

.433 

1.44222 

.458 

1.48889 

.483 

1.53714 

.384 

1.35575 

.409 

1.39900 

.434 

1.44405 

.459 

1.49079 

.484 

1.53910 

.385 

1.35744 

.410 

1.40077 

.435 

1.44589 

.460 

1.49269 

.485 

1.54106 

.386 

1.35914 

.411 

1.40254 

.436 

1.44773 

.461 

1.49460 

.486 

1.5480J 

.387 

1.36084 

.412 

1.40432 

.437 

1.44957 

.462 

1.49651 

.487 

1.5449» 

.388 

1.36254 

.413 

1.40610 

.438 

1.45142 

.463 

1.49842 

.488 

1.54696 

.389 

1.36425 

.414 

1.40788 

.439 

1.45327 

.464 

1.50033 

.489 

1.54893 

.390 

1.36596 

.415 

1.40966 

.440 

1.45512 

.465 

1.50224 

.490 

1.55091 

.391 

1.36767 

.416 

1.41145 

.441 

1.45697 

.466 

1.50416 

.491 

1.55289 

.392 

1.36939 

.417 

1.41324 

.442 

1.45883 

.467 

1.50608 

.492 

1.55487 

.393 

1.37111 

.418 

1.41503 

.443 

1.46069 

.468 

1.50800 

.493 

1.55685 

.394 

1.37283 

.419 

1.41682 

.444 

1.46255 

.469 

1.50992 

.494 

1.55884 

.395 

1.37455 

.420 

1.41861 

.445 

1.46441 

.470 

1.51185 

.495 

1.56083 

.396 

1.37628 

.421 

1.42041 

.446 

1.46628 

.471 

1.51378 

.496 

1.56282 

.397 

1.37801 

.422 

1.42221 

.447 

1.46815- 

.472 

1.51571 

.497 

1.56481 

.398 

1.37974 

.423 

1.42402 

.448 

1.47002 

.473 

1.51764 

.498 

1.56681 

.399 

1.38148 

.424 

1.425S3 

.449 

1.47189 

.474 

1.51958 

.499 

1.56881 

.400 

1.38322 

.425 

1.42764 

.450 

1.47377 

.475 

1.52152 

.500 

1.57080 

If  tbe  arc  is  g^reater  tlian  a  semicircle,  then,  as  directed  at  top  of 

p  187. find  the  diam  of  the  circle.     Then  find  its  circumf.    From  diam   take   ht  of  arc.     The  rem 
will  be  ht  of  the  smaller  arc  of  the  circle.     By  rule  at  top  of  p  183  find  the  length  of  this  smaller  aix>> 
Subtract  it  from  circumf. 
The  length  of  1  degree  of  a  circular  arc  is  equal  to  .017453  292  520  X  its  radius. 
"         "        "   1  minute        "         "         "         "         "  .000290  888  209  X    " 
"        "         "   1  second        "         "         '<         "         "  .000004  848  137  X    " 

An  arc  of  1°  of  the  earth's  g-reat  circle  is  but  4.6356  feet  longer  than  its 

chord.  Its  length  is  69.16  land  or  statute  miles.     Earth's  equatorial  rad  =  3962.5705  miles.  Polar  3949.67. 
An  arc  of  1°,  rad  1  mile,  is  92.1534  feet ;  a  minute  is  1.5859  feet ;  a  second  is  .0266  of  a  foot: 
•r  Tery  nearly  5-8ixte«nths  of  an  inch.     Arc  of  1°,  rad  100  ft  =  1.74633  fe«t. 


MENSURATION. 


185 


To  find  the  leiig^tli  of  a  circular  arc  by  the  following:  table. 

Knowing  the  rad  of  the  circle,  and  the  measure  of  the  arc  in  deg  min   &c 
fh?«^^  ^fni/^hfi^f^  the  lengths  in  the  table  found  respectively  opposite  to  the  deg,  min,  &c,  of 
the  arc.     Mult  the  sum  by  the  rad  of  the  circle.  •>     rt-  »>        .       ,  »»» 

Ex.  In  a  circle  of  12.43  feet  rad,  is  an  arc  of  13  deg,  27  min,  8  sec.    How  long  is  the  arc  ? 
Here,  opposite  13  deg  in  the  table,  we  And,  .2268928 
27  min  "  "        "      .0078540 

"  8  sec    "  »        '<     .0000388 


And  .2347856  X  12.43  or  rad  =  2.918385  feet,  the  reqd  length  of  arc' 

I.ENOTHS  OF  CIRCUI.AR  ARCS  TO  RAD  1. 


Deg. 

Length. 

Deg. 

Length. 

Deg. 

Length. 

Min. 

Length. 

Sec. 

Length. 

1 

.0174533 

61 

1.0646508 

121 

2.1118484 

1 

.0002909 

1 

,0000048 

2 

.0349066 

62 

1.0821041 

122 

2.1293017 

2 

.0005818 

2 

.0000097 

3 

.0523599 

63 

1.0995574 

123 

2.1467550 

3 

.0008727 

3 

.0000145 

4 

.0698132 

64 

1.1170107 

124 

2.1642083 

4 

.0011636 

4 

.0000194 

5 

.0872665 

65 

1.1344640 

125 

2.1816616 

5 

.0014544 

5 

.0000242 

6 

.1047198 

66 

1.1519173 

126 

2.1991149 

6 

.0017453 

6 

.0000291 

7 

.1221730 

67 

1.1693706 

127 

2.2165682 

7 

.0020362 

7 

.0000339 

8 

.1396263 

68 

L 1868239 

128 

2.2340214 

8 

.0023271 

8 

.0000388 

9 

.1570796 

69 

1.2042772 

129 

2.2514747 

9 

.0026180 

9 

.0000436 

10 

.1745329 

70 

1.2217305 

130 

2.2689280 

10 

.0029089 

10 

.0000485 

11 

.1919862 

71 

1.2391838 

131 

2.2863813 

11 

.0031998 

11 

.0000533 

12 

.2094395 

72 

1.2566371 

132 

2.3038346 

12 

.0034907 

12 

.0000582 

IS 

.2268928 

73 

1.2740904 

133 

2.3212879 

13 

.0037815 

13 

.0000630 

u 

.2443461 

74 

1.2915436 

134 

2.3387412 

14 

.0040724 

14 

.0000679 

15 

.2617994 

75 

1.3089969 

135 

2.3561945 

15 

.0043633 

15 

.0000727 

16 

.2792527 

76 

1.3264502 

136 

2.3736478 

16 

.0046542 

16 

.0000776 

17 

.2967060 

77 

1.3439035 

137 

2.3911011 

17 

.0049451 

17 

.0000824 

18 

.3141593 

78 

1.3613568 

138 

2.4085544 

18 

.0052360 

18 

.0000873 

19 

.3316126 

79 

1.3788101 

139 

2.4260077 

19 

.0055269 

19 

.0000921 

20 

.3490659 

80 

1.3962634 

140 

2.4434610 

20 

.0058178 

20 

.0000970 

21 

.3665191 

81 

1.4137167 

141 

2.4609142 

21 

.0061087 

21 

.0001018 

22 

.3839724 

82 

1.4311700 

142 

2.4783675 

22 

.0063996 

22 

.0001067 

23 

.4014257 

83 

1.4486233 

143 

2.4958208 

28 

.0066904 

23 

.0001115 

24 

.4188790 

84 

1.4660766 

144 

2.5132741 

24 

.0069818 

24 

.0001164 

25 

.4363323 

85 

1.4835299 

145 

2.5307274 

25 

.0072722 

25 

.0001212 

26 

.4537856 

86 

1.5009832 

146 

2.5481807 

26 

.0075631 

26 

.0001261 

2T 

.4712389 

87 

1.5184364 

147 

2.5656340 

27 

.0078540 

27 

.0001309 

28 

.4886922 

88 

1.5358897 

148 

2.5830873 

28 

.0081449 

28 

.0001357 

29 

.5061455 

89 

1.5533430 

149 

2.6005406 

29 

.0084358 

29 

.0001406 

80 

.5235988 

90 

1.5707963 

150 

2.6179939 

30 

.0087266 

30 

.0001454 

SI 

.5410521 

91 

1.5882496 

151 

2.6354472 

31 

.0090175 

31 

.0001503 

32 

.5585054 

92 

1 .6057029 

152 

2.6529005 

32 

.0093084 

32 

.0001551 

38 

.5759587 

93 

1.6231562 

153 

2.6703538 

33 

.0095993 

33 

.0001600 

34 

.5934119 

94 

1.6406095 

154 

2.6878070 

34 

.0098902 

34 

.0001648 

85 

.6108652 

95 

1.6580628 

155 

2.7052603 

35 

.0101811 

35 

.0001697 

86 

.6283185 

96 

1.6755161 

156 

2.7227136 

36 

.0104720 

36 

.0001745 

87 

.6457718 

97 

1.6929694 

157 

2.7401669 

37 

.0107629 

37 

.0001794 

38 

.6632251 

98 

1.7104227 

158 

2.7576202 

38 

.0110538 

38 

.0001842 

39 

.6806784 

99 

1.7278760 

159 

2.7750735 

39 

.0113446 

39 

.0001891 

40 

.6981317 

100 

1.7453293 

160 

2.7925268 

40 

.0116355 

40 

.0001939 

41 

.7155850 

101 

1.7627825 

161 

2.8099801 

41 

.0119264 

41 

.0001988 

42 

.7330383 

102 

1.7802358 

162 

2.8274334 

42 

.0122173 

42 

.0002036 

43 

.7504916 

103 

1.7976891 

163 

2.8448867 

43 

.0125082 

43 

.0002085 

44 

.7679449 

104 

1.8151424 

164 

2.8623400 

44 

.0127991 

44 

.0002133 

45 

.7853982 

106 

1.8325957 

165 

2.8797933 

45 

.0130900 

45 

.0002182 

46 

.8028515 

106 

1.8500490 

166 

2.8972466 

46 

.0133809 

46 

.0002230 

47 

.8203047 

107 

1.8675023 

167 

2.9146999 

47 

.0136717 

47 

.0002279 

48 

.8377580 

108 

1.8849556 

168 

2.9321531 

48 

.0139626 

48 

.0002327 

49 

.8552113 

109 

1.9024089 

169 

2.9496064 

49 

.0142535 

49 

.0002876 

50 

.8726646 

110 

1.9198622 

170 

2.9670597 

50 

.0145444 

80 

.0002424 

51 

.8901179 

'  111 

1.9373155 

171 

2.98451.30 

51 

.0148353 

51 

.0002473 

62 

.9075712 

112 

1.9547688 

172 

3.0019663 

52 

.0151262 

52 

.0002521 

S3 

.9250245 

113 

1.9722221 

173 

3.019419'' 

53 

.0154171 

53 

.0002570 

»4 

.9424778 

114 

1.9896753 

174 

8.0.368729 

54 

.0157080 

54 

.0002618 

65 

.9599311 

115 

2.0071286 

175 

3.0543262 

65 

.0159989 

55 

.0002666 

66 

.9773844 

116 

2.0245819 

176 

3.0717795 

56 

.0162897 

56 

.0002715 

67 

.9948377 

117 

2.0420352 

177 

3.0892328 

57 

.0165806 

57 

.0002768 

68 

1.0122910 

118 

2.0594885 

178 

3.1066861 

58 

.0168715 

58 

.0002812 

69 

1.0297443 

119 

2.0769418 

179 

3.1241394 

59 

.0171624 

69 

.0002860 

•0 

1.0471976 

120 

2.0943951 

180 

3.1415927 

60 

.0174533 

60 

.000290» 

186 


MENSURATION. 


CIRCUIiAR  SECTORS,  RINGS,  SSGMKNTS,  ETC. 

Area  of  a  circular  sector,  a  db  c.  Fig.  A, 
arc  adb  , 


X  radius  c  o. 
-=.  area  of  entire  circle  X 


arc  adb  in  degrees. 
360 


Area  of  a  circular  ring,  Fig.  B, 
-=  area  of  larger  circle,  c  d,  —  area  of  smaller  one,  a  b. 
I ,  =  .7854  X  (sum of  diams,  cd  -^  ah)  X  (diff-  of diams.  cd — ab.) 
=  1^708  X  thickness  ea  \  sum  of  diameters  c  d and  a  b. 


To  ftiid.  tl&e  radius  ot  a  circle  wlilcli  sliall  liave  tlie  same  area 
as  a  given  circular  ring  c  a  d  ah,  Fig.  B, 

Draw  any  radius  n  r  of  the  outer  circle  ;   and  from  where  said  radius  cuts  th« 
Inner  circle  at  t^  draw  (  s  at  right  angles  to  it.    Then  will  t  she  the  required  radius. 

Breadtli,  e  a  =  b  d,  of  a  circular  rlngy  Fig.  B, 
M>  \^  difference  of  diameters  c  d  and  a  6, 
•a  34  (diameter  cd—  >/  1.2732  area  of  circle  a  6.) 

Area  of  a  circular  zone  abed, 

«»  area  of  circle  m  n — areas  of  segments  amb  and  end, 
(for  areas  of  segments,  see  below.) 

A  circular  lune  is  a  crescent-shaped 
figure,  comprised  between  two  arcs  abc 
and  a  o  c  of  circles  of  different  radii,  a  d 
and  an. 

Area  of  a  circular  lune  ab  eo 

m=  area  of  segment  abc — area  of  segment  aoe, 
i&tr  areas  of  segments  see  below.) 


To  And  tlte  area  f%t  a  circular  •Mf^ment,  abed;  Figs.  iO,  D. 

Ar«a  of  Segment  adbn^  Fig.  A  (at  top  of  page) 
—  Area  of  Sector  adbe  —  Area  of  Triangle  abc, 
■-J^(Arcad6  X  radius  a o  —  en  X  chord  a  6). 

Hairing  tlie  area  of  a  segment  required  to  be  out  off  from  a 
given  circle,  to  find  its  cliord  and  rise. 

<"  Diride  the  area  by  the  square  of  the  diameter  of  the  circle ;  look  for  the  quotient 
In  tha  column  of  areas  in  the  table  of  areas,  opposite ;  take  out  from  the  table 
the  corresponding  number  in  the  column  of  rises.  Multiply  this  number  by  the 
diameter.    The  product  will  be  the  required  rise.    Then 

shord  —  2  X  %/  (diameter  —  rise)  X  riss^ 


MENSURATION.  187 

TABIii:  OF  AREAS  OF  CIRCUIiAR  I^EGMEXTS,  Figs  C,  D. 

If  the  Se$^ineilt  excc^eds  a  semicircle,  its  area  is  ^  area  of  circle-  ar« 
of  a  segment  whose  rise  is  =  (diam  of  circle  —  rise  of  given  segment).  Diam  of  circle  =  (squar 
of  half  choid  -r  rise)  -\~  rise,  whether  the  segment  exceeds  a  semicircle  or  not. 


Rise 

Area  = 

Rise 

Area  = 

Rise 

Area  = 

Rise 

Area= 

Rise 

Area  =» 

div  by 

(square 

div  by 

(square 

div  by 

(square 

div  by 

(square 

div  by 

(square 

diam  of 

of  diam) 

diam  of 

of  diam) 

diam  of 

of  diam) 

liam  of 

of  diam) 

diam  of 

of  diam 

circle. 

mult  bj 

circle. 

mult  by 

circle. 

mult  by 

mult  by 

circle. 
.253 

mult  by 

.001 

.000042 

.064 

.021168 

.127 

.057991 

.190 

.103900 

.156l4y 

.002 

.000119 

.065 

.021660 

.128 

.058658 

.191 

.104686 

.254 

.157019 

.003 

.000219 

.066 

.022155 

.129 

.059328 

.192 

.105472 

.255 

,157891 

.004 

.000337 

.067 

.022653 

.130 

.059999 

.193 

.106261 

.256 

.158763 

.005 

.000471 

.068 

.023155 

.131 

.060673 

.194 

.107051 

.257 

,159636 

.006 

.U00619 

.069 

.023660 

.132 

.061349 

.195 

.107843 

.258 

.160511 

.007 

.000779 

.070 

.024168 

.133 

.062027 

.196 

.108636 

.259 

.161386 

.008 

.000952 

.071 

.024680 

.134 

,062707 

.197 

.109431 

.260 

.162263 

.009 

.  .00113". 

.072 

.025196 

.135 

.063389 

.198 

.110227 

.261 

.163141 

.010 

.001329 

.073 

.025714 

.136 

.064074 

.199 

.111025 

.262 

,164020 

.011 

.001533 

.074 

.026236 

.137 

.064761 

.200 

.111824 

.263 

,164900 

.012 

.001746 

.075 

.026761 

.138 

.065449 

.201 

.112625 

.264 

,165781 

.013 

.001969 

.076 

.027290 

.139 

.066140 

.202 

.113427 

.265 

.166663 

.014 

.002199 

.077 

.027821 

.140 

.066833 

.203 

.114231 

.266 

.167546 

.015 

=002438 

.078 

.028356 

.141 

.067528 

.204 

.115036 

.267 

,168431 

,016 

.002685 

.079 

.028894 

.142 

.068225 

.205 

.115842 

.268 

,169316 

.017 

.002940 

.080 

.029435 

.143 

.068924 

.206 

.116651 

.269 

.170202 

.bl8 

.003202 

.081 

.029979 

.144 

.069626 

.207 

.117460 

.270 

.171090 

.019 

.003472 

.082 

.030526 

.145 

.070329 

.208 

,118271 

.271 

.171978 

020 

.003749 

.083 

.031077 

.146 

.071034 

.209 

.119084 

.272 

.172868 

.021 

.004032 

.084 

.031630 

.147 

.071741 

.210 

.119898 

.273 

.173758 

.022 

.004322 

.085 

.032186 

.148 

.072450 

.211 

.120713 

.274 

.  .174650 

.023 

.004619 

.086 

.032746 

.149 

.073162 

.212 

.121530 

.275 

,175542 

.024 

.004922 

.087 

.033308 

.150 

.073875 

.213 

.122348 

.276 

,176436 

.025 

.005231 

.088 

.033873 

.151 

.074590 

.214 

.123167 

.277 

.177330 

.026 

.005546 

.089 

.034441 

.152 

.075307 

.215 

.123988 

.278 

,178226 

.027 

.005867 

.090 

.035012 

.153 

.076026 

.216 

,124811 

.279 

.179122 

.028 

.006194 

.091 

.035586 

.154 

.076747 

.217 

,125634 

.280 

.180020 

.029 

.006527 

.092 

.036162 

.155 

.077470 

.218 

.126459 

.281 

.180918 

.030 

.006866 

.093 

.036742 

.156 

.078194 

.219 

.127286 

.282 

.181818 

X)31 

.007209 

.094 

.037324 

.157 

.078921 

.220 

.128114 

.283 

.182718 

.032 

.007559 

.095 

.037909 

.158 

.079650 

.221 

.128943 

,284 

.183619 

ms 

.007913 

.096 

.038497 

.159 

.080380 

.222 

.129773 

.285 

.184522 

.034 

.008273 

.097 

.039087 

.160 

.081112 

,223 

.130605 

.286 

.185425 

.035 

.008638 

.098 

.039681 

.161 

.081847 

.224 

.131438 

,287 

.186329 

.036 

.009008 

.099 

.040277 

.162 

.082582 

.225 

.132273 

.288 

.187235 

.037 

.009383 

.100 

.040875 

.163 

.083320 

.226 

.133109 

.289 

.188141 

.038 

.009764 

.101 

.041477 

.164 

.084060 

.227 

.133946 

.290 

.189048 

.039 

.010148 

.102 

.042081 

M65 

.084801 

.228 

.134784 

.291 

.189956 

.040 

.010538 

.103 

.042687 

.166 

.085545 

.229 

.135624 

,292 

.190865 

041 

.010932 

.104 

.043296 

.167 

.086290 

.230 

.136465 

.293 

.191774 

.042 

.011331 

.105 

.043908 

.168 

.087037 

.231 

.137307 

.294 

.192685 

.043 

.011734 

.106 

.044523 

.169 

.087785 

.232 

.138151 

.295 

.193597 

.044 

.012142 

.107 

.045140 

.170 

.088536 

.233 

.138996 

.296 

,194509 

.045 

.012555 

.108 

.045759 

.171 

.089288 

.234 

.139842 

.297 

,195423 

.046 

.012971 

.109 

.046381 

.172 

.090042 

.235 

140689 

:298 

.196337 

.047 

.013393 

.110 

.047006 

.173 

.090797 

.236 

.141538 

.299 

.197252 

.048 

.013818 

.111 

.047633 

.174 

.091555 

.237 

,142388 

.300 

.198168 

.049 

.014248 

.112 

.048262 

.175 

.092314 

.238 

.143239 

.301 

.199085 

.050 

.014681 

.113 

.048894 

.176 

.093074 

.2.39 

.144091 

.302 

.20000S 

.051 

.015119 

.114 

.049529 

.177 

.093837 

.240 

.144945 

.303 

.200922 

.052 

.015561 

.115 

.050165 

.178 

.094601 

.241 

.145800 

.304 

.201841 

.053 

.016008 

.116 

.050805 

.179 

.095367 

.242 

,146656 

.305 

.202762 

.054 

.016458 

.117 

.051446 

.180 

.096135 

.243 

.147513 

.306 

,203683 

.055 

.016912 

.118 

.052090 

.181 

.096904 

.244 

,148371 

.307 

.204605 

.066 

.017369 

.119 

.052737 

.182 

.097675 

.245 

.149231 

.308 

.205528 

.057 

.017831 

.120 

.053385 

.183 

.098447 

.246 

.150091 

.309 

.206452 

.058 

.018297 

.121 

.0540^7 

.184 

.099221 

.247 

.150953 

.310 

.207376 

.059 

.018766 

.122 

.054690 

.185 

.099997 

.248 

.151816 

.311 

.208302 

.060 

.019239 

,123 

.055346 

.186 

.100774 

.249 

.152681 

.312 

.209228 

.061 

.019716 

.124 

.056004 

.187 

.101553 

.250 

.153546 

.313 

.210155 

.062 

.020197 

.125 

.056664 

.188 

.102:^34 

.251 

.154413 

.314 

.211083 

J063 

.020681 

.126 

.067327 

.18© 

.iu;iii6 

.252 

.15.'i281 

315 

iJ12011 

188 


MENSURATION. 


TABLE  OF  AREAS  OF  CIRCVIiAR  JSEOMENTS— (Continued.) 


Rise 

Area  = 

Rise 

Area  = 

Rise 

Area  = 

Riise 

Area  = 

Rise 

Area  =- 

divby 

(square 

divby 

(square 

divby 

(square 

divby 

(square 

divby 

(square 

diam  of 

of  diam) 

diam  of 

of  diam) 

diam  of 

of  diam) 

diam  of 

of  diam) 

diam  of 

of  diam 

oircle. 

mult  by 

circle. 

mult  by 

circle. 

mult  by 

circle. 
.427 

mult  by 

circle. 
.464 

mult  by 

.316 

.212941 

.353 

.247845 

.390 

;283593 

.319959 

.356730 

.317 

.213871 

.354 

.248801 

•391 

.284569 

.428 

.320949 

.465 

.357728 

^18 

.214802 

.355 

.249758 

•392 

.285545 

.429 

.321938 

.466 

.358726 

.319 

.215734 

.356 

.250715 

•393 

.286521 

.430 

.322928 

.467 

.359723 

.320 

.216666 

.357 

.251673 

•394 

.287499 

.431 

.323919 

.468 

.360721 

.321 

.217600 

.358 

.252632 

•395 

.288476 

.432 

.324909 

.469 

.361719 

.322 

.218534 

.359 

.253591 

.396 

.289454 

.433 

.325900 

.470 

.362717 

.323 

.219469 

.360 

.254551 

•397 

.290432 

.434 

.326891 

.471 

.363716 

.324 

.220404 

.361 

.255511 

.398 

.291411 

.435 

.327883 

.472 

.364714 

.325 

.221341 

.362 

.256472 

•399 

.292390 

.436 

.328874 

.473 

.365712 

.326 

.222278 

.363 

.257433 

•400 

.293370 

.437 

.329866 

.474 

.366711 

.327 

.223216 

.364 

.258395 

•401 

.294350 

.438 

.330858 

.475 

.367710 

.328 

.224154 

.365 

.259358 

•402 

.295330 

.439 

.331851 

.476 

.368708 

.329 

.225094 

.366 

.260321 

•403 

.296311 

.440 

.332843 

.477 

.369707 

.330 

.226034 

.367 

.261285 

•404 

.297292 

.441 

.333836 

.478 

.370706 

.331 

.226974 

.368 

.262249 

•405 

.298274 

.442 

.334829 

.479 

.371705 

.332 

.227916 

.369 

.263214 

•406 

.299256 

.443 

.335823 

.480 

.372704 

.333 

.228858 

.370 

.264179 

•407 

.300238 

.444 

.336816 

.481 

.373704 

.334 

.229801 

.371 

.265145 

.408 

.301221 

.445 

.337810 

.482 

.374703 

.335 

.230745 

.372 

.266111 

•409 

.302204 

.446 

.338804 

.483 

.375702 

.336 

.231689 

.373 

.267078 

.410 

.303187 

.447 

.339799 

.484 

.376702 

.337 

.232634 

.374 

.268046 

•411 

.304171 

.448 

.340793 

.485 

.377701 

.338 

.233580 

.375 

.269014 

•412 

.305156 

.449 

.341788 

.486 

.378701 

.339 

.234526 

.376 

.269982 

•413 

.306140 

.450 

.342783 

.487 

.379701 

.340 

.235473 

.377 

.270951 

•414 

.307125 

.451 

.343778 

.488 

.380700 

.341 

.236421 

.378 

.271921 

.415 

.308110 

.452 

.344773 

.489 

.381700 

.342 

.237369 

.379 

.272891 

•416 

.309096 

.453 

.345768 

.490 

.382700 

.343 

.238319 

.380 

.273861 

•417 

.310082 

.454 

.346764 

.491 

.383700 

.344 

.239268 

.381 

.274832 

•41 P 

.311068 

.455 

.347760 

.492 

384699 

.345 

.240219 

.382 

.275804 

•419 

.312055 

.456 

.348756 

.493 

.385699 

.346 

.241170 

.383 

.276776 

.420 

.313042 

.457 

.349752 

.494 

.386699 

.347 

.242122 

.384 

.277748 

.421 

.314029 

.458 

.350749 

.495 

.387699 

.348 

.243074 

.385 

.278721 

.422 

.315017 

.459 

.351745 

.496 

.388699 

.349 

.244027 

.386 

.279695 

.423 

.316005 

.460 

.352742 

.497 

.389699 

.350 

.244980 

.387 

.280669 

.424 

.316993 

.461 

.353739 

.498  . 

.390699 

.361 

.245935 

.388 

.281643 

.425 

.317981 

.462 

.354736 

.499 

.391699 

.352 

.246890 

.389 

.282618 

.426 

.318970 

.463 

.355733 

.500 

.392699 

EI.L.IPSE  (page  189). 

Focal  distance  ^fg  =  2  yco'^—bo'^ ;  * 


cf=gw  = 


cw — fgc  w 


■  y/cO^  —  bO^. 


*  Because  c  o 


c  w 
2 


fc+cg  _/b-{-bg 
2  2 


=  /&. 


MEJJSURATION. 
THE  ElililPSB. 


189 


^><:^^^^Tn 


\     eCy:::^::!^ 


Fig.  1. 


Fig.  3. 


Jin  ellipteU  a  curve,  eeee,  Pigl,  formed  by  an  oblique  section  of  either  a  cone  or  a  cylinder,  pass, 
ing  through  its  curved  surface,  without  cutting  the  base.  Its  nature  is  such  that  if  two  lines,  as 
»/  and  n  g,  Fig.  2,  be  drawn  from  any  point  n  in  its  periphery  or  circumf,  to  two  certain  points/ 
and  g,  in  its  long  diam  c  w,  (and  called  the  foci  of  the  ellipse.)  their  sum  will  be  equal  to  that  of  any 
other  two  lines,  as  b /,  and  b  g,  drawn  from  any  other  point,  a«  b,  in  the  cireumf,  to  the  foci /and  g; 
also  the  sum  of  any  two  such  lines  will  be  equal  to  the  long  diam  c  w.  The  line  c  t<;  dividing  the  el  lipae 
into  two  equal  parts  lengthwise,  is  called  its  transverse,  or  major  axis,  or  long  diam  ;  and  a  6,  which 
divides  it  equally  at  right-angles  to  c  w,  is  called  the  conjugate,  or  minor  axis,  or  short  diam.  To 
find  the  position  of  the  foci  of  an  ellipse,  from  either  end,  as  6,  of  the  short  diam,  measure  off  the 
dists  &/and  b  g,  Fig  2,  each  equal  to  o  c,  or  one-half  the  long  diam. 

The  parameter  of  an  ellipse  is  a  certain  length  obtained  thus ;  as  the  long  diam  i  short  diam  :  : 
short  diam  :  parameter.  Any  line  r  v,  or  s  d,  Fig  3,  drawn  from  the  circumf,  to,  and  at  right  angles 
to,  either  diam,  is  called  an  ordinate;  and  the  parts  c  v  and  vw,bs  and  a  a,  of  that  diam,  between 
the  ord  and  the  circumf,  are  called  abacissce,  or  abscisaet. 

To  find  tbe  leng^tb  of  any  ordinate,  rvorsd,  drawn  to  eitbev 

diam 9  C  to  or  b  <*,    Knowing  the  absciss,  e  o  or  «  a,  and  the  two  diams,  cw,bai 

e  w^  i  b  cfi '.'.  e  V  X  V  w,  r  ^.  6a*:cw*::6<X«a:«d*. 

To  find  the  circumf  of  an  ellipse. 

Mathematicians  have  furnished  practical  men  with  no  simple  working  rule  for  this  purpose.  The 
to-called  approximate  rules  do  not  deserve  the  name.  They  are  as  follows,  D  being  the  long  diam: 
and  d  the  short  one.  

BuLE  1.  Circumf  =  3.1416  J5_±A  •    Rule  2.  3.1416  /  /P^<^-\  .  Eulb  8.  2.2215 ;/  B^-d2; 

this  is  the  same  as  Rule  2,  but  in  adiflf  shape.  EiTiJe4.2X|/  D2-f- 1.4674  d2.    Now,  in  an  ellipse 

whose  long  and  short  diams  are  10  and  2,  the  circumf  la  actnallv  21,  very  approximately;  but  rule  1 
fives  it  =  18.85 ;  rule  2,  or  3,  =:  22.65 ;  and  rule  4,  =  20.58.  Again,  if  the  diams-be  10  and  6,  the  cir- 
cumf actually  =  25.59;  but  rule  4  gives  24.72.  These  examples  show  that  none  of  the  rules  usuall/ 
given  are  reliable.  The  following  one  by  the  writer,  is  sufficiently  exact  for  ordinary  purposes;  fi«^ 
being  in  error  probably  more  than  1  part  in  1000.    When  D  is  not  more  than  5  times  as  long  as  d. 


S        ""        8.8 

If  I>  exceeds  5  times  il,  then  in*      p    «^^«^ 
stead  of  dividing  (D  —  d)^  by  8.8,  div  it  by       -     •^  • 
the  number  in  this  table. 

The  following  rule  originated  with  Mr.  M. 
Arnold  Pears,  of  New  South  Wales,  Australia, 
and  was  by  him  kindly  communicated  to  the  author, 
rate  than  our  own,  it  is  much  neater. 


sssas 


I  tto>o»oia>aiako>a>o»0>a>aioooc 


Q  —  —  —  - 


•e '8 13 -8  73 "«  T3  •« -a  M 


Circumf  =  3.1416  d+  2(D  —  d)- 


Although  not  more  accu* 
d(D  —  d) 


^(D  +d)  X  (D  +  2d) 


following  table  of  semi-elliptic  arcs  was  prepared  by  our  rota. 


To  use  this  table,  div  the  height  or  rise  of  the  are,  by  its  span  or  chord.  The  quot 
will  be  the  height  of  an  arc  whose  span  is  1.  Find  thia  quot  in  the  column  of 
heights  ;  and  take  out  the  corresponding  number  from  the^<Jol.  of  lengths.  Mult  this 
number  by  the  actual  span.     The  prod  will  be  the  reqd  length. 

When  the  height  becomes  .500  of  the  chord  (as  at  the  ehd  of  the  table)  the  ellipse 
becomes  a  circle.  When  the  height  exceeds  .500  of  the  cljord,  as  in  a  6  c,  then  take 
a  0,  or  half  the  chord,  as  the  rise ;  and  div  this  rise  by. the  long  diam  6  d,  for  the 
quot  to  be  looked  lor  in  the  col  of  heights ;  and  to  be  molt  by  long  diam.  We  thoa 
get  the  arc  bad,  which  is  evidently  equal  to  a  &  c. 


190 


MENSURATIOIS. 


TABI.I:  OF  I^ENOTHS  OF  SEMI-FI.I.IPTIC  ARCS. 

(trriginal.! 

Height 

Length  = 

Height 

Length  = 

Height 

Length  = 

Height 

Length  = 

♦  span. 

span X  by 

■rspan. 

span  X  by 

■f  span. 

span  X  by 

+span. 

span  X  by 

.005 

1.000 

.130 

1.079 

.255 

1.219     . 

.380 

1.390 

.01 

1.001 

.135 

1.084 

.260 

1.226 

.385 

1.397 

.015 

1.002 

.140 

1.089 

.265 

1.233 

.390 

1.404 

.02 

1.003 

.145 

1.094 

.270 

1.239 

.395 

1.412 

025 

1.004 

.150 

1.099 

.275 

1.245 

.400 

1.419 

,03 

1.006 

.155 

1.104 

.280 

1.252 

.405 

1.426 

.0"^5 

1.008 

.160 

1.109 

.285 

1.259 

.410 

1.434 

,04 

1.011 

.165 

1.115 

.290 

1.265 

.415 

1.441 

.045 

1.014 

.170 

1.120 

.295 

1.272 

.420 

1.449 

.05 

1.017 

.175 

1.125 

.300 

1.279 

.425 

1.456 

.055 

1.020 

.180 

1.131 

.305 

1.285 

.430 

1.464 

.06 

1.023 

.185 

1.137 

.310 

1.292 

.435 

1.471 

.065 

1.026 

.190 

1.142 

.315 

1.298 

.440 

1.479 

.07 

1.029 

.196 

1147 

.320 

1.305 

.445 

1.486 

.075 

1.032 

.200 

1.153 

.325 

1.312 

.450 

1.494 

.08 

1.036 

.205 

1.159 

.330 

1.319 

.455 

1.501 

.085 

1.039 

.210 

1.165 

.335 

1.325 

.460 

1.509 

.09 

1.043 

.215 

1.171 

.340 

1.332 

.465 

1.517 

.095 

1.046 

.220 

1.177 

•345 

1.339 

.470 

1.524 

.100 

1.051 

.225 

1.183 

.350 

1.346 

.475 

1.532 

.105 

1.055 

.230 

1.189 

.355 

1.353 

,480 

1.540 

.110 

1.059 

.235 

1.196 

.360 

1.361 

.485 

1.547 

.115 

1.064 

.240 

1.202 

.365 

1.368 

.490 

1.555 

.120 

1.069 

.245 

1.207 

.370 

1.375 

.495 

1.563 

.125 

1.074 

.250 

1.213 

.375 

1.382 

.500 

1.571 

Area  of  an  ellipse  =  prod  of  diams  X  .7854.  Ex.  D  =  10 ;  d  =  6.  Then  10  X  6  X  .7854 
r:  47.124  area.  The  area  of  an  ellipse  is  a  mean  proportional  between  the  areas  of  two  circles,  de- 
scribed on  its  two  diams  ;  therefore  it  may  be  found  by  mult  together  the  areas  of  those  two  circles  ; 
and  taking  the  sq  rt  of  the  prod.  The  area  of  an  ellipse  is  therefore  always  greater  than  that  of  the 
circular  section  of  the  cylinder  from  which  it  may  be  supposed  to  be  derived. 

Diam  of  cire  of  same  area  as  a  griven  ellipse  =  y^Long  diam  x  short  diam. 
To  find  tbe  area  of  an  elliptic  segment  whose  base  is  paral- 
lel to  eitliei*  diam.  Div  the  height  of  the  segment,  by  that  diam  of  which  said  height 
is  a  part.  Prom  the  table  of  circular  segments  take  out  the  tabular  area  opposite  the  quot.  Malt 
together  this  area,  the  long  diam,  and  the  short  diam. 

To  draiV  an  ellipse.      Having  its  long  and  short  diams  a  h  and  c  d,  Fig.  4. 

RcLE  1.  From  either  end  of  the  short 
diam.,  as  c,  lay  off  the  dists.  c/,  cf,  each 
equal  to  e  o,  or  to  one-half  of  the  long  d;am. 
The  points/,  /'  are  the  foci  of  the  ellipse. 
Prepare  a  string,/*  n  /,  or/'  g  /,  with  a  loop 
at  each  end  ;  the  total  length  of  string  from 
end  to  end  of  loop,  being  equal  to  the  long 
diam.  Place  pins  at /and/';  and  placing 
the  loops  over  them,  trace  the  curve  by  a 
peuci],  which  in  every  position,  as  at  n,  or  y, 
keeps  the  string/'  n/,  or/'  gf  stretched  all 
the  time. 

Note.  Owing  to  the  difficulty  of  keeping 
the  string  equally  stretched,  this  method  is 
not  as  satisfactory  as  the  following. 

Rule  2.  On  the  edge  of  a  strip  of  paper 
w  8,  mark  w  I  equal  to  half  the  short  diam. ; 
and  w  s  equal  half  the  loug  diam.  Then  in 
whatever  position  this  strip  be  placed,  keep- 
ing I  on  the  long  diam.,  and  a  on  the  short 

diam.,  w  will  mark  a  point  in  the  circumf.  of  the  ellipse.     We  may  thus  obtain  a 
as  we  please  ;  and  then  draw  the  curve  through  them  by  hand. 

Rule  3.  From  the  two  foci  /  and  /',  Fig.  4,  with  a  rad.  equal  to  any  part  whatever  of  the  long 
diam.  describe  4  short  arcs,  o  o  o  o ;  also  with  a  rad.  equal  to  the  remaining  part  of  the  long  diam.^ 
describe  4  other  arcs,  i  i  i  i.  The  intersections  of  these  four  pairs  of  arcs,  will  give  four  points  in  the 
circumf.    In  this  manner  any  number  of  such  points  may  be  found,  and  the  curve  be  drawn  by  hand. 

To  draw  a  tangent  1 1,  at  any  point  n  of  an  ellipse.    Draw  «  / 

and  n  /',  to  the  foci ;  bisect  the  angle  /  n  f  by  the  line  xp  ;  draw  «  n  «  at  right  angles  to  xp. 

To  draw  a  joint  »t  p,  of  an  elliptic  areli,  from  any  point  n,  in 

the  arcll.     Proceed  as  in  the  foregoiug  rule  for  a  tangent,  only  omitting  tt;  np  will  be  tlM 
required  joiat  " 


Fig.  4. 


3  many  such  points 


MENSURATIOH. 


191 


To  draw  an  oiral,  or  false  ellipse. 

"When  only  the  long  diam  o  b  is  given,  the  following 
will  give  agreeable  curves,  of  which  the  span  a  b  will 
not  exceed  about  three  times  the  rise  co.  On  a  5  de- 
scribe two  intersecting  circles  of  aay  rad;  througU 
their  intersections  s,  v,  draw  e  g;  make  a  g  and  v  « 
each  equal  to  the  diam  of  one  of  the  circles.  Through 
the  centers  of  the  circles,  draw  ey,  eh,  gd,  gt.  Prom 
e  describe  hiy;  and  from  g  describe  dot. 


When  the  span,9nn^  and  tli« 
rise,  s  t,  are  both  given. 

Make  any  tw  and  mr,  equal  to  each  other, 
but  each  less  than  t  s.  Draw  r  w;  and  through 
its  center  o  draw  the  perp  i  o  y.  Draw  y  r  z. 
Make  n  x  equal  m  r,  and  draw  yxh.  From  x  and 
r  describe  n  c  and  m  a;  and  from  y  describe 
ate.  By  making  s  d  equal  to  »  y,  we  obtain 
the  center  for  the  other  side  of  the  oval. 

The  beauty  of  the  curve  will  depend  upon 
what  portion  of  «  «  is  taken  for  m  r  and  t  w. 
When  an  oval  is  very  flat,  more  than  three  cen- 
ters are  required  for  drawing  a  graceful  curve; 
but  the  findingof  these  centers  is  quite  as  trou« 
blesome  as  to  draw  the  correct  ellipse. 


On  the  griven  line,  a  «,  to  draw  a 
eyma  recta,  acs. 

Find  the  center  c,  of  a  s.  From  a,  c,  and  s,  with  one-half 
of  a  a  as  rad,  draw  the  four  small  arcs  at  o.  o.  The  inter- 
sections 0,0,  are  the  centers  for  drawing  thecyma,  witn 
the  same  rad.  By  reversing  the  position  of  the  arcs,  wo 
»bi&in  the  q/ma  revers;  or  oget,  def. 


192 


MENSURATION. 


THi:  PARABOIiA. 


I 

) 

1 

1 

■  \ 

o  a  c 

Fig.l.  *    Fig.  2. 

Tbe  commcii  or  conic  parabola, 

ob  c,  Fig  1,  is  a  curve  formed  by  cutting  a  cone  in  a  direction  6  a,  parallel  to  its  side.  Th* 
•urved  line  o  b  c  itself  is  called  the  perimeter  of  the  parabola ;  the  line  o  c  is  called  its  base ;  6  a  it» 
height  or  axis;  b  its  apex  or  vertex;  any  line  e  s,  or  o  a,  Fig  2,  drawn  from  the  curve,  to,  and  at  right 
angles  to,  the  axis,  is  an  ordinate  ;  and  the  part  «  6,  or  a  b,  of  the  axis,  between  the  ordinate  and  tha 
apex  b,  is  an  abscissa.  The  foctts  of  a  parabola  is  that  point  in  the  axis,  where  the  abscissa  b  s,  is 
equal  to  one-half  of  the  ord  e  s.  The  dist  from  apex  to  focus,  called  the  focal  dist,  is  found  thus: 
square  any  ord,  as  o  a;  div  this  square  by  the  abscissa  6  a  of  that  ord;  dlv  the  quot  by  4.  The 
nature  of  the  parabola  is  such  that  its  abscissas,  as  6  «,  6  a,  &c,  are  to  each  other  as,  or  in  proportion 
to,  the  squares  of  their  respective  ords  e  s,  oa,  &g;  that  is,  as  6  s  :  &  a  :  :  es'^  :  o  a^;  or  bs  :  es^  ::  b  a: 
©  a2  .  If  the  square  of  any  ord  be  divided  by  its  abscissa,  the  quot  will  be  a  constant  quantity ;  that 
Is,  it  will  be  equal  to  the  square  of  any  other  ord  divided  by  its  abscissa.  This  quot  or  constant  quan- 
tity is  also  equal  to  a  certain  quantity  called  the  parameter  of  the  parabola.  Therefore  the  parameter 
may  be  found  by  squaring  c  s,  or  o  a,  {one-half  of  the  base,)  and  dividing  said  square  by  the  height 
bs,  or  b  a,  as  the  case  may  be.  If  the  square  of  any  ord  be  divided  by  tbe  parameter,  the  quot  wiy 
he  the  abscissa  of  that  ord. 

To  find  tbe  len^tb  of  a  parabolic  curve. 

The  approximate  rule  given  by  various  pocket-books,  is  as  follows : 

Length  =  2  X  Vi/i  base)2   +  1]^  times  the  (Height^) 

Where  the  height  does  not  exceed  1-lOth  of  the  base,  this  rule  may,  for  practical 

purposes,  be  called  exact.    With   ht  —  14  base,  it  gives  about  }4  per  cent  to« 

muoh  ;  ht  =  }^  base,  about  3J^  percent;  ht=:ba3e,  about  8}^  per  cent;  ht  = 

ftwice  the  bMe,  about  12>^  per  cent ;  ht=  10  X  base,  or  more,  about  15H  per  cent 

The  following  by  the  write*  is  correct 

within  perhaps  1  part  in  800,  in  ail  cases  ;  and  will 

therefore  answer  for  many  purposes. 

Let  adb,  Fig  3,  or  n  o  d,  Fig  4,  be  the  parabola, 
in  which  are  given  the  base  ab  or  n  d;  and  the 
height  c  d  or  c  a.  Imagine  the  complete  fig  a  d  6  «, 
or  n  a  d  6,  to  be  drawn  ;  and  in  et'tAer  case,  assume 
its  long  diam  a  6  to  be  the  chord  or  base ;  and  one- 
half  the  short  diam,  or  c  d,  to  be  the  height,  of  » 
circular  arc.  Find  the  length  of  this  circular  arc, 
by  means  of  the  rule  and  table  given  for  that  pur- 
pose. Then  div  tbe  chord  or  base  o  6,  or  n  d  of 
the  parabola,  by  its  height  c  d  or  c  a.  Look  for 
the  quot  in  the  column  of  bases  in  the  following 
table,  and  take  from  the  table  the  correspondiag 
multiplier.  Mult  the  length  of  the  circular  arc  by 
this;  the  prod  will  be  the  length  of  arc  adi,OT 
n  a  d,  3,3  the  case  may  be.  For  bases  of  parabolas 
less  than  .05  of  the  height,  or  greater  than  10  times 
the  height,  the  multiplier  is  1,  and  is  very  approx- 
imate; or  in  other  words,  the  parabola  will  be 
of  almost  exactly  the  same  length  as  the  circular 
arc. 

To  find  tbe  area  of  a  parabola  m  anl>. 

Mult  its  base  m  n,  Fig  5,  by  its  height  a  b ;  and  take  %ds  of  the  prod. 
The  area  of  any  segment,  as  ub  v,  whose  base  «  v  is  parallel  to  mn,  is 
found  in  the  same  way,  using  u  v  and  a  b,  instead  of  m  n  and  a  b. 

To  find  tbe  area  of  a  parabolic  zone,  or  frus« 
tnm,  as  rn.  ti  t*  v. 

RuLK  1.  First  find  by  the  preceding  rule  the  area  of  the  whole  pnrabola 
mbn\  then  that  of  the  segment  ubv,  and  subtract  the  last  from  the 
first. 

Rule  2.  From  the  cube  of  m  n,  take  the  cube  of  «  « ;  call  the  diff  «.' 
From  the  square  of  m  n,  take  the  square  of  t«  v;  call  the  differ.  Div  c  by 
«.    Mult  the  quot  by  ^ds  of  the  height  a  a. 


Fig.  5. 


MENSURATION, 


193 


Tabic  lor  Lengths  of  Parabolic  Carves.   See  opp  page.    (Orvgiaal.) 


Base. 

Mult. 

Base. 

Mult. 

Base. 

Mult. 

Base. 

Mult. 

.05 

1.000 

1.10 

.^99 

2.15 

.949 

3.20 

.983 

.10 

1.001 

1.15 

.997 

2.20 

.951 

3.30 

.984 

.15 

1.002 

1.20 

.995 

2.25 

.954 

3.40 

.985 

.20 

1.004 

1.25 

.993 

2.30 

.956 

3.50 

.986 

.25 

1.006 

1.30 

.990 

2.35 

.958 

3.60 

.987 

.30 

1.007 

'    1.35 

.987 

2.40 

.960 

3.70 

.988 

.35 

1.007 

1.40 

.984 

2.45 

.962 

3.80 

.989 

.40 

1.008 

1.45 

.980 

2.50 

.963 

3.90 

.990 

.45 

1.009 

1.50 

.977 

2.55 

.965 

4.00 

.991 

.50 

1.010 

1.55 

.974 

2.60 

.967 

4.25 

.992 

.55 

1.010 

1.60 

.970 

2.65 

.969 

4.50 

.993 

.60 

1.010 

1.65 

.966 

2.70 

.970 

4.75 

.994 

.65 

1.011 

1.70 

.963 

2.75 

.972 

5.00 

.995 

.70 

1.011 

1.75 

.960 

2.80 

.973 

5.25 

.996 

.75 

1.010 

1.80 

.957 

2.85 

.975    • 

6.50 

.997 

.80 

1.009 

1.85 

.953 

2.90 

.976 

5.75 

.998 

.85 

1.008 

1.90 

.950 

2.95 

.978 

6.00 

.998 

.90 

1.006 

1.95 

.946 

3.00 

.979 

7.00 

.999 

.95 

1.004 

2.00 

.942 

3.05 

.980 

8.00 

]  .000 

1.00 

1.002 

2.05 

.944 

3.10 

.981 

10.00 

1.000 

1.05 

1.001 

2.10 

.946 

3.15 

.982 

To  draw  a  parabola,  having  base  c  s  and  height  e  o. 


e««,  Fig6.  Make  o  <  equal  to  the  height  CO.  Drawciand 
•  t ;  and  divide  ench  of  them  into  any  number  of  equal  parts  ; 
numbering  them  as  in  the  Pig.  Join  1, 1 ;  2,  2 ;  3,  3,  &c; 
then  draw  the  curve  by  hand.  It  will  be  observed  that  the 
intersections  of  the  lines  1, 1  ;  2,  2,  &c,  do  not  give  points  la 
the  curve  ;  but  a  portion  of  each  of  those  lines  forms  a  tan- 
gent to  the  curve.  By  increasing  the  number  of  divisions 
on  ct  and  s  t,  an  almost  perfect  curve  is  formed,  scarcely 
requiring  to  be  touched  up  by  hand.  In  practice  it  is  best 
first  to  draw  only  the  center  portions  of  the  two  lines  which 
cross  each  other  just  above  o  ;  and  from  them  to  work  down- 
ward; actually  drawing  only  that  small  portion  of  each 
successive  lower  line,  which  is  necessary  to  indicate  the 
ourve. 


Or  the  parabola  may  be  drawn 
thas: 

Let  J  c,  Fig  7,  be  the  base  ;  and  a  d  the  height.   Draw  the 
rectangle  b  nm  c;  div  each  half  of  the  base  into  any  num- 
ber of  equal  parts,  and  number  them  from  the  center  each  C' 
way.  Div  n  h',  and  m  c  into  the  same  number  of  equal  parts ; 
and  number  them  from  the  top,  downward.    From  the  points 
•n  6  c  draw  vert  lines ;  and  from  those  at  the  sides  draw  lines 
to  d.  Then  the  intersections  of  lines  1,1;  2.  2,  &c, 
will  form  points  in  the  parabola.    As  in  the  pre- 
ceding case,  it  is  not  necessarj-  to  draw  the  entire 
lines ;  but  merely  portions  of'them,  as  shown  be- 
tween d  and  c. 

Or  a  parabola  may  be  drawn  by  first  div  the 
height  a  h.  Fig  5,  into  any  number  of  parts,  either 
equal  or  unequal;  and  then  calculating  the  ordi- 
sates  us,  &c;  thus,  as  the  height  a  b  :  square  of 
half  base  am::  any  absciss  b  s  :  square  of  its 
ord  u  8.    Take  the  sq  rt  for  ms. 

Rem. —When  the  height  of  a  parabola  is  not 
greater  than  1-lOth  part  its  base,  the  curve  coin- 
cides so  very  closely  with  that  of  a  circular  arc, 
that  in  the  preparation  of  drawings  for  suspen- 
sion bridges,  &c.,  the  circular  arc  may  be  em- 
ployed ;  or  if  no  great  accuracy  is  reqd,  the  circle 
may  be  used  even  when  the  height  is  as  great  as 
•ne-eighth  of  the  base. 


To  draw  a  tangent  tv  v.  Fig.  5,  to  a  parabola,  from  any  point  v. 

Draw  V  s  perp  to  axis  a  b  ;  prolong  a  b  until  b  w  equals  a  b.    Join  w  v. 

13 


194 


MENSURATION. 


Tlie  Cycloid, 

ach   is  the  curve  described  by  a  point  a  in  the  circumference  of  a  circle, 

an  during  one  complete  revolution  of  the  circle,  rolled  along  a  straight  line 

'  ^  a6;  which  is  called  the  base  of  the 

b  cyclcfid. 

The  vertex  of  the  cycloid  is  at  c. 
Base,  a  &,  =  circumference  of  generat- 
ing circle  a  n 
=  diameter,  cd^  of  generat- 
ing circleXn"  =  3.1416cci. 
Axis,  or  lieig^lit,    cd=  an. 
lien^tli,  ac6,  =  4cd 

Area,  a  c6  d  =  3  X  area  of  generating  circle,  an 

=  3^^  =cd^Xin  =  cd^X  2.3562. 
4 
Center  of  gravity  of  swface  at  g.    c  g  =  -^^  c  d.    Center  of  gravity  of 

cycloid  (curved  line  ac  b)  in  axis  c  d  at  a  point  (as  5)  distant  ^  cd  from  c. 

To  draw  a  tangent,  eo,  from  any  point  e  in  a  cycloid ;  draw  e  s  at  right 
angles  to  the  axis  cd;  on  cd  describe  the  generating  circle  dct;  join  t  c ;  frum 
e  draw  e  0  parallel  to  t  c.    The  cycloid  is  the  curve  of  quickest  descent ; 

so  that  a  body  would  fall  from  b  to  c  along  the  curve  b  m  c,  in  less  time  than 
along  the  inclined  plane  b  i  c,  or  any  other  line. 


THE    REGUIiAR    BODIES. 

A  regular  body,  or  regular  polyhedron,  is  one  which  has  all  its 
sides,  and  its  solid  angles,  respectively  similar  and  equal  to  each  other.  There 
are  but  five  such  bodies,  as  follows : 


Name. 

Bounded  by 

Surface 

(=sum  of  surfaces 
of  all  the  faces). 

Multiply  the  square 
of  the  length  of 
one  edge  by 

Volume. 

Multiply  the 

cube  of  the 

length  of  one 

edge  by 

4  equilateral  triangles. 
6  squares. 

8  equilateral  triangles. 
12         "       pentagons. 
20         "       triangles. 

1.7320 
6. 

3.4641 
20.6458 
8.6602 

.1178 

Hexahedron  or  cube 
Octahedron 

1. 
.4714 

Dodecahedron 

7.6631 

Icosahedron 

2.1817 

Ouldinns- 

Fig.  A. 


Theorem. 

Fig.  B. 


To  find  the  volume  of  any  body  Cas  the 

irregularmassa  6  cm.  Fig  A,  or  the  ring 
ab  cm,  Fig  B),  generated  by  a  complete 
or  partial  revolution  of  any  figure  (as 
abca)  around  one  of  its  sides  (as  <ic, 
Fig  A),  or  around  any  other  axis  (as 
a;  2/,  Fig  B). 

Volume  =  surface  abca  X  length 
of  arc  described  by  its  center  of  grav- 
ity G. 

If  the  revolution  is  complete,  the  arc 
described  is  =  circumference  =  radius 
0  G*  X  277  =  radius  0  G*  X  6.283186 ;  and 
Volume  =surface  abca  X  radius 
0  G  *  X  6.283186. 
If  the  revolution  is  incomplete, 

complete   .  incomplete  . .   circumference  .       arc 
revolution  '  revolution    '  *  found  as  above  "  described 

*  Measured  perpendicularly  to  the  axis  of  revolution. 


MENSURATION. 
PARAl.Ii£l40PIP£l>IS. 


195 


Fig.  1. 


Fig.  2. 


Fig.  3. 


Fig.  4. 


A  parallelopiped  is  any  solid  contained  within  six  sides,  all  of  which  are 
parallelograms ;  and  those  of  each  opposite  pair,  parallel  to  each  other.  We 
show  but  four  of  them  ;  corresponding  to  the  four  paralh  lograras ;  namely,  th« 
cube.  Fig  1,  which  has  all  its  sides  equal  squares,  aud  all  its  angles  right  angles; 
the  right  rectangular  prism,  Fig  2,  has  all  its  angles  right  angles,  each  pair  of 
opposite  faces  equal,  but  not  all  of  its  faces  equal ;  the  Hhombohedron,  Fig  3, 
which  has  all  its  sides  equal  rhombuses,  aud  which,  like  the  rhombus,  p  157,  is 
sometimes  called  "  rhomb  "  ;  the  Rhombic  prism,  Fig  4  ;  its  faces,  rhombuses,  or 
rhomboids,  each  pair  of  opposite  faces  equal,  but  not  all  its  faces  equal.  All 
parallelepipeds  are  prisms. 

Volume  of  any  ^  area  of  any  face,  y,  perpendicular  distance,  p, 
as  a,  ^       to  the  opposite  face. 

=  cube  of  length  of  one  edge, 
=  1.90985  X  volume  of  inscribed  sphere, 
=  1.27324  X  "  "  cylinder, 

=  3.81972  X  "  "  cone. 

]>iag;oual  of  a  cube  =  diameter  of  circumscribing  sphere, 

=  1.7320508  X  length  of  one  edge  of  cub«. 


parallelopiped 
Volume  of  a  cube 


h 


P 


§ 

J 


PRISMS. 

A  prism  is  any  solid  whose 
twoends  are  parallel,  similar, 
and  equal ;  and  whose  sides 
are  parallelograms,  as  Figs  5 
i     to  10.  Consequently  the  fore- 
\p  going    parallelopipeds    are 
i      prisms.  A  right  prism  is  one 
whose  sides  are  perpendic- 
ular to  its  ends,  as  5,  6,  7  ; 
when  ^ot  so,  the  prism  is 
oblique,  as  8,  9, 10.   When  all 
the  sides  of  the  15  gures  which 
form  the  ends  are  equal,  and  the  angles  included  between  those  sides  are  alsv 
equal,  the  prism  is  said  to  be  regular :  otherwise,  irregular. 
Volume  of  any  prism  (whether  regular  or  irregular,  right  or  oblique) 
=  area  of  one  end  X  perpendicular  distance,  jt),  to  the  other  end, 
«=  area  of  cross  section  perpendicular  to  the  sides  X  actual  length,  a  b,  Figs 

5  to  10, 
=*=  3  X  volume  of  pyramid  whose  base  and  height  are  —  those  of  the  prism. 


To  find  the  volume  of  any  frustum 
of  any  prism. 

Whose  cross  section,  perpendicular  to  its  sides, 
is  either  any  triangle ;  any  parallelogram ;  a 
square,  (as  in  Fig  loX^)  or  a  regular  polygon  of 
any  number  of  sides;  no  matter  how  the  two 
ends  of  the  frustum  may  be  inclined  with  regard 
to  each  other;  or  whether  one,  or  neither  of 
them,  is  parallel  to  the  base  of  the  original 
prism. 

sum  of  lengths  of  parallel  edges, 
Volume  rr  +  2^  +  3^  +  T4 

of  frustum  " 


number  of  such  edges 
(4  in  Fig  10^4) 


Figs.  101^. 

area  of  cross  section 

X        perpendicular 
to  such  edges. 


196 


MENSURATION. 


This  rule  may  be  used  for  ascertaining  beforehand,  the  quantity  of  earth  to 
be  removed  from  a  "borrow  pit."  The  irregular  surface  of  the  ground  is  first 
staked  oat  iu  squares;  (the  tape-line  being  stretched  Aoriswito^,  when  meas- 
uring off  their  sides).  These  squares  should  be  of  such 
a  size  that  without  material  error  each  of  them  may  be 
considered  to  be  a  plane  surface,  either  horizontal  or  in- 
clined. The  depth  of  the  horizontal  bottom  of  the  pit 
being  determined  on,  and  the  levels  being  taken  at  every 
corner  of  the  squares,  we  are  thereby  furnished  with  the 
lengths  of  the  four  parallel  vertical  edges  of  each  of  the 
resulting  frustums  of  earth.  In  Figs  103^  y  may  be  sup- 
posed to  represent  one  of  these  frustums. 
If  the  frustum  is  that  of  an  irregular  4-sided,  or  polyg- 
onal prism,  first  divide  its  cross  section  perpendicular  to  its  sides,  into  tri- 
angles, by  lines  drawn  from  any  one  of  its  angles,  as  a.  Fig  10^.  Calculate  the 
area  of  each  of  these  triangles  separately ;  then  consider  the  entire  frustum  to 
be  made  up  of  so  many  triangular  ones;  calculate  the  volume 
V,cr\  of  each  of  these  by  the  preceding  rule  for  triangular  frustums ; 

and  add  them  together,  for  the  volume  of  the  entire  frustum. 


Fig.  101^. 


Volume  of  any  frnstum  of  any  prism. 

Or  of  a  cylinder.  Consider  either  end  to  be  the  base ;  and  find  its 

^ area.'  Also  find  the  center  of  gravity  c  of  the  other  end,  and  the 

n    ^ — ^       perpendicular  distance  n  c,  from  the  base  to  said  center  of  gravity. 

Fig.  10^.       Then   Tolume  of  frustum  =  area  of  base  Xw  c,  Fig  10%;. 

The  slant  end,  c,  is  an  ellipse.    Its  area  is  greater  than  that  of  the  circular  end. 

Surface  of  any  prism.  Figs  5  to  10,  whether  right  or  oblique,  regular 

or  irregular 

_  (  circumference  measured    ^      ,     ,  ipntrfh  ah\A.  ^^^  ^^  *^^  ^^^^ 

-  Vperpendicular  to  the  sides  ^  ^^^^^^  Jengin,  a  oj  +  ^^  ^^^  ^^^  ^^^^^ 

CYIilNDERS. 

A  cylinder  is  any  solid  whose  ends  are 
parallel,  similar,  and  eqxxi^i  curved  figures  ; 
and  whose  sections  parallel  to  the  ends 
are  everywhere  the  same  as  the  ends. 
Hence  there  are  circular  cylindf  rs,  ellip- 
tic cylinders  (or  cylindroids)  -^nd  many 
others ;  but  when  not  otherwise  expressed, 
the  circular  one  is  understood.  A  right 
cylinder  is  one  whose  ends  are  perpen- 
dicular to  its  sides,  as  Fig.  11 ;  when  other- 
wise, it  is  obliqve,  as  Fig  12.  If  the  ends 
of  a  right  circular  cylinder  be  cut  so  as  to 
make  it  oblique,  it  becomes  an  elliptic  one ;  because  theni)oth  its  ends,  and  all 
sections  parallel  to  them,  are  ellipses.  An  oblique  circular  cylinder  seldom 
occurs ;  it  may  be  conceived  of  by  imagining  the  two  ends  of  Fig  12  to  be  circles, 
united  by  straight  lines  forming  its  curved  sides. 

A  cylinder  is  a  prism  having  an  infinite  number  of  sides. 
Volume  of  any  cylinder  (whether  circular  or  elliptic,  &c,  right  or  oblique) 

—  area  of  one  end  X  perpendicular  distance,  p,  to  the  other  end, 

_  f       area  of  cross  section        ^  ^     j  1  ^^1^      ^  Pi      j ^    ^  ^2, 
( measured  perp  to  the  sides  '^  b     >       >      g  » 

^  3  X  volume  of  a  cone  whose  base  and  height  are  —  those  of  the  cylinder. 
Surface  of  any  cylinder  (whether  circular  or  elliptic,  &c,  right  or  oblioue) 
/circumference       ....  ...        .,       .  \  ,  sum  of  the  areas 


Fig.  11. 


Fig.  12. 


=  (  measured  perpendicularly    X  actual  length,  ah\ 
\to  the  sides,  as  at  c  o,  Fig  12,  / 


of  the  two  ends. 
=  diameter. 


Right  circular  cylinder  whose  height  = 

Volume  =  1^  X  volume  of  inscribed  sphere. 

Curved  surface     =  surface  of  inscribed  sphere. 

Area  of  one  end  =  I  snrface  of  inscribed  sphere  =  a  curved  surface. 

Entire  surface      =  1^  X  surface  of  inscribed  sphere  =  1^  X  curved  surface. 


CONTENTS  OF  CYLINDERS,  OR  PIPES. 


197 


Contents  for  one  foot  in  length,  in  Cub  Ft,  and  in  U.  S.  Gallons  of 

231  cub  ins,  or  7.4805  Galls  to  a  Cub  Ft.    A  cub  ft  of  water  weighs  about  623^  lbs ;  and  a  gallon 
about  8M  lbs.    Diams  3*  8»  or  10  times  as  great,  give  4,  9,  or  100  times  the  content. 


For  1ft.  in 

For  1  ft  in 

For  1  ft.  in 

length. 

length. 

length. 

Diam. 

Diam. 

in  deci- 

Diam. 

Diam. 
in  deci- 

Diam. 

Diam. 
in  deci- 

in 

^.2 

»-  -i, 

Ts 

*-  rn 

■        •   " 

Ins. 

mals  of 

SS- 

°5 

Ins. 

mals  of 

1'^^ 

0  a 

in 

mals  of 

I'S   . 

0  a 

a  foot. 

'^t^ 

li 

a  foot. 

^l  . 

i-2 

Ins. 

a  foot. 

^i" 

Cx5 

il^ 

^^ 

,0-6" 
s  §  " 

■|5 

il^ 

|5 

0:5 

^i 

«5 

^S 

^__ 

^a 

Va. 

.0208 

.0003 

.0025 

% 

.5625" 

.2485 

1.859 

19. 

1.583 

1.969 

14.73 

5-16 

.0260 

.0005 

.0040 

7. 

.5833 

.2673 

1.999 

3^ 

1.625 

2.074 

15.51 

% 

.0313 

.0008 

.0057 

1 

74: 

.6042 

.2867 

2.145 

20 

1.667 

2.182 

16.32 

7-16 

.0365 

.0010 

.0078 

.6250 

.3068 

2.295 

K 

1.708 

2.292 

17.15 

9-f^ 

0417 

.0014 

.0102 

.6458 

.3276 

2.450 

21.^ 

1.750 

2.405 

17.99 

.0469 

.0017 

.0129 

8-. 

.6667 

.3491 

2.611 

y2 

1.792 

2.521 

18.86 

% 

.0521 

.0021 

.0159 

1^ 

.6875 

.3712 

2.777 

22.^ 

1.833 

2.640 

19.75 

11-16 

.0573 

.0026 

.0193 

74 

.7083 

.3941 

2.948 

'A 

1.875 

2.761 

20.66 

13-?l 

.0625 

.0031 

.0230 

.7292 

.4176 

3.125 

23 

1.917 

2.885 

21.58 

.0677 

.0036 

.0269 

9. 

.7500 

.4418 

3.305 

M 

1.958 

3.012 

22.53 

% 

.0729 

.0042 

.0312 

1 

.7708 

.4667 

S.491 

24:^ 

2.000 

3.142 

23.50 

15-16 

.0781 

.0048 

.03r)9 

.7917 

.4922 

3.682 

25. 

2.083 

3.409 

25.50 

1. 

.0833 

.0055 

,  .040S 

74 

.8125 

.5185 

3.879 

26. 

2.167 

3.687 

27.58 

.1042 

.0085 

'  .0638 

10. 

.8333 

.5454 

4.080 

27. 

2.250 

3.976 

29.74 

.1250 

.0123 

.0918 

K 

.8542 

.5730 

4.286 

28. 

2.333 

4.276 

31.99 

74 

.1458 

.0167 

.1249 

U;  .8750 

.6013 

4.498 

29. 

2.417 

4.587 

34.31 

0 

.1667 

.0218  • 

.1632 

%    .8958 

.6303 

4.715 

30. 

2.500 

4.909 

36.72 

"'   Va. 

.1875 

.0276 

.2066 

11.        .9167 

.6600 

4.937 

31. 

2.583 

5.241 

39.21 

I 

.2083 

.0341 

.2550 

%    .9375 

.6903 

5.164 

32. 

2.667 

5  585 

41.78 

.2292 

.0412 

.3085 

K!   .9583 
%\  .9792 

.7213 

5.S96 

33. 

2.750 

5.940 

44.43 

3. 

.2500 

.0491 

.3672 

.7530 

5.633 

34. 

2.833 

6.305 

47.13 

Va 

.2708 

.0576 

.4309 

12.     ilFoot. 

.7854 

5.875 

35. 

2.917 

6.681 

49.98 

y 

74 

.2917 

.0668 

.4998 

1^;  1.042 

.8.')22 

6.375 

36. 

3.000 

7.069 

52.88 

.3126 

.0767 

.5738 

13.     j  1.083 

.9218 

6.895 

37. 

3.083 

7.467 

55.86 

4. 

.3333 

.0873 

.6528 

M 1-125 

.9940 

7.436 

38. 

3.167 

7.876 

58.92 

34 

.3542 

.0985 

.7369 

14.     il.167  • 

1.0R9 

7.997 

39. 

3.250 

8.296 

62.06 

.3750 

.1104 

.8263 

M1I.2O8 

1.147 

8.578 

40. 

3.333 

8.727 

65.28 

/a 

.3958 

.1231 

.9206 

15.     11.250 

1.227 

9.180 

41. 

3.417 

9.168 

68.58 

5. 

.4167 

.1364 

1.020 

l^!l.292 

1.310 

9.801 

42. 

3.500 

9.621 

71.97 

Y^ 

.4375 

.1503 

1.125 

16.     11.333 

1.396 

10.44 

43. 

3.583 

10.085 

75.41 

'i 

.4583 

.1650 

1.234 

141.375 

1.485 

11.11 

44. 

3.667 

10.559 

78.99 

.4792 

.180.S 

1.349 

17.     11.417 

1.576 

11.79 

45. 

3.750 

11.045 

82.62 

6. 

.5000 

.1963 

1.469 

■     1^11.458 

1.670 

12.49 

46. 

3.833  !ll.541 

86.33 

'i 

.5208 

.2131 

1.594 

18.     11.500 

1.767 

13.22 

47. 

3.917    12.048 

90.13 

.5417 

.2304 

1.724 

)^jl.542 

1.867 

13.96 

48. 

4.000    12.566 

94.00 

Table  continued,  but  witb  tbe  diams  in  feet. 


IMam. 

Cub. 

u.  s. 

Diam. 

Cub. 

u.  s. 

Dia. 

Cub. 

U.S. 

Dia. 

Cub. 

U.S. 

Feet. 

Feet. 

GaUa. 

Feet. 

Feet. 

Galls. 

Feet. 

Feet. 

Galls. 

Feet. 

Feet. 

Galls. 

4 

12.57 

94.0 

7 

38.48 

287.9 

12 

113.1 

846.0 

24 

452.4 

3384 

'4 

14.19 

106.1 

8 

74 

41.28 

308.8 

13 

132.7 

992.9 

25 

490.9 

8672 

15.90 

119.0 

44.18 

330.5 

14 

153.9 

1152. 

26 

530.9 

3972 

!^ 

17.72 

132.6 

47.17 

352.9 

15 

176.7 

1322. 

27 

572.6 

4283 

5 

19.63 

146.9 

8 

50.27 

376.0 

16 

201.1 

1504. 

28 

615.8 

4606 

1^ 

74 

21.65 

161.9 

}4 

56.75 

424.5 

17 

227.0 

1698. 

29 

660.5 

4941 

23.76 

177.7 

9 

63.62 

475.9 

18 

254.5 

1904. 

30 

706.9 

5288 

25.97 

194.2 

U 

70.88 

530.2 

19 

283.5 

2121. 

31 

754.8 

5646 

6 

28.27 

211.5 

10 

78.54 

587.5 

20 

SI  4.2 

2350. 

32 

804.2 

6016 

1^ 

30.68 

229.5 

y2 

86.59 

647.7 

21 

346.4 

2591. 

33 

855.3 

6398 

33.18 

248.2 

11 

95.03 

710.9 

22 

380.1 

2844. 

34 

907.9 

6792 

35.78 

267.7 

y. 

103.87 

777.0 

23 

415.5 

3108. 

35 

962.1 

7197 

198 


CONTENTS   AND    LININGS   OF   WELLS. 


CONTENTS  AWD  I.ININOS  OF  WEI.I.S. 

For  diams  twice  as  great  as  those  in  the  table,  for  the  cub  yds  of  digging,  take  out  those  opposlt* 
one  half  of  the  greater  diam  ;  and  mult  them  by  4.  Thus,  for  the  cub  ?ds  in  each  foot  of  depth  of  a 
well  31  feet  in  diam,  farst  take  out  from  the  table  those  opposite  the  diam  of  lo>6  feet  •  namelv  6  989 
Then  6  989  X  4  :=  27  956  cub  yds  reqd  for  the  31  ft  diam-."^  But  for  the  stone  Uning  or  waH  n|.'bHcks 
or  plastering,  mult  the  tabular  quantity  opposite  half  the  greater  diam,  by  2.  Thus,  the  perches  of 
stone  walling  for  each  foot  of  depth  of  a  well  of  31  ft  diam,  will  be  2.073  X  2  =  4  146  If  the  wall  is 
more  or  less  than  one  foot  thick  within  usual  moderate  limits,  it  will  generally  be  near  enough  for 
^oSr      ^^^"""^  number  of  perches,  or  of  bricks,  will  increase  or  decrease  inthe  same  pro- 

The  size  of  the  bricks  is  taken  at  8>^  X  4  X  2  inches :  and  to  be  laid  dry,  or  without  mortar.  Ic 
K bdcrthSk  or'st^  fna°"'     ^^''  °^"'  ^^^"^"^  ^®  ^^^^  ^*^^  ^''^*'^-    ^^®  ^^^^^  "'^^"^  ^^  supposed  to 

CAUTIOX.  —  Be  careful  to  observe  that  the  diams  to  be  used  for  the  digeinff 
are  greater  than  those  for  the  walling,  bricks,  or  plastering.  No  errors. 


For  each  foot  of  depth. 

For  each  foot  of  depth. 

For  this 

For  these  three  cols  use  the 

For  this 

For  these  three  cols 

nso  tKo 

Diam. 

col  use  the 

Diameter 

of  the 

diam  in 

clear  of  the  lining. 

Diam. 
in 

col  use  the 

Diameter 

of  the 

diam  in 

clear  of  the  lining. 

in 

Feet. 

Digging. 

Stone 

No.  of 

Square 
Yards  of 
Plaster- 

Feet. 

Digging. 

Stone 

No.  of 

Lining 
1ft  thick. 
Perches  of 

Bricks  in 
a  Lining 
1  Brick 

Lining 
1  ft  thick. 
Perches  of 

Bricks  in 
a  Lining 
1  Brick 

Square 

Cub  Yds. 

of 
Digging. 

Cub  Yds. 

Yards 
of  Plas- 

25 Cub  Ft. 

thick. 

ing. 

of 
Digging. 

25  Cub  Ft. 

thick. 

tering. 

1. 

.0291 

.2513 

57 

.3491 

H 

5.107 

1.791 

750 

4.625 

Ya 

.0455 

.2827 

71 

.4364 

% 

5.301 

1.822 

764 

4.713 

]^ 

.0654 

.3142 

85 

.5236 

% 

5.500 

1>G54 

778 

4.800 

H 

.0891 

.3456 

99 

.6109 

14. 

5.701 

1.885 

792 

4.887 

2. 

.1164 

.3770 

114 

.6982 

Va. 

5.907 

1.916 

806 

4.974 

M 

.1473 

.4084 

128 

.7855 

^ 

6.116 

1.948 

820 

5.062 

^ 

.1818  • 

.4398 

142 

.8727 

?i 

6.329 

1.979 

834 

5.149 

.    ^ 

.2200 

.4712 

156 

.9600 

15. 

6.545 

2.011 

849 

5.236 

3. 

.2618 

.5027 

170 

1.047 

34 

6.765 

2.042 

S&^i 

5.323 

M 

.3073 

.5341 

184 

1.135 

3^ 

6.989 

2.073 

877 

5.411 

^ 

.3563 

.5655 

198 

1.222 

% 

7.216 

2.105 

891 

5.498 

H 

.4091 

.5869 

212 

1.309 

16. 

7.447 

2.136 

905 

5.585 

4. 

.4654 

.6283 

227 

1.396 

M 

7.681 

2.168 

919 

5.673 

H 

.5254 

.6597 

241 

1.484 

% 

7.919 

2.199 

933 

5.760 

Yl 

.5890 

.6912 

255 

1.571 

H 

8.161 

2.231 

948 

5.847 

H 

.6563 

.7226 

269 

1.658 

17. 

8.407 

2.262 

962 

5.934 

5. 

.7272 

.7540 

283 

1.745 

]4. 

8.656 

2.293 

976 

6.022 

34 

.8018 

.7854 

297 

1.833 

8.908 

2.325 

990 

6.109 

}4 

.8799 

.8168 

311 

1.920 

'     % 

9.165 

2.356 

1004 

6.196 

H 

.9617 

.8482 

326 

2.007 

18. 

9.425 

2.388 

1018 

6.283 

e. 

1.047 

.8796 

340 

2.095 

34 

9.688 

2.419 

1032 

6.371 

H 

1.136 

.9111 

354 

2.182 

3^ 

9.956 

2.450 

1046 

6.458 

1.229 

.9425 

368 

2.269 

% 

10.23 

2.482 

1061 

6.545 

V 

1.325 

.9739 

382 

2.356 

19. 

10.50 

2.513 

1075 

6.633 

». 

1.425 

1.005 

396 

2.444 

34 

10.78 

2.545 

1089 

6.720 

H 

1.529 

1.037 

410 

2.531 

3^ 

11.06 

2.576 

1103 

6.807 

H 

1.636 

1.068 

425 

2.618 

H 

11.35 

2.608 

1117 

6.894 

H 

1.747 

1.100 

439 

2.705 

20. 

11.64 

2.639 

1131 

6.982 

8. 

1.862 

1.131 

455 

2.793 

3^ 

11.93 

2.670 

IU5 

7.069 

34 

1.980 

1.162 

467 

2.880 

12.22 

2.702 

1160 

7.156 

3^ 

2.102 

1.194 

481 

2.967 

% 

12.52 

2.733 

1174 

7.243 

% 

2.227 

1.225 

495 

3.054 

21. 

12.83 

2.765 

1188 

7.331 

9. 

2:356 

1.257 

509 

3.142 

y* 

13.14 

2.796 

1202 

7.418 

Va 

2.489 

1.288 

523 

3.229 

H 

13.45 

2.827 

1216 

7.505 

H 

2.625 

1.319 

538 

3.316 

% 

13.76 

2.859 

1230 

7  593 

% 

2.765 

1.351 

552 

3.404 

22. 

14.08 

2.890 

1244 

7.HS0 

M. 

2.909 

1.382 

566 

3.491 

34 

14.40 

2.922, 

1259 

7.767 

M 

3.056 

1.414 

580 

3.578 

3^ 

14.73 

2.953 

1278 

7.854 

M 

3.207 

1.445 

594 

3.665 

% 

15.06 

2.985 

1287 

7.942 

% 

3.362 

1.477 

608 

3.75S 

23. 

15.39 

3.016 

1301 

8.029 

11. 

3.520 

1.508 

622 

8.840 

H 

15.72 

3.047 

1315 

8.116 

Vi 

3.682 

1.539 

637 

8.927 

H 

16.06 

3.079 

1329 

8.203 

yi 

3.847 

1.571 

651 

4.014 

% 

16.41 

3.110 

1343 

8.291 

H 

4.016 

1.602 

666 

4.102 

24. 

16.76 

3.142 

1357 

8.378 

J2. 

4.189 

1.634 

679 

4.189 

34 

17.11 

3.173 

1372 

8.465 

1 

4.365 

1.665 

693 

4.276 

H 

17.46 

3.204 

1386 

8.552 

4.545 

1.696 

707 

4.364 

H 

17.82 

3.236 

1400 

8.640 

H 

4.729 

1.728 

721 

4.451 

25. 

18.18 

3.267 

1414 

8.72T 

18. 

4.916 

1.759 

736 

4.538 

A4 

nbyd  = 

=  202  1] 

r.s.  gra 

Is. 

If  perclies  are  named  in  a  contract,  it  is  necessary,  in  order  to  prevent  fraud, 

to  specify  the  number  of  cub  feet  contained  in  the  perch  ;  for  stone-quarriers  have  one  perch,  stone- 
masons  another,  &c.  Engineers,  on  this  account,  contract  by  the  cubic  yard.  The  perch  should  be 
4one  away  with  entirely ;  Perches  of  25  cub  ft  X  .926  =  cub  yds  ;  and  cub  yds-r  .926—  pers  of  25  cub  ft 


CYLINDRIC   UNGULAS,    ETC. 


199 


€IR€IJI.AR  CYIilNDRIC  UNOUIiAS. 

I.  When  the  catting  plane  does  not  cnt  the  base.    Figs  13,  U 


area  of  base  0  5^  X  3^  sum  of  greatest  &  least  perp  heights,  on,  cm, 
area  of  cross  sec  measd  w  yi  sum  of  greatest  and  least  lengths, 
perp  to  sides,  as  a;,  


Volume 

of 

ungula 

Area  of 

ciirved 

surface 

Add  areas  of  ends  if  required. 

For  areas  of  sections  perpendicular  to  the  sides,  see  Circles. 
For  areas  of  sections  oblique  to  the  sides,  see  The  Ellipse. 


_  r  circumf  measd  perp 
~  I      to  sides,  as  at  x, 


gm,oty  meabd  along  the  sides. 

,  half  sum  of  greatest  and  least  lengths, 
gm,  0 1,  measd  along  the  sides. 


II.  When  the  catting  plane  toaches  the  base.    Figs  A  to  D. 


Tolame     Fig  A  = 

FigB: 
FigC  = 


=  (?^a68 — a  cX  area  adm&    of  base) • 

^^^  am 


(whether 
right  or 
oblique)v 


Figl> 


Carved  „.     ^ 

sarface  *  ig  ^  = 

{right  ^'^^  = 

ungula  Fig  C  = 

"■"^^  Flg»  = 


.  (^ab^  +  aeX&Te&admb    of  base) 

» l^  area  of  circle  ym   X^^ 

.  }4  volume  of  cylinder  xymn. 

1  (ah  X  my  ~~aeX  length  of  arc  dmb  )  - 

■  my  X  'rnn. 

:  {abXmy  -\-  aeX  length  of  arc  dm&  )  - 

•  14  circumference  of  bate    myXnin 

" "%  curved  surface  of  cylinder  xymn. 


m  n 

•meas'd  perp 

to  base 


JOO 


PYllAMIDS  AND  CONES, 


PYRAMIDS   AlTD   CONES. 

^  d        d 


'  '  '  T  5 

A  pyramid,  Figs.  1,  2,  3,  is  any  solid  which  has,  for  its  base,  a  plane  figure 
>f  any  number  of  sides,  and,  for  its  sides,  plane  triangles  all  terminating  at  one 
)oint  d,  called  its  apex^  or  top.  When  the  base  is  a  regular  figure,  the  pyramid 
s  regular ;  otherwise  irregular. 

A  cone,  Figs.  4  and  5,  is  a  solid,  of  which  the  base  is  a  curved  figure ;  and 
rhich  may  be  considered  as  made  or  generated  by  a  line,  of  which  one  end  is 
tationary  at  a  certain  point  d,  called  the  apex  or  top,  while  the  line  is  being 
arried  around  the  circumference  of  the  base,  which  may  be  a  circle,  ellipse, 
)r  other  curve.  A  cone  may  also  be  regarded  as  a  pyramid  with  an  infinite 
lumber  of  sides. 

The  axis  of  a  pyramid  or  cone,  is  a  straight  line  d  o  in  Figs.  1,  2,  4 ;  and  d  t  ia 
rigs.  3  and  5,  from  the  apex  d,  to  the  center  of  gravity  of  the  base.  When  the 
.xis  is  perpendicular  to  the  base,  as  in  Figs.  1,  2,  4,  the  solid  is  said  to  be  a  right 
>ne;  when  otherwise,  as  Figs.  3,  5,  an  oblique  one.  When  the  word  cone  is  used 
ilone,  the  right  circular  cone.  Fig.  4,  is  understood.  If  such  a  cone  be  cut,  as  at 
i,  obliquely  to  its  base,  the  new  base  1 1  will  be  an  ellipse ;  and  the  cone  dtt 
)ecomes  an  oblique  elliptic  one.  Fig.  5  will  represent  either  an  oblique  circular 
tone,  or  an  oblique  elliptic  one,  according  as  its  base  is  a  circle  or  an  ellipse. 

Volume  of  pyramid  or  cone,  regular  or  irregular,  right  or  oblique. 
Volume  =  3^  area  of  base  X  perpendicular  height  d  o.  Figs.  1  to  5. 

=  %  volume  of  prism  or  cylinder  having  same  area  of  base  and 

same  perpendicular  height. 
*=  ^  volume  of  hemisphere  (\f  same  base  and  same  height. 
Or,  a  cone,  hemisphere  and  cylinder,  of  the  same  base  and  same  height,  have 
rolumes  as  1,  2  and  3. 

Area  o^  surface  of  sides  of  right  regular  pyramid  or  right  circular  cone. 
Area  =.  }^  circumference  of  base  X  slant  height.*  ^ 
In  the  cone,  this  becomes  I  Add  area  of  base 

Area  =   area  of  base  ^  ^j^^^  ^^^^  t    if  required, 

radius  of  base  ^  "^  J 


Area  of  surface  of  oblique  elliptic  cone,  d  t  ty 

Fig.  5i,  cut  from  a  right  circular  cone,  d  s  s.  From  the  point 
c  where  the  axis  d  o  of  the  right  circular  cone  cuts  the  elliptic 
base  tt,  measure  a  perpendicular,  r,  in  any  direction,  to  the 
curved  surface  of  the  cone.  Let  v  =  the  voriume  of  oblique 
elliptic  cone,  dtt;  let  a  =  the  area  of  its  elliptic  base  t  i,  and 
let  h  =  the  height  d  u  measured  perpendicularly  to  said  base. 
Then 

,  ^  a  h         S  V 

Curved  surface  = = . 

r  r 

Add.  area  of  base  if  required 

No  measurement  has  been  devised  for  the  surface  of  an 
oblique  circular  cone. 

*In  the  pyramid,  this  slant  height  must  be  measured  along  the  middle  of  one 
)f  the  sides,  and  not  along  one  of  the  edges. 


PYRAMIDS  AND  CONES.  201 

To  find  tli«  BTurface  ot  an  irregular  pyramid. 

Whether  right  or  oblique,  each  side  must  be  calculated  as  a  separate  triangle  (see 
p.  148);  and  the  several  areas  added  together.    Add  the  area  of  base  if  required. 


FRUSTUMS  OF  PYRAMIDS  AND  CONES. 


Fig.  6.  Fig,  7. 

Frustum  of  pyramid  (Fig.  6)  or  of  eone  (Fig.  7)  with  hase  and  tcp 
parallel. 

Volume  (regular  or  irregular,  right  or  oblique) 

—  V  V  perpendicular  ^    /  area      i      area       i        /    area    v^     area   \ 
^  ^      height  oo      ^    ^of  top  "T"  ©f  base    •      V    of  top  ^  of  base/ 

—  !•  NX  perpendicular  ^  (  area      t      area       ,      *  ^,  ^^ea  of  a  section  \ 

—  >&  X      height  0  0      X  \  of  top  "T  of  base  "T  parallel  to,  and  midway  I 

^  between,  base  and  top  / 

•*■  (for  frustum  of  right  or  oblique  circular  cone  only;  See  Fig.  7) 
H  X  "^KfgM^i"  X  S.U1«  X   (o««  +  0,*  +  ot .  «,) 

Surface  of  frustum  of  right  regular  pyramid  or  cone,  with  top  and  base  parallel^ 
figs.  6  and  7. 

—  Xd^  /circumference     i    circumferenceX  y.      slant     * 
^V       of  top         T        of  base      )  ^  height  «< 

Add  areas  of  top  and  base  if  required. 

In  tile  frustum  of  a  right  circular  cone,  this  becomes 

Surface-  ^  1''^]!^^  -L    radius \    ^  , slant* 
V  of  top    I    of  base^    ^  height  tt 

(rr  —  3.1416)  .  Add  areas  of  top  and  base  if  required. 

Frustum  of  irregular  or  oblique  pyraniid  or  eone.     Surface  •■ 
rom  of  surfaces  of  sides,  each  of  which  must  be  treated  as  a  trapezoid. 

*  In  the  frustum  of  the  pvramid  (Fig  6)»  this  slant  height  must  be  measured  aloof 
th«  middle  of  one  of  the  ndet  (as  at  i<),  and  not  along  one  of  the  edges. 


202 


PRISMOIDS. 


PRISMOIl>gi. 


¥is,  1. 

,  A  prlsmold  is  sometimes  fteltnMl  at  a  mlid  haTing  for  Ita  ends  two  parallel 
plane  figures,  connected  by  other  plane  figures  on  which,  and  through  every  point 
of  which,  a  straight  line  may  be  drawn  from  one  of  the  two  parallel  ends  to  the 
other.  These  connecting  planes  may  be  parallelograms  or  not,  and  parallel  to  each 
other  or  not. 

Tills  definition  vroiQd  include  the  cube  and  all  other  parallelopipedi; 
the  prism;  the  cylinder  (considered  as  a  prism  having  an  infinite  number  of  sides); 
the  pyramid  and  cone  (in  which  one  of  the  two  parallel  ends,  i  e  the  one  forming  the 
apex,  is  considered  to  be  infinitely  small),  and  their  frustums  with  top  and  base 
parallel ;  and  the  wedge. 

But  the  use  of  the  term  prismoid  is  frequently  restricted  to  six-sided  eolida, 
in  which  the  two  parallel  ends  are  unequal  quadrangles;  and  the  connecting  planes, 
trapezoids ;  as  in  Figs.  1  and  2 ;  and,  by  some  writers,  to  cases  where  the  parallel 
quadrangular  ends  are  rectangles. 

The  following  ^prismoidal  fbrmnla"  applies  to  all  the  foregoing  solidi, 
and  to  others,  as  noted  below. 

Let  A  =-  the  area  of  one  of  the  two  parallel  ends. 

a  =    "  '*      the  other  of  the  two  parallel  ends. 

M  =    "         "a  cross  section  midway  between,  and  parallel  to^  th«  two 

parallel  ends. 
L  —  the  perpendicular  distance  between  the  two  parallel  ends; 
Then 
Volume  =  L  X  ^  +  ft^+'*M 

=  L  X  mean  area  of  cross  section. 


The  following  six  figures  represent  a  few  of  the  irregular  solids  which  fall  nnder  tht 
above  broad  definition  of  "  prismoid,'*  and  to  which  the  prismoidal  formula  applies 
They  may  be  regarded  as  one-chain  lengths  of  railroad  cuttings ;  a  o  being  the  length, 
er  perpendicular  (horizontal)  distance  between  the  two  parallel  (vertical)  ends. 


WEDGES. 


203 


Tine  prlsmoidal  formula  applies  also  to  the  sphere,  hemisphere,  and 
other  spherical  segments;  also  to  any  sections  such  as  a  6c  d,  and  onidbe,  of  the 


cone,  in  which  the  sides  ad,  ac,  or  od,  ic,  are  straight;  as  they  are  only  when  the 
•utting  plane  adc  passes  through  the  apex  or  top  a.    Also  to  tlie  cylinder 

when  a  plane  parallel  to  the  sides  passes  through  both  ends;  but  not  if  the  plane 
wz  is  oblique,  as  in  the  fig.,  though  never  erring  more  than  1  in  142.  In  tills  last 
case  we  must  imagine  the  plane  to  be  extended  until  it  cuts  the  side  of  the  cylinder 
likewise  extended;  and  then  by  page  199  find  the  solidity  of  the  ungula  thus  formed 
Then  find  the  solidity  of  the  small  ungula  above  w,  also  thus  formed,  and  subtract 
it  from  the  large  one. 

This  very  extended  applicability  of  the  prismoidal  formula  was  first  discovered. 
and  made  known,  by  Ellwood  Morris,  C.  E.,  of  Philadelphia,  in  1840. 


WEDGES. 

_b      a b        a 


A  xvedjge 

Is  usually  defined  to  be  a  solid,  Figs.  8  and  9,  generated  by  a  plane  triangle,  one, 
moving  parallel  to  itself,  in  a  straight  line.  This  definition  requires  that  the  two 
triangular  ends  of  the  wedge  should  be  parallel;  but  a  wedge  may  be  shaped  as  in 
Fig.  10  or  11.  We  would  therefore  propose  the  following  definition,  which  embraces 
ftll  the  figs.;  besides  various  modifications  of  them.  A  solid  of  five  plane  faces ;  one 
of  which  is  a  parallelogram  abed,  two  opposite  sides  of  which,  as  a c  and  b d,  are 
anited  by  means  of  two  triangular  faces  a  en,  and  bdm,  to  an  edge  or  line  nm, 
parallel  to  the  other  opposite  sides  ab  and  ed.  The  parallelogram  abed  may  be 
either  rectangular,  or  not ;  the  two  triangular  faces  may  be  similar,  or  not ;  and  the 
same  with  regard  to  the  other  two  faces.    The  following  rule  applies  equally  to  all: 

Volume        ^    ^     ^^'^^^IiZ^^}!f    v    Perphtpfrom    w        v  "^H^V^^ 
QiwOze   "">^    X      of  the  3  edges     X     edgetobiMsk      X        back  (a  6  cd) 
^*  ub+cd  +  nm  ougo  w  u«.».  meas'd  nerp  to  a  & 


202 


PmSMOIDS. 


PRISMOID 


Fig,  1. 

.  A  prlsmold  is  sometimes  clelin«cl  bm  a  solid  haTing  for  Ita  ends  two  parallil 
plane  figures,  connected  by  other  plane  figures  on  which,  and  through  every  point 
of  which,  a  straight  line  may  be  drawn  from  one  of  the  two  parallel  ends  to  the 
other.  These  connecting  planes  may  be  parallelograms  or  not,  and  parallel  to  each 
other  or  not. 

Tills  definition  -woiild  Include  the  cube  and  all  other  parallelopipedi; 
the  prism;  the  cylinder  (considered  as  a  prism  having  an  infinite  number  of  sides); 
the  pyramid  and  cone  (in  which  one  of  the  two  parallel  ends,  i  e  the  one  forming  the 
apex,  is  considered  to  be  infinitely  small),  and  their  frustums  with  top  and  base 
parallel ;  and  the  wedge. 

But  the  use  of  the  term  prisnioid  is  frequently  restricted  to  six-sided  solids, 
in  which  the  two  parallel  ends  are  unequal  quadrangles;  and  the  connecting  planes, 
trapezoids ;  as  in  Figs.  1  and  2 ;  and,  by  some  writers,  to  cases  where  the  parallel 
quadrangular  ends  are  rectangles. 

The  following  ^prlsmoldal  formitla"  applies  to  all  the  foregoing  solidi, 
and  to  others,  as  noted  below. 

Let  A  =-  the  area  of  one  of  the  two  parallel  ends. 

a  =    "  '*      the  other  of  the  two  parallel  ends. 

M  -=-    «         "a  cross  section  midway  between,  and  parallel  to,  th«  two 

parallel  ends. 
L  —  the  perpendicular  distance  between  the  two  parallel  eodflL 
Then 
Volume  -  L  X  ^  + '/ *" 

=»  L  X  mean  area  of  cross  section. 


The  following  six  figures  represent  a  few  of  the  irregular  solids  which  fall  nnder  fht 
above  broad  definition  of  "  prismoid,"  and  to  which  the  prismoidal  formula  applies 
They  may  be  regarded  as  one-chain  lengths  of  railroad  cuttings ;  a  o  being  the  length, 
er  perpendicular  (horizontal)  distance  between  the  two  parallel  (vertical)  ends. 


WEDGES. 


203 


T9ie  prismoidal  formula  applies  also  to  the  sphere,  hemisphere,  and 
other  spherical  segments;  also  to  any  sections  such  as  abed,  and  onidbc,  of  the 


eone,  in  which  the  sides  ad,  ac,  or  od,  ic,  are  straight;  as  they  are  only  when  the 
•utting  plane  adc  passes  through  the  apex  or  top  a.    Also  to   tlie  cylinder 

when  a  plane  parallel  to  the  sides  passes  through  both  ends ;  but  not  if  the  plane 
wz  is  oblique,  as  in  the  fig.,  though  never  erring  more  than  1  in  142.  In  this  last 
case  we  must  imagine  the  plane  to  be  extended  until  it  cuts  the  side  of  the  cylinder 
likewise  extended;  and  then  by  page  199  find  the  solidity  of  the  ungula  thus  formed. 
Then  find  the  solidity  of  the  small  ungula  above  w,  also  thus  formed,  and  subtract 
it  from  the  large  one. 

This  very  extended  applicability  of  the  prismoidal  formula  was  first  discovered, 
•nd  made  known,  by  Ellwood  Morris,  C.  E.,  of  Philadelphia,  in  1840. 


WEDGES. 

b       a b         a 


n  rn     ii  m  n  m    n  m 

Flgr.8.  Fig.  9.  Fig.KX  ¥ig.lL 


A  -wedge 

Is  usually  defined  to  be  a  solid,  Figs.  8  and  9,  generated  by  a  plane  triangle,  ane, 
moving  parallel  to  itself,  in  a  straight  line.  This  definition  requires  that  the  two 
triangular  ends  of  the  wedge  should  be  parallel ;  but  a  wedge  may  be  shaped  as  in 
Fig.  10  or  11.  We  would  therefore  propose  the  following  definition,  which  embraces 
all  the  figs.;  besides  various  modifications  of  them.  A  solid  of  five  plane  faces ;  one 
of  which  is  a  parallelogram  abed,  two  opposite  sides  of  which,  as  a c  and  6 d,  are 
nnited  by  means  of  two  triangular  faces  a  en,  and  bdm,  to  an  edge  or  line  nm, 
parallel  to  the  other  opposite  sides  ab  and  cd.  The  parallelogram  abed  maybe 
either  rectangular,  or  not ;  the  two  triangular  faces  may  be  similar,  or  not ;  and  ths 
same  with  regard  to  the  other  two  faces.    The  following  rule  applies  equally  to  all: 


Volume 

of  w«dse 


H   X 


Snm  of  Isngths 

of  the  3  edges 

ab  +ed  +  nm 


perp  ht  p  from 
edge  to  back 


width  of 

back  {abed) 

meas'd  nerp  to  a  & 


204  MENSURATION. 

SPHERES   OR   GLOBES. 
A  Spliere 

Ib  a  solid  generated  by  the  revolution  of  a  semicircle  around  its  diameter.  Every 
point  in  the  surface  of  a  sphere  is  equidistant  from  a  certain  point  called  the  center. 
Any  line  passing  entirely  through  a  sphere,  and  throagh  its  center,  is  called  its  oxt*, 
or  diameter.  Any  circle  described  on  the  surface  of  a  sphere,  from  the  center  oi 
the  sphere  as  the  center  of  the  circle,  is  called  a  great  circle  of  that  sphere ;  in  other 
words  any  entire  circumference  of  a  sphere  is  a  great  circle.  A  sphere  has  a  greater 
content  or  solidity  than  any  other  solid  with  the  same  amount  of  surface ;  so  that  i| 
the  shape  of  a  sphere  be  any  way  changed,  its  content  will  be  reduced.  The  inter- 
section of  a  sphere  with  any  plane  is  a  circle. 

Volume  of  sphere 

=  I  TT  radius  3  ^    4.1888    radius  3 

=  3^  TT  diameter*  =    0.5236   diameter* 

, ,  circumference  *  „ „    ,  ,  , 

"^   % 5 ""    0.01689  circumference* 

—  %  diameter  X  area  of  surface 

"^  ^  diameter  X  area  of  great  circle 
-=  %  volume  of  circumscribing  cylinder 
«=  0.5236  volume  of  circumscribing  cube. 

Axea  of  surface  of  sphere 

=  4  TT  radius*  -=    12.5664  radius* 

==»  TT  diameter*  =      3.1416  diameter* 

circumference*  «„,oo  x.  • 

—  ==      0.3183  circumference* 

TT 

=  diameter  X  circumference 

—  4  X  area  of  great  circle 

=  area  of  circle  whose  diameter  is  equal  to  twice  diameter  of  spheres 
=  curved  surface  of  circumscribing  cylinder 
6  X  volume 
diameter. 

Radius  of  sphere 


=  / 


Area  of  surface 


=    \/ .  07958  X  area  of  surface 


Circumference  of  sphere 

=    'v/e  7r2  volume  =•    '^^59.2176  volume 

=5    x/tt  »rea  of  surface  =    v^34416  area  of  surfaee 

area  of  surface 

"""         diameter. 


MENSURATION. 


SPHERES.    (Original.) 
Some  errors  of  1  ia  the  last  figure  only. 


i 

1 

is 

i 

1 

1 

a 

1 

2 

a 

.5 

1 

^ 

s 

a 

3 

o 

H 

a 

o 

5 

a 

s 

3 

o 

OQ 

OQ 

OQ 

QQ 

m 

OQ 

OQ 

OQ 

1-64 

.00077 

13-82 

18.190 

7.2949 

9i    170.87 

210.03 

H 

921.33 

2629.6 

1-32 

.00307 

.00002 

7-16 

18.666 

7.5829 

M    176.71 

220.89 

Ya 

934.83 

2687.6 

3-64 

.00690 

.00005 

15-32 

19.147 

7.87«3 

182.66 

232.13 

% 

948.43 

2746.5 

1-16 

.01227 

.00013 

H 

19.635 

8.1813 

H 

188.69 

243.73 

}4 

962.12 

2806.2 

3-32 

.02761 

.00043 

17-32 

20.129 

8.4919 

Ve 

194.83 

255.72 

% 

975.91 

2866.8 

H 

.04909 

.00102 

9-16 

20.629 

8.8103 

8. 

201.06 

268.08 

Ya 

989.80 

2928.2 

5-32 

.07670 

.00200 

19-32 

21.135 

9.1366 

3^ 

■207.39 

280.85 

% 

1003.8 

2990.5 

3-16 

.11045 

.00345 

% 

21.648 

9.4708 

H 

213.82 

294.01. 

18. 

1017.9 

3053.6 

7-32 

.15033 

.00548 

21-32 

22.166 

9.8131 

% 

220.36 

307.58 

% 

1032.1 

3117.7 

H 

.19635 

.00818 

11-16 

22.691 

10.164 

^ 

226.98 

321.56 

% 

1046.4 

3182.6 

9-32 

.24851 

.01165 

23-32 

23.222 

10.522 

H 

233.71 

335.95 

% 

1060.8 

3248.5 

5-16 

.30680 

.01598 

H 

23.758 

10.889 

H 

240.53 

350.77 

H 

1075.2 

3315.3 

11-32 

.37123 

.02127 

25-32 

24.302 

11.265 

K 

247.45 

366.02 

% 

1089.8 

3382.9 

H 

.44179 

.02761 

1316 

24.850 

11.649 

9. 

254.47 

381.70 

Ya 

1104.5 

3451.5 

13-82 

.51848 

.03511 

27-32 

25.405 

12.041 

H 

261.59 

397.83 

% 

1119.3 

3521.0 

7-16 

.60132 

.04385 

K 

25.967 

12.443 

268.81 

414.41 

19. 

1134.1 

3591.4 

15-32 

.69028 

.05393 

29-32 

26.535 

12.853 

% 

276.12 

431.44 

^ 

1149.1 

3662.8 

34 

.78540 

•06545 

15-16 

27.109 

13.272 

^ 

283.53 

448.92 

M 

1164.2 

3735.0 

17-32 

.88664 

.07850 

31-32 

27.688 

13.700 

% 

291.04 

466.87 

% 

1179.3 

3808.2 

9-16 

.99403 

.09319 

3. 

28.274 

14.137 

H 

298.65 

485.31 

yi 

1194.6 

3882.5 

19-32 

1.1075 

.10960 

1-16 

29.465 

15.039 

% 

306.36 

504.21 

% 

1210.0 

3957.6 

% 

1.2272 

.12783 

H 

30.680 

15.979 

10. 

314.16 

523.60 

Ya 

1225.4 

4033.7 

21-32 

1.3530 

.14798 

3-16 

31.919 

16.957 

H 

322.06 

543.48 

% 

1241.0 

4110.8 

11-16 

1.4849 

.17014 

H 

33.183 

17.974 

Ya 

330.06 

563.86 

20. 

1256.7 

4188.8 

23-32 

1.6230 

.19442 

5-16 

34.472 

19.031 

Vs 

338.16 

584.74 

H 

1272.4 

4267.8 

H 

1.7671 

.22089 

Vs 

35.784 

20.129 

H 

346.36 

606.13 

% 

1288.3 

4347.8 

25-32 

1.9175 

.24967 

716 

37.122 

21.268 

H 

354.66 

628.04 

% 

1304.2 

4428.8 

13-16 

2.0739 

.28084 

J4 

38.484 

22.449 

H 

363.05 

650.46 

H 

1320.3 

4510.9 

27-32 

2.2365 

.31451 

916 

39.872 

23.674 

K 

371.54 

673.42 

H 

1336.4 

4593.9 

M 

2.4053 

.35077 

H 

41.283 

24.942 

11. 

380.13 

696.91 

Ya 

1352.7 

4677.9 

29-32 

2.5802 

.38971 

11-16 

42.719 

26.254 

H 

388.83 

720.95 

% 

1369.0 

4763.0 

15-16 

2.7611 

.43143 

H 

44.179 

27.611 

H 

397.61 

745.5 1 

21. 

1885.5 

4849.1 

31-32 

2.9483 

.47603 

13-16 

45.664 

29.016 

% 

406.49 

770.64 

H 

1402.0 

4936.2 

1. 

3.1416 

.52360 

% 

47.173 

30.466 

34 

415.48 

796.33 

Ya 

1418.6 

5024.S 

1-32 

3.3410 

.57424 

15-16 

48.708 

31.965 

Vs 

424.56 

822.58 

% 

1435.4 

5113.5 

1-16 

3.5466 

.62804 

4. 

50.265 

33.510 

H 

433.73 

849.40 

H 

1452.2 

5203.7 

3-32 

3.7583 

.68511 

1-16 

51.848 

35.106 

% 

443.01 

876.79 

H 

1469.2 

5295.1 

H 

3.9761 

.74551 

^ 

53.456 

36.751 

12. 

452.39 

904.78 

Ya 

1486.2 

.5387.4 

5-32 

4.2000 

.80939 

316 

55.089 

38.448 

M. 

461.87 

933.34 

% 

1503.3 

5480.8 

3-16 

4.4301 

.87681 

H 

56.745 

40.195 

34 

471.44 

962.52 

22. 

1520.5 

5575.S 

7-32 

4.6664 

.94786 

5-16 

58.427 

41.994 

% 

481.11 

992.28 

H 

15379 

5670.8 

H 

4.9088 

1.0227 

% 

60.133 

43.847 

H 

490.87 

1022.7 

Ya 

1555.3 

5767.6 

9-32 

5.1573 

1.1013 

7-16 

61.863 

45.752 

% 

500.73 

1053.6 

% 

1572.8 

5865.1 

5-16 

5.4119 

1.1839 

H 

63.617 

47.713 

H 

510.71 

1085.3 

H 

1590.4 

5964.1 

11-32 

5.6728 

1.2704 

9-16 

65.397 

49.729 

% 

520.77 

1117.5 

% 

1608.2 

60«4.1 

^ 

5.9396 

1.3611 

% 

67.201 

51.801 

13. 

530.93 

1150.3 

Ya 

1626.0 

6165.2 

18-32 

6.2126 

1.4561 

11-16 

69.030 

53.929 

H 

541.19 

1183.8 

K 

1643.9 

6267.3 

7-16 

6.4919 

1.5553 

H 

70.888 

56.116 

Ya 

551.55 

1218.0 

23. 

1661.9 

6370.6 

15-32 

67771 

1.6590 

13-16 

72.759 

58.359 

% 

562  00 

1252.7 

H 

1680.0 

6475.0 

M 

7.0686 

1.7671 

% 

74.663 

60.663 

'%. 

572.55 

1288.3' 

1698.2 

6580.6 

17-32 

7.3663 

1.8799 

15-16 

76.589 

63.026 

% 

583.20 

1324.4 

% 

1716.5 

6687.3 

9-16 

7.6699 

1.9974 

6. 

78.540 

6.S.450 

H 

593.95 

1361.2 

}i 

1735.0 

6795.2 

19-32 

7.9798 

2.1196 

1-16 

80.516 

67.9.35 

Ji 

604.80 

1398.6 

% 

1753.5 

69042 

H 

8.2957 

2.2468 

H 

82.516 

70.482 

14. 

615.75 

1436.8 

Ya 

1772.1 

7014.3 

21-32 

8.6180 

2.3789 

3-16 

84.541 

73.092 

H 

626.80 

1475.6 

% 

1790.8 

7125.6 

11-16 

8.9461 

2.5161 

H 

86.591 

75.767 

Ya 

637.95 

1515.1 

24. 

1809.6 

7238.2 

23-32 

9.2805 

2.6586 

5-16 

88.664 

78.505 

% 

649.17 

1555.3 

H 

1828.5 

7351.9 

H 

9.6211 

2.8062 

H 

90.763 

81.308 

H 

660.52 

ft96.3 

Ya 

1847.5 

7466.7 

25-32 

9.9678 

2.9592 

7-16 

92.887 

84.178 

% 

671.95 

1637.9 

% 

1866.6 

7588.0 

13-16 

10.321 

3.1177 

34 

95.033 

87.113 

Ya 

683.49 

1680.3 

^ 

1885.8 

7700.1 

27-32 

10.680 

3.2818 

9-16 

97.205 

90.118 

% 

695.13 

1723.3 

1905.1 

7818.6 

% 

11.044 

3.4514 

% 

99.401 

93.189 

15. 

706.85 

1767.2 

?i 

1924.4 

7938.3 

29-32 

11.416 

3.6270 

1116 

101.62 

96.331 

M 

718.69 

1811.7 

=    ^ 

1943.9 

8059.2 

15-16 

11.793 

3.8083 

H 

103.87 

99.541 

730.63 

1857.0 

25. 

1963.5 

8181.3 

31-32 

12.177 

3.9956 

13-16 

106.14 

102.82 

yk 

742.65 

1903.0 

^ 

1983.2 

8304.7 

2. 

12.566 

4.1888 

Vb 

108.44 

106.18 

H 

754.77 

1949.8 

Ya 

2002.9 

8429.3 

*l-32 

12.962 

4.3882 

15-16 

110.75 

109.60 

% 

767.00 

1997.4 

% 

2022.9 

8554.9 

1-16 

13.364 

4.5939 

6. 

113.10 

113.10 

Ya 

779.32 

2045.7 

H 

2042.8 

8682.0 

3-32 

13.772 

4.8060 

H 

117.87 

120.31 

K 

791.73 

2094.8 

% 

2062.9 

8810.3 

H 
5-32 

14.186 

5.0243 

M 

122.72 

127.83 

16. 

804.25 

2144.7 

Ya 

2083.0 

8939.9 

14.607 

5.2493 

% 

127.68 

135.66 

% 

816.85 

2195.3 

% 

2103.4 

9070.6 

3-16 

15.033 

5.4809 

^ 

132.73 

143.79 

829.57 

2246.8 

26. 

2123.7 

9202.8 

7-32 

15.466 

5.7190 

% 

137.89 

152.25 

% 

842.40 

2299.1 

^ 

2144.2 

9336.2 

9.,1 

15.904 

5.9641 

H 

143.14 

161.03 

H 

855.29 

2352.1 

Ya 

2164.7 

9470.8 

16.349 

6.2161 

y» 

148.49 

170.14 

^ 

868.81 

2406.0 

% 

2185.5 

9606.7 

5-16 

16.800 

6.4751 

7. 

153.94 

170.59 

H 

881.42 

2460.6 

H 

2206.2 

9744.0 

11-82 

17.258 

6.7412 

'  Ji 

159.49 

189.39 

% 

894.6S 

2516.1 

% 

2227.1 

9882.1 

H 

17.721 

7.0144 

3 

166.18 

199.53 

17. 

907.93 

2572.4 

H 

2248.0 

10022 

206 


MENSURATION. 


SPHERES  —  (Continued.) 

a" 

1 

is 

i 

1 

2 

a 

.2 

1 

i 

a 

i 

2 

o 

S 

"o 

Q 

3 

o 

Q 

a 

o 

3 

cc 

02 

QQ 

02 

OS 

02 

a 

m 

Va 

2269.1 

10164 

% 

4214.1 

25724 

% 

6756.5 

52222 

Y 

9896.0 

92570 

27. 

2290.2 

10306 

4243.0 

25988 

P 

6792.9 

52645 

Y 

9940.2 

93190 

% 

2311.5 

104:0 

•u 

4271.8 

26254 

6829.5 

53071 

% 

9984.4 

93812 

M 

233-2.8 

10595 

37. 

4300.9 

26522 

6866.1 

53499 

Y 

10029 

94438 

% 

2354.3 

10741 

H 

4330.0 

26792 

% 

6902.9 

53929 

10073 

96066 

H 

2375.8 

10889 

M 

4359.2 

27063 

47. 

6939.9 

54362 

% 

10118 

95697 

% 

2397.5 

11038 

% 

4388.6 

27337 

Y. 

6976.8 

547S7 

Va 

10163 

96330 

Vi 

2419.2 

11189 

3^ 

4417.9 

27612 

Yi 

7013.9 

55234 

57. 

10207 

96967 

K 

2441. 1 

11341 

% 

4447.5 

27889 

% 

7050.9 

55674 

Y 

10252 

97606 

28. 

2463.0 

11494 

H 

4477.1 

28168 

H 

7088.3 

66115 

10297 

98248 

14 

2485.1 

11649 

% 

4506.8 

28449 

7125.6 

5655^ 

% 

10342 

98893 

M 

2507.2 

11805 

38. 

4536.5 

28731 

% 

7163.1 

57006 

Y 

10387 

99641 

% 

2529.5 

11962 

M 

4566.5 

29016 

% 

7200.7 

67455 

10432 

100191 

"A 

2551.8 

12121 

Yi 

4596.4 

29302 

48. 

7238.3 

57906 

H 

10478 

100845 

y% 

2574.3 

12281 

% 

4626.5 

29590 

% 

7276.0 

58360 

Vb 

10523 

101501 

H 

2596.7 

12443 

^ 

4656.7 

29880 

M 

7313.9 

58815 

58. 

10568 

102161 

Ve 

2619.4 

12606 

% 

4686.9 

30173 

% 

7351.9 

69274 

Y 

10614 

102825 

29. 

2642. 1 

12770 

% 

4717.3 

30466 

Y. 

7389.9 

59734 

Y 

10660 

103488 

H 

2665.0 

12936 

% 

4747.9 

30762 

% 

7428.0 

60197 

% 

10706 

104155 

2687.8 

13103 

39. 

4778.4 

31059 

% 

7466.3 

60663 

Y 

10751 

104836 

y 

2710.9 

13272 

H 

4809.0 

31359 

% 

7504.6 

61131 

10798 

105499 

■L/ 

2734.0 

13442 

4839.9 

31661 

49. 

7543.1 

61601 

% 

10844 

106175 

% 

2757.3 

13614 

% 

4870.8 

31964 

Y 

7581.6 

62074 

K 

10890 

106854 

% 

2780.5 

137fc7 

}i 

4901.7 

32270 

Y 

7620.1 

62549 

59. 

10936 

107536 

% 

2804.0 

13961 

% 

4932.7 

32577 

H 

7658.9 

63026 

Y 

10983 

108221 

80. 

2827.4 

14137 

H 

4S64.0 

32886 

Y 

7697.7 

63506 

11029 

108909 

% 

2851.1 

14315 

% 

4995.5 

33197 

% 

7736.7 

63969 

% 

11076 

109600 

Vi 

2874.8 

14494 

40. 

5026.6 

33510 

H 

7775.7 

64474 

Y 

11122 

110294 

% 

2898.7 

14674 

^ 

5058.1 

33826 

H 

7814.8 

64861 

Y 

11169 

110990 

yi 

2922.5 

14856 

M 

5089.6 

34143 

50. 

7854.0 

65450 

H 

11216 

111690 

% 

2946.6 

15039 

% 

5121.3 

3-1462 

Y 

7893.8 

65941 

'    % 

11263 

112892 

,  % 

2970.6 

15224 

H 

5153. 1 

34783 

7932.8 

66436 

60. 

11310 

113098 

% 

2994.9 

15411 

% 

5184.9 

351C6 

% 

7972.2 

66234 

Y 

11357 

113806 

SI. 

3019.1 

15599 

% 

6216.8 

35431 

Y 

8011.8 

67433 

11404 

114618 

^ 

3043.6 

15788 

% 

5248.9 

35758 

% 

8051.6 

67935 

% 

11452 

115232 

yi 

3068.0 

15979 

41. 

6281.1 

36087 

% 

8091.4 

68439 

Y 

11499 

115949 

% 

3092.7 

16172 

H 

5313.3 

36418 

H 

8131.8 

68946 

Y 

11547 

116669 

^ 

3117.3 

16366 

M 

5345.6 

36751 

51. 

8171.2 

69456 

H 

11596 

117892 

% 

3142.1 

16561 

Ys 

5378.1 

37086 

Y 

8211.4 

69967 

Y 

11642 

118118 

% 

3166.9 

16758 

^ 

5410.7 

37428 

Y 

8251.6 

70482 

61. 

11690 

118847 

K 

3192.0 

16957 

% 

6443.3 

37763 

% 

8292.0 

70969 

Y 

11738 

n957» 

«2. 

3217.0 

17157 

H 

5476.0 

38104 

Y 

8332.3 

71519 

Y 

11786 

120315 

H 

3242.2 

17359 

% 

6508.9 

88448 

Y 

8372.8 

72040 

% 

11884 

121063 

U 

3267.4 

17563 

^2. 

5541.9 

38792 

H 

8418.4 

72565 

Y 

11882 

121794 

% 

3292.9 

17768 

^ 

6574.9 

39140 

K 

8454.1 

73092 

Y 

11931 

122588 

^. 

3318.3 

17974 

5608.0 

39490 

52. 

8494.8 

73622 

H 

119F0 

128286 

3343.9 

18182 

% 

5641.3 

39841 

Y 

8535.8 

74154 

Y 

12028 

124036 

^ 

3369.6 

18392 

y4 

5674.5 

40194 

8576.8 

74689 

62. 

12076 

124789 

K 

3395.4 

18604 

% 

5708.0 

40551 

% 

8617.8 

75226 

Y 

12126 

125645 

33. 

3421.2 

18817 

% 

6741.5 

40908 

Y 

8658.9 

75767 

12174 

126305 

^ 

3447.3 

19032 

K 

5775.2 

41268 

% 

8700.4 

76309 

% 

12223 

127067 

H 

3478.3 

19248 

43. 

5808.8 

41630 

% 

8741.7 

76854 

Y 

12272 

12788J 

% 

8499.5 

19466 

H 

5842.7 

41994 

% 

8783.2 

77401 

% 

12322 

128601 

'A 

3525.7 

19685 

5876.5 

42360 

53. 

8824.8 

77952 

% 

12371 

129873 

H 

3552. 1 

19907 

% 

5910.7 

42729 

Y 

8866.4 

78505 

Y 

12420 

130147 

H 

3578.5 

20129 

r^ 

5944.7 

43099 

Y 

8908.2 

79060 

63. 

12469 

130925 

Yh 

3605.1 

20354 

% 

5978.9 

43472 

% 

8950.1 

79617 

Y 

12519 

131706 

84. 

3631.7 

20580 

% 

6013.2 
6047.7 

43846 

Y 

8992.0 

80178 

12568 

132490 

% 

3658.5 

20808 

% 

44224 

% 

9034.1 

80741 

% 

12618 

133277 

3685.3 

21037 

44. 

6082.1 

44602 

H 

9076.4 

81308 

Y 

12668 

134067 

% 

3712.3 

21268 

H 

6116.8 

449S4 

Vs 

9118.5 

81876 

Y 

12718 

134860 

^ 

3739.3 

21501 

Yi 

6151.5 

45367 

54. 

9160.8 

82448 

% 

12768 

135657 

% 

3766.5 

21736 

% 

6186.3 

45753 

Y 

9203.8 

83021 

Y 

12818 

136466 

% 

3793.7 

21972 

Yl 

6221.2 

46141 

Y 

9246.0 

83598 

64. 

12868 

137259 

K 

3821.1 

22210 

% 

62,56.1 

46530 

% 

9288.5 

84177 

Y 

12918 

188065 

86. 

3848.5 

22449 

% 

6291.2 

46922 

Y 

9381.2 

84760 

Ya. 

12969 

138874 

H 

3876.1 

22691 

% 

6326.5 

47317 

% 

9374.1 

85344 

% 

13019 

139686 

}4. 

3903.7 

22934 

45. 

6361.7 

47713 

% 

9417.2 

86931 

Y 

13070 

140501 

% 

3931.5 

23179 

H 

6397.2 

48112 

% 

9460.2 

86521 

% 

13121 

141320 

H 

3959.2 

23+25 

M 

6432.7 

48513 

55. 

9503.2 

87114 

H     13172 

142142 

H 

3987.2 

23674 

% 

6168.3 

48916 

Y 

9546.5 

87709 

K  13222 

142966 

H 

4015.2 

23924 

A 

6503.9 

49321 

9590.0 

88307 

65.    13273 

143794 

% 

4043.8 

24  J  76 

% 

6539.7 

49729 

% 

9633.8 

88908 

Y     13324 

144625 

36. 

4071.5 

24429 

H 

6575.5 

50139 

Y 

9676.8 

89511 

Y\    13376 

145460 

H 

4099.9 

246R5 

% 

6611.6 

.50551 

H 

9720.6 

90117 

%\    13427 

146297 

4128.3 

24942 

46. 

6647.6 

60965 

H 

9764.4 

90726 

H'    13478 

147138 

% 

4IS6.9 

25201 

H 

6683.7 

51382 

% 

9808.1 

91338 

%     13530 

147983 

H 

4186.5 

25461 

Ya 

6720.0 

61801 

56. 

9852.0 

91953 

?i' 

1358?   1 

148829 

MENSURATION. 


207 


SPHERES— (Continued.) 


a 

1 

^ 

S3 

1 

j3 

i 

1 

1 

1 

s 

o 

3 

9 

p 

m 

m 

s 

OQ 

CO 

5 

QQ 

m 

5 

CO 

CO 

Va 

15633 

149680 

34 

17437 

216505 

H 

21708 

300743 

% 

26446 

40440S 

66/* 

13685 

150533 

y% 

17496 

217597 
218693 

M 

21773 

.302100 

% 

26518 

406060 

H 

13737 

151390 

% 

17554 

% 

21839 

303463 

92. 

26590 

407721 

13789 

152251 

% 

17613 

219792 

^ 

21904 

304831 

34 

26663 

409384 

% 

13841 

153114 

75. 

17672 

220894 

% 

21970 

306201 

Ya 

26735 

411054 

3^ 

13893 

153980 

H 

17731 

222001 

% 

22036 

307576 

% 

26808 

412736 

% 

13946 

154850 

M 

17790 

223111 

K 

22102 

308957 

34 

26880 

414405 

% 

13998 

155724 

% 

17849 

224224 

84. 

22167 

310340 

26953 

416086 

Va 

14050 

156600 

^ 

17908 

225341 

^ 

22234 

311728 

% 

27026 

417774 

67. 

14103 

157480 

% 

17968 

226463 

H 

22300 

318118 

Ya 

27099 

419464 

H 

14156 

158363 

H 

18027 

227588 

% 

22366 

314514 

93. 

27172 

421161 

34 

14208 

159250 

% 

18087 

228716 

H 

22432 

315915 

^ 

27245 

422862 

% 

14261 

160139 

76. 

18146 

229848 

% 

22499 

317318 

27318 

424567 

M 

14314 

161032 

% 

18206 

230984 

% 

22565 

318726 

H 

27391 

426277 

H 

14367 

161927 

34 

18266 

232124 

% 

22632 

320140 

34 

27464 

427991 

H 

14420 

162827 

% 

18326 

233267 

85. 

22698 

321556 

% 

27538 

429710 

H 

14474 

163731 

34 

18386 

234414 

H 

22765 

322977 

H 

27612 

431433 

68. 

14527 

164637 

% 

18446 

235566 

22832 

324482 

Ys 

27686 

433160 

H 

14580 

165547 

% 

18506 

236719 

% 

22899 

325831 

94. 

27759 

434894 

H 

14634 

166460 

Vh 

18566 

237879 

34 

22966 

327264 

% 

27833 

436630 

% 

14688 

167376 

77. 

18626 

239041 

H 

23034 

328702 

Ya 

27907 

438373 

^ 

14741 

168295 

H 

18687 

24020C 

H 

23101 

330142 

Ys 

27981 

440118 

% 

14795 

169218 

18748 

241376 

% 

23168 

331588 

H 

28055 

441871 

% 

14849 

170145 

% 

18809 

242551 

86. 

23235 

333039 

% 

28130 

443625 

% 

14903 

171074 

J4 

18869 

243728 

H 

23303 

33449* 

Yi 

28204 

445387 

69. 

14957 

172007 

% 

18930 

244908 

Ya 

23371 

335951 

X 

•28278 

447151 

H 

15012 

172944 

H 

18992 

24609S 

% 

23439 

337414 

95. 

28353 

448920 

H 

15066 

173888 

Vb 

19053 

247283 

23506 

338882 

H 

28428 

45069J 

% 

15120 

174838 

78. 

19114 

248475 

% 

23575 

340352 

Ya 

28503 

452475 

14 

15175 

175774 

ii 

19175 

249672 

H 

23643 

341829 

% 

28577 

454259 

% 

15230 

176728 

34 

19237 

250873 

K 

23711 

343307 

H 

28652 

456047 

% 

15284 

17767T 

% 

19298 

252077 

87. 

23779 

344792 

% 

28727 

457839 

K 

15339 

178635 

34 

19360 

253284 

H 

23847 

346281 

Ya 

28802 

459638 

70. 

15394 

179595 

H 

19422 

254496 

23916 

347772 

Ys 

28878 

461439 

H 

15449 

180559 

H 

19483 

255713 

% 

23984 

349269 

96. 

28953 

463248 

H 

15504 

181526 

% 

19545 

256932 

34 

24053 

350771 

H 

29028 

465059 

15560 

182497 

79. 

19607 

258155 

24122 

352277 

Ya 

29104 

466875 

14 

15615 

183471 

^ 

19669 

259383 

H 

24191 

353785 

% 

29180 

466697 

% 

15670 

184449 

34 

19732 

260618 

% 

24260 

355301 

^ 

29255 

470524 

%/ 

15726 

185430 

% 

19794 

261848 

88. 

24328 

356819 

Ya 

29331 

472354 

h 

15782 

186414 

34 

19856 

263088 

34 

24398 

358342 

Ya 

29407 

474189 

71. 

15837 

187402 

% 

19919 

264330 

% 

24467 

359869 

Yb 

29483 

476029 

H 

15893 

188394 

% 

19981 

265577 

% 

24536 

361400 

97. 

29559 

477874 

15949 

189389 

% 

20044 

266829 

34 

24606 

362935 

^ 

29636 

479725 

% 

16005 

190387 

80. 

20106 

268083 

% 

24676 

364476 

M 

29712 

481579 

1^ 

16061 

191389 

H 

20170 

269342 

H 

24745 

366019 

h 

29788 

483433 

k£ 

16117 

192395 

20232 

270604 

Vs 

24815 

367568 

34 

29865 

485302 

H 

16174 

193404 

Yi 

20296 

271871 

89. 

24885 

369122 

% 

29942 

487171 

% 

16230 

194417 

H 

20358 

273141 

H 

24955 

370678 

Ya 

30018 

489046 

n. 

16286 

195433 

% 

20422 

274416 

u 

25025 

372240 

Ys 

30095 

490924 

H 

16343 

196453 

H 

20485 

275694 

H 

25095 

373806 

98. 

30172 

492808 

H 

16400 

197476 

% 

20549 

276977 

}4 

25165 

375378 

^ 

30249 

494695 

% 

16456 

198502 

81. 

20612 

278263 

% 

25236 

376954 

Ya 

30326 

496588 

^ 

16513 

199532 

34 

20676 

279553 

% 

25306 

378531 

Yb 

30404 

498486 

% 

16570 

200566 

34 

20740 

280847 

Ys 

25376 

380115 

^ 

•  30481 

500388 

% 

16628 

201604 

y% 

20804 

282145 

90. 

25447 

381704 

% 

30558 

502296 

3 

16685 

202645 

34 

20867 

283447 

H 

25518 

383297 

Ya 

30636 

504208 

T3.^ 

16742 

203689 

% 

20932 

284754 

25589 

384894 

K 

.80713 

506125 

H 

16799 

204737 

H 

20996 

286064 

% 

25660 

386496 

99. 

30791 

508047 

16857 

205789 

% 

21060 

287378 

H 

25730 

388102 

H 

30869 

509975 

az 

16914 

206844 

82. 

21124 

188696 

Yi 

25802 

389711 

34 

30947 

511906 

\Z 

16972 

207903 

34 

21189 

290019 

H 

25873 

391327 

% 

31025 

51384S 

% 

17030 

208966 

M 

21253 

291345 

% 

25944 

392946 

H 

31103 

515785 

H 

17088 

210032 

% 
34 

21318 

292674 

91. 

26016 

894570 

% 

31181 

517730 

% 

17146 

211102 

21382 

294010 

H 

26087 

396197 

% 

31259 

519682 

U. 

17204 

212175 

% 

21448 

295347 

34 

26159 

397831 

% 

31338 

521638 

H 

17262 

213252 

% 

21512 

296691 

% 

26230 

399468 

100. 

81416 

523596 

34 

17320 

214333 

% 

21578 

298036 

Y, 

26302 

401109 

% 

17379 

215417 

83. 

21642 

299388 

Ya 

26374 

402756 

208 


SEGMENTS,  ETC.,  OF  SPHERES. 


To  find  tlie  solidity  of  a  spberical  seg^ment. 

RuLK  1.'  Square  the  rad  on,  of  its  base;  mult  this  square  by  3  ;  i 
the  prod  add  the  square  of  its  height  o  s  ;  mult  the  sum  by  the  heigl 
0  s ;  and  mult  this  last  prod  by  .5236. 

Rule  2.  Mult  the  diam  a'b  of  the  sphere  by  3 ;  from  the  pro 
take  twice  the  height  o  s  of  the  segment;  mult  the  rem  by  the  squai 
ef  the  height  o  s ;  and  mult  this  prod  by  .5236. 

The  solidity  of  a  sphere  being  %Aa  that  of  its  circumscribing  cy  lii 
der,  if  we  add  to  any  solidity  in  the  table,  its  half,  we  obtain  ths 
of  a  cylinder  of  the  same  diam  aa  the  sphere,  and  whose  heigi 
equals'its  diam. 


To  find  tbe  curved  surface  of  a  spherical  segment. 

Rule  1.  Mult  the  diam  a  6  of  the  sphere  from  which  the  segment  is  cut,  by  3.1416 
malt  the  prod  by  the  height  o  «  of  the  seg.  Add  area  of  base  if  reqd.  Rem.  Having  the  diam  n 
•f  the  seg,  and  its  height  o  s,  the  diam  a  6  of  the  sphere  may  be  found  thus:  Div  the  square  of  hal 
the  diam  n  r,  by  its  height  o  s  ;  to  the  quot  add  the  height  o  s.  Rule  2.  The  curved  surf  of  eithe 
•  segment,  lait  Fig,  or  of  a  zone,  (next  Pig,)  bears  the  same  proportion  to  the  surf  of  the  whol 
■phere,  that  the  height  of  the  seg  or  zone  bears  to  the  diam  of  the  sphere.  Therefore,  first  find  th 
•urf  of  the  whole  sphere,  either  by  rule  or  from  the  preceding  table ;  mult  it  by  the  height  of  the  ae 
or  zone ;  div  the  prod  bv  dianl  of  sphere.  Rule  3.  Mult  the  circum.f  of  the  sphere  by  the  height  o 
9£  the  seg. 

To  find  the  solidity  of  a  spherical  zone 

Add  together  the  square  of  the  rad  e  d,  the  square  of  rad  o  I 
and  3^d  of  the  square  of  the  perp  height  eo;  mult  the  sum  b 
1.5708;  and  mult  this  prod  by  the  height  0  0.  * 

To  find  the  curved  surface  of  a  spher- 
ical zone. 

Bulk  1.  Mult  together  the  diam  mn  of  the  sphere ;  the  heigh 
e  0  of  the  zone,  and  the  number  3.1416.  Or  see  preceding  Rule 
for  surf  of  segments.  Rule  2.  Mult  the  circumf  of  the  sphere,  b 
the  height  of  the  zone.  * 

To  find  the  solidity  of  a  hollow  spher 
ieal  shell. 

Take  from  the  foregoing  table  the  solidities  of  two  aphtres  havin 
the  diams  a  b,  and  c  d.  Subtract  the  least  from  th*sr«atest.  fier 
aooebcliBthe  thickac«s  of  th«  akalL 


THE  EI.I.IPSOID,  OR  SPHEROID, 

Is  a  solid  generated  by  the  revolution  of  an  ellipse  around  either  its  long  or  its  short  diam.     Win 

around  the  long  (or  transverse)  diam,  as  at  a.  Fig  1,  it  is  an  oblongs  or  pre 
late  spheroid;  when  around  the  short  (or  conjug'ate)  one,  as  at  w,  in  Fig 
it  is  oblate. 


Flg.l. 


Fig.  2. 


For  the  solidity  in  either  case,  mult  the  fixed  diam  or  axis  by  the  squar 

of  the  revolving  one ;  and  mult  the  prod  by  .5236. 

*  This  rule  applies,  whether  the  zone  includes  the  equator  (as  in  our  figure),  o 
not,  as  in  the  earth's  temperate  zones. 


PARABOLOIDS,   ETC. 


20y 


THE   PARABOI.OID,  OR  PARABOLIC  CONOID, 

Qext  Fig,  is  a  solid  generated  by  the  revolution  of  a  parabola  a  cb,  around  its  axis,  c  r. 

For  its  solidity  mult  the  area  of  its  base,  by  half  its  height,  r  c.    Or  miilt 

togeilier  the  square  of  the  rad  o  r  of  the  base;  the  height  r  c;  and  the  number  1.5708. 


For  tlie  solidity  of  a  frnstmii, 

ui  g  h,  the  ends  of  which  are  perp  to  the  axis  r  c ;  add  together  tlM 
squares  of  the  two  diams  a  b  and  g  h;  mult  the  sum  by  the  height  r  I; 
mult  the  prod  by  the  deciiaal  .3927. 

To  find  tbe  surface  of  a  paraboloid, 

Mult  the  rad  a  r  of  its  base,  by  6.2832;  div  the  proi  by  12  times  the 
square  of  the  height  re,  call  the  quotp.  Then  add  together  the  square 
«f  the  rad  a  r,  and  4  times  the  square  of  the  height  r  c.  Cube  the 
aam ;  take  the  sq  rt  of  this  cube ;  from  the  sq  rt  subtract  the  cube  of 
the  rad  a  r.     Mult  the  rem  by  ^. 

Either  the  solidity,  or  the  surface  of  a  frustum,  ah  g  h,  when  ghis 
parallel  to  a  h,  may  be  found  by  calculating  for  the  whole  paraboloid, 
aad  for  the  upper  portion  c  g  h,  &a  two  separate  paraboloids,  and  taking  their  diff. 

THE    CIRCUIiAR    SPINDI.E, 

|a  «  solid  ab  ny  generated  by  the  revolution  of  a  circular  segment 
mh  ne  a,  around  its  chord  a  n  as  an  axis. 

To  find  its  solidity. 

RlTLS  1.   First  find  the  area  of  a  5  e,  or  half  the  generating  circular  i 
segment.    Then  to  the  square  of  a  e,  add  the  square  of  6  e;  div  the  sum 
by  6  y;  from  the  quot  take  6  e;  mult  the  rem  by  the  area  of  a  e  6  ;  call  , 
the  prod  p.   Cube  a  e ;  div  the  cube  by  3 ;  from  the  quot  take  p.   Mult  the  / 
rem  by  12.5664. 

Rule  2.   When  th«  dlst  o  e  is  known,  from  the  center  of  the  circle  to  \ 
the  cen  of  the  spindle,  then  mult  that  dist  o  e,  by  tbe  area  of  a  6  e ;  call  \ 
the  prod  p ;  cube  a  « ;  div  the  cube  by  3 ;  from  the  quot  take  p ;  mult  the 
rem  by  12.5664. 

To  find  its  surface. 

RuLB  1.  First  find  the  length  of  the  circular  arc  a  5  n ;  and  mult  it  by 
the  dist  o  e  from  the  center  of  the  circle  to  the  center  of  the  spindle.  Call 

the  prod  p.    Next  mult  the  length  a  n  of  the  spindle,  by  the  rad  o  6  of  the  circle.    Prom  the  prod 
take  p ;  mult  the  rem  bv  6.2832. 

RuLK  2.  First  find  the  length  of  the  arc  abn.  Square  a  e;  also  square  b  e;  add  these  squares 
together;  div  their  sum  hjby;  call  the  quot  «;  and  mult  it  by  a  n;  call  the  prod  p.  Next  from  stake 
6  e  ;  malt  the  rem  by  the  length  of  the  arc  abn.    Subtract  the  prod  from  p ;  mult  the  rem  by  6.2832. 

To  find  tlie  solidity  of  a  middle  zone  of  a  circular  spindle, 

AMhekp 

/  (a  e«  — ^)  X  S  «)  -  (o  e  X  area  of  f  ft  i  *)  J  X  6.2882. 


CIRCriiAR  RIUTGS. 

^^1^^^      area  of  cross  section  of  bar  ^  i  sura  of  inner  and  outer  ^  g  j^^g^^ 
Volume  =>     of  which  ring  is  made     ^      diameters,  «  a  and  6  &     a  o.i*io»* 

tUnrface  = 
14 


circumference  of  bar  ^  J  sum  of  inner  and  outer  ^  3  ia^km 
of  which  rin«  is  made  ^     diameters,  a  a  and  66     ^    ' 


210  SPECIFIC   GRAVITY. 


SPEOIFIO  GRAVITY. 

1.  The  specific  gravity,  or  relative  density,  D*,  of  a  substance. 
is  the  ratio  between  the  weight,  W,  of  any  given  volume  of  that  substance  and 
the  weight,  A,  of  an  equal  volume  of  some  substance  adopted  as  a  standard  ol 

comparison.    Or:    D  =  — . 

2.  For  gaseous  substances,  the  standard  substance  is  air,  at  a  temper- 
ature  of  0°  Cent.  =  32°  Fahr.,  with  barometer  at  760  millimeters  =  29.922  inches. 

3.  For  solids  and  liquids,  the  standard  substance  is  distilled  water,  at  its 
temperature  (4°  Cent.  =  39.2°  Fahr.)  of  maximum  density. 

4.  For  all  ordinary  purposes  of  civil  engineering,  any  clear  fresh 
water,  at  any  ordinary  temperature,  may  be  used.  Even  with  water  a1 
30°  Cent.,  ^  86°  Fahr.,  the  result  is  only  4  parts  in  1000  too  great. 

5.  When  a  body  is  immersed  in  water,  the  upward  force,  or  **  buoyancy," 
exerted  upon  it  by  the  water,  or  the  **loss  of  weight "  of  the  body,  due  to  its 
immersion,  is  equal  to  the  weight  of  the  water  displaced  by  the  immersion  ol 
the  body  f ;  or,  if 

W  =  the  weight  of  the  body  in  air, 
w  =  its  weight  in  water, 
D  =  its  relative  density  or  specific  gravity, 
A  =  the  weight  of  water  displaced  ; 
W  W 

then  A  =  W  —  w,  and  D  =  —  =  .=: . 

AW  —  w 

6.  Since  the  volume,  V,  of  a  body,  of  given  weight,  W,  is  inversely  as  ita 
density,  or  specific  gravity,  D ;  the  specific  gravity  is  equal  also  to  the  ratio 
between  the  volume  Vg  of  an  equal  weight  of  the  standard  substance,  to  the 

y 
volume,  V,  of  the  body  in  question ;  or  D  =:  ^^ 

7.  The  specific  gravities  of  substances  heavier  than  water  are  ordi- 
narily determined  by  weighing  a  mass  of  the  substance,  first  in  air  (obtain- 
ing its  weight,  W),  and  then  when  the  mass  is  completely  submerged  in  water 

(obtaining  its  diminished  weight,  w).    Then  D  =       ,  as  in  ^5. 

8.  If  the  body  is  lighter  than  water,  it  must  be  entirely  immersed, 
and  held  down  against  its  tendency  to  rise.     Its  weight,  w,  in  water,  or  ita 

,  upward  tendency,  is  then  a  negative  quantity,  and  means  must  be  provided  for 
measuring  it,  as  by  making  it  act  upward  against  the  scale  pan.  We  then  have, 
A  =  W  —  {—w)=W+w;  or 

Loss  due  to  immersion  =  weight  of  body  in  air,  plus  its  buoyancy. 

9.  Or,  first  allow  the  body  to  float  upon  the  water,  and  note  the  resulting  dis- 
placement, V,  of  water,  as  by  the  rise  of  its  surface  level  in  a  prismatic  vessel. 
Then  immerse  the  body  completely,  and  again  note  the  displacement,  V.*  Now 
r,  the  volume  displaced  by  the  body  when  floating,  and  V,  the  volume  displaced 
by  the  body  when  completely  immersed,  are  proportional  respectively  to  the 
weight,  W,  of  the  body,  and  to  the  weight,  W  —  w?,  of  a  mass  of  water  of  equal 

W  V 

volume  with  the  body.    Hence  D  =  — - —  =  -,. 

W  —  w       V 

10.  Or,  attach  to  the  light  body,  b,  a  heavier  body,  or  sinker,  S,  of  such  den- 
sity and  mass  that  both  bodies  together  will  sink  in  water.  Let  W  be  the 
weight  of  the  light  body,  6,  in  air ;  Q  the  weight  of  both  bodies  in  air,  and  q 
their  combined  weight  in  water.  Then  Q  —  g  =  the  weight  of  a  mass  of  water 
of  equal  volume  with  the  two  bodies,  and  Q  —  W  =  the  weight,  S,  of  the  sinker 
in  air.  By  immersing  the  sinker  alone,  find  the  weight,  k,  of  water  equal  in 
volume  to  the  sinker  alone,  =  loss  of  weight  in  sinker,  due  to  immersion. 
Then,  for  the  weight.  A,  of  water  of  equal  volume  with  the  light  body,  b,  or  for 

♦Strictly  speaking,  "  specific  gravity  "  refers  to  weight,  and  "  relative  density  " 
to  mass  (see  Mechanics,  Art.  14  a);  but,  as  specific  gravity  and  density  are 
numerically  equal,  they  are  often  treated  as  identical. 

t  See  Hydrostatics,  Art.  18. 


SPECIFIC   GRAVITY.  211 

the  loss  of  weight  of  6,  due  to  immersion,  we  have  A  =  Q  —  q  —  k;  and,  for 

W                   W 
the  specific  gravity,  D,  of  the  light  body,  h.  we  have  D  =  — -,   =  ~ 

where  w  —  the  (unknown)  buoyancy  of  b. 

11.  A  granular  body,  as  a  mass  of  saw-dust,  gravel,  sand,  cement,  etc., 
or  a  porous  body,  as  a  mass  of  wood,  cinder,  concrete,  sandstone,  etc.,  is  a  com- 
posite body,  consisting  partly  of  solid  matter  and  partly  of  air.  Thus,  a  cubic 
foot  of  quartz  sand  weighs  about  100  lbs.;  while  a  cubic  foot  of  quartz  weighs 
about  165  ft)s. 

12.  The  specific  gravity  of  porous  substances  is  usually  taken 
as  that  of  the  composite  mass  of  solid  and  air.  Thus,^  a  wood,  weighing  (with 
its  contained  air)  62.5  lbs.  per  cubic  foot,  or  the  same  as  water,  is  said  to  have  a 
specific  gravity  of  1.  The  absorption  of  water,  when  such  bodies  are  immersed 
for  the  purpose  of  determining  their  specific  gravities,  may  be  prevented  by  a 
thin  coat  of  varnish. 

13.  The  specific  gravity  of  granular  substances  is  sometimes  taken 
as  that  of  tbe  solid  part  alone.  Thus,  Portland  cements  ordinarily  weigh  (in 
air)  from  75  to  90  lbs.  per  cubic  foot,  which  would  correspond  to  specific  gravities 
of  from  1.20  to  1.44;  but  the  specific  gravity  of  the  solid  portion  ranges  from 
3.00  to  3.25  ;  and  the  latter  figures  are  usually  taken  as  representing  the  specific 
gravities. 

14.  In  determining  the  specific  gravities  of  substances  (such  as  cement) 
which  are  soluble  in  water  or  otherwise  affected  by  it,  the  substances  are 
weighed  in  some  liquid  (such  as  benzine,  turpentine  or  alcohol)  which  will  not 
affect  them,  instead  of  in  water.  The  result,  so  obtained,  must  then  be  multi- 
plied by  the  ratio  between  the  density  of  the  liquid  and  that  of  water. 

15.  The  specific  gravity  of  a  liquid  is  most  directly  determined  by 
weighing  equal  volumes  of  the  liquid  and  of  water. 

16.  Or  weigh,  in  the  liquid,  some  body,  whose  weight,  W,  in  air,  and  whose 
specific  gravity,  d,  are  known.  Let  u/  =  its  weight  in  the  liquid.  Then,  for 
the  specific  gravity,  D,  of  the  liquid,  we  have 

W'.W  —  v/^d'.T);   orD=  ~^-y- ' . 

17.  Or,  let  the  body,  in  ^  16  (weighing  W  in  air),  weigh  w  in  water,  and  (as 
before)  w^  in  the  liquid  in  question.  Then,  since  specific  gravity  of  water  =  1, 
we  have 

W v/ 

W  —  w:W  —  w'  =  1:1)',   orD  =  ^^ —. 

W  —  w 

18.  The  specific  gravities  of  liquids  are  commonly  obtained  by  observing  the 
depth  to  which  some  standard  instrument  (called  a  hydremeter)  sinks  when 
allowed  to  float  upon  the  surface  of  the  liquid.  The  greater  the  depth,  the  less 
the  specific  gravity  of  the  liquid.  In  Beaum^'s  hydrometer  the  depth 
of  immersion  is  shown  by  a  scale  upon  the  instrument.  The  graduations  of  the 
scale  are  arbitrary.  For  liquids  heavier  than  water,  0°  corresponds  to  a  specific 
gravity  of  1,  and  76°  to  a  specific  gravity  of  2.  For  liquids  lighter  than  water, 
10°  correspond  to  a  specific  gravity  of  1,  and  60°  to  a  specific  gravity  of  0.745. 

19.  In  Twaddell's  hydrometer,  used  for  liquids  heavier  than  water, 

5  X  No.  of  degrees  4- 1,000 
specific  gravity  = ^m 

Thus,  if  the  reading  be  90° 

5  X  90  +  1,000        1,450 
specific  gravity  =  ^^ =  ITOOO  =  ^•^^• 

30.  In  Nicholson's  hydrometer,  largely  used  also  for  solids,  the  specific 
gravity  is  deduced  from  the  weights  required  to  produce  a  standard  depth  of 
immersion.  It  consists  of  a  hollow  metal  float,  from  which  rises  a  thin  but  stiff 
wire  carrying  a  shallow  dish,  which  always  remains  above  water.  From  the 
float  is  suspended  a  loaded  dish,  which,  like  the  float,  is  always  submerged.  On 
the  wire  supporting  the  upper  dish  is  a  standard  mark,  which,  in  observations, 
is  always  brought  to  the  surface  of  the  water.  The  specific  gravity  is  then  deter- 
mined by  means  of  the  weights  carried  in  the  two  dishes  respectively. 

31.  The  determination  of  the  specific  gravities  of  gaseous  sub- 
stances requires  the  skill  of  expert  chemists. 


212 


SPECIFIC   GRAVITY. 


Table  of  specific  gravities,  and  weig-tits. 

In  this  table,  the  sp  gr  of  air,  and  gases  also,  are  compared  with  that  of  water 
instead  of  that  of  air ;  which  last  is  usual. 


The  specific  gravity  of  any  substance  is  =  its  wcigrlit 
in  grams  per  cubic  centimetre. 


Air,  atmospheric ;  at  60°  Fah,  and  under  the  pressure  of  one  atmosphere  or 

14.7  fts  per  sq  inch,  weighs  -g-^-j-  part  as  much  as  water  at  60° 

Alcohol,  pure 

' '        of  commerce 

'*        proof  spirit 

Ash,  perfectly  dry.  , average.. 

1000  ft  board  measure  weighs  1.748  tons. 

Ash,  American  white,  dry «' 

1000  ft  board  measure  weighs  1.414  tons. 

Alabaster,  falsely  so  called;  but  really  Marbles 

*'  real;  a  compact  white  plaster  of  Paris average.. 

Aluminium 

Antimony,  cast,  6.66  to  6.74 average  .. 

"  native " 

Anthracite.     See  Coal,  below. 

Asphaltum,  1  to  1.8 "     •• 

Basalt.    See  Limestones,  quarried "     •• 

Bath  Stone,  Oolite "     .. 

Bismuth,  cast.     Also  native "     •• 

Bitumen,  solid.     See  Asphaltum. 

Brass,  (Copper  and  Zinc,)  cast,   7.8  to  8.4 "     .. 

"     rolledi "     .. 

Bronze.    Copper  8  parts;  Tin  1.    (Gun  metal.)    8.4  to  8.6 **     •• 

Brick,  best  pressed 

"■       common  hard 

"       soft,  in  ferior 

Brickwork.     See  Masonry. 

Boxwood,  dry •'     .. 

Calcite,  transparent **     .. 

Carbonic  Acid  Gas,  is  13.^  times  as  heavy  as  air "     .. 

Cement.    (See  IT  13.) 

«*    Portland,  3.00  to  3.25.- ~ 

"    Natural,  2.75  to  3.00 

Chalk,  2.2  to  2.8.    See  Limestones,  quarried 

Charcoal,  of  pines  and  oaks 

Cheriy,  perfectly  dry 

Chestnut,  perfectly  dry 

Coal.    See  also  page  215. 

Anthracite,  1.3  to  1.7 

"  piled  loose 

Bituminous,  1.2  to  1.4 

piled  loose 


Coke .. 


piled  loose 

In  coking,  coals  swell  from  25  to  50  per  cent. 

Copper,  cast 8.6  to  8.8 

rolled, 8.8  to  9.0 

Crystal,  pure  Quartz.     See  Quartz. 
Cork.. 


Diamond,  3.44  to  3.55  ;  usually  3.51  to  3.55 

Barth  j  common  loam,  perfectly  dry,  loose 

"  "  *'  "  "     shaken 

"  **  "  '*  "     moderately  rammed.. 

•*  "  "    slightly  moist,  loose 

"  "  "    more  moist. 


shaken 

*•  moderately  packed 

as  a  soft  flowing  mud 

as  a  soft  mud,  well  pressed  into  a  box.. 


Ether  

Elm,  perfectly  dry.  . . . . , 

1000  ft  board  measure  weighs  1 

Ebony,  dry 

Emerald,  2.63  to  2.76 

Fat. 


Flint 

Feldspar,   2.5  to  2.8 

Garnet,  3.5  to  4.3 ;  Precious,  4.1  to  4.3 

Glass,  2.5  to  3.45 

"     common  window 

"  Millville,  New  Jersey.  Thick  flooring  glass  . 
Granite,  2.56  to  2.88.    See  Limestone,  160  to  180 


Average 
SpGr. 


.716 
.56 


2.6 
2.65 
4.2 

2.98 
2.52 
2.53 
2.72 


SPECIFIC  GRAVITY. 


213 


Table  of  specific  gravities,  and  iFeiglits  —  (Continued.) 


The  specific  gravity  of  any  substance  is  =  its  weigbt 
ill  g^rams  per  cubic  centimetre. 


Gneiss,  common,  2.62  to  2.76 

"      in  loose  piles '• 

'•      Hornblendic «♦ 

"  "  quarried,  in  loose  piles " 

Gypsum,  Plaster  of  Paris,    2.24  to  2.30 ♦• 

"        in  irregular  lumps «'-    .. 

"         ground,  loose,  per  struck  bushel,  70 " 

"  "        well  shaken,    "         "        80 " 

"  "       Calcined,  loose,  per  struck  bush,  65  to  76- " 

Greenstone,  trap,  2.8  to  3. 2 ♦' 

"  "      quarried,  in  loose  piles " 

Gravel,  about  the  same  as  sand,  which  see. 

Gold,      cast,  pure,  or  24  carat " 

"         native,  pure,  19.3  to  19.34 ' " 

"  "        frequently  containing  silver,  15.6  to  19.3.. " 

"         pure,  hammered,  19.4  to  19.6. " 

Gutta  Percha " 

Hornblende,  black,  3.1  to  3.4 " 

Hydrogen  Gas,  is  1434  times  lighter  than  air;   and  16  times  lighter  than 

oxygen average.. 

Hemlock,  perfectly  dry.  " 

1000  feet  board  measure  weighs  .930  ton. 

Hickory,  perfectly  dry.  " 

1000  feet  board  measure  weighs  1.971  tons. 
Iron,  and  steel. 

««    Pig  and  cast  iron  and  cast  steel m..m.m.....*... 

«<    Wrought  iron  and  steel,  and  wire,  7.6  to  7.9 

Ivory average . . 

Ice,  .917  to  .922 "       .. 

India  rubber '* 

Lignum  vitae,  dry " 

Lard ....   " 

Lead,  of  commerce,  11.30  to  11.47 ;  either  rolled  or  cast " 

Limestones  and  Marbles,  2.4  to  2.86,  150  to  178.8 

"  '*  •'        ordinarily  about 

"  "  "        quarried  "in  irregular  fragments,  1  cub  yard  solid, 

makes  about  1.9  cub  yds  perfectly  loose ;  or  about 
1^  yds  piled.  In  this  last  case,  571  of  the  pile 
is  solid;  and  the  remaining  .429  part  of  it  is 

voids piled . . 

Lime,  quick,  ground,  loose,  per  struck  bushel  62  to  70  Tbs 

"  "  "        well  shaken,     "        *•     ....80      *'    

"  "  "        thoroughly  shaken,  "     ....93Ji  "    

Mahogany,  Spanish ,  dry  * average . . 

"  Honduras,  dry " 

Maple,  dry» , *' 

Marbles,  see  Limestones. 

Masonry,  of  granite  or  limestone?,  well  dressed  throughout 

"         •'        "      well-scabbled  mortar  rubble.     About  4  of  the  mass 

will  be  mortar 

"  "        '*      well-scabbled  dry  rubble ". 

"  "        "      roughly  scabbled  mortar  rubble.    About  %tf>  %  part 

will  be  mortar 

"  *'        "      roughly  scabbled  dry  rubble 

At  155  fts  per  cub  ft,  a  cub  yard  weighs  1.868  tons  j  and  14.45  cub  ft, 
1  ton. 
Masonry  of  sandstone ;  about  %  part  less  than  the  foregoing. 

"        "  brickwork,  pressed  brick,  fine  joints average.. 

"        "  "  medium  quality " 

"        "  "  coarse ;  inferior  soft  bricks " 

At  125  fts  per  cub  ft,  a  cub  yard  weighs  1.507  tons;  and  17.92  cub 
ft,  1  ton. 

Mercury,  at  32°  Fah 

60O     "  

212°     " 

Mica.  2.75  to  3.1 

Mortar,  hardened,  1.4  to  1.9 

M ud ,  dry,  close 

' '     wet,  moderately  pressed 

"    wet,  fluid 


Average 
Sp  Gr. 


19.5 
.98 
3.25 


7.2 
7.75 
1.82 


*  Green  timbers  usually  weigh  from  one-fifth  to  nearly  one-half  more  than 
dry ;  and  ordinary  building  timbers  when  tolerably  seasoned  about  one-sixth  more  than  perfectly  dry. 


214 


SPECIFIC   GRAVITY. 


Table  of  specific  griravities,  and  weights  — (Continued.) 


The  specific  gravity  of  any  substance  is  =  its  iveig'lit 
ill  g^rams  per  cubic  centimetre. 


Naphtha 

Nitrogen  Gas  is  about  ^^  part  lighter  than  air 

Oak,  liTe,  perfectly  dry,  .'88  to  1.02* average.. 

"    white,      "         "      .66  to    .88 " 

"    red,  black,  &c* " 

Oils,  whale;  olive " 

"     ofturpentiae " 

Oolites,  or  Roestones,  1.9  to  2.5 " 

Oxygen  Gas,  a  little  more  than  J^  part  heavier  than  air 

Petroleum 

Peat,  dry,  unpressed 

Pine,  white,  perfectly  dry,  .35  to  .45* 

1000  ft  board  measure  weighs  .930  ton.* 

"     yellow,  Northern,  .48  to  .62 

1000  ft  board  measure  weighs  1.276  tons.* 

"  "        Southern,  .64  to  .80 

1000  ft  board  measure  weighs  1.674  tons.* 
Pine,  heart  of  long-leafed  Southern  yellow,  unseas.  ... 

1000  ft  board  measure  weighs  2.418  t*ns. 

Pitch 

Plaster  of  Paris  ;  see  Gypsum. 

Powder,  slightly  shaken 

Porphyry,  2.66  to  2.8 .•. .    

Platinum 21  to  22 

"        native,  in  grains 16  to  19 

Quartz,  common,  pure 2.64  to  2.67 

•'  "         finely  pulverized,  loose 

"  "  "  "         well  shaken 

"  "  "  "         well  packed 

"    quarried,  loose.    One  measure  solid,  makes  full  1^   broken  and 

piled 

Ruby  and  Sapphire,  3.8  to  4.(^^ 

Rosin 

Salt.., 


Average 
SpQr. 


.95 

.77 


.92 
.87 
2.2 
.00136 

.878 


.40 

.55 

.72 

1.04 

1.15 

1. 

2.73 
21.5 
17.5 

2,65 


3.41 
*2.*6** 

.59 
2.6 


Sand,  pure  quartz,  perfectly  dry,  loose 

*«  "         "  '♦  "      slightly  shaken 

"  '*         •*       rammed,  dry 

Natural  sand  consists  of  grains  of  different  sizes,  and  weighs  more,  per 
unit  of  volume,  than  a  sand  sifted  from  it  and  having  grains  of 
uniform  size.    Sharp  sand  with  very  large  and  very  small  grains 

may  weigh  as  much  as 

Sand  is  very  retentive  of  moisture,  and,  when  in  large  bulk,  its  natural 
moisture  may  diminish  its  weight  from  5  to  10  per  cent. 

"    perfectly  wet,  voids  full  of  water 

Sandstones,  fit  for  building,  dry,  2.1  to  2.73, 131  to  171. 

"■  quarried,  and  piled,  1  measure  solid,  makes  about  1^  piled... 

Serpentines,  good 2.5  to  2.66 

Snow,  fresh  fallen = 

"      moistened,  and  compacted  by  rain 

Sycamore,  perfectly  dry,  

1000  ft  board  measure  weighs  1.376  tons. 

Shales,  red  or  black 2.4  to  2.8 average.. 

"        quarried,  in  piles " 

Slate 2.7  to  2.9 "      .. 

Silver "      .. 

Soapstone, or  Steatite 2.65  to  2.8 "      .. 

Steel,  7. 7  to  7.9.     The  heaviest  contains  least  carbon " 

Steel  is  not  heavier  than  the  iron  from  which  It  is  made;  unless  the 
iron  had  impurities  which  were  expelled  during  its  conversion  into 
steel. 

Bnlph  ur average . . 

Spruce,  perfectly  dry.  •* 

1000  ft  board  measure  weighs  .930  ton. 

Spelter,  or  Zinc 6.8  to  7.2 "      .. 

Sapphire;  and  Ruby,  3.8  to  4.... " 

Tallow " 

Tar '* 

Trap,  compact,  2.8  to  3.2 "      .. 

"     quarried  ;  in  piles ; " 

Topaz.  3.45to3.65 "      .. 

♦Green  timbers  usually  weigh  from  one-fifth  to  nearly  one-half  more  than 

dry ;  and  ordinary  building  timbers  when  tolerably  seasoned  about  one-sixth  more  than  perfectly  dry. 


2.8 
10.5 
2.73 
7.85 


7.00 
3.9 
.94 


WEIGHT   OF    COAL. 


215 


Table  of  speeific  g^ravities,  and  wei^bts  —  (Continued.) 


The  specific  gravity  of  any  substance  is  =  its  iveig^llt 
in  g^rams  per  cubic  centimetre. 


Tin,  cast,  7.2  to  7.5 average.. 

Turf,  or  Peat,  dry,  unpressed 

Water.    See  page  3'26.  

W.ix,  bees average., 

Wiues,  .993  to  l.Oi <'      .. 

Walnut,  black,  perfectly  dry. " 

1000  ft  board  measure  weighs  1.414  tons. 

Zinc,  or  Spelter,  6.8  to  7.2 ««      .. 

Zircon,  4.0  to  4.9 " 


.97 

.998 
.61 

7.00 
4.45 


Average 

Wt  of  a 

Cub  Ft. 

Lbs. 


459. 

20  to  30 

62.417 

60.5 

62.3 

38. 


Space  occupied  by  coal.    In  cubic  feet  per  ton  of  2240  pounds. 
Pennsylvania  Anthracite. 


Bro- 
ken. 

Egg. 

Stove. 

Nut. 

Pea. 

Buck- 
wheat. 

38.6 

39.2 

39.8 

40.5 

41.1 

39.4 

39.6 

39.6 

39.6 

39.8 

39.8 

39.0 

39.6 

40.2 

40.8 

41.5' 

39.6 

39.6 

39.6 

41.2 

41.9 

42.4 

39.3 

39.9 

40.5 

41.2 

41.9 

"  39.0 

39.9 

42.6 

45.7 

46.5 

47.7 

39.6 

40.3 

40.9 

41.6 

42.3 

40.0 

40.5 

41.1 

41.7 

42.3 

( 

44.8 

45.2 

45.7 

46.2 

46.7 

144.2 

44.3 

44.3 

45.0 

46.1 

46.5 

40.0 

39.8 

39.4 

39.4 

38.8 

38.5 

38.4 

42.1 

41.4 

38.5 

38.8 

40.1 

40.3 

40.3 

40.5 

.3:  di 

St,  39.] 

Aver- 
age. 


Hard  white  ash* 

Free-burning  white  ash  *.. 
Shamokiu  * 


Schuylkill  white  ash  *.. 
"         red        "   *.. 

Lykens  Valley* 


Wyoming  free-burningf 

Lehigh  t 

Lehigh  ;  Reading  C.  &  I.  Co.  *. 
Lehigh:!  Lump,  40.5;  cupola,  40.3 


39.6 
40.2 
40.7 
40.6 
43.6 
40.9 
41.1 
45.7 
45.1 
39.7 
40.0 
39.7 


Bituminous. 


From  Coxe  Bros. 

&Co.t 

From  Jour.  U.  S.  Ass'n  Charcoal  Iron  Workers. 
Vol.  Ill,  1882.§ 

Pittsburg 

Erie 

48.2 

46.6 

Pittsburg 47.1 

Cumberland,  max 42.3 

min 41.2 

Blossburg,  Pa 42.2 

Clover  Hill,  Va 49.0 

Ricbmond,  Va. 

45  4 

(Midlothian) 41.0 

Ohio  Cannel 

Indiana  Block 

Illinois 

45.5 

51.1 

47.4 

Caunelton,  Ind 47.0 

Pictou,N.  S 45.0 

Sydney,  Cape  Breton  .47.0 

Logarithm. 
1  cubic  foot  per  ton  of  2240  pounds  =  _ 

0.89286  cubic  foot  per  ton  of  2000  pounds 1.950  7820 

2240  (exact)  pounds  per  cubic  foot 3.350  2480 

1  cubic  foot  per  ton  of  2000  pounds  = 

1.12  (exact)  cubic  feet  per  ton  of  2240  pounds 0.049  2180 

2000  (exact)  pounds  per  cubic  foot 3.301  0300 

1  pound  per  cubic  foot  = 

2240  (exact)  cubic  feet  per  ton  of  2240  pounds 3.350  2480 

2000        "  "  "  2000        "      3.301  0300 


*From  Edwin  F.  Smith,  Sup't  &  Eng'r,  Canal  Div.,  Phila.  and  Reading  R.  R. 

fFrom  very  careful  weighings  in  the  Chicago  yards  of  Coxe  Bros.  &  Co. 
Note  the  irregular  variation  with  size  of  anthracite  in  Coxe  Bros,'  figures. 

g  Quoted  from  The  Mining  Record.  On  the  authority  of  "  many  years'  experi- 
ence" of  "a  prominent  retail  dealer  in  Philadelphia,"  the  Journal  gives  also 
figures  requiring  from  4  to  13  per  cent,  less  volume  per  ton  than  those  here 
quoted  from  the  Journal  and  from  other  authorities. 


216  WEIGHTS   AND  MEASURES, 

WEIGHTS  AND  MEASURES. 

United  States  and  Britisb  measures  of  leng-tli  and  weigbt, 

of  the  same  denomination,  may, /or  all  ordinary  purposes  ^  be  considered  as  equal ; 
but  the  liquid  and  dry  measures  of  the  same  denomination  differ  widely 
in  the  two  countries.  Tlie  standard  measure  of  leng-tli  of  both  coun- 
tries is  theoretically  that  of  a  pendulum  vibrating  seconds  at  the  level  of  the 
sea,  in  the  latitude  of  London,  in  a  vacuum,  with  Faiirenheit's  thermometer  at 
62°.  The  length  of  such  a  pendulum  is  supposed  lo  be  divided  into  39.1393 
equal  parts,  called  inches;  and  36  of  these  inches  were  adopted  as  the  standard 
yard  of  both  countries.  But  the  Parliamentary  standard  having  been  destroyed 
Dy  fire,  in  1834,  it  was  found  to  be  impossible. to  restore  it  by  measurement  of  a 
pendulum.  The  present  British  Imperial  yard,  as  determined,  at  a  temperature 
of  62°  Fahrenheit,  by  the  standard  preserved  in  the  Houses  of  Parliament,  is 
the  standard  of  the  United  States  Coast  and  Geodetic  Survey,  and  is  recognized 
as  standard  throughout  the  country  and  by  the  Departments  of  the  Govern- 
ment, although  not  so  declared  by  Act  of  Congress.  The  yard  between  the  27th 
and  63d  inches  of  a  scale  made  for  the  U.  S.  Coast  Survey  by  Troughton,  of  Lon- 
don, in  1814,  is  found  to  be  of  this  standard  length  when  at  a  temperature  of 
59°.62  Fahrenheit ;  but  at  62°  is  too  long  by  0.00083  inch,  or  about  1  part  in  43373, 
or  1  46  inch  per  mile,  or  0.0277  inch  in  100  feet. 

The  Coast  Survey  now  uses,  for  purposes  of  comparison,  two  measures  pre- 
sented by  the  British  Government  in  1855,  as  copies  of  the  Imperial  standard, 
namely : 

"  Bronze  standard.  No.  11 ;"  of  standard  length  at  62°. 25  Fahr. 
"  Malleable  iron  standard,  No.  57  ;"  "        "        "        62°  10      " 

See  Appendix  No.  12,  Report  of  U.  S.  Coast  and  Geodetic  Survey  for  1877. 
The  legral  standard  of  iveiffht  of  the  United  States  is  the  Troy 

pound  of  the  Mint  at  Philadelphia.  This  standard,  containing  5760 
grains,  is    an    exact   copy  of  the  Imperial   Troy   pound    of  Oreat 

Britain.  The  avoirdupois  or  commercial  pound  of  the  United  States,  con- 
taining 7000  grains,  and  derived  from  the  standard  Troy  pound  of  the  Mint,  is 
found  to  agree  within  one  thousandth  of  a  grain  with  the  British  avoirdupois 
pound.  The  U.  S.  Coast  Survey  therefore  declares  the  weights  of  the  two  coun- 
tries identical. 

The  Ton.  In  Revised  Statutes  of  the  United  States,  2d  Edition,  1878,  Title 
XXXIV,  Collection  of  Duties  upon  Imports,  Chapter  Six,  Appraisal,  says : 

"Sec.  2951,  Wherever  the  word  *ton'  is  used  in  this  chapter,  in  reference  to 
weight,  it  shall  be  construed  as  meaning  twenty-hundredweight,  each  hundred- 
weight being  one  hundred  and  twelve  pounds  avoirdupois." 

This  appears  to  be  the  only  U,  S.  Government  regulation  on  the  subject. 

The  ton  of  2240  ft)s  (often  called  a  gross  ton  or  long  ton)  is  commonly 
used  in  buying  and  selling  iron  ore,  pig  iron,  steel  rails  and  other  manufactured 
iron  and  steel.  Coke  and  many  other  articles  are  bought  and  sold  by  the  net 
ton  or  short  ton  of  2000  lbs.  The  bloom  ton  had  2464  ft)s,  =  22*40  lbs  -4-  2 
hundredweight  of  112  ft)s  each  ;  and  the  pig  iron  ton  had  2268  lbs,  =  2240  lbs  +  » 
"sandage"  of  28  tbs,  or  one  "quarter,"  to  allow  for  sand  adhering  to  the  pigs, 
but  some  furnace  men  allowed  only  14  ft)s.  In  electric  traction  woi'k  the  ton 
means  2000  fts. 

As  a  measure,  the  ton,  or  tun,  is  defined  as  252  gallons,  as  40  cubic  feet  of 
round  or  rough  timber  or  in  ship  measurement,  or  as  50  feet  of  hewn  timber.  252 
U.  S.  gallons  of  water  weigh  about  2100  lbs  ;  252  Imperial  gallons  about  2500  ft)s ; 
50  cub  ft  yellow  pine  about  2500  lbs. 

The  metric  system  *  ivas  legalized  in  the  United  States  in 

*  The  metric  system,  as  compared  with  the  English,  has  much  the  same  advantages 
and  disadvantages  that  our  American  decimal  coinage  has  in  comparison  with  the 
English  monetary  system  of  pounds,  shillings  and  pence.  It  will  enormously  facili- 
tate all  calculations,  but,  like  all  other  improvements,  it  will  necessarily  cause  some 
inconvenience  while  the  change  is  being  made.  The  metric  system  has  also  tMs  fur- 
ther and  very  great  advantage,  that  it  bids  fair  to  become  universal  among  c^ilizeo 
rations. 


WEIGHTS   AND   MEASURES.  217 

1866,  but  has  not  been  made  obligatory.  The  government  has  since  furnished 
very  exact  metric  standards  to  the  several  States.  The  use  of  the  metric  system 
has  been  permitted  in  Great  Britain,  beginning  with  August  6,  1897,  and  in 
Russia,  beginning  with  1900.  Its  use  is  now  at  least  permissive  in  most  civil- 
ized nations. 

The  metric  unit  of  lengftli  is  tbe  metre,  or  meter^  which  was 

Intended  to  be  one  ten-millionth  f  J  of  the  earth's  quadrant,  i.e.,  of 

that  portion  of  a  meridian  embraced  between  either  pole  and  the  equator.  This 
length  was  measured,  and  a  set  of  metrical  standards  of  weight  and  measure 
were  prepared  in  accordance  with  the  result,  and  deposited  among  the  archives 
^f  France  at  Paris  (Mfetre  des  Archives.  Kilogramme  des  Archives,  etc.).  It  has 
since  been  discovered  that  errors  occurred  in  the  calculations  for  ascertaining 
the  length  of  the  quadrant ;  but  the  standards  nevertheless  remain  as  originally 
prepared. 

Tbe  metric  measures  of  surface  and  of  capacity  are  the  squares 
and  cubes  of  the  meter  and  of  its  (decimal)  fractions  and  multiples. 

Tlie  metric  unit  of  weig-lit  is  tlie  g-ramme  or  gram,  which  is 
the  weight  of  a  milliliter  or  cubic  centimeter  *  of  pure  water  at  its  tempera- 
ture of  maximum  density,  about  4.5°  Centigrade  or  40°  Fahrenheit. 

By  the  concurrent  action  of  the  principal  governments  of  the  world,  an  In- 
ternational Bureau  of  Weiglits  and  Measures  has  been  estab- 
lished, with  its  seat  near  Paris.  It  has  prepared  two  ingots  of  pure  platinum- 
iridium,  from  one  of  which  a  number  of  standard  kilograms  (1000  grams)  havf 
been  made,  and  from  the  other  a  number  of  standard  meter  bars,  both  derived 
from  the  standards  of  the  Archives  of  Frauce.  Of  these  copies,  certain  ones 
were  selected  as  international  standards,  and  the  others  were  distributed  to  the 
different  governments.  Those  sent  to  the  United  States  are  in  the  keeping  of 
the  U.  S.  Coast  Survey. 

The  determination  of  the  equivalent  of  the  meter  in  Eng-lish 
measure  is  a  very  difficult  matter.  The  standard  meter  is  measured  /row  end 
to  end  of  a  platinum  bar  and  at  the  freezing  point ;  whereas  the  standard  yard  is 
measured  between  two  lines  drawn  on  a  silver  scale  inlaid  in  a  bronze  bar,  and  at 
62°  Fahrenheit.  The  United  States  Coast  Survey  f  adopts,  as  the 
length  of  the  meter  at  62°  Fahrenheit,  the  value  determined  by  Capt.  A.  R. 
Clarke  and  Col.  Sir  Henry  James,  at  the  office  of  the  British  Ordnance  Survey, 
in  1866,  viz. :  39.370432  inches  {=  3.2808666  +  feet  =  1.0936222  +  yards) ;  but  the 
lawful  equivalent,  established  by  Congress,  is  39.37  inches  (=  3.28083  feet 
=  1.093611  yards).  This  value  is  as  accurate  as  any  that  can  be  deduced  from 
existing  data. 

The  gram  weighs,  by  Prof.  W.  H.  Miller's  determination,^  15.43234874 
grains.  An  examination  made  at  the  International  Bureau  of  Weights  and 
Measures  in  1884  makes  it  15.43235639  grains.    The  legal  value  in  the  United 

States  is  15.432  grains. 

*  1  centimeter  =  y^^j  meter  =  0.3937  inch.    1  milliliter  {yi^xj  liter)  or  cubic  centi^ 
meter  =  0.061  +  cubic  inches, 
t  Appendix  No.  22  to  report  of  1876,  page  6. 
j  Philosophical  Transactions,  1866,  pp.  893,  etc. 


218  FOREIGN  COINS. 

Approximate  Talnes  of  Foreign  Coins,  in  U.  S.  Money. 
The  references  (i,  2,  3^  and  *)  are  to  foot-notes  on  next  page. 

From  Circular  of  U.  S.  Treasury  Department,  Bureau  of  the  Mint,  Jan.  1, 1887; 

from  "Question  Monetaire,"  by  H.  Costes,  Paris,  1884;  and  from  our  10th  edition. 

Argentine  Repub. — Peso  =  100  Centavos,  96.5  ctg.2  3    Argentino  =  5  Pesos,  $4.82. 

Austria.— Florin  =  100  Kreutzer,  47.7  cts.,2  35.9  cts.3  Ducat,  $2.29.  Maria  Theresa 
Thaler,  or  Levantin,  1780,  $1.00.2    Rix  Thaler,  97  cts.4    Souverain,  $3.57.'* 

Belgium. 1— Franc  =  100  centimes,  17.9  cts.,2  19.3  cts.3 

Bolivia.— Boliviano  =  100  Centavos,  96.5  cts.,2  72.7  cts.s  Once,  $14.95.  Dollar, 
96  cts.4 

Brazil.— Mil  reis  =  1000  Reis,  50.2  cts.,2  54.6  cts.s 

Canada. — English  and  U.  S.  coins.     Also  Pound,  $4.4 

Central  America.'*— Doubloon,  $14.50  to  $15.65.  Reale,  average  5%  cts.  See 
Honduras. 

Ceylon. — Rupee,  same  as  India. 

Chili.— Peso  =  10  Dineros  or  Decimos  =  100  Centavos,  96.5  cts.,2  91,2  cts.*  Con- 
dor =  2  Doubloons  =  5  Escudos  =  10  Pesos.    Dollar,  93  cts.^ 

Cuba.— Peso,  93.2  cts.3    Doubloon,  $5.02. 

Denmark.— Crown  =  100  Ore,  25.7  ct8.,2  26.8  cts.3    Ducat,  $1.81.4    Skilling,  %  ct.* 

Ecuador.— Sucre,  72.7  cts.^  Doubloon,  $3.86.  Condor,  $9.65.  Dollar,  93  cts.* 
Reale,  9  cts.* 

Egypt.— Pound  -=  100  Piastres  -=  4000  Paras,  $4.94,3.3 

Finland.— Markka  =  100  Penni,  19.1  cts.2    10  Markkaa,  $1.93. 

France.i— Franc  =100  Centimes,  17.9  cts.,2  19.3  cts.3  Napoleon,  $3.84.4  Livre, 
18.5  cts.4    Sous,  1  ct.4 

Germany.— Mark  =  100  Pfennigs,  21.4  ct8.,2  23.8  cts.s  Augustus  (Saxony),  $3.98.* 
Carolin  (Bavaria),  $4.93.4  Crown  (Baden,  Br.varia,  N.  Germany),  $1.06.'^ 
Ducat  (Hamburg,  Hanover),  $2.28.4  Florin  (Prussia,  Hanover),  55  cts.4 
Groschen,  2.4  cts.4  Kreutzer  (Prussia),  .7  ct.  Maximilian  (Bavaria),  $3.30.4 
Rix  Thaler  (Hamburg,  Hanover),  $1,104  (Baden,  Brunswick),  $1,004  (Prussia, 
N.  Germany,  Bremen,  Saxony,  Hanover),  69  ct8.4 

Great  Britain.— Pound  Sterling  or  Sovereign  (£)  =  20  Shillings  =  240  Pence, 
$4.86.65.3  Guinea  =  21  Shillings  Crown  =  5  Shillings.  Shilling  (s),  22.4 
cts.,2  24.3  cts.  (^  pound  sterling).    Penny  (d),  2  cts. 

Greece.i— Drachma  =  100  Lepta,  17  cts.,2  19.3  cts.3 

Hay ti.— Gourde  of  100  cents,  96.5  cts.2  3 

Honduras.— Dollar  or  Piastre  of  100  cents,  $1.01.    See  Central  America. 

India.— Rupee  =  16  Annas,  45.9  cts.,2  345  cts.^  Mohur  =  15  Rupees,  $7.10.  Star 
Pagoda  (Madras),  $1.81.4 

Italy,  etc.i— Lira  =  100  Centesimi,  17.9  cts.,2  19.3  cts.s  Carlin  (Sardinia),  $8.21.* 
Crown  (Sicily),  96  cts.4  Livre  (Sardinia),  18.5  cts.4  (Tuscany,  Venice),  16 
5ts.4  Ounce  (Sicily),  $2.50.4  Paolo  (Rome),  10  cts.4  Pistola  (Rome),  $3.37.* 
Scudo*  (Piedmont),  $1.36  (Genoa),  $1.28  (Rome),  $1.00  (Naples,  Sicily),  95 
cts.  (Sardinia),  92  cts.    Teston  (Rome),  30  cts.4    Zecchino  (Rome),  $2.27.4 

Japan.— Yen  =  100  Sen  (gold),  99.7  cts.3  (silver),  $1,042,  78.4  cts.3 

Liberia.— Dollar,  $1.00.3  4 

Mexico.— Dollar,  Peso,  or  Piastre  =  100  Centavos  (gold),  98.3  cts.  (silver),  $1.05,2 
79  ct8.3    Once  or  Doubloon  =  16  Pesos,  $15.74. 

Netherlands.— Florin  of  100  cents,  40.5  cts.,2  40.2  cts.a  Ducatoon,  $1.32.4  Guilder, " 
40  cts.4    Rix  Dollar,  $1.05.4    Stiver,  2  cts.4 

New  Granada.— Doubloon,  $15.34.4 

Norway.— Crown  =  100  Ore  =  30  Skillings,  25.7  cts.,*  26.8  cts.* 

Paraguay. — Piastre  =  8  Reals,  90  cts. 

Persia.— Thoman  =  5  Sachib-Kerans  =  10  Banabats  =  25  Abassis  =  100  Scahls, 
$2.29. 

Peru.— Sor=  10  Dineros  =  100  Centavos,  96.5  cts.,2  72.7  cts.3    Dollar,  93  ct8.4 

Portugal.— Milreis  =  10  Testoons  =  1000  Reis,  $1.08.3  Crown  =  10  Milreis. 
Moidore,  $6.50.4 

Russia.— Rouble  =  2  Poltinniks  =  4  Tchetvertaks  =  5  Abassis  =  10  Griviniks  = 
20  Pietaks  =  100  Kopecks,  77  cts.,2  58.2  cts.3  Imperial  =-  10  Roubles,  $7.72. 
Ducat  =  3  Roubles,  $2.39. 

Sandwich  Islands.— Dollar,  $1.00.4 

Sicily.— See  Italy. 

Spain.— Peseta  or  Pistareen  =  100  Centimes,  17.9  cts.,*  19.3  cts.s  Doubloon  (new) 
=  10  Escudos  =  100  Reals,  $5.02.  Duro  =  2  Escudos,4  $1.00.2  Doubloon  (old), 
$15.65.4  Pistole  =  2  Crowns,  $3.90.*  Piastre,  $1.04.4  Reale  Plate,  10  ct«.4 
Beale  vellon,  5  cts.* 

1,  2, 3, 4.    See  ibot-aotes,  next  page. 


FOREIGN  COINS. 


219 


(Foreigrn  Coins  Untinued.    Small  figures  0, 2,  3^  <)  refer  to  foot  notes.) 

Sweden.— Crown  ^  100  Ore,  25.7  cts.,2  26.8  cts.3   Ducat,  $2.20.*  Rix  Dollar,  81.05  * 

Switzerland.!— Franc  =  100  Centimes,  17.9  cts.,-  19.3  cts.3 

Tripoli.— Mahbub  =  20  Piastre  s,  65.6  cts.3 

Tunis.— Piastre  =  16  Karobs,  12  cts.2    10  Piastres,  Si.  16.6. 

Turkey.— Piastre  =  40  Paras,  4.4  cts.3    Zecchin,  ^1.40.4 

United  States  of  Colombia.— Peso  =  10  Dineros  or  Decimos  =  100  Centayos,  96.5 

cts.,2  72.7  cts.3    Condor  =  10  Pesos,  S9.65.    Dollar,  93.5  cts.* 
Uruguay.— Peso  =  100  Centavos  or  Centesiraos  (gold),  $1.03  (silver),  96.5  cts.2 
Venezuela. — Bolivar  =  2  Decimos,  17.9  cts.,2  19.3  cts.3    Venezolano  =  5  Bolivars. 

Sizes  and  Weights  of  United  States  Coins.* 


Gold,  10  per  cent,  alloy : 

Double  eagle ,    .   .  $20 

Eagle 10 

Half  eagle 5 

Three  dollars 3 

Quarter  eagle 2.50 

Dollar ........  .  .      1.00 

Silver,  10  per  cent,  alloy : 

Trade  dollar 100  cts. 

Standard  dollar 100  " 

Half  dollar 50  " 

Quarter  dollar 25  " 

Twenty  cents 20  " 

Dime 10  " 

Half  dime 5  " 

Three  cents 3  " 

Minor. 
5  cents,  75^  copper,  25%  nickel     .   . 

2     *'       95%       "         5  %  tin  and  zinc. 
1     " 


Diameter. 

Thickness. 

Legal  weight  of  coin. 

Inch. 

Inch. 

Grains. 

Grams. 

1.35 

.077 

516 

33.436 

1.05 

.060 

258 

16.718 

.85 

.046 

129 

8.359 

.8 

.034 

77.4 

5.015 

.75 

.034 

64.5 

4.179 

.55 

.018 

25.8 

1.672 

1.5 

.082 

420. 

27.215 

1.5 

.080 

412.5 

26.729 

1.2 

.057 

192.9 

12.6 

.95 

.045 

96.45 

6.25 

.875 

.047 

77.16 

5. 

.7 

.032 

38.58 

2.5 

.6 

.023 

19.2 

1.244 

.55 

.018 

11.52 

.746 

.8 

.062 

77.16 

5.0 

.725 

.034 

30. 

1.944 

.9 

.060 

96. 

6.22 

.75 

.043 

48. 

3.11 

Perfectly  pnre  g-old  is  worth  SI  per  23.22  grs  =  $20.67183  per  troy  oz  = 
S18.84151  per  avoir  oz.  Standard  (U.  S.  coin)  is  worth  $18.60465  per  troy  oz  = 
$16.95736  per  avoir  oz.  It  consists  of  9  parts  by  weight  of  pure  gold,  to  1  part 
alloy.  Its  value  is  that  of  the  pure  gold  only ;  the  cost  of  the  alloy  and  of  the 
coinage  being  borne  by  Government.  A  cubic  foot  of  pure  g'oid  ireighs 
about  1204  avoir  5)s;  and  is- worth  $362963.  A  cubic  inch  weighs  about  11.148 
avoir  oz  ;  and  is  worth  $210.04. 

Pure  gold  is  called  fine,  or  24  carat  gold ;  and  when  alloyed,  the  alloy  is  sup- 
posed to  be  divided  into  24  parts  by  weight,  and  according  as  10,  15,  or  20,  &c,  of 
these  parts  are  pure  gold,  the  alloy  is  said  to  be  10,  15,  or  20,  &c,  carat. 

The  averag-e  fineness  of  California  native  grold,  by  some  thou- 
sands of  assays  at  the  U.  S.  Mint  in  Philada.,  is  88.5  parts  gold,  11.5  silver.  Some 
from  Georgia,  99  per  cent.  gold. 

Pure  silver  fluctuates  in  value:  thus,  during  1878-1879  it  ranged  between 
$1.05  and  $1.18  per  troy  oz.,  or  $.957  and  $1,076  per  avoir,  oz.  A  cubic  inch  weighs 
about  5.528  troy,  or  6.065  avoir,  ounces. 


1  France    Belgium,  Italy,  Switzerland,  and  Greece  form  the  Latin  Union. 
Their  coins  are  alike  in  diameter,  weight,  and  fineness. 

2  =  19  3  times  the  value  of  a  single  coin  in  francs  as  given  by  Costes. 

3  Par  of  exchange,  or  equivalent  value  iu  terms  of  U.  S.  gold  dollar.— Treasury 
Circular. 

*  From  our  10th  edition. 

*  Thirteenth  Anuual  Report  of  the  Director  of  the  Mint.  1885 ;  pp.  106  and  148. 


220  WEIGHTS   AND    MEASURES. 

Troy  Weight.    U.  S.  and  British. 

24  grains 1  pennyweight,  dwt. 

20  pennyweights 1  ounce  ~  480  grains. 

12  ounces  1  pound  =  240  dwts.  =  5760  grains. 

Troy  weigrlit  Is  used  for  g-old  and  silver. 

A  carat  of  the  jewellers,  for  precious  stones  is,  in  the  U.  S.  =  3.2  grs. ;  in 
London,  3.17  grs. ;  in  Paris,  3.18  grains.,  divided  into  4  jewellers'  grs.  In  troy, 
apothecaries'  and  avoirdupois,  the  grain  is  the  same. 

Apothecaries'  liVeight.    U.  S.  and  British. 

20  grains 1  scruple. 

3  scruples  1  dram  =  60  grains. 

8  drams 1  ounce  =  24  scruples  =  480  grains. 

12  ounces 1  pound  =  96  drams  =  288  scruples  =  5760  grains. 

In  troy  and  apothecaries'  weights,  the  grain,  ounce  and  pound  are  the  same. 

Avoirdupois  or  Commercial  liVeight.    U.  S.  and  British. 

27.34375  grains - 1  dram. 

16  drams 1  ounce  =  437^^  grains. 

16  ounces 1  pound  =  256  drams  =  7000  grains. 

28  pounds 1  quarter  =  448  ounces. 

4  quarters 1  hundredweight  =  112  lbs. 

20  hundredweights 1  ton  =  80  quarters  =  2240  lbs. 

A  stone  =  14  pounds.    A  quintal  =  100  pounds  avoir. 

The  standard  of  the  avoirdupois  pound,  which  is  the  one  in 
common  commercial  use,  is  the  weight  of  27.7015  cub  ins  of  pure  distilled  water» 
at  its  maximum  density  at  about  39°.2  Fahr,  in  latitude  of  London,  at  the  level 
of  the  sea ;  barometer  at  30  ins.  But  this  involves  an  error  of  about  1  part  in 
1362,  for  the  lib  of  water  =  27.68122  cub  ins. 

A  troy  lb  =  .82286  avoir  lb.    An  avoir  ft)  =  1.21528  troy  ft),  or  apoth. 

A  troy  oz.  =  1.09714  avoir,  oz.    An  avoir,  oz.  =  .911458  troy  oz.,  or  apoth. 

liOng  Measure.    U.  S.  and  British. 

12  inches 1  foot  =  .3047973  metre. 

3  feet 1  yard  =  36  ins  =  .9143919  metre. 

514  vards 1  rod,  pole,  or  perch  =  16^  feet  =  198  ins. 

40  rods 1  furlong  =  220  yards  =  660  feet." 

8  furlongs 1  statute,  or  land  mile  =  320  rods  =  1760  yds  =[5280  ft  =  63360  ins. 

3  miles 1  league  =  24  furlongs  =  960  rods  =  5280  yds  =  15840  ft. 

A  point  =  7^5  inch.  A  line  =  6  points  =  yV  ii^ch.  A  palm  =  3  ins.  A 
hand  =  4  ins.    A  span  =  9  ins.   A  fathom  =  6  feet.  A  cable's  length 

=  120  fathoms  =  720  feet.    A  Ounter's  surveying  chain  is  66  feet,  or  4 
rods  long.    It  has  100  links,  7.92  inches  long.    80  Gunter's  chains  =  1  mile. 
A  nautical  mile,  geographical  mile,  sea  mile,  or  knot,  is 

variously  defined  as  being  =  the  length  of 

1  min  of  longitude  at  the  equator 
1        "    latitude        "  " 

1        "  "  "        pole 

1        ♦'  "        atlat45° 

1        "a  great  circle  of  a  true') 
sphere  whose  surface  area  is  > 
equal  to  that  of  the  earth  J 
British  Admiralty  knot 
The  above  lengths  of  minutes,  in  metres  and  feet,  are  those  published  by  the  U.  8. 
Soast  and  Geodetic  Survey  in  Appendix  No  12,  Keport  for  1881,  and  are  calculated 
from  Clarke's  spheroid,  which  is  now  the  standard  of  that  Survey. 
At  the  equator  1°  of  lat  =  68.70  land  miles ;  at  lat  20°  =  68.78 ;  at  40°  = 

69.00 ;  at  60°  -  69.23  ;  »t  80°  =  69.89 ;  at  90°  =  69.41. 


metres 

feet 

statute  miles 

=    1855.345 

6087.15 

1.15287 

=     1842.787 

6045.95 

1.14607 

=     1861.665 

6107.85 

1.15679 

=     1852.181 

6076.76 

1.15090 

(  value  adopted  by  U. 

.  S.  Coast 

=^     and  Oeodetic  Survey 

1  1853.248 

6080.27 

1.15157 

=     1853.169 

6080.00 

1.15152 

WEIGHTS  AND  MEASURES. 


liengrths  of  a  Begree  of  L.on^itiic1e  in  different  liatitndefi, 

And  at  the  level  of  tlie  Siea.  These  lengths  are  iu  common  land  or  statute  miles, 
of  5280  ft.  Since  the  figure  of  the  earth  has  never  been  precisely  ascertained,  these  are  but  close  ap 
proximations.  Intermediate  ones  may  be  found  correctly  by  simple  proportion.  1°  of  longitude 
••rresponds  to  i  mins  of  civil  or  clock  time ;  1  min  of  longitude  to  4  sees  of  lime. 


Degof 
Lat. 

Miles. 

Degof 
Lat. 

Miles. 

Degof 
Lat. 

MileB. 

Degof 
Lat. 

Miles. 

Degof 
Lat. 

Miles. 

Degof 
Lat. 

Miles. 

0 

69.16 

14 

67.1-2 

28 

61.11 

42 

51.47 

56 

38.76 

70 

23.72 

2 

69.12 

16 

66.50 

30 

59.94 

44 

49.83 

58 

36.74 

72 

21.43 

4 

68.99 

18 

65.80 

32 

58.70 

46 

48.12 

60 

34.67 

74 

19.12 

6 

68.78 

20 

65.02 

34 

57.39 

48 

46.36 

62 

32.55 

76 

16.78 

8 

68.49 

•22 

64.15 

36 

56.01 

50 

44.54 

64 

30.40 

78 

14.42 

10 

68.12 

24 

63.21 

38 

54.56 

52 

42.67 

66 

28.21 

80 

12.05 

12 

«7.66 

26 

62.20 

40 

53.05 

54 

40.74 

68 

26.98 

82 

9.66 

Indies  reduced  to  Decimals  of  a 

Foot. 

No  errors. 

Ins. 

Foot. 

Ins. 

Foot. 

Ins. 

Foot. 

Ins. 

Foot. 

Ins. 

Foot. 

Ins. 

Foot. 

o 

.0000 

2 

.1667 

4 

.3333 

6 

.5000 

8 

.6667 

10 

,8333 

1-32 

.00-26 

.1693 

.3359 

.5026 

.6693 

,8359 

1-16 

.0052 

.1719 

.3385 

.5052 

.6719 

,8385 

3-32 

.0078 

.1745 

.3411 

.5078 

.6745 

,8411 

c,^ 

.0104 

H 

.1771 

H 

.3438 

\i 

.5104 

H 

.6771 

^ 

.8438 

5-32 

.0130 

.1797 

.3464 

.5130 

.6797 

,8464 

3-16 

.0156 

.1823 

.3490 

.5156 

.6823 

.8490 

7-32 

.0182 

.1849 

.3516 

.5182 

.6849 

.8516 

H 

.0208 

H 

.1875 

34 

.3542 

34 

.5208 

H 

.6875 

y*. 

.8542 

9-32 

.0234 

.1901 

.3568 

.5234 

.6901 

.8568 

5-16 

.0260 

.1927 

.3594 

.5260 

.6927 

,8594 

11-32 

.0286 

.1953 

.3620 

.5286 

.6953 

,8«20 

H 

.0313 

% 

.1979 

% 

.3646 

% 

.5313 

% 

.6979 

Va 

.8646 

13-32 

.0339 

.'2005 

.3672 

.5339 

.7005 

-  ,8672 

7-16 

.0365 

.2031 

.3698 

.5365 

.7031 

.8698 

16-32 

.0391 

.2057 

.3724 

.5391 

.7057 

,8724 

H 

.0417 

% 

.2083 

}4 

.3750 

14 

.5417 

H 

.7083 

H 

.8750 

17-32 

.0443 

.2109 

.3776 

.5443 

.7109 

,8776 

9-16 

.0469 

,2135 

.3802 

.5469 

.7135 

.8802 

19-32 

.0495 

.2161 

.3828 

.5495 

.7161 

.8828 

% 

.0521 

% 

.2188 

% 

.3854 

H 

.5521 

% 

.7188 

H 

.8854 

21-32 

.0547 

.2214 

.3880 

.5547 

.7214 

,8880 

11-16 

.0573 

.2240 

.3906 

.5573 

.7240 

,8906 

23-32 

.0599 

.2-266 

.3932 

,5599 

.7266 

,8932 

H 

.0625 

H 

.2292 

H 

.3958 

H 

,5625 

h 

.7292 

H 

.8958 

25-32 

.0651 

.'2318 

,3984 

,6651 

.7318 

,8984 

13-16 

.0677 

.2344 

.4010 

.5677 

.7344 

.9010 

27-32 

.0703 

.'2370 

.4036 

.5703 

.7370 

,9036 

H 

.0729 

Ji 

.'2396 

}i 

.4063 

H 

.57'29 

% 

.7396 

% 

.9063 

29-32 

.0755 

.2422 

.4089 

.5755 

.74-22 

,9089 

15-16 

.0781 

.2448 

.4115 

.5781 

.7448 

.9116 

31-32 

.0807 

.2474 

.4141 

.5807 

.7474 

.9141 

1 

.0833 

S 

.-2500 

5 

.4167 

7 

.5833 

9 

.7500 

11 

.9167 

1-32 

.0859 

.2526 

.4193 

,5859 

.7526 

.9193 

1-16 

.0885 

.2552 

.4219 

.5885 

.7552 

.9219 

3-32 

.0911 

.2578  " 

.4245 

.5911 

.7578 

.9246 

5-M 

.0938 

H 

.2604 

H 

.4-271 

H 

.5938 

H 

.7604 

H 

.9271 

.0964 

.2630 

.4297 

.5964 

.7630 

.9297 

3-16 

.0990 

.'2656 

.43-23 

.6990 

.7656 

.9823 

7-32 

.1016 

.2682 

.4349 

.6016 

.7682 

,9349 

}4 

.1042 

% 

.2708 

H 

,4375 

H 

.6043 

H 

,7708 

yi 

,9376 

9-32 

.1068 

.2734 

.4401 

.6068 

.7734 

.9401 

5-16 

.1094 

.2760 

.4427 

.6094 

.7760 

.9427 

11-32 

.1120 

.2786 

.4453 

.6120 

.7786 

,9463 

% 

.1146 

% 

.2813 

% 

,4479 

% 

.6146 

% 

.7813 

% 

,9479 

13-.(2 

.1172 

.2839 

.4505 

.6172 

.7839 

.9605 

7-16 

.1198 

.2865 

.4531 

.6198 

.7865 

.9531 

15-32 

.1224 

.2891 

.4557 

.6224 

,7891 

,9557 

3^ 

.1250 

% 

.2917 

H 

.4583 

H 

.6250 

^ 

.7917 

M 

.9583 

17-32 

.1276 

.2943 

.4609 

.6276 

.7943 

.9609 

9-16 

.1302 

.2969 

.4635 

.6302 

.7969 

.9635 

19-32 

.1328 

.2995 

.4661 

.6328 

.7995 

.9661 

% 

.1354 

H 

.3021 

% 

.4688 

H 

,6354 

% 

.8021 

% 

.9688 

21-3-2 

.1380 

.3047 

.4714 

.6380 

.8047 

.9714 

11-16 

.1406 

.3073 

.4740 

.6406 

.8073 

.9740 

23-32 

.1432 

.3099 

.4766 

.6432 

.8099 

.9766 

H 

.1458 

H 

.3125 

H 

.4792 

H 

.6458 

H 

,8125 

.  % 

.9792 

25-3-2 

.1484 

.3151 

.4818 

.6484 

,8151 

.9818 

13-16 

.1510 

.3177 

.4844 

.6510 

,8177 

.9844 

27-32 

.1536 

.3203 

.4870 

,6536 

.8203 

.9870 

29-32 

.1563 

% 

.3229 

% 

.4896 

% 

,6563 

K 

.8229 

K 

.9896 

.1589 

.3255 

.4922 

.6589 

.8255 

,9922 

15-16 

.1615 

.3-281 

.4948 

.6615 

.8281 

.9948 

81-32 

.1641 

.3307 

.4974 

.6641 

.8307 

.9974 

222 


WEIGHTS   AND    MEASURES. 


Square,  or  liaiid  Pleasure. 

U.  S.  and  British. 

144  square  inches 1  sq  foot.     100  sq  ft  =  1  square. 

9  sq  feet 1  sq  yard  =  1296  aq  ins. 

30J4  sq  yards 1  sq  rod  =  272}^  sq  feet. 

40  sq  rods 1  rood  =  1210  sq  yds  =r  10890  sq  feet. 

.    4  roods 1  acre  ~  160  rods  =  4840  sq  yds  =  43560  sq  feet. 

A  section  of  land  is  1  mile  sq,  or  27878400  sq  ft ;  or  3097600  sq  yds ;  or  640  acres.  An  acr« 
contains  10  sq  Gunter's  chains.  A  sq  acre  is  208.710  feet;  a  sq  half  acre,  147.581  ft;  and  a  sq 
quarter  acre,  104.355  ft  on  each  side.  A  circular  acre  is  235.504  feet:  a  circular  half  acre  = 
166.527  ft;  and  a  circular  quarter  acre  =  117.752  ft  diaui.  A  circular  inch  is  a  circle  of  1  inch 
diani;  a  sq  ft  =  183.346  cir  ins.  Also  1  sq  inch  =  1.27324  cir  ins  ;  and  1  cir  inch  —  .7854  of  a  sq 
inch. 

Cubic,  or  $$olicl  Measure. 

U.  S.  and  British. 

1728  cubic  inches 1  cubic,  or  solid  foot. 

27  cubic  feet 1  cubic,  or  solid  yard. 

A  cord  of  wood  =  128  cub  ft ;  being  4  ft  X  4  ft  X  8  ft.  A  perch  of  masonry  actually  con- 
taiDS  24?^  cub  ft;  being  16}^  ft  X  134  ft  X  1  ft.  It  is  generally  taken  at  25  cub  ft;  but  by  some  at  22, 
&c ;  and  there  is  every  probability  that  a  payer  will  be  cheated  unless  the  number  of  cubic  ft  be  dis- 
tinctly agreed  upon  in  his  contract.  It  is  gradually  falling  into  disuse  among  engineers ;  and  the  cub 
yd  is  very  properly  taking  its  place.  To  reduce  cub  yds  to  perches  of  25  cub  ft,  mult  by  1.080; 
and  to  reduce  perches  to  cub  yds,  mult  by  .926.  The  Brit  rod  of  brickwork,  of  house-builders,  is 
16K  feet  square,  by  14  inches'ci)^  English  bricks;  thick  r:  272>4  sq  ft  of  14  inch  wall.  It  is  conven- 
tionally taken  at  272  sq  ft  j  which  gives  317}^  cub  ft.  In  Brit  engineering  works  the  rod  is  306  cub 
ft,  or  UH  cub  yds.  The  Montreal,  (Canada,)  toise  =  261>^  cub  ft;  or  9.6852  cub  yds,  or  10.46 
perches  of  25  cub  ft.  The  Canadian  chaldron  =  58.64  cub  ft.  A  ton  (2240  fts)  of  Pennsylvania 
anthracite,  when  broken  for  domestic  use,  occupies  from  11  to  43  cub  ft  of  space;  the  mean  of  which 
is  equal  to  1.556  cub  yds:  or  a  cube  of  3.476  ft  on  each  edge.  Bituminous  coal  44  to  43  cub  ft;  meaa 
equal  to  1.704  cub  yd ;  or  a  cube  of  3.583  ft  on  each  edge.     Coke  80  cub  ft. 

A  cubic  foot  is  equal  to 

1728  cub  ioB,  or  3300.23  spherical  ins. 
.037037  cub  yard,  or  1.90985  spherical  fU 
.002832  myriolitre,  or  decastere. 


r  Htere. 


.028316  kilolitre,  or  cubic  metre,  o 

.283161  hectolitre,  or  decistere. 

2.83161  decalitres,  or  centisteres. 

28.3161  litres,  or  cub  decimetres. 

283.161  decilitres. 

2831.61  centilitres. 

28316.1  millilitres,  or  cub  centimetres. 

.803564  U.S.  struck  bushel  of  2150.42  cub  ins,  or 

1.24445  cub  ft. 
.779013  Brit  bushel  of  2218.191  cub  ins,  or  1.28368 

cub  ft. 
3.21426  U.  S.  peeks. 

A  cubic  iuch  is  equal  to 

1^38«63  millilitres;  or  1.638663  centilitres;  or  .1638663  decilitre;  or  .01638663  litre;  or  to  .0005783 
f«b  ti;  or  to  .138528  U.  S.  gill;  or  1.90985  spherical  ins. 

A  cubic  yard  is  equal  to 

76.4534  decalitres. 

764.534  litre.s,  or  cub  decimetres. 

7645.34  decilitres. 


3.11605  Brit  pecks. 

7.48052  U.  S.  liquid  galls  of  231  cub  ins. 

6.42851  U.  S.  dry  galls. 

6.23210  Brit  galls  of  277.274  cub  ins. 

29.92208  U.  S.  liquid  quarts. 

25.71405  U.  S.  drv  quarts. 

24.92842  Brit  quarts. 

59.84416  U.  S.  liquid  pints. 

51.42809  U.  S.  dry  pints. 

49.85684  Brit  pints. 

239.37662  U.  S.  gills. 

199.42737  Brit  gills. 

.26667  flour  barrel  of  3  struck  bushels. 

.23748  U.  S.  liquid  barrel  of  313^  galls. 


n  eub  feet,  or  to  201.974  U.  S.  galls. 

46656  cub  ins. 

.0764534  myriolitre. 

.764534  kilolitre,  or  cub  metre. 

7.64534    hectolitres. 

f.2  flour  barrels  of  3  struck  bushels. 


21.69623  U.  S.  bushels  (struck). 
21.03336  Brit  bushels. 


A  spliere  1  foot  in  diameter,  contains 


.01939  cub  yard. 
.5236  cub  foot. 
904.781  cub  inches. 
.42075  U.  S.  bushel. 
1.6830  U.  S.  pecks. 
13.4639  U.  S.  drv  quarts. 
26.9278  U.  S.  drV  pints. 
3.9168  U.  S.  liquid  gallons, 
11.6672  U.  S.  liquid  quarts. 

A  sphere  1  incb 

.000303  cub  foot. 
.5236  cub  inch. 
.07263  U.  S.  gill. 


31.3344  U.  S.  liquid  pints. 
125.3376  U.  S.  liquid  gills. 
3.2631  Brit  imp  gallons. 
13.0625  Brit  imp  quarts. 
26.1050  Brit  imp  pints. 
104.4201  Brit  imp  gills. 
14.826;5  litres. 
1.48263  decalitres. 
.148263  hectolitres. 

in  diameter,  contains 

.06043  Brit  gill. 
8.580  mlllilitre. 
.8580  centilitre. 
.08580  deciUtr*. 


WEIGHTS    AND    MEASUBES. 


223 


A  cylinder  1  foot  in  diameter,  and  1  foot  bigrli,  contains 

.02909  cub  yard. 
.7854  cub  foot. 
i357. 1712  cub  inches. 

.63112  U.  S.  dry  bushels. 
2.5245  U.  S.  dry  pecks. 
20.1958  U.  S.  dry  quarts. 
40.3916  U.  S.  dry  pints. 
5.8752  U.  S.  liquid  gallons. 
23.5008  U.  S.  liquid  quarts. 


47.0016  U.  S.  liquid  pints. 
188.0064  U.  S.  liquid  gills. 
4.8947  Brit  imp  gallons. 
19.5788  Brit  imp  quarts. 
39.1575  Brit  imp  pints. 
156.6302  Brit  imp  gills. 
222.395  decilitres. 
22.2395  litres. 
2.22395  decalitres. 
.222395  hectolitre. 


A  cylinder  1  inch  in  diameter,  and  1  foot  higb,  contains 


.005454  cub  foot. 
9.4248  cub  inches. 

.2805  U.  S.  dry  pint. 

.3264  liquid  pint. 
1.3056  U.  S.  gill. 


.2719  Brit  imp  pint. 
1.0877  Brit  imp  gill. 
15.4441  centilitres. 
1.54441  decilitres. 

.154441  litres. 


liiqnid  Measure,    v.  S.  only. 

The  1>asi8  of  this  measure  in  the  U.  S.  is  the  old  Brit  wine  gallon  of  231  cub  ins ;  or  8.33888  B>« 
avoir  of  pure  water,  at  its  max  density  of  about  39°. 2  Fahr ;  the  barom  at  30  ins.  A  cylinder  7  ins 
diam,  and  6  ins  high,  contains  230.904'  cub  ins,  or  almost  precisely  a  gallon  ;  as  does  also  a  cube  of 
6.1358  ins  on  fn  edge.  Also  a  gallon  —  .13.308  of  a  cnh  ft ;  and  a  cub  ft  contains  7.48052  galls  ;  nearly 
7 Ji  galls.  This  basis  however  involves  an  error  of  about  1  part  in  1362,  for  the  water  actu- 
ally weighs  8.34o008  lbs. 

cub  ins. 

4  gills 1  pint     =28.875.  63  gallons 1  hogshead. 

2pints 1  quart  =  57.750  =  8  gills.  2  hogsheads 1  pipe,  or  butt. 

4  quarts 1  gallon  =:  231.  =8  pints ~ 32  gills.      2  pipes 1  tun. 

In  the  U.  S.  and  Great  Brit.  1  barrel  of  wine  or  brandy  =  31}^  galls  :  in  Pennsylvania,  a  half 
barrel,  16  galls;  a  double  barrel,  64  galls;  a  puncheon,  84  galls;  a  tierce,  42  galls.  A  liquid 
measure  barrel  of  31^^  galls  contains  4.211  cub  ft  =  a  cube  of  1.615  ft  on  an  edge;  or  3.38i  U.  S.  struck 
bushels.  A  still  =  7.21H75  cub  ins.  The  following  cylinders  contain  some  o."  these  measures 
rery  approximately. 


Diam. 
cub  ins.  Ins. 

Gill  v7. 21875) 1% 

J^pint 2% 

Pint SVjj' 

Quart 33^ 


Height. 
Ins, 


Diam. 
Ins. 

7     .. 


Gallon 

3^  2  gallons 

8  gallons 14 

10  gallons 14 

Apothecaries'  or  "Wine  Measure. 


Height. 
Ins. 


Symbol. 

Pints. 

Fluid 
ounces. 

Fluid 
drachms. 

Minims. 

Cubic 
inches. 

Weight  of  water.J 

Pounds,  av. 

Grains. 

1  Gallon 

1  Pint 

Cong* 

ot 

m 

8 

128 
16 

1024 
128 

8 

61440 
7680 

480 
60 

1 

231 

28.875 

1.8047 
0.2256 
0.0038 

8.345 

1.043 

Ounces,  av. 

1.043 

68415 
7301.9 

1  Fluid  ounce  ... 
1  Fluid  drachm.. 
IMinira 

456.4 
67.05 
0.91 

To  reduce  U.  S.  liquid  measures  to  Brit  ones  of  the  same  denomina- 
tion, divide  by  1.20032;  or  near  enough  for  common  use,  by  1.2;  or  to  reduce  Brit  to  U.  S.  multiply 
by  1.2. 

I>ry  Measure, 

U.  8.  only. 
The  basis  of  this  is  the  old  British  Winchester  struck  bushel  of  2150.42  cub 

ins;  or  77.627413  pounds  avoir  of  pure  water  at  its  max  density.  Its  dimensions  by  law  are  181.^  ins 
inner  diam ;  19}^  ins  outer  diam ;  and  8  ins  deep  ;  and  when  heaped,  the  cone  is  not  to  be  less  than  < 
ins  high ;  which  makes  a  heaped  bushel  equal  to  1J4  struck  ones  :  or  to  1.55556  cub  tt. 

Edge  of  a  cube  of     • 
equal  capacity. 

2  pints      1  quart,   =  67.2006  cub  ins  :=  1.16365  liquid  qt 4.066  ms. 

4  quarts   1  gallon,  =  8  pints,  =  268.8025  cub  ins,  =r  1.16365  liq  gal 6.454    " 

2  gallons  1  peck,     rr  16  pints,  =  8  quarts.  =  537.6050  cub  ins 8.131    " 

4  pecks     1  struck  bushel,  =  64  pmts,  =  32  quarts.  =  8  gals,  =r  2150.4200  cub  ins.  12.908    •« 

*  Abbreviation  of  Latin,  Congius. 
t  Abbreviation  of  Latin,  Octariu^. 

t  At  its  maximum  density,  62,425  pounds  per  cubic  foot,  corresponding  to  a  temperature  of  4° 
Centigrade  =  39.2°  p^ahrenheit. 


224 


WEIGHTS    AND   MEASURES. 


A  struck  bushel  =  1.24445  cub  ft.    A  cub  ft  =  .80356  of  a  struck  bushel 
The  dry  flour  barrel  =  3.75  cub  ft;  =  3  struck  bushels.    The  dry  barrel  h 

not,  however,  a  legalized  measure;  and  no  great  attention  is  given  to  its  capacity;  consequently 
barrels  varj  considerably.  A  barrel  of  flour  contains  by  law,  196  H>s.  In  ordering  by  the  barrel,  th< 
amount  of  its  contents  should  be  specified  in  pounds  or  galls. 

To  reduce  U.  S.  dry  measures  to  Brit  imp  ones  of  the  same  name,  dii 

by  1.031516;  and  to  reduce  Brit  ones  to  U.  S.  mult  by  1.031516 ;  or  for  common  purposes  use  1.0S2. 

British  Imperial  Measure,  both  liquid  and  dry. 

This  system  is  established  throughout  Great  Britain,  to  the  exclusion  of  the  old  ones.     Its  basis  ii 
the  imperial  gallon  of  277.274  cub  ins,  or  10  fts  avoir  of  pure  water  at  the  temp  of  62°  Fahr,  whei 

the  barom  is  at  30  ins.  This  basis  involves  an  error  of  about  1  part  in 
1836,  for  10  as  of  the  water  =  only  277.123  cub  ins. 


igills       1  pint 1.25 

2  pints     1  quart 2.50 

3  quarts  1  pottle 5. 

2  pottles  1  gallon 

2  gallons  1  peck 

♦  peeks    1  bushel 80.      1     Dry 

4  bushelsl  coomb 320.     j   meas. 

2  coombs  1  quarter 1640. 


Avoir  lbs. 
of  water. 


Cub.  ins. 


34.6592 
69.3185 
138.637 
277.274 
554.548 
2218.192 
8872.768 
17745.586 


Cub.  ft. 


1.2837 
5.1347 
10.2694 


Edge  of  a  cube  ol 

equal  capacity. 

Inches. 


3.2605 
4.1079 
5.1756 
6.5208 
8.2157 
13.0417 


The  imp  gall  =  .16046  cub  ft;  and  1  cub  (1=6.23310 galls. 


Measure. 

Symbol. 

Pints. 

Fluid 
ounces. 

Fluid 
drachms. 

Minims. 

Cubic 
inches. 

Weight  of  water  .J 

Pounds,  av. 

Oraios. 

1  Gallon 

1  Pint 

c* 
or 

fl.  oz. 
fl.dr. 

8 

1 

160 
20 

1 

1280 
160 

8 

1 

76800 
9600 

480 
60 

1 

277.274 
36.659 

1.733 
0.217 
0  0036 

10 
1.25 
Ounces,  av. 

1 

70000 
8750 

1  Fluid  ounce  ... 
I  Fluid  drachm.. 
1  Minim 

437.5 
54.6875 
0  9114 

The  weight  of  water  affords  an  easy  way  to  find  the  cubic  contents  of  a  vessel. 


To  obtain  the  size  of  commercial  measures  by  means  of  the 
weig^ht  of  water. 

At  the  common  temperature  of  from  70°  to  75°  Fah,  a  cub  foot  of  fresh  water  weighs  very  approxl 
mately  623^  lbs  avoir.  A  cubic  half  foot,  (6  ins  on  each  edge,)  7.78125  H)s.  A  cub  quarter  foot,  (3  ini 
on  each  edge,)  .97266  lb.  A  cub  yard,  1680.75  B)s ;  or  .75034  ton.  A  cub  half  yd,  (18  ins  on  each  edge,] 
210.094  tts;  or  .0938  ton.  A  cub  inch,  .036024  lb  ;  or  .576384  ounce ;  or  9. 2222  drams  ;  or  252.170  grains, 
An  inch  square,  and  one  foot  long,  .432292  S).  Also  1  9>  =  27.75903  cub  ins,  or  a  cube  of  3.028  ins  on  an 
edge.    An  ounce,  1.735  cub  ins  ;  a  ton,  35.984  cub  ft,  all  near  enough  for  common  use. 

Original. 


liiquid  Measures.    Lbs  Avoir, 
of  Water. 

U.  S.  Gill 26005* 

U.  S.  Pint 1.0402 

U.  S.  Quart 2.0804 

U.  S.  Gallon  8  fts  5^  oz 8.3216 

U.  S.  Wme  Barrel,  31  Ji  Gall 262.1310 

JDry  Measures. 

tJ.  S.  Pint 1.2104 

tJ.  S.  Quart 2.4208 

U.  S.  Gallon 9.6834 

U.  S.  Peck 19.3668 

U.  S.  Bushel,  struck 77.4670 


*  Or  4  ounces ;  2  drams ;  15.6625  grs. 


liiquid  and  Dry.    Lbs  Avoir, 

•'of  Water. 

British  Imp  Gill 31214» 

"     Pint 1.24858 

"        "     Quart 2.49715 

"         "      Gallon 9.9886 

"        "     Peck 19.9772 

"        "     Bushel 79.9088 

*  4.9942 ;  or  very  nearly  5  ounces. 

Metric  Measures. 

Centilitre 021981 

Decilitre .2198J 

Litre 2.1981 

Decalitre,  or  Centistere 21.9808 

Metre,  or  Stere 2198.0786 

t  Or  5.6271  drams ;  or  153.866  gr». 
}  3.5169  ounces. 


♦  Abbreviation  of  Latin,  Congius. 
t  Abbreviation  of  Latin,  Octarius. 
t  At  the  standard  temperature,  62°  Fahrenheit  = 


about  16.7°  Centigrade. 


WEIGHTS   AND   MEASURES. 


225 


]^Ietric  Measures  of  lieng-tb. 

By  U.  8.  and  British  Standard. 


Ins. 

Ft. 

Yds. 

Miles. 

Millimetre* 

.039370 
.39370428 
3  9370428 

.003281 

.032809 

.3280869 

3.280869 

32.80869 

328.0869 

3280.869 

32808.69 

Centimetref 

Decimetre 

.1093623 
1.093623 
10.93623 
109.3623 
1093.623 
10936.23 

MetreJ 

39.370428 
393.70428 

Road 
measures. 

Decametre "I 

Hectometre 

.0621375 
.6213760 
6.213750 

Kilometre [ 

Myriametre J 

*  Nearly  the  -^-^  part  of  an  inch.  t  Full  %  inch. 

X  YtTj  nearly  3  ft,  3%  ins,  which  is  too  long  by  only  1  part  in  8616. 

Metric  Square  Measure. 

By  U.  S.  and  British  Standard. 


Sq.  Ins. 

Sq.  Feet. 

Sq.  Yds. 

Acres. 

Sq  Millimetre 

.001550 
.155003 
15.5003 
1550.03 
155003 

.00001076 
.00107641 
.10764101 
10.764101 
1076.4101 
10764.101 
107641.01 
10764101 

.0000012 

.0001196 

.0119601 

1.19601 

119.6011 

1196.011 

11960.11 

1196011. 

Sq  Centimetre 

Sq  Decimetre 

Sq  Metre,  or  Centiare 

Sq  Decametre,  or  Are 

Decare  (not  used) 

.000247 
.024711 
.247110 
2  47110 

Hectare 

Sq  Kilometre 

.3861090  sq  miles. 

38.61090           " 

247  110 

8q  Myriametre 

24711.0 

Metric  Cubic  or  Solid  MLeasure. 

Aocordinc  to  U.  S.  Standard. 

Only  those  marked  "  Brit"  are  British. 


Millilitre,  or  cub 
Centimetre- 


Centilitre. 


Decilitre .. 


Litre,  or    cubic 
Decimetre .. 


Decalitre, 
Centistere.. 


Hectolitre, 
Decistere.. 


Kilolitre,  or 
Cubic  Metre, 
or  Stere 


Myriolitre,      or 
Decastere 


Cub  Ins. 

.0610254 


.610254 


61.0254 

610.254 
Cub  Ft. 

.353156 

.  3.53156 
35.3156 


{Liquid. 
Dry. 
( Liquid. 
iDry. 
■  (  Liquid. 
(Dry. 

{Liquid. 
Dry. 
r  Liquid. 

iDry. 

{Liquid. 
Dry. 
I  Liquid. 
(Dry. 

( Liquid. 
(Dry- 


.0084537  gill. 
.0070428  Brit  gill. 
.0018162  dry  pint. 

.084537  gill. 
.070428  Brit  gill. 
.018162  dry  pint. 

.84537  gill  =  .21134  pint. 

.70428  Brit  gill  =  .17607  Brit  pint. 

.18162  dry  pint. 

1.05671  quart  =  2.1134  pints. 

.88036  Brit  quart  =  1.7607  Brit  pints. 

.11351  peck  =  .9081  dry  qt  =  1.8162  dry  pt. 

2.64179  U.  S.  liquid  gal. 

2.20090  Brit  gal. 

.283783  bush  =  1.1351  peck  ^  9.081  dry  qts. 

26.4179  U.  S.  liquid  gal. 
22.0090  Brit  gal. 
2.83783  bush. 

264.179U.S.  liquid  gal.) 

220.090  Brit  gal.  yf»h  yds,  1.3080. 

28.3783  bush.  j 


2641.79  U.  S.  liquid  gal. 
283.783  bush. 


}  Cub  yds, 


13.060. 


15 


226 


WEIGHTS    AND    301ASUBES. 


Metric   Weisrhts,  reduced  to  eommoii  Commercial  or  ATOii 
Wel^bt,  of  1  pound  =  16  ounces,  or  7000  g^rains. 


Milligramme 

Centigramme 

Decigramme 

Gramme 

By  law  a  5-cent  nickel  =  5  grammes- 
Decagramme 

Hectogramme 

Kilogramme 

Myriogramme 

Quintal* 

Tonneau;  Millier;  or  Tonne 


Grains. 

.015432 

.15432 

1.5432 

15.432 

Pounds  av. 

.022046 
.22046 
2.2046 
22.046 
220.46 
2204.6 


The  gramme  !■  the  basis  of  French  weights ;  and  la  the  weight  of  a  cub  centimetre  of  distillal 
Vater  at  Its  max  density,  at  sea  level,  in  lat  of  Paris  ;  barom  29.922  ins. 

Frencli  Measures  of  the  "  Systeme  Usuel." 

This  system  was  in  use  from  about  1812  to  1840,  when  it  was  forbidden  by  law  to  use  even  its  names, 
This  was  done  in  order  to  expedite  the  general  use  of  the  tables  which  we'have  before  given.  But  a£ 
the  Systeme  Usuel  appears  in  books  published  during  the  above  interval,  we  add  a  table  of  some  of  ita 


Measures  of  liCng^th. 

Yards. 

Feet. 

Inches. 

.09113 

.09113 
1.09362 
3.93708 
6.56181 

1.09362 

.36454 
1.31236 
2.18727 

13.12344 

Aune  nanel,  or  ell <, 

Toi8«nBuel,=6pieds 

47.245 
78.74172 

Weights,  TJeuel. 

Cubic,  or  Solid,  TTsuel. 

.8375  grains. 
60.297        " 

1.10258  avoir  OS. 
.66129  avoir  lb. 

1.10258  avoir  lb. 

Litron  usuel,  or  1  litre 
Bolsseaa  usuel 

=  1.7608  British  pint. 

2.7512  British  gals. 

Livre  usuel,  > 
or  pound,  J  " 

Before  1812,  or  before  the  "Systeme  usuel,"  the  Old  System,  •'  Systeme  Ancien,"  was  in  use. 

Frencb  Measures  of  tbe  *'  Systeme  Ancien.'' 


LlneaU 

SquaM. 

Cubic. 

Point  ancien,  .0148  ins 

Ligne  ancien,  .0888  ins 

Sq.  ins. 
.00789 
1.1359 

Sq.ft. 

'i."i359 
40.8908 

Sq.  yds. 
4.5434 

C.  ins. 
.0007 
1.2106 

C.  ft. 

1.2106 
261.482 

C.yds. 

Pouce  ancien,  1.06577  ins  —  .0888  ft 

Pied  ancien  12  7892  ins  —  1  06577  ft 

Aune  ancien,  46.8939  ins  =3.90782  ft=  1.30261  yds 
Toise  ancien  "~  6  3946  ft — 2  1315  yds 

9.684S 

There  is,  however,  much  confusion  about  these  old  measures.     Different  measures  had  the  j 
lame  in  difereat  provinces. 


•  Tke  av9ir4ujfoi$  quintal  is  100  avoirdupois  pwamds. 


WEIGHTS   AND   MEASURES. 


Rassian. 


227 


Foot ;  same  as  U.  S.  or  British  foot.  Sactiine  =  7  feet.  Terst-  —  50f 
sachine  =  3500  feet  =  1166^  yards  =  .6629  mile.  Pood  =  36.114  lbs  avoirdupois, 

Spanish. 

Tlie  castellano  of  Spain  and  New  Granada,  for  weighing  gold,  is  variously 
estimated,  from  71.07  to  71.04  grains.  At  71.055  grains,  (the  mean  between  the 
two,)  an  avoirdupois,  or  common  commercial  ounce  contains  6.1572  castellano; 
and  a  ft)  avoirdupois  contains  98.515.  Also  a  troy  ounce  =  6.7553  castellano  ;  ana 
a  troy  ft)  =  81.064  castellano.    Three  U.  S.  gold  dollars  weigh  about  1.1  castellano. 

Tbe  ISpanisli  mark,  or  marco,  for  precious  metals,  in  South  America, 
may  be  taKen  in  practice,  as  .5065  of  a  ft)  avoirdupois.  In  Spain,  .5076  ft).  In 
other  parts  of  Europe,  it  has  a  great  number  of  values;  most  of  them,  however, 
being  between  .5  and  .54  of  a  pound  avoirdupois.  The  .5065  of  a  ft)  =  3545^ 
grains ;  and  .5076  ft)  =  3553.2  grains.  1  marco  =  50  castellanos  =  400  tomine  =« 
4800  Spanish  gold-grsAns. 

The  arroba  has  various  values  in  different  parts  of  Spain.  That  of  Cas- 
tile, or  Madrid,  is  25.4025  ft)s  avoirdupois;  the  tonelada  of  Castile  =  2032.2 
ft)s  avoirdupois;  the  quintal  =  101.61  ft)s  avoirdupois;  the  libra  =  1.0161 
ft)s  avoirdupois;  the  cantara  of  wine,  &c,  of  Castile  =  4.263  U.  S.  gallons; 
that  of  Havana  =  4.1  gallons. 

The  vara  of  Castile  =  32.8748  inches,  or  almost  preciselv  32'^  inches;  or  2 
feet  %%  inches.  The  faneg^ada  of  land  since  1801  =  1.5871  acres  =  69134.08 
square  feet.  The  fane^a  of  corn,  &c  =  1.59914  U.  S.  struck  bushels.  In 
California,  the  vara  by  law  =  33.372  U.  S.  inches ;  and  (he  le§^aa  =  5000 
taras;  or  2.6335  U.  S.  mile*. 


228 


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CONVERSION   TABLES. 


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CONVERSION   TABLES. 


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»    •Of0TttI>O 

u    -t^ooi-HOO 
4i '*.'-;  p  CO  cop -H- 
"Sco-^d»0'-Hdo 

*^(NCD{NCO  O 

gp<N(N  p^ 


CO  Tj*  05  i-H  Ti< 

i-HC^Q0<NfO 

Ot^OI>r-t 

C0f0  05I>'-l 
rHcoeOOOS 
lO  rt|  CO  Csj  p 
»0  »0  ^'  CO  CO 


CQ  ft 


w  ^3  ;3  3  3 
bObOo  o  o 


CO 
CO 

•coos 
•  ■»-*  "^ 
OrH(NOcoeo 

«H  lO  t^  CD  rH  CO 

•  OC^^iCCO^fN 

tilOi-HCO'T-TrH 


T-lO'-< 

oooco 
05  00^0 

OiOiN 

C01>Tt< 


QOOCOCO 
05(MC0O 

Oi-Hcoco 
lo-^coco 

"^OkOiO 
00  Ol  p  (£> 
CO  T-i  <m"  ,-J 


o  :        :  :  :  fl 
•  :       : •  :s 


OQ'd   DQ  TO   CD   M    S 

03  S§  c^SfSO 


>• 

<!    § 

•  ; 

•CO 

i 

o 

•h-    • 

•OJ 

t^ 

lOCO 

•lO 

'eS 

Tf 

05  03 

COCO 

> 

CO 

H-COTt* 

05IO 

n*^ 

II  »0  O  CO 

II  -KH-iO  ^, 

1 

Oo 

t>do6 

©CO       (M 

_  o  CO  CO  d 
'dO'-i'o 

C3 

flp    ^ 

H 

S 

0 

0 

tf 

o 

Ph 

«CDO 
Oo^ph' 

t-3 


«a 


CO     ^t^05 


o 


2§ 

OSO 

Tj«0 

III 

2^ 

poo 

f-Jr-i 

lOO 

ooo 
t^p 

lOCO 

oooo 
t>:00 
(n'coco 

i   Is    I|Ss 


X 

nCOrfH 
CO  00 

I   b-l003t^ 


^.•a 


d 

Q 


(fc.^  : 


is 


oo 
oo 
oo 
oo 

oo 
oo 
oo 


C<JC^OiOOO 
CO  '^t^  I^  CI  O  o 
OSOOOOiOOO 
'^OOiOOSOO 

kOi-iiMl^OO 

OOIN'^'^OO 

t^T}<co»opp 

THr-irHCicOCO 


•I 


as 


a 


:COH-o 
Cdod 


^00  Tin 

oc^co 

'-'OIN 

'd 


•»o 

00^ 
OOr-l 
OCO 

o»o 
c^co 
c^o  + 
ddc 


oo 
oo 
oo 
oo 


oo 


a    ^ 


l«_+J  3 


t^     o 


ft 


S-i-o 
;Sod 


236 


CONVERSION   TABLES. 


S(NrH 

Scoco 
Oeo(N' 


H  -< 


O  03 


si 

»0(N 

l>  »0  CI  O  r-<  OS 

CO  <N  1-1  (M  t^  00 

i-HCO^OOi-H 

»OC0(N05  1>iO 

3025 
1513 
7888 
4959 

COI^iO 

t-iOCO 

§8 

COiOOOiOCOCO 
C5  1-H  CO  T-1  Tt  rjl 

lOt^i-HlO 

OOi-Hi-l 

t^<MOO 

T-HCOCO 

S5 

(MOO 

00  (N 

rtl  CO  rH  T-H  CO  CO 

COr-KN'-t 

T-l(N'* 

(MtJIo 

wo 

is 


«    II 

>;  '^ooco 
p,J>co 
'^    c«od 


•CO"* 
OCOCO 

oor^t^ 

<N(N(M 

TjlOO 

oo 
ort^ddd 

TtH(N 


g3>> 


S  O^  •?  O  c3  O  o3 


e«oo 


SoO'^'* 

SocOcO 
<!»000 
Cjcopp 
Ococod 
1-^ 


si 


5  s*^ 


lO 


s 


<MO 

o 
H 


go      »hO 


oo 
c^io 

coo 
^o 


1-ICO  iH 


fac^       feb 


CSIr-H 

coo 

rtO 

1  »o6    ^^o    bto 


11^ 

So 


O         r 


bco 


Q      M      >$ 


1 

II  M 

Ton 

poun 
tons 
ton  ( 
kilogi 

1 

'«  • 

!  '.  '. 

»H      • 

V 

.iO    . 

a 

^ 

,-iO     • 

0) 

4)    . 

^  : 

(NTH     . 

« 

^<<NO0C     • 

>]( 

'"'O 

-"^i-iOO 

CJO            O 

fl'"' 

C(M           p 

fiCSJ             r-« 

s 

o 

c 

H 

CONVERSION   TABLES. 


237 


Koooco 
Sosooo 

^00  05  CO 

Oeco<M 


3  -S 


o  II 


btB^ 


4h  p^m 


•OOM 


CO      *'      J^r' 


1^  ,-4  lO  ic  «o  t^ 

rH(M(NC0»OO5 

CO  (M  (d  T}<  Tf  CO 

CO  d  CO  (N  th  CO 


OOOGOOi 
(M(NC0iO 
00C0(£)Oi 
OOCOO 


T-HCO 

COO 


^  CO  Tti  H-O)  00  o 
^CO      <©co 


So 

HO 


X  00  00  uo  05  CO 

r-lOr-IOCOX 

CDCOOCO'^O 

ROfNOiOOO 
3  rti  CO  Tf  CO  ^  t^ 

H  ^  ,_.  ,rH  C<J  lO  lO 
0  ■>^'  d  rH  CO  -^  ■^" 


(MCOOOrH 

tOCOOCOO 

0500^  COO 
(N  Tf  ■<*  '^  00 

dcocbiocsi 


swsg 


3  G  (U  bc'iH  faC^l' 


S^  "x 

■  V 


si      00 

O       CO 

o     ^ 


.     .     .(NO 

•   -oo-* 

"^      •  ■^  C5  t^ 

TtllCCOCOO 
XidTt^fNt^ 
C^(MXCOCO 
OX'-HOO 
lOOOOOO 

lOCOr-lOOO 


.,  --       .-£0 

Wa-^  M  S  t: 

o 


■a 


0 


O  O  --ti  lO  oi  o 

O  05  1— •  CO  05  >— ' 

00005  TfiCO. 

005COCO  coco 

OO'^^  t^x 

OOiCOO  OO 

o  00  p  -*_  00  p 


a 

II   0)  cc  c)*^ 

s^§aa 
'^^  s  ??  s 


W  p  '*'  CO  O  rH 

*> — 'O  ''f  ''^  '^  ■'^ 
O"0Tf<t>O 
P^o'fN'iOCO 

Kx     xt^ 


(NlOO 

x»ox 

r-HUOCO 

COO)  CO 


Ot^T-i»0 

05  X  C-l  o 

ocococo 
ooii>co 

Oi-HTf  05»0 

DOiiMOCO 

ftpT-HCO-^ 

6d<N  CO  CO 


fl  c  :  a  a  fl    JO 


0? 


O  03 


^:5 


Xn    C  a>  t-,  bc^L 


tA'i^-*^    -•-•■ 

HH'd'^t-o 
.     o  o 


xr^ 

OCO 

coos 


X 


OJ      '0505 

^     -oo     •       (NiO 
aiO»COi     -COrt^W 

SOriC^  -cocot^ 
C0COO5  •*  (MC^ 
coo  fi  CO  .-I  O '-I  r-"  O  O 
l^rH     S(NOOXOOO 

dd  '^t>'d>d>d>dd>d 


238 


CONVERSION  TABLES. 


mOOOO 

.'  Sos«oeo 

'-'      tJcOOSrH 


» 


» 
H 
O 

1-9 
«   II- 

Pd        -l-QOCO 


coco  OOS 

t^tJ*  i-HiO 
0505  ^CO 

coGOoeo 


"  $  cr  c  o* 


t>l>C0CO 

cooo»oo 

O5O5  5O00 
C0COr-t(N 
OC0  05r-J 


m 

^^  S  ^. 
ft  m  P, 


CO  CO  CO  o> 

irjlCr-HCO 

«OCO'^J<0 
O  0)0  ^ 

ddcoiH 


w  o 
ft  *>  -T 

J.a| 

Sees 


co^os 

l>«Or-l 

wscoeo 
cowt> 


II  « o  ^ 


—I  CO 
(N«0 

»CCO 


li  O'er 


1^*0  00  o 

00  t^"^   TP 

^COCO  CO 
00«CO  00 

O'-HTt^  Tj< 

IN'^OO 
0)000''^ 
WC0c4»H 

Cflfl 


O  O     - 

ftft«-  «- 

■g-gftft 
«2»2  0)  g 


WOS 

coos 
^^ 

lOCO 

coco 
ooo 

0<N 


o 
6 


^a 

o^ 


II  2  2* 


^Si 


:  11  : 

„ 

9  ^^^ 

5t^ 

S  -05^ 
.  (NOO(N 
2^<MOt- 

P.oo»o 
^ooo 
c66d 

o 

1^ 


M.<c,-l 


.    CO*    r-H      . 
5^*    COr-O 

M-XOOO 
OO*   "* 


P-rHO 

*   CO 


en 


CO 


CONVERSION  TABLES. 


239 


HO) 

3^ 


H 


2    ^^    S  ^ 

o      oj  o      So 
ft    -Ma    2ft 


Toco 


a    I 

o   3   a 


w 

P 


1^ 

p^ 


OCfNOi 

OSr-,<N 

OOOOJ 
^CCOO 


3  « 


t*      ©CJCOO 


Or-<<N 

O-^Oi 
0<NCO 


.  ft  o 
O  o  S^"-! 


5  -coo 
a,  'COth 
•^  •cci'O 

^4-*  o 

o-icod 


00© 

eot^ 

rHOO 
COCO 


ft^ 

II  rC  +:> 


4)     -OJ 
O 


iOCC-^ 
,-icOcO 


O    M    O 

03  ;h  a 

fe  fto- 


o 


OJCO 
qGOtHCO 

•^OCOb- 

ri<MOS 
HI       ^ 


Ob-© 

ooco 
oc:5  00 

8»oo 


O   03 


fl.SJ 


CO 


i  -2 


w-too 

^  s 

CO 


R(N00O> 
H'-i'-HCO 

S«><N05 

O-^coco 
oddco 


ma 


1;^  ft 

"3  3^  m 

>^2 


O 
O 


a,  .. 

i^+-(N  CO 
^-CONOO 


D.5 


cooco 

l^COrH 


H     •     MOO 


"^00 


J3 


'd-»- 

CO" 


00© 
»H©COI> 
CJiOOO 
ftOCOlC 

O*  I> 

^dodd 

Co 


ocso 

cooo 

Tj*OOTt< 

00-^05 

wooo 


o»ooi 

C^OirH 

T-tTj^OO 

©<M<N 
CO  10  CO 
CO(Nr-< 


Sa 


!»    5^    Si 
©    ft  « 

«    <1J    W 


DOST}* 

3©»0 


©  O  © 

5  c  © 


g;i7 


OiOCOtH  O     -OCO  .0     -CO 

SfNoorji  S-f-©o  Ijqcqoq 

pSdiOfN  c^i-ii-!d  cd  06  © 

^^HOi-H  ^co        -"^  ^       tH»0 


O  »H  ,-1 


H 

s 

Ah       I 

2b^ 


sfe©  5. 


^? 


©      '-' 

"oi 

00 

^  ^00 

Ml^ 

h^ 

C  000 

©10 

©00 

ft^. 

p.«? 

ft-^ 

i-« 

N. 

0 

•d 

•OrH 

•d 

a 

C3 

a 

?3 

J3 

0 

0 

0 

0 

P4 

A< 

04 

240 


CONVERSION  TABLES. 


ii 


•  cooooooco 
S  CO  a>  05  00 -^  Tt< 

<< « i>eo  CO  1-t  T-i 
O  X  "^.  lO  '^  p  o 

o  c^'  »H  r-1  .-^  d  d 


S  o  ^ 
O  O)  o 


o3  a 


obc2 


pCj      .0505 

&odfo 
wcocsjec 


005 

di-5 


P     C     JH 

a>  o  C 


coo 

t^eoeo 

OiCOCO 
COOO 


1  §.o 


ooooo 

,-((NCO^ 
00l>.05i0 

(NC^COOO 

cocoTt^o 

CO  OC  00  CO 

dcM'cid 


QJ  Qj  B   rj 
Vs  m  Vi  cc 

II 5  lis 


O*     -r-MCO 

COOCOGO 
^*COCX)0»0 
OJOcot^co 
p,coooo 
_^(Nddc<i 

o 


Ort< 

rH(N 


^  o  o 
©  •  • 

fe  :  : 

Coo  00 

(«COt;H 
'-HO 

;i^TticD 

CJi-HrH 

ftOO 
S3 

o 


OCOiO'*!"^ 

oot^x»o 

OOt^O>t>" 

OfO'^'OTH 

o^oX'-ico 


O)  O  <n       fn  Q) 

1?>  S  ^  £  i^  S 
c      ■  ■  '  " 


4) 

P(      -Tt^KCOO 

•coi^oot^ 

S-  (M  CO  O  »0 
•  (N  05  00  05 

g  1-H  TtJ  d  (m"  00 

gj       rH        CO(N 


ooooo 
r-ioa>io 

i>t>00 

CO'*  CO  00 
rt<©cO-<# 

C^r-ii-i,-; 


his 


®  ^  C  ?! 


Si>c<Jo 

.♦aOlT-HCO 
Ce^t^COlOrH 

^C^  00  CO  00 
O00TfT}< 

ooodd 

•♦a 

o 
o 


s§^ 

00 1- 

»Cl^ 

CO(N 

W(N 

T-Hoo       ooo 

coos 

t:-rH 

00  CO 

co-t 

OC^iC 

(NCO          0>? 

0<NCC 

W:::;S 

S^ 

(Nt-I 

-^OS 

iO« 

»oo       Or: 

O^OO 

'^ 

HOO<N 

coco 

t-rH 

(N05 

ooS 

(NO          00> 

oc^o 

QC 

§S^ 

r:::!^ 

(NO 

(NO 

ICO 

o^oo 

i-tOi          GO- 

OrHOl 

Ah 

-iJC^t^ 

CO  1-1 

COt^ 

cot^ 

COtH 

o^ox 

CO(M 

OQC 

OCO<N 

0O<N 

'^Os 

OOTt* 

<35  05 

l>cO 

050        occ 

ooo 

3°: 

i-id 

1-ir-; 

i-ir-; 

CO^ 

CO<N<N 

d«H       di- 

COr-id 

S 

!  s 

C 

:  cj 

fi     Ifl 

c 

fi  :  : 

e 

1 

• 

«s  ■ 

If 

a 

w 

2: 

03 

a 

2 

n 

1 

2: 

1 

''A  ■ 

1 

-0 

5 

ot 

a 

o 

a 

« 

^1 

•B  a 

^1    ft 
11^^ 

^"  : 

11^ 

aft 

IS 

1 

1 

CO 

B 

U 

ft 

;|s 

g 

» 
^ 
P 

T 

«5= 

O  S  3 

1 

3  d 

t3   CJ 

g  :  : 

Sq  Meter 

It mc 

3 cu 

45423... U 

d  :  . 

^ap 

a*  .*  ! 

igp 

•CO 

S  :S 

s 

•05 

ft  :^; 

-^co 

ft  •  • 

"OCO 

«   .   . 

ft    'CO 

+-"«^o 

s 

A^SS 

ft    •© 

ft 

*iO 

^  -N 

t^OO 

'Cq 

H-Oi 

Oioco 

©COTt< 

gOOSO 

aoiNd 

t.^O 

t^      •Tji 

J^OCOCO 

o 

tsq-i 

-C-f-(M, 

'd 

05(M 

a*  05 

S^t^ 

P,05  0 

S-J-iO 

0) 

OtJ^O 

k 

fe-^^ 

!SS° 

died 

^00 

cScod 

r^ 

/-«s 

H 

H 

h 

^1     ^ 

»^ 

^ 

iH 

GO 

QC»0 

o 

« 

lH 

S3 

P 

^ 

f4 

s 

«4 

0 

CONVERSION   TABLES. 


241 


a  s 


OrH(NrH 


a 


2       « 

.Si    ^3 


S  2  * 

S  O  Ui 

o  aw) 


a  A 


t-fOQO 

iqqco 

qOOCO 


p 


0:75 


"10  CO 

peot> 

o 


03 


JS  o3 

'd  S 

a  be 

»  or;3 


CO 

flfH 


c8 


10  <N 

05»0 


iij  bO 

^.2 


-S3  : 
be  : 


O(N06 


coco 

(NOO 

CNOO 


-t,-H         eo<N 


fl  bC 


00 

100 


:§      g 


Ssa 


P 
U 


eco 
coo 


^ 


«  fl  bC 

2  0:3 


2J 


|o8 


WO(N 


-<! 
ST 

i« 

2  H 

H    pi? 

Si 


<»  a 


9-2 


«coi^ 


2gi 

(N(Mr 

0<NC 

csio 


1-100 

OiCO 


o  (U  !«  tn 

5  <»  o  s 


o 
o 


tH  T-H  o 


go  "^ 


a    a 


^  Qc  <j  p  : 

a 
o 


*>    fl  ."ti   II  ."t; 


Is. 


«-  »- 


5  If  W)  '?  bO 


'5 


^      PM 


t<00     ^|C0 
OJi-H     dJOi 

o      o 


D  ^^  5r! 
<  '^  u% 
•    P  0.  S  ^ 

!  -2-^  f^ 

-  S  o  ^  o 
ft-g  ft 

^c^o  : 
S?5  Si  : 

(M         00 

SCO     ^^<N 
O     <^^ 

go  ft<N 

5    i 
w   5 


16 


242 


CONVERSION   TABLES. 


o 


w  ^      •  : 

M    ;3    J3    -co 


OtNOOO 

CO  OS  CO '^ 

cco<©»o 


00  CO  t^  00 
O0COO5CO 

COCOrt^OO 

t^coooo 
0'<t  ooo 


:  c 

2 

,^3 

-»j 

T 

5    -OOOOO 
J     •OO'OiO 

SiOco'^co 


mu 


'^  •  ■  :^ 

oqco 

O^CO 

•  O'-ico 

S.,-(rH|> 
cocooi> 

^COt^rHO 


H 

o 


H 

s 


coo 

C0Tf< 

ooo 


coca 
MOO 

l-HO) 


5^00  00 

QCQOt>-  i 

*    OS 


CO"* 
COCO 

coo 
cOiO 

T-HCO 


Tig 


rHOO 
O5  05 


"^00 

COiO 


(NCO 


•i    s' 


<D  g         £    II   o^ 

fl  •  o 

o  :  « 


2^  00  CO 


4)00  *     -OS 
QCO     .       -co 

^^co   ^!»cco 

h^    2^00(N 

fta> 


■^CO(N 


'  o 

o 


11^^ 


^  s 


s(nocomco 
Kco>o»Ot-ico 

Hr-tiOINcOCO 

WNOOOS-^CO 
<;^i>050»o 

O'^OOiCNcO 

O  ,-;  rH  C<i  C4  T-J 

:  c3  fl  fl  c 


(NO 

coo 
OiO 

lOO 
OOO 

i>o 

w6 


N(M.-HiOO 

COTf  t^c^io 

OSOOOOkOO 
'^XOOiO 


In.  "-t  CO  top 
r-JrHrH(NO 


:a2j 


2fl 


'«  S  e5  o 
o  be bco;:^ 


,  ,  J<    .  «co 

^     'lOOOOOC?  t^ 

05     -(NC^i-tO  SCSJ 

c«ooooooo»o  !«0 

fljO5O5O5COC0  CwrH 

^  1>  T-H  05 '-H  lO  flJCO 

gcOi-JOO'*.  HO  + 

s  s 

®  Js 


fljpl— ( 

o  fl  fl  o 

s 


I>     'OfOTtt 
H     -t^OOi-i 

?;^T-ioco 

coTt^oio 

aoicocqco 


o 
p 

O  - 

o 

w 

M 


00  00 
»-(OS 
»Or-t 

""^co 

1-1  <N 
lOCO 


lOt^-co 

05  0SC0 
005CO 

cqcoco 

»OOCO 


005 

0*0 

lOi-i 


1-1^  W(Nr- 


coco 

i-H> 

CO(N 

coo 
00  »o 

CC05 

coos 

6(N     bo 


^.-i     I 


'  -=5  : 


^  ^ 


'  «o 

a 

M 


.  y,     .     .     .  95     .     .  ^     .     • 

•  ^  :  •  :  ^  :  :  be  :  : 

:  bi)  .^   .  be  .   .  "S   .(N 

iO  fT     -t^TjH  jTiO     •  ^OO 

OS  S'-iO^O  So5»o  oco 

00  '^COX'-'  !>'-•  ^h"50> 

i-i^aOpOT^i-iCO  0)00 <N 

<N  ©CO      CO  d      00  i2 


s 
o 


;3 


S3 


CONVERSION   TABLES. 


243 


»H 

H 

tH 

tH 

tH 

-H 

r 

-1 

H 

tH 

y^ 

lO 

r^ 

«C 

■^ 

»OC^ 

CT 

t^OTH 

oeox 

•iC 

If! 

u: 

COO 

K> 

Soo: 

»0 

«c 

Utlr-H> 

(N  M 

(N 

rt^O^ 

lO 

ffit-cc 

00  «3 

cc 

t^ 

OOCO-H 

<^^^^ 

■^ 

1-Hrt^CC 

®S5 

M^^ 

COiC 

oc 

t-ooo 

T-irt* 

<N 

(N(NCO 

05005 

o 

^-« 

pJoscc 

05CC 

-^ 

oc 

(NTt^O 

occ 

OC 

l>^TtH 

1^05  "5 

d 

t 

H 

<Jt^l-H 

»Ot>i 

IC 

oot-io 

lOCC 

Tt^ 

rH05(M 

05CC50 

II 

CO-* 

(M 

o»oq 

l>rH 

ec 

05-^  in 

COt-hCO 

0(N(N 

c 

C 

CC 

'^ 

(NO 

c 

"^ 

C 

<NCO(N 

(NO 

CI  : 

i 

•-0 

•  a 

o 
d 

H 

^i 

> 

'.  ^ 

^1 

3 

m 

H 

-d 

13 

H 

c 

^ 

^1 

o 

O 

3 
5 

.s 

6 
T 

O   C 

II '^ 

II 

1 
5 

w  m  M 

II    g'i 

^  1 

per  Second  = 

826446.  .  .acre  per  ho 
0309917.  .sq  mile  per 
Ft2 sn  mftfprs  n 

per  Second  = 

sq  feet  per 

137741.  ,  .acre  per  da 
2259 sq  meters  p( 

•3 

H 

P 

11 

.11 

o 

1^ 

o 
o 

S 

0 

o 

2^ 

1 

^?c 

^-co(N 

.  <MCC 

ft(N 

%a 

1^ 

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0 

s- 

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o 

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iH 

c^ 

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t4 

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s 

H 

1 

s 

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CO 

H 

244 


CONVERSION   TABLES. 


n 


HOCO 
OQOOS 

Ococo 


^3  O) 


r|Dg 

Ah 

'    o    •    • 

iU  :t^ 

U 

•lO 

^ 

S  :§ 

P 

A  :g 

H^ 

^^o>o 

o 

Sqq 

2^^ 

^^. 

o 

CO 

w 

S 

U 

12 
9§ 


GO  to  CO 
oocdtj^ 

iOI>iO 

(MlO  rH 

ec6<M* 


•T3  O 

•  C3  O 

^  o  <i> 

5   O  M 

O  <u  j_ 


OcOTt< 
cOiOCO 
Q0O5  1O 

Csir-ir-J 


1'^ 

Hi 

•3    "^    (h 


go- 


ill! 


•CO 

rHTt^ 


o 


Oiocod  o 


coco 

Q-J— lOCO 

Ocodd 


cocoes 

TJHIOOO 


o 

Plh  < 


^^    s   4   ^ 

H 

gl    i    s    s 

g 

03        -g         ft        ft      2 
H    „           0           <»      II    r/j        ^ 

§|ll^lll1 

P^ 

3|,5|asa  ^ 

^•5|:W:^ 

P^ 
0 

^|iO        0 

p,<M   Boi   Sec 

w 

t>l    i^fM'     =^CC 

w 

'S^  bi'^  bcS; 

■!3    2    St- 

(£3   3 

eo  CO  CO  CO 

»HGOOOI> 
t>-C0O3l> 


»0  00  O  C<J  OS  lO  o 


t>.iOt> 


00  00O<N 
CSi-id'-< 


t;  !=!  o  c  5 
as  1  8  S 

O  o  O  P  g  oTl 


•OSOOi-H 
•CO  CO  CO  00 
•»OOt-H> 
•H— »— 05I>0'-' 

q  o  o  «^  6 1-i  6 

(MOiOTt^ 

T-<  O   10  t^ 

Pi*'*' co- 
os CO 


I 


^ 

2!; 

d 


CO 


CC 


P^ 
P 

o 


g     ^ 

^    w 


Sooco 
Wt^oso 

t^OSCOGO 

Sgo<no 

«- 05  05  1^ 
0,-ir-(r-4 


^H 

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0) 

ft 

Ui 

'n^ 

£ 

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W) 

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CC 

a» 

g^ 

P 

O  o3 


C3iO 


CO^  00 

"^(M  05l^ 

too  1^<N 

lOcO  C<JCO 

05^  cooo 

rH^O  lOOS 

d^*  d<N 


73 
5?  t 


05l>iM 

(NO5C0 

TjHt^lO 

OTjHTt^ 

05C0C0    , 

dcoco 


00 1-1  CO 

t^03^ 

co-*-^ 

(NCO^ 

CX)(M  CO 


^(  ft 


2     1^     fH 

h  ^  ft 
II  a»  a> 
"  aa-g 
«+3-+e  o 

-M    0)    0«4H 

S3  «  o  ^ 

M  O  O  o3 


ft05       ftCO 


10        lJ 


© 
© 

© 

© 
© 


OJ     .-^  O)     •     'i-t  4>  "^ 

ftCOO  ft    -0005  ft  -lO 

coo  '^T}H(M.-I  "^  -1-1 

-*   00  r-OON^  -.       lOCO 

i-w.,  Sooo   Soc^i'*   S^-*  i> 

TtH— f     ©005     0(NOO     OiOCOfN 

tqoq  S<Nq  Sqqq  a<^.<^^'^ 
*co  «^cod   ^oddd  oSi-Idco 

tH    ^  ^  ^00         tH 

CO  CC      '  QC 


2  u,  01 
II  fe^M 

ft"S| 

«  3  o  d         s 

4)  c)  c3  -^ 
JK    -     ■ 


*d 

fl  o 
2-^ 


gft 


10  »H         10 


ft  .  TS    . 


«  : 

ftrt^ 


I 


210 


CO 


'^       rt 


.5q  s-i 

flco     ftt- 


■1^  s 


J3       JS 

U        Hi 


CONVERSION   TABLES. 


245 


wo 

OrH 

t-:i  . 


I 

•      -2 

W    II  ® 


g 


ftco 


03 


O  bO 

t)  :  : 

h    '.    '. 

ft  — 


CO 


&4 


OOICO 


OOr-llO 


j3  bC    . 

•^  ^^  « 

II  <tl  ^'^ 

Ho  ^  6 

O  C>  bC 


Weo<No 
CO     ^ 
Pnco 


CO  l>-<*i-^ 

00  cooco 

'*  0  0  05 

1-5  <N(Nd 


PW 

o  : 


00  00 


O5  00 

ooo 

OOi 


S 


bcS 


08     • 


p^oooo 
^6'*od 

'U     o 

O 
Ph 


5wj  bC 

^.^   • 
S^cooi 

"d     «o 

a 

!3 


II  J5  o 


P.||      ft| 


'O  o"      «-'  *-> 

d  OM    ee  o 
bfi 


d    -00 

o 


Qo 


&- 


t^CO">if« 
COCOIO 

coo-* 

1-HCOCO 

eo'sJ^io 
cico(N 


IS 

_^  o  ^ 

ft  •    ;    ; 

h    '•<£>    • 

«    -OSOJ 

Soooo 
o 


*>l8 


M    M 


SCO       »-j 
t^       OS 


o        W 


n:co»c 


7303 
3795 
0613 
0168 

OOiCO 

«ocoi>05 
oot^ooos 

00  "i* 
1-1  (N 

W<N.-(^ 

cocoes 

(NC^ 

M  5 
0,^ 


•   02   £ 


CO 

W>i3t>.     -00 

«COTt<t>. 


Ph  O 
.2  a 


73    ;    ; 

gcoco 

OlOr-t 

OOOJ 

aQcoco 

T— (    T-l    CO 

ftddd 

CO 
«  CO 

o 
1^ 


ft 


OiOCO 

S.-<co 
CO* 


coco 
00^ 
00  CO 
OCO 

>ooo 
CO  00 

(NCO 


C3  O 

bCg 


I 

o 
> 

Ph  0-73  a 

^§a 
o 


o 
d 


w 


^D.-i<J 


'"^    2^00     . 

^>.     ft-"-!       • 

CO        t^'-' 

^^  Sooo 
gooo 

§,°^ 

I 

>** 

o 
o 


;3W) 
S  : 

ft    ; 

I 
•♦a 

o 
o 


248 


CONVERSION   TABLES. 


.2  ^ 


o 


COCC(N 


coooo 

OOJ 

eo(N 

ooo 

r^Oi-H 

l-HO--^ 

i-HO 

t-HrH 

oo 

^;S;S 


OOP, 


?3  3  O 

«  O  fH 

^,  .  a> 

«  QJ  ft 

CJ  O  01  rj 


B  5? -ft  ft 


.     I'd 

:3  3  o 
«  o  p. 
f^  .    a» 

^ftbo 
ti  b  <i5  S 

11    o3  o*^ 

S-2  ^^ 

42^  ft  to 


o 


ftdooo 


H 
S     . 

3  ■§ 

^^ 

e    ft 

51 

00  I 


ft^ 


03  e3 


OJCOO 

o 
Q 


occoo 


n  2-^ 

O 


WO 


OCOrHO 
ooo  CO  "^ 
•-     <00«> 


HiH  0<N(N(N  tH,-i.-(tH         (NCOOO 


^ 

% 

s 
? 


5cQ  s  S 


^  en  ««*-(  P 

i;^    •    -coob 
o    -coootji 

P^l-HCCr-lTH 

i-^ecdd 


Aco  , 


«  u.  (I 

^H     ^     «     Si 

rj^  OT  «  ft 


ftg,ftft 

a3;S  <JM 
C3 


-^^  o  S  2 

O  3 
•«  o  3  3 


_  5  "-^ 


10+-00 
^       CO 


Q-t—  -coco 
.  oO'-i'-iko 

^(MrHCO(N 

2i>^r-;66 


4)00 


•^  C    OQ    M 

'SS:ag§ 


too  .HrHTHr-l 


.      QJ   ftft        03 

s ft§  M   a 

w  2i  n.  S  o. 


CO 


^  Tjioooo 

>>Tt<00 

flOJoq 


00  Tj* 

_       COIN 

005     ftfOl^CO 

•■     ■         i-HOO 

M)d66 


(^ 

coco 

P,-K-_  1-1 1^  "^ 

co66»o 


ooTt<»oco 

ftO-^CDCO 

^(6<6d>d) 


A 


o 


CONVERSION   TABLES. 


249 


S0i050 
KOiOO 


^    < 


a 
o 

T 


e8  >>>> 
QJ  u.  f-i 


»h05(NO 

^  ^  _;  _r 


'O  O  CI 

5  S^  ?? 


O  0)  o 

5  M  0) 

4>   L.  W 

M  5  fc, 

II  no  A 


>>-0  o3  (S 


"     -00 
^1— l'«T  1— I 

ooo 


^  dc<l(N(N 


O  fl 
w  «  o3 


C5COO 
CCiDQO 


oooo 

Miodo 


a 
m 


>>>>o3 
^  o3T3 

S  <»  ft 


I 


^   >>o3 

I,    t^H^D     M 


»O»Ot>t>G0 

lO  <N  r-J  0«  r-J 
rH  CO  C<i  (N  (N 


S   t-   o   fa   ^^ 


ii-sa^ls 


03   g 


ftdioco 


CO  ....  .  g 
fci-j-  -(Nioeo  ^ 
Scoooo6di-J      2 


CO  00 
COtJH 


flfl 


^   o 


.-0 

a  3 
;3  o 
o  ft 


3  o 


a    -.S':^       ^ii^ 


1^3 

c8-c 


"lOCO 

•^dd 

V 

eg 


add 
;4 


c8 


O  CO  Tt<  CO 
(N^COrt^ 
OCOCOC^ 


M 

H 

05    d 


^  o  :3  s 

o  2  o  o 


05C010 


o  s 
h'^  2 


^Is 


toosoos 

005<N(N 
OOl>cO 

d  th  (n  csi 


3  :3  Jj  3 
o  o  3  o 

11  cj  ft«i3  B 


t-CO00<©iO 

P^T-H  O  (N  rH 

.  ^'(Ndd 


o 
0 


"co 

0)00  CO  lo 
p^O-^co 


.  lO  CD 

o»ooo 


C<irH,-l 


l>050 
(HrH,H 

C  ti  a 


O   3    O 

ftoj  ft 
Soft 


m 

•     •  M       _rt 

fH     tH     3  ^ 

3  3  O 

i « g  '^ 

^  o  ft      bl 


HoOfM 
.    CO'^t^ 

Sroooi 
*  ooo  CO 


OJOOS-^ 

ftddd 


250 


CONVERSION   TABLES. 


""a 


o  • 


CCOiOCO 

ooo5i>oo 


o 
© 

W 
H 

;?; 


cS  c3  ^  O) 
$  0)  a»  03 

teg  bCfl 
P  op  o 


CO  "5 

•  ceo 
•coo 


O5(M00 
OOI>i-l 

o^o 

CC(NO0 

COOOTfH 

""^rH  CO 


2^-s 


'    ?^   CO 


H-COO 

flot^oo 
^o    o 

Oco"     co" 


0<MiO 
01>r-^ 
0005 

0^05 

OiOCX) 

OrHCO 


S  So* 

o* 

«  fe  ^ 

»H  P.O. 


i^     -CO^ 

SooJd 
.-KM 


OrHOS 

0<M»0 

Ot-rH 

OOOi 

o^o> 

OiOOO 
OrHCO 


CJ    .  o 


ft 
Q 

•"oco-^ 

ViOt-KM 

S005 


'3 


^:coi>t^«o 

Scot^t>oo 


«0>0(N«0  l>COCCt^ 

»ocOTt<io        Tt^ioeo-^ 


o 
o 


P    O    fH  in 

^7]  O  O 

a>  P.  ^  (h 

P<  m  5?  9^ 


P^ 


TO   ' 


5,0.0. 


^ 


„   fl    CD    TO 

H  fl  - 
r  o 
P  « 

OJO 
S    .    C<J  O  GO  CD 

"-^  p-     cococo 


tH     ^H     P     O 

^H  fH  a>  O. 


•   •  fa  ^ 

Pi  h!  3  O 

2  5  2-^ 

!h    ;h  (U   0. 

<1^      <1^  O.    r« 

^^toS 

„     TO    TO  t,    rt 


cc 


ft    ■      -O  CD 

3f"<^«Do6id 

(H^tCO       CO 

0 


-^  CO  OOi 

1^00  0000 

lO'^         oot^ 


OSOO  «00  t^TH 

»OCO  <N(N  1-HrH 

O500  "^r-t  lOOl 


p^ 
o 

pi; 

p^ 


.  o 
Xoo 


rH,-(  OO 


bCbC 

II  a  ft 
k  ft 

I 

o  o 


as 

bObC 


S  f^  b 

P  c3  03 

©     tH     C 


BB 

he  be 

II     Sh    ^ 

^3  0.0. 

?S    TO   TO 


ill 


fi^ooko 

ft>>«^ 


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:J'2  ^..  „. 

2  a  ft  "S  ft  ^    .§ 

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H    gj    O)  C    CJ    0) 

re  o  p.  c3  ^  ft 

3  :  :  S 

t^OCOO 

g^CDCO  S<NCO 

^(N  r-l  ftCO  iH 

Eh  S 


CONVERSION   TABLES. 


251 


i-iO 

00"^ 

OSiO 

oo 

l>l> 

t- 

O-^J* 

ooo 

00  00 

i>o 

kOl^ 

c^t> 

U5C0 

05  10 

(Nl>. 

l>00 

Kloo 

CO  00 

Tt^O 

00^ 

Soo 

1-1  CO 

rJHlO 

OSrH 

, 

^0<N 

ot- 

0  04 

(N^ 

OS  00 

00 1> 

CO 

C<>(N 

COCO 

Tt<(N 

H 

^TtHlO 

(NO 

lO-H 

COOS 

00  (M 

00  CO 

CO 

Tt*O0 

Tj^OO 

(NCO 

<!Cqt^ 

IOCS 

t^»o 

T^-^ 

0*0 

CO 

ooo 

CO-* 

lO-^ 

^ 

(JCD05 

io»o 

coco 

oi> 

coc^ 

c^o 

OICO 

Oi-l 

-^Oi 

p 

'^"l 

dc^ 

(Ni-i 

<Ni-i 

^o 

6c^ 

(N 

t-tr-i 

d<N 

Wr-i 

;  fl 

fl  : 

fl  : 

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c5 

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11  : 

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1 

1  i-^ 

pa 

to   02 

?.  ^a 

11 

6 

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OJ  o 

Sis 

wis 

11-21 

1 

T3 

S^t^-S 

.  SIS 

^a 

Ksa 

0,0*  o" 

o 

3 

0)  w 

02    CQ 

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02  &<  C 

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tt 

t'i^ 

02  M    02 

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fl 

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Soscc 
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ojoocT 
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252 


CONVERSION   TABLES. 


Hi 

io 


B       C0<£) 
El       0(N 

(5  II  T-iO> 


0)  ^  ^ 


ooco 

oo»o 

05  05.0 
05  05X 

CO  C0 1-1 


OJ5 


«J  ft  ^^  ft 
o  o  c3  c) 


00 

II  '"' 

«0  050 

T-llO 

lOfO 

II  '^'^ 

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l-HTt^05 

11  o>eo 
11  xo 

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«o 

II  »oeo 

"  --t  <N 

rHrHO 

>>  ^ 


:  ^ 

ft  ft       ftft  Jh 


Ol>       (5COI>. 
^fO     --hOO 


TOi— I 


^li      ^11 


^05  : 

CJiOCO 


aj  0)  '-I 

^x  : 

^00  05 
Rl0»0 


111! 

S  5f?  »^  ^ 

fH   U    O    0) 

«  CO  fl 


p.-H(NO 

,0606 


© 
o 

o<o  o 

>^  p !! 

5^  o  s 


S(NCO 
C8!>0> 


OS          M 

CO 

OS 

coo 

COtH 

rH(N          COTt 

ooco 

0          M 

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CO  10 

rfiiO 

00  Tt 

Oiir 

10 1-1 

RO 

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t^ 

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10  »o 

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g 

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CONVERSION  TABLES. 


253 


,:;  OS  ^  t>.  00  00 

K  T-i  00  o  CO -^ 

fHi-Ht>OS»00 

<.T^Ot^I^00 

53  OS  QO  '-H  00  1> 

o  Tj^'  csi  T-I  o  CO 


Q  ^  a 

H^P    g 
»  QC  O 

CO  On 


OT3 

t,  C  m  «  o 


3  3 


:Dt)i 


oo 


a  • 
o  : 
«  . 


So "O  Tt^"  I>  CO 

»!NTf<T-H        OS 


o 


ft.OSt^ 
»CS>OS 

^oso 


CCOSrHTjteO 

»O(NI>iOC0 

CO'* --too —I 
OS  <-<  t>.  i>  t>. 

00'<^CO(Nt^ 
OOiOioXfN 

oowosoos 

CO  (N*  r-i  r-i  rH 


O-rt 


'  s 


01 


«  "  ^1  «  u, 
C+5   0)  O  (U 

.«  "^   ft  K    Q 

_^gbM^        -Oft 


OSI> 

«oc^ 

l>rH 

ooo 

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»OI> 


-+3         u  ly 


o 

©t>r 

o 


„§p 


t^  ft 


SOO*      ^CJO 
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t^         t^         1-* 


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o   . 

As 

0) 

Q,tH 

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po 


ft-a 


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d 


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lOCO 


io«o 
coos 

COfH 


II  Pc 

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fl  • 
o  : 


cc  -o 


o 
6 


at. 

_  o 
CIS  ft 


254 


WEIGHTS    AND   MEASURES. 


TABIiE  OF  ACRES  REQUIRED  per  mile,  and  per  100  feet, 
for  different  widtlis. 


Width. 

Feet. 

Acres 
Mile. 

Acres 

per 

100  Ft. 

Width. 
Feet. 

Acres 
per 
Mile. 

Acres 

per 

100  Ft. 

Width. 
Feet. 

Acres 
per 
Mile. 

Acres 
KxTpt. 

Width. 
Feet. 

Acres 

Acres 
lOO^FU 

1 

.121 

.002 

26 

3.15 

.060 

52 

6.30 

.119 

78 

9.45 

.179 

2 

.242 

.005 

27 

3.27 

.062 

53 

6.42 

.122 

79 

9.58 

.181 

3 

.364 

.007 

28 

3.39 

.064 

54 

6.55 

.124 

80 

9.70 

.184 

4 

.485 

.009 

29 

3.52 

.067 

55 

6.67 

.126 

81 

9.82 

.186 

5 

.606 

.011 

30 

3.64 

.069 

56 

6.79 

.129 

82 

9.94 

.188 

6 

.727 

.014 

31 

3.76 

.071 

67 

6.91 

.131 

K 

10. 

.189 

7 

.848 

.016 

32 

3.88 

.073 

M 

7. 

.133 

83 

10.1 

.190 

8 

.970 

.018 

33 

4.00 

.076 

58 

7.03 

.133 

84 

10.2 

.193 

H 

1. 

.019 

34 

4.12 

.078 

59 

7.15 

.135 

85 

10.3 

.195 

9* 

1.09 

.021 

35 

4.24 

.080 

60 

7.27 

.138 

86 

10.4 

.197 

10 

1.21 

.023 

36 

4.36 

.083 

61 

7.39 

.140 

87 

10.5 

.200 

11 

1.33 

.025 

37 

4.48 

.085 

62 

7.52 

.142 

88 

10.7 

.202 

12 

1.46 

.028 

38 

4.61 

.087 

63 

7.64 

.145 

89 

10.8 

.204 

13 

1.58 

.030 

39 

4.73 

.090 

64 

7.76 

.147 

90 

10.9 

.207 

14 

1.70 

.032 

40 

4.85 

.092 

65 

7.88 

.149 

% 

11. 

.209 

15 

1.82 

.034 

41 

4.97 

.094 

66 

8. 

.151 

91 

11.0 

.209 

16 

1.94 

.037 

% 

5. 

.094 

67 

8.12 

.154 

92 

11.2 

.211 

IT^ 

2. 

.038 

42^ 

5.09 

.096 

68 

8.24 

.156 

93 

11.3 

.213 

2.06 

.039 

43 

5.21 

.099 

69 

8.36 

.158 

94 

11.4 

.216 

18 

2.18 

.041 

44 

5.33 

.101 

70 

8.48 

.161 

95 

11.5 

.218 

19 

2.30 

.044 

45 

5.45 

.103 

71 

8.61 

.163 

96 

11.6 

.220 

20 

2.42 

.046 

46 

5.58 

.106 

72 

8.73 

.165 

97 

11.8 

.223 

21 

2.55 

.048 

47 

5.70 

.108 

73 

8.85 

.168 

98 

11.9 

.225 

22 

2.67 

.051 

48 

5.82 

.110 

74 

8.97 

.170 

99 

12. 

.227 

23 

2.79 

.053 

49 

5.94 

.112 

H 

9. 

.170 

100 

12.1 

.230 

24 

2.91 

.055 

H 

6. 

.114 

75 

9.09 

.172 

% 

3. 

.057 

50^^ 

6.06 

.115 

76 

9.21 

.174 

36* 

3.03 

.057 

51 

6.18 

.117 

77 

9.33 

.177 

GRADES. 


255 


Table  of  g>radeN  per  mile,  and  per  100  feet  measured  hori- 
zontally, ana  corresponding^  to  different  ang^les  of  incli* 
nation. 


^  .9 

Feet  per 

Feet  per 

ti   d    F 

eet  per 

Feet  per 

«  a 

Feet  per 

Feet  per 

lA 

Feet  per 

Feet  per 

o  S 

mile. 

100  ft. 

Q   S 

mile. 

100  ft. 

Q  a 

mile. 

100  ft. 

mile. 

100  ft. 

0     1 

1.536 

.0291 

0  45 

69.11 

1.3090 

1  58 

181.3 

3.4341 

3  26 

316.8 

5.9994 

2 

3.072 

.0582 

46 

70.64 

1.3381 

2    0 

184.4 

3.4924 

28 

319.8 

6.0579 

3 

4.608 

.0873 

47 

72.18 

1.3672 

2 

187.5 

3.5506 

30 

322.9 

6.1163 

4 

6.144 

.1164 

48 

73.72 

1.3963 

4 

190.6 

8.6087 

32 

326.0 

6.1747 

5 

7.680 

.1455 

49 

75.26 

1.4254 

6 

193.6 

S.6669 

84 

329.1 

6.2330 

6 

9.216 

.1746 

60 

76.80 

1.4545 

8 

196.7 

3.7250 

36 

332.2 

6.2914 

7 

10.75 

.2037 

51 

78.33 

1.4837 

10 

199.8 

3.7833 

38 

335.3 

6.3498 

8 

12.29 

.2328 

52 

79.87 

1.5128 

12 

202.8 

3.8416 

40 

338.4 

6.4083 

9 

13.82 

.2619 

53 

81.40 

1.5419 

14 

205.9 

8.8999 

42 

341.4 

6.4664 

30 

15.36 

.2909 

54 

82.94 

1.5710 

16 

208.9 

3.9581 

44 

344.5 

6.5246 

11 

16.90 

.3200 

55 

84.47 

1.6000 

18 

212.0 

4.0163 

46 

347.6 

6.5832 

12 

18.43 

.3491 

56 

86.01 

1.6291 

20 

215.1 

4.0746 

48 

350.7 

6.6418 

13 

19.96 

.3782 

57 

87.54 

1.6583 

22 

218.1 

4.1329 

50 

353.8 

6.7004 

14 

21.50 

.4073 

58 

89.08 

1.6873 

24 

221.2 

4.1911 

62 

356.8 

6.7583 

15 

23.04 

.4364 

59 

90.62 

1.7164 

26 

224.3 

4.2494 

54 

359.9 

6.8163 

16 

24.58 

.4655 

1 

92.16 

1.7455 

28 

227.4 

4.3076 

66 

363.0 

6.8751 

17 

26.11 

.4946 

2 

95.23 

1.8038 

30 

230.5 

4.3659 

68 

366.1 

6.9339 

18 

27.64 

.5237 

4 

98.30 

1.8620 

32 

233.5 

4.4242 

4 

369.2 

6.9926 

19 

29.17 

.5528 

6      ] 

01,4 

1.9202 

34 

236.6 

4.4826 

6 

376.9 

7.1384 

20 

30.72 

.5818 

8      ] 

04.5 

1.9784 

36 

239.7 

4.5409 

10 

?84.6 

7.2842 

21 

32.26 

.6109 

10      ] 

07.5 

2.0366 

38 

242.8 

4.5993 

15 

392.3 

7.4300 

22 

33.80 

.6400 

12      1 

10.6 

2.0948 

40 

245.9 

4.6576 

20 

400.1 

7.5767 

23 

35.33 

.6691 

14      1 

13.6 

2.1530 

42 

248.9 

4.7159 

25 

407.8 

7.7234 

24 

86.86 

.6982 

16      1 

16.7 

2.2112 

44 

252.0 

4.7742 

80 

415.5 

7.8701 

25 

38.40 

.7273 

18      1 

19.8 

2.2694 

46 

255.1 

4.8325 

35 

423.2 

8.0163 

26 

39.94 

.7564 

20      1 

22.9 

2.3277 

48 

258.2 

4.8908 

40 

431.0 

8.1625 

27 

41.47 

.7855 

22      1 

26.0 

2.3859 

50 

261.3 

4.9492 

45 

438.7 

8.3087 

28 

43.01 

.8146 

24      1 

29.1 

2.4441 

52 

264.3 

5.0075 

60 

446.5 

8.4554 

29 

44.54 

.8436 

26      1 

32.1 

2.5023 

64 

267.4 

5.0658 

55 

454.2 

8.6021 

30 

46.08 

.8727 

28      1 

35.2 

2.5604 

56 

270.5 

5.1241 

5 

461.9 

8.7489 

31 

47.62 

.9018 

30      1 

38.3 

2.6186 

68 

273.6 

6.1824 

6 

469.6 

8.8951 

32 

49.16 

.9309 

32      1 

41.3 

2.6768 

3 

276.7 

6.2407 

10 

477.4 

9.0413 

33 

50.69 

.9600 

34      I 

44.4 

2.7350 

a 

279.7 

6.2990 

15 

485.1 

9.1875 

34 

52.23 

.9891 

36      1 

47.4 

2.7932 

4 

282.8 

6.3573 

20 

492.9 

9.3347 

35 

53  76 

1.0182 

38      1 

50.5 

2.8514 

6 

285.9 

6.4158 

25 

500.6 

9.4819 

36 

55.30 

1.0472 

40      1 

53.6 

2.9097 

8 

289.0 

6.4742 

80 

508.4 

9.6292 

37 

56  83 

1.0763 

42      1 

56.6 

2.9679 

10 

292.1 

6.5326 

35 

516.1 

9.7755 

38 

58.37 

1.1054 

44      1 

59.7 

3.0262 

12 

295.1 

6.5909 

40 

523.9 

9.9218 

39 

59.90 

1.1345 

46      1 

62.8 

3.0844 

14 

298.2 

6.6493 

45 

531.6 

10.068 

40 

61.44 

1.1636 

48      1 

65.9 

3.1427 

16 

301.3 

6.7077 

50 

539.4 

10.215 

41 

62.97 

1.1927 

50      1 

69.0 

3.2010 

18 

304.4 

J.  7660 

55 

547.2 

10.362 

42 

6*.51 

1.2218 

52      1 

72.0 

3.2592 

20 

307.5 

5.8244 

6 

555. 

10.510 

4S 

66.04 

1.2509 

54      1 

75.1 

3.3175 

22 

310.5 

5.8827 

44 

67.57 

1.2800 

56      1 

78.2 

3.3758 

24 

313.6 

5.9410 

On  a  turnpike  road  1°  38',  or  about  1  in  35,  or  151  feet  per  mile,  is  the 
greatest  slope  that  will  allow  horses  to  trot  down  rapidly  with  safety.  In  crossing 
mountains,  this  is  often  increased  to  3°,  or  eren  to  6°.  It  should  never  exceed  2^^°, 
except  when  abaolutely  necessary. 

Any  iior  dist  is  =  sloping  dist  X  cosine  ang  of  slope. 

**     sloping*  dist  is  =  hor  dist        -*-  cosine     "    "       " 
"     vert  beigbt    is  =  hor  dist        X  tangent "    "       " 
or  =  sloping  dist  X  sine        "    "       " 

A  grade  of  n  feet  rise  per  100  feet  is  usually  called  a  grade  of  n  per  cent. 


256 


WEIGHTS   AND   MEASURES. 


SLOPES    IN    FEKT    PER   100    FT.   HORIZONTAI<. 


The  fractions  of  minutes  are  given  onlj  to  34  feet  in  100. 

A  clinometer  graduated  by  the  3d  columr,  and  numbered  by  the  first  one, 
will  give  at  sight  the  slopes  in  feet  per  100  feet.     No  errors.  Original. 


(6 

Length  of 
slope  per 
100  ft  hor. 

Angle  of 
slope. 

Length  of 
slope  per 
100  ft  hor. 

Angle  of 
slope. 

1§g 

Length  of 
100  ft  hor. 

Angle  of 
slope. 

Feet. 

Deg. 

Min. 

Feet. 

Deg. 

Min. 

Feet. 

Deg. 

Min. 

1 

100.005 

0 

34.4 

35 

105.948 

19 

17 

69 

121.495 

34 

36 

2 

100.020 

1 

8.7 

36 

106.283 

19 

48 

70 

122.066 

35 

0 

3 

100.045 

1 

43.1 

37 

106.626 

20 

18 

71 

122.642 

35 

23 

4 

100.080 

2 

17.5 

38 

106.977 

20 

48 

72 

123.223 

35 

45 

5 

100.125 

2 

51.8 

39 

107.336 

21 

18 

73 

123.810 

36 

8 

6 

100.180 

3 

26.0 

40 

107.703 

21 

48 

74 

124.403 

36 

30 

7 

100.245 

4 

0.3 

41 

108.079 

22 

18 

75 

125.000 

36 

52 

8 

100.319 

4 

34.4 

42 

108.462 

22 

47 

76 

125.603 

37 

14 

9 

100.404 

5 

8.6 

43 

108.853 

23 

16 

77 

126.210 

37 

36 

10 

100.499 

5 

42.6 

44 

109.252 

23 

45 

78 

126.823 

37 

57 

11 

100.603 

6 

16.6 

45 

109.659 

24 

14 

79 

127.440 

38 

19 

12 

100.717 

6 

50.6 

46 

110.073 

24 

42 

60 

128.062 

38 

40 

IS 

100.841 

7 

24.4 

47 

110.494 

25 

10 

81 

128.690 

39 

1 

14 

100.975 

7 

58.2 

48 

110.923 

25 

38 

82 

129.321 

39 

21 

15 

101.119 

8 

31.9 

49 

111.359 

26 

6 

83 

129.958 

39 

42 

16 

101.272 

9 

5.4 

50 

111.803 

26 

34 

84 

130.599 

40 

2 

17 

101.435 

9 

38.9 

51 

.  112.254 

•27 

1 

85 

131.244 

40 

22 

18 

101.607 

10 

12.2 

52 

112.712 

27 

28 

86 

131.894 

40 

42 

19 

101.789 

10 

45.5 

53 

113.177 

27 

55 

87 

132.548 

41 

1 

20 

101.980 

11 

18.6 

54 

113.649 

28 

22 

88 

133.207 

41 

21 

21 

102.181 

11 

51.6 

55 

114.127 

28 

49 

89 

133.869 

41 

40 

22 

102.391 

12 

24.5 

56 

114.612 

29 

15 

90 

134.536 

41 

59 

23 

102.611 

12 

57.2 

57 

115.104 

29 

41 

91 

135.207 

42 

18 

24 

102.840 

13 

29.8 

58 

115.603 

30 

7 

92 

135.882 

42 

3T 

25 

103.078 

14 

2.2 

59 

116.108 

30 

32 

93 

136.561 

42 

55 

26 

103.325 

14 

34.5 

CO 

116.619 

30 

58 

94 

137.244 

43 

14 

27 

103.581 

15 

6.6 

61 

117.137 

31 

23 

95 

137.931 

43 

32 

28 

103.846 

15 

38.5 

62 

117.661 

31 

48 

96 

138.622 

43 

50 

29 

104.120 

16 

10.3 

63 

118.191 

32 

13 

97 

139.316 

44 

8 

30 

104.403 

16 

42.0 

64 

118.727 

32 

37 

98 

140.014 

44 

25 

31 

104.695 

17 

13.4 

65 

119.269 

33 

1 

99 

140.716 

44 

43 

32 

104.995 

17 

44.7 

66 

119.817 

33 

25 

100 

141.421 

45 

00 

33 

105.304 

18 

15.8 

67 

1J0.370 

33 

49 

101 

142.180 

45 

17 

34 

105.622 

18 

46.7 

69 

120.930 

34 

13 

102 

142.843 

45 

34 

In  describing  railroad  g'rades,  it  is  usual,  as  in  our  tables,  to  refer  the 

rise,  A,  to  the  corresponding  horizontal  length,  B.    We  then  have  , r  =  ^  = 

length       B 
the  tangent  of  the  angle,  a,  between  the  plane  and  the 
horizontal.    If  the  rise,  A,  be  referred  to  the  sloping 

length,  C,  we  have  .  ^  ^  r  ^  "**^  -^ '  ^^^  *^^^ 
fraction  is  proportional  to  the  component,  S,  of  the 
weight,  W,  in  the  direction  of  the  slope.  Thus,  on  a 
grade  where  rise,  A,  =  0.1  X  sloping  length,  C,  we 
have  sin  a  =  0.1,  and  S  =  0.1  W.  The  tangent  of  a  is 
only  approximately  proportional  to  S ;  but  the  steepest 
grades,  surmounted  by  traction  only,  even  on  electric 
railways,  rarely,  if  ever,  exceed  from  13  to  15  per 
cent ;  and,  on  these,  the  error,  due  to  using  tan  a  in- 
stead of  sin  a,  is  less  than  a  difference  of  0.2  per  cent 

in  the  grade,  and  about  =  1  per  (Tent  of  the  true  vakie  of  S.     For  steeper  grades, 

such  as  those  of  rack  railways,  it  should  always  be  specified  whether  the  rise 

refers  to  the  horizontal  or  to  the  sloping  measurement. 
Transvense  slopes,  such  as  those  of  earthwork,  are  sometimes,  like  railroad 
^         i.  X  J  .      ft  vertical  A    ,    ,  „     .       ft  horizontal         B     „ 

grades,  stated  in  -^-r — -. —;-,   =,,  but  usually  in  -^^ — ,  as  -r-    Here 

ft  horizontal '   B'  •'ft  vertical      '       A 

-  is  the  cotangent  of  the  angle,  a,  with  the  horizontal,  or  the  tangent  of  the 

angle  (90°-^)  with  the  vertical.    Thus  stated,  a  slope  of  2  to  1  means  a  slope  of  2 
horizontal  to  1  vertical. 


GRADES. 


257 


Table  of  srrades  per  mile;  or  per  100  feet  measured  liorl* 
zontally. 


Grade 

Grade 

Grade 

Grade 

Grade 

Grader 

Grade 

Grade 

in  ft. 

in  ft. 

in  ft. 

in  ft. 

in  ft. 

in  ft. 

in  ft. 

in  ft. 

per  mile. 

per  100  ft. 

per  mile. 

per  100  ft. 

per  mile. 

per  100  "ft. 

per  mile. 

per  100  ft. 

1 

.01894 

39 

.73S64 

77 

1.45833 

115 

2.17803 

2 

.03788 

40 

.75758 

78 

1.47727 

116 

2.19697 

3 

.05682 

41 

.77652 

79 

1.49621 

117 

2.21591 

4 

.07576 

42 

.79545 

80 

1.51515 

118 

2.23485 

5 

.09470 

43 

.81439 

81 

1.53409 

119 

2.25379 

6 

-11364 

44 

.83333 

82 

.  1.55303 

120 

2.27273 

7 

.13258 

45 

.85227 

83 

1.57197 

121 

2.29167 

8 

.15152 

46 

.87121 

84 

1.59091 

122 

2.31061 

9 

.17045 

47 

.89015 

85 

1.60985 

123 

2.32955 

10 

.18939 

48 

.90909 

86 

1.62879 

124 

2.34848 

11 

.20833 

49 

.92803 

87 

1.64773 

125 

2.36742 

12 

.22727 

50 

.94697 

88 

1.66666 

126 

2.38636 

13 

.24621 

51 

.96591 

89 

1.68561 

127 

2.40530 

14 

.26515 

52 

.98485 

90 

1.70455 

128 

2.42424 

15 

.28409 

63 

1.00379 

91 

1.72348 

129 

2.44318 

16 

.30303 

54 

1.02273 

92 

1.74242 

130 

2.46212 

17 

.32197 

65 

1.04167 

93 

1.76136 

131 

2.48106 

18 

.34091 

66 

1.06061 

94 

1.78030 

132 

2.50000 

19 

.35985 

67 

1.07955 

95 

1.79924 

133 

2.51894 

20 

.37879 

68 

1.09848 

96 

1.81818 

134 

2.53788 

21 

.39773 

59 

1.11742 

97 

1.83712 

135 

2.55682 

22 

.41667 

60 

1.13636 

98 

1.85606 

1.36 

2.57576 

23 

.43561 

61 

1.15530 

99 

1.87500 

137 

2.59470 

24 

.45455 

62 

1.17424 

100 

1.89394 

138 

2.61364 

25 

.47348 

63 

1.19318 

101 

1.91288 

139 

2.63258 

26 

.49242 

64 

1.21212 

102 

1.93182 

140 

^65152 

27 

.51136 

65 

1.23106 

103 

1.95076 

141 

2.67045 

28 

.53030 

66 

1.25000 

104 

1.96970 

142 

2.68939 

29 

.54924 

67 

1.26894 

105 

1.98864 

143 

270833 

30 

.56818 

68 

1.28788 

106 

2.00758 

144 

2.72727 

31 

.58712 

69 

1.30682 

107 

2.02652 

145 

2.74621 

32 

.60606 

70 

1.32576 

108 

2.04545 

146 

2.76515 

33 

.62500 

71 

1.34470 

109 

2.06439 

147 

2.78409 

34 

.64394 

72 

1.36364 

110 

2.08333 

148 

2.80303 

35 

.66288 

73 

1.38258 

111 

2.10227 

149 

2.82197 

36 

.68182 

74 

1.40152 

112 

2.12121 

150 

2.84091 

37 

.70076 

75 

1.42045 

113 

2.14015 

151 

2.85986 

38 

.71970 

76 

1.43939 

114 

2.15909 

152 

2.87879 

If  the  grade  per  mile  should  consist  of  feet  and  tenths,  add  to  the  grade  per  100 
feet  in  the  foregoing  table,  that  corresponding  to  the  number  of  tenths  taken  from 
the  table  below ;  thus,  for  a  grade  of  43.7  feet  per  mile,  we  have  .81439  +  .01326  = 
.82765  feet  per  100  feet. 


Ft.  per  Mile. 

Per  100  Feet. 

Ft.,  per  Mile. 

Per  100  Feet. 

Ft.  per  Mile. 

Per  100  Feet. 

.05 

.00094 

.4 

.00758 

.7 

.01326 

.1 

.00189 

.45 

.00852 

.75 

.01420 

.16 

.00283 

.6 

.00947 

.8 

.01515 

.2 

.00379 

.65 

.01041 

.85 

.01609 

.26 

.00473 

.6 

.01136 

.9 

.01705 

A 

.00568 

.65 

.01230 

.95 

.0179i 

.35 

.00662 

17 


258 


WEIGHTS  AND   MEASURES. 


TABIiJG  OF  HEABiS  OF  WATFR  CORRESiPOlTDINO  TO 
OITEN  PRESSURES. 

Water  at  maximum  density,  62.425  lbs.  per  cubic  foot  =  1  gram  per  cubia 
centimeter ;  correspon^iing  to  a  temperature  of  4°  Centigrade  =  39.2"^  Fahrenheit. 

Head  in  feet  =  2.306768   X  pressure  in  lbs.  per  square  inch. 
"  "     =  0.0160192  X  pressure  in  lbs.  per  square  foot. 

Heads  corresponding  to  pressures  not  given  in  the  table  can  be  found  by  these 
formulae,  or  taken  from  the  table  by  simple  proportion. 


Pregsure. 

Head. 

Pressure, 

Head. 

Pressure. 

Head. 

lb«.  per 

lbs.  per 

Feet. 

lbs.  per 

lbs.  per 

Feet. 

lbs.  pel 

lbs.  per 

Feet. 

aq.  in. 

sq.  ft. 

sq.  in. 

sq.  ft. 

sq.  in. 

sq.  ft. 

1 

144 

2.3068 

51 

7344 

117.645 

101 

14544 

232.984 

2 

288 

4.6135 

52 

7488 

119.952 

102 

14688 

235.290 

3 

432 

6.9203 

53 

7632 

122.259 

103 

14832 

237.597 

4 

576 

9.2271 

54 

7776 

124.565 

104 

14976 

239.904 

5 

720 

11.5338 

55 

7920 

126.872 

105 

15120 

242.211 

6 

864 

13.8406 

56 

8064 

129.179 

106 

15264 

244.517 

7 

1008 

16.1474 

57 

8208 

131.486 

107 

15408 

246.824 

8 

1152 

18.4541 

58 

8352 

133.793 

108 

15552 

249.131 

9 

1296 

20.7609 

59 

8496 

136.099 

109 

15696 

251.438 

10 

1440 

23.0677 

60 

8640 

138.406 

110 

15840 

253.744 

11 

1584 

25.3744 

61 

8784 

140.713 

111 

15984 

256.061 

12 

1728 

27.6812 

62 

8928 

143.020 

112 

16128 

258.358 

13 

1872 

29.9880 

63 

9072 

145.326 

113 

16272 

260.665 

14 

2016 

32.2948 

64 

9216 

147.633 

114 

16416 

262.972 

15 

2160 

34.6015 

65 

9360 

149.940 

115 

16560 

265.278 

16 

2304 

36.9083 

66 

9504 

152.247 

116 

16704 

267.585 

17 

2448 

39.2151 

67 

9648 

154.553 

117 

16848 

269.892 

18 

2592 

41.5218 

68 

9792 

156.860 

118 

16992 

272.199 

19 

2736 

43.8286 

69 

9936 

159.167 

119 

17136 

274.505 

20 

2880 

46.1354 

70 

10080 

161.474 

120 

17280 

276.812 

21 

3024 

48.4421 

71 

10224 

163.781 

121 

17424 

279.119 

22 

3168 

50.7489 

72 

10368 

166.087 

122 

17568 

281.426 

23 

3312 

53.0557 

73 

10512 

168.394 

123 

17712 

283.732 

24 

3456 

55.3624 

74 

1065S 

170.701 

124 

17856 

286.039 

25 

3600 

57.6692 

75 

10800 

173.008 

125 

18000 

288.346 

26 

3744 

59.9760 

76 

10944 

175.314 

126 

18144 

290.653 

27 

3888 

62.2827 

77 

11088 

177.621 

127 

18288 

292.960 

28 

4032 

64.5895 

78 

11232 

179.928 

128 

18432 

295.266 

29 

4176 

66.8963 

79 

11376 

182.235 

129 

18576 

297.573 

30 

4320 

69.2030 

80 

11520 

184.541 

130 

18720 

299.880 

31 

4464 

71.5098 

81 

11664 

186.848 

131 

18864 

302.187 

32 

4608 

73.8166 

82 

11808 

189.155 

132 

19008 

304.493 

33 

4752 

76.1233 

83 

11952 

191.462 

133 

19152 

306.800 

34 

4896 

78.4301 

84 

12096 

193.769 

134 

19296 

309.107 

35 

5040 

80.7369 

85 

12240  t 

196.075 

135 

19440 

311.414 

36 

5184 

83.0436 

86 

12384 

198.382 

136 

19584 

313.720 

87 

5328 

85.3504 

87 

12528 

200.689 

137 

19728 

316.027 

38 

5472 

87.6572 

88 

12672 

202.996 

138 

19872 

318.334 

39 

5616 

89.9640 

89 

12816 

205.302 

139 

20016 

320.641 

40 

5760 

92.2707 

90 

12960 

207.609 

140 

20160 

322.948 

41 

5904 

94.5775 

91 

13104 

209.916 

141 

20304 

325.254 

42 

6048 

96.8843 

92 

13248 

212.223 

142 

20448 

327.561 

43 

6192 

99.1910 

93 

13392 

214.529 

143 

20592 

329.868 

44 

6336 

101.4978 

94 

13536 

216.836 

144 

20736 

332.175 

45 

6480 

103.8046 

95 

13680 

219.143 

145 

20880 

334.481 

46 

6624 

106.1113 

96 

13824 

221.450 

146 

21024 

336.788 

47 

6768 

108.4181 

97 

13968 

223.756 

147 

21168 

339.095 

48 

6912 

110.7249 

98 

14112 

226.063 

148 

21312 

341.402 

49 

7056 

113.0316 

99 

14256 

228.370 

149 

21456 

343.708 

50 

7200 

115.3384 

100 

14400 

230.677 

150 

21600 

346.015 

WEIGHTS  AND  MEASURES. 


259 


TABIi£  OF  PRESSURES  CORRESPONDINO  TO  OIVEIT 
HEADS  OF  WATER. 

Water  at  maximum  density,  62.425  lbs.  per  cubic  foot  =  1  gram  per  cubic 
•entimeter ;  corresponding  to  a  temperature  of  4°  Centigrade  =  39.2^  Fahrenheit. 

Pressure  in  lbs.  per  square  inch  =   0.433507  X  head  in  feet. 
Pressure  in  lbs.  per  square  foot  =  62.425       X  head  in  feet. 
Pressures  carresponding  to  heads  not  given  in  the  table  can  be  found  by  thes« 
formulae,  or  taken  from  the  table  by  simple  proportion. 


Head. 

Inches. 


Pressure. 


lbs.  per  sq.  in.    lbs.  per  sq.  ft. 


0.036126 
0.072251 
0.108377 
0.144502 
0.180628 
0.216753 


5.202083 
10.404167 
15.606250 
20.808333 
26.010417 
31.212500 


Head. 

Inches. 


9 
10 
11 
12 


Pressure. 


lbs.  per  sq.  in.    lbs.  per  sq.  ft. 


0.252879 
0.289005 
0.325130 
0.361256 
0.397381 
0.433507 


36.414583 
41.616667 
46.818750 
52.020833 
57.222917 
62.425000 


Pressure. 

Pressure. 

Pressure. 

Head. 

Feet. 

Head. 

Feet. 

Head. 

lbs.  per 

lbs.  per 

lbs.  pel 

lbs.  per 

Feet. 

lbs.  per 

IbB.  per 

sq.  in. 

sq.  ft. 

sq.  in. 

sq.  ft. 

sq.  in. 

sq.  ft. 

1 

0.4335 

62.425 

38 

16.4733 

2372.150 

75 

32.5130 

4681.875 

2 

0.8670 

124.850 

39 

16.9068 

2434.575 

76 

32.9465 

4744.300 

3 

1.3005 

187.275 

40 

17.3403 

2497.000 

77 

33.3800 

4806.725 

4 

1.7340 

249.700 

41 

17.7738 

2559.425 

78 

33.8135 

4869.150 

5 

2.1675 

312.125 

42 

18.2073 

2621.850 

79 

34.2471 

4931.575 

6 

2.6010 

374.550 

43 

18.6408 

2684.275 

80 

34.6806 

4994.000 

7 

3.0345 

436.975 

44 

19.0743 

2746.700 

81 

35.1141 

5056.425 

8 

3.4681 

499.400 

45 

19.5078 

2809.125 

82 

35.5476 

5118.850 

9 

3.9016 

561.825 

46 

19.9413 

2871.550 

83 

35.9811 

5181.275 

10 

4.3351 

624.250 

47 

20.3748 

2933.975 

84 

36.4146 

5243.700 

11 

4.7686 

686.675 

48 

20.8083 

2996.400 

85 

36.8481 

5306.125 

12 

5.2021 

749.100 

49 

21.2418 

3058.825 

86 

37.2816 

5368.550 

13 

5.6356 

811.525 

50  ■ 

21.6753 

3121.250 

87 

37.7151 

5430.975 

14 

6.0691 

873.950 

51 

22.1089 

3183.675 

88 

38.1486 

5493.400 

15 

6.5026 

936.375 

52 

22.5424 

3246.100 

89 

38.5821 

5555.825 

16 

6.9361 

998.800 

53 

22.9759 

3308.525 

90 

39.0156 

5618.250 

17 

7.3696 

1061.225 

54 

23.4094 

3370.950 

91 

39.4491 

5680.675 

18 

7.8031 

1123.650 

55 

23.8429 

3433.375 

92 

39.8826 

5743.100 

19 

8.2366 

1186.075 

56 

24.2764 

3495.800 

93 

40.3162 

5805.525 

20 

8.6701 

1248.500 

57 

24.7099 

1  3558.225 

94 

40.7497 

5867.950 

21 

9.1036 

1310.925 

58 

25.1434 

3620.650 

95 

41.1832 

5930.375 

22 

9.5372 

1373.350 

59 

25.5769 

3683.075 

96 

41.6167 

5992.800 

23 

9.9707 

1435.775 

60 

26.0104 

3745.500 

97 

42.0502 

6055.225 

24 

10.4042 

1498.200 

61 

26.4439 

3807.925 

98 

42.4837 

6117.650 

25 

10.8377 

1560.625 

62 

26.8774 

3870.350 

99 

42.9172 

6180.075 

26 

11.2712 

1623.050 

63 

27.3109 

3932.775 

100 

43.3507 

6242.500 

27 

11.7047 

1685.475 

64 

27.7444 

3995.200 

101 

43.7842 

6304925 

28 

12.1382 

1747.900 

65 

28.1780 

4057.625 

102 

44.2177 

6367.350 

29 

12.5717 

1810.325 

66 

28.6115 

4120.050 

103 

^4.6512 

6429.775 

30 

13.0052 

1872.750 

67 

29.0450 

4182.475 

104 

45.0847 

6492.200 

31 

13.4387 

1935.175 

68 

29.4785 

4244.900 

105 

45.5182 

6554.625 

32 

13.8722 

1997.600 

69 

29.9120 

4307.325 

106 

45.9517 

6617.050 

33 

14.3057 

2060.025 

70 

30.3455 

4369.750 

107 

46.3852 

6679.475 

34 

14.7392 

2122.450 

71 

30.7790 

4432.175 

108 

46.8188 

6741.900 

36 

16.1727 

2184.875 

72 

31.2125 

4494.600 

109 

47.2523 

6804.325 

36 

15.6063 

2247.300 

73 

31.6460 

4557.025 

110 

47.6868 

6866.750 

87 

16.0398 

2309.725 

74 

32.0795 

4619.450 

111 

48.1193 

6929.175 

260 


WEIGHTS   AND  MEASIJRES. 


TABI.I: 

OF  PRESISURES  (Continued). 

Pressure. 

Pressure. 

Pressure. 

Head. 

Feet. 

Head. 

Feet. 

Head. 

Feet. 

lbs.  per 

lbs.  per 

lbs.  per 

lbs.  per 

lbs.  per 

lbs.  per 

sq.  in. 

sq.  ft. 

sq.  in. 

sq.  ft. 

sq.  in. 

sq.  ft. 

112 

48.5528 

6991.600 

144 

62.4250 

8989.200 

176 

76,2972 

10986.800 

113 

48.9863 

7054.025 

145 

62.8585 

9051.625 

177 

76.7307 

11049.225 

114 

49.4198 

7116.450 

146 

63.2920 

9114.050 

178 

77.1642 

11111.650 

115 

49.8533 

7178.875 

147 

63.7255 

9176.475 

179 

77.5978 

11174.075 

116 

50.2868 

7241.300 

148 

64.1590 

9238.900 

180 

78.0313 

11236.500 

117 

50.7203 

7303.725 

149 

64.5925 

9301.325 

181 

78.4648 

11298.925 

118 

51.1538 

7366.150 

150 

65.0260 

9363.750 

182 

78.8983 

11361.350 

119 

51.5873 

7428.575 

151 

65.4596 

9426.175 

183 

79.3318 

11423.775 

120 

52.0208 

7491.000 

152 

65.8931 

9488.600 

184 

79.7653 

11486.200 

121 

52.4543 

7553.425 

153 

66.3266  • 

9551.025 

185 

80.1988 

11548.625 

122 

52.8879 

7615.850 

154 

66.7601 

9613.450 

186 

80.6323 

11611.050 

123 

53.3214 

7678.275 

155 

67.1936 

9675.875 

187 

81.0658 

11673.475 

124 

53.7549 

7740.700 

156 

67.6271 

9738.300 

188 

81.4993 

11735.900 

125 

54.1884 

7803.125 

157 

68.0606 

9800.725 

189 

81.9328 

11798.325 

126 

54.6219 

7865.550 

158 

68.4941 

9863.150 

190 

82.3663 

11860.750 

127 

55.0554 

7927.975 

159 

68.9276 

9925.575 

191 

82.7998 

11923.175 

128 

55.4889 

7990.400 

160 

69.3611 

9988.000' 

192 

83.2333 

11985.600 

129 

55.9224 

8052.825 

161 

69.7946 

10050.425 

193 

83.6669 

12048.025 

130 

56.3559 

8115.250 

162 

70.2281 

10112.850 

194 

84.1004 

12110.450 

131 

56.7894 

8177.675 

163 

70.6616 

10175.275 

195 

84.5339 

12172.875 

132 

57.2229 

8240.100 

164 

71.0951 

10237.700 

196 

84.9674 

12235.300 

133 

57.6564 

8302.525 

165 

71.5287 

10300.125 

197 

85.4009 

12297.725 

134 

58.0899 

8364.950 

166 

71.9622 

10362.550 

198 

85.8344 

12360.150 

135 

58.5234 

8427.375 

167 

72.3957 

10424.975 

199 

86.2679 

12422.575 

136 

58.9570 

8489.800 

168 

72.8292 

10487.400 

200 

86.7014 

12485.000 

137 

59.3905 

8552.225 

169 

73.2627 

10549.825 

201 

87.1349 

12547.425 

138 

59.8240 

8614.650 

170 

73.6962 

10612.250 

202 

87.5684 

12609.850 

139 

60.2575 

8677.075 

171 

74.1297 

10674.675 

203 

88.0019 

12672.275 

140 

60.6910 

8739.500 

172 

74.5632 

10737.100 

204 

88.4354 

12734.700 

141 

61.1245 

8801.925 

173 

74.9967 

10799.525 

205 

88.8689 

12797.125 

142 

61.5580 

8864.350 

174 

75.4302 

10861.950 

206 

89.3024 

12859.550 

143 

61.9915 

8926.775 

175 

75.8637 

10924.375 

207 

89.7359 

12921.975 

Table  showing  the  total  pressure  against  a  vertical  plane 

one  foot  wide,  extending  from  the  surface  of  the  water  to  the  depth  named  in 
the  first  column. 

Water  at  its  maximum  density.  62.425  lbs  per  cubic  foot  =  1  gram  per  cv^M 
centimeter,  corresponding  to  a  temperature  of  4°  Cent.  -=  39.2°  Fahr. 

Total  pressure  in  poun^-^  =  31.2125  X  square  of  depth  in  feet. 


Depth. 

Total 
pressure. 

Depth. 

Total 
pressure. 

Depth. 

Total 
pressure. 

Depth. 

Total 
pi-essurt 

Feet. 

Pounds. 

Feet. 

Pounds. 

* 

Feet. 

Pounds. 

Feet. 

Pounds. 

1 

31.21 

17 

9020 

33 

33990 

49 

74941 

2 

124.85 

18 

10113 

34 

36082 

50 

78031 

3 

280.9 

19 

11268 

35 

38235 

51 

81184 

'   4 

499.4 

20 

12485 

36 

40451 

52 

84399 

5 

780.3 

21 

13765 

37 

42730 

53 

87676 

6 

1124 

22 

15107 

38 

45071 

54 

9ioie 

7 

1529 

23 

16511 

39 

47474 

55 

94418 

8 

1998 

24 

17978 

40 

49940 

60 

112365 

9 

2528 

25 

19508 

41 

52468 

65 

131873 

10 

3121 

26 

21100 

42 

55059 

70 

152941 

11 

3777 

27 

22754 

43 

57712 

75 

175570. 

12 

4495 

28 

24471 

44 

60427 

80 

1997G0 

13 

5275 

29 

26250 

45 

63205 

85 

225510 

14 

6118 

30 

28091 

46 

66046 

90 

252821 

15 

7023 

31 

29995 

47 

68948 

95 

281(593 

16 

7990 

32 

31962 

48 

71914 

100 

312125 

WEIGHTS  AND  MEASURES. 


261 


TABIiE  OF  I>ISCHARO£S  IN  CUBIC  FEET  PER  SECOND 
CORRESPONOINO  TO  GIVEX  OUSCHAROESi  IN  IJ.  S. 
OAEEONS  PER  24  HOURS. 

U.  S.  gallon  =  231  cubic  inches. 

Discharge  in  cubic  feet  per  second  =  1.54728  X  discharge  in  millions  of  U.  S.  gal' 

Ions  per  24  hours. 


Millions 

Millions 

Millions 

Millions 

of  U.S. 

Cubic  feet 

of  U.  S. 

Cubic  feet 

of  U.  S. 

Cubic  feet 

of  U.  S. 

Cubic  feet 

gals,  per 

'  per  second. 

gals,  per 

per  second. 

gals,  per 

per  second. 

gals,  per 

per  second. 

24  hrs. 

24  hrs. 

24  hrs. 

24  hrs. 

.010 

.0154723 

13 

20.1140 

43 

66.5308 

72 

111.400 

.020 

.0309446 

14 

21.6612 

44 

68.0781 

73 

112.948 

.030 

.0464169 

15 

23.2084 

45 

69.6253 

74 

114.496 

.040 

.0618891 

16 

24.7557 

46 

71.1725 

75 

116.042 

.050 

.0773614 

17 

26.3029 

47 

72.7197 

76 

117.589 

.060 

.0928337 

18 

27.8501 

48 

74.2670 

77 

119.137 

.070 

.108306 

19 

29.3973 

49 

75.8142 

78 

120.684 

.080 

.123778 

20 

30.9446 

50 

77.3614 

79 

122.231 

.090 

.139251 

21 

32.4918 

51 

78.9087 

80 

123.778 

.100 

.154723 

22 

34.0390 

52 

80.4559 

81 

125.326 

.200 

.309446 

23 

35.5863 

53 

82.0031 

82 

126.873  ; 

.300 

.464169 

24 

37.1335 

54 

83.5503 

83 

128.420 

.400 

.618891 

25 

38.6807 

56 

85.0976 

84 

129.967 

.500 

.773614 

26 

40.2279 

56 

86.6448 

85 

131.514 

.600 

.928337 

27 

41.7752 

57 

88.1920 

86 

133.062 

.700 

1.08306 

28 

43.3224 

58 

89.7393 

87 

134.609 

.800 

1.23778 

29 

44.8696 

59 

91.2865 

88 

136.156 

.900 

1.39251 

30 

46.4169 

60 

92.8337 

89 

137.703 

1 

1.54723 

31 

47.9641 

61 

94.3809 

90 

139.251 

2 

3.09446 

32 

49.5113 

62 

95.9282 

91 

140.798 

3 

4.64169 

33 

51.0685 

63 

97.4754 

92 

142.345 

4 

6.18891 

34 

52.6058 

64 

99.0226 

93 

143.892 

5 

7.73614 

35 

54.1530 

.65 

100.570 

94 

145.439 

6 

9.28337 

36 

55.7002 

66 

102.117 

95 

146.987 

7 

10.8306 

37 

57.2475 

67 

103.664 

96 

148.534 

8 

12.3778 

38 

58.7947 

68 

105.212 

97 

150.081 

9 

13.9251 

39 

60.3419 

69 

106.759 

98 

151.628 

19 

15.4723 

40   • 

61.8891 

70 

108.306 

99 

153.176 

11 

17.0195 

41 

63.4364 

71 

109.853 

100 

154.723 

12 

18.5667 

42 

64.9836 

262 


WEIGHTS  AND  MEASURES. 


TABIiE  OF  I>I8CHAROES  i:sr  CUBIC  FEET  PER  SECOND 
CORRESPOIVDINO  TO  GIVEN  DISCHARGES  IN  IM- 
PERIAI.   GAI.EONS   PER  24   HOURS. 

Imperial  gallon  =  277.274  cubic  inches. 

Discharge  in  cubic  feet  per  second  =  1.85717  X  discharge  in  Imperial  gallons  per 

24  hours. 


Millions 

Millions 

Millions 

Millions 

of  Imp. 

Cubic  feet 

of  Imp. 

Cubic  feet 

of  Imp. 

Cubic  feet 

of  Imp. 

Cubic  feet 

•gals,  per 

per  second. 

gals,  per 

per  second. 

gals,  per 

per  second. 

gals,  per 

per  second. 

24hrs. 

24  hrs. 

24  hrs. 

24  hrs. 

.010 

.0185717 

13 

24.1432 

43 

79.8583 

72 

133.7162 

.020 

.0371434 

14 

26.0004 

44 

81.7155 

73 

135.5734 

.030 

.0557151 

15 

27.8576 

45 

83.5727 

74 

137.4306 

.040 

.0742868 

16 

29.7147 

46 

85.4298 

75 

139.2878 

.050 

.0928585 

17 

31.5719 

47 

87.28/0 

76 

141.1449 

.060 

.111430 

18 

33.4291 

48 

89.1442 

77 

143.0021 

.070 

.130002 

19 

35.2862 

49 

91.0013 

78 

144.8593 

.080 

.148574 

20 

37.1434 

50 

92.8585 

79 

146.7164 

.090 

.167145 

21 

39.0006 

51 

94.7157 

80 

148.5736 

.100 

.185717 

22 

40.8577 

52 

96.5728 

81 

150.4308 

.200 

.371434 

23 

42.7149 

53 

98.4300 

82 

152.2879 

.300 

.557151 

24 

44.5721 

54 

100.2872 

83 

154.1451 

.400 

.742868 

25 

46.4293 

55 

102.1444 

84 

156.0023 

.500 

.928585 

26 

48.2864 

56 

104.0015 

85 

157.8595 

.600 

1.11430 

27 

50.1436 

57 

105.8587 

86 

159.7166 

.700 

1.30002 

28 

52.0008 

58 

107.7159 

87 

161.5738 

.800 

1.48574 

29 

53.8579 

59 

109.5730 

88 

163.4310 

.900 

1.67145 

30 

55.7151 

60 

111.4302 

89 

165.2881 

1 

1.85717 

31 

57.5723 

61 

113.2874 

90 

167.1453 

2 

3.71434 

32 

59.4294 

62 

115.1445 

91 

169.0025 

3 

5.57151 

33 

61.2866 

63 

117.0017 

92 

170.8596 

4 

7.42868 

34 

63.1438 

64 

118.8589 

93 

172.7168 

6 

9.28585 

35 

65.0010 

65 

120.7160 

94 

174.5740 

6 

11.1430 

36 

66.8581 

66 

122.5732 

95 

176.4312 

7 

13.0002 

37 

68.715"3 

67 

124.4804 

96 

178.2883 

8 

14.8574 

38 

70.5725 

68 

126.2876 

97 

180.1455 

9 

16.7145 

39 

72.4296 

69 

128.1447 

98 

182.0027 

10 

18.5717 

40 

74.2868 

70 

130.0019 

99 

183.8598 

11 

20.4289 

41 

76.1440 

71 

131.8591 

1»0 

185.7170 

12 

22.2860 

42 

78.0011 

WEIGHTS  AND   MEASURES. 


263 


TABL.I:  OF  DlfSCHAROES  IN  OAT.I.ONS  PER  34  HOURS 
€ORR£SPONI>INO  TO  GIVEN  DISCHARGES  IN  CUBIC 
FEET   PER  SECOND. 

U.  S,  gallon  =  231  cubic  inches.    Imperial  gallon  =  277.274  cubic  inches. 
Discharge    in  U.  S.  gallons  per  24  hours  =  646317  X  discharge  in  cubic  feet 

per  second. 
Discharge  in  Imperial  gallons  per  24  hours  =  538454  X  discharge  in  cubic  feet 

per  second. 


Cub.  ft. 
per  sec. 

Millions  of 

Millions  of 

Cub.  ft. 

Millions  of 

Millions  of 

U.  S.  gallons 

Imperial  gallons 

per  sec. 

U.  S.  gallons 

Imperial  galloni 

per  24  hours. 

per  24  hours. 

per  24  hours. 

per  24  hours. 

1 

0.646317 

0.538454 

53 

34.254795 

28.538W4 

2 

1.292634 

1.076907 

54 

34.901112 

29.076498 

3 

1.938951 

1.615361 

55 

35.547428 

29.614951 

4 

2.585268 

2.153815 

56 

36.193745 

30.153405 

5 

3.231584 

2.692268 

57 

36.840062 

30.691859 

6 

3.877901 

3.230722 

58 

37.486379 

31.230312 

7 

4.524218 

3.769176 

59 

38.132696 

31.768766 

8 

5.170535 

4.307629 

60 

38.779013 

32.307220 

9 

5.816852 

4.846083 

61 

39.425330 

32.845673 

10 

6.463169 

5.384537 

62 

40.071647 

33.384127 

11 

7.109486 

5.922990 

63 

40.717963 

33.922581 

12 

7.755803 

6.461444 

64 

41.364280 

34.461034 

13 

8.402119 

6.999898 

65 

42.010597 

34.999488 

14 

9.0484:^6 

7.538351 

66 

42.656914 

35.537942 

15 

9.694753 

8.076805 

67 

43.303231 

36.076395 

16 

10.341070 

8.615259 

68 

43.949548 

36.614849 

17 

10.987387 

9.153712 

69 

44.595865 

37.153303 

18 

11.633704 

9.692166 

70 

45.242182 

37.691756 

19 

12.280021 

10.230620 

71 

45.888498 

38.230210 

20 

12.926338 

10.769073 

72 

46.534815 

38.768664 

21 

13.572654 

11.307527 

73 

47.181132 

39.307117 

22 

14.218971 

11.845981 

74 

47.827449 

39.845571 

23 

14.865288 

12.384434 

75 

48.473766 

40.384025 

24 

15.511605 

12.922888 

76 

•  49.120083 

40.922478 

25 

16.157922 

13.461342 

77 

49.766400 

41.460932 

26 

16.804239 

13.999795 

78 

50.412717 

41.999385 

27 

17.450556 

14.538249 

79 

51.059034 

42.5378.39 

28 

18.096873 

15.076702 

80 

51.705350 

43.076293 

29 

18.743190 

15.615156 

81 

52.351667 

43.614746 

30 

19.389506 

16.153610 

82 

52.997984 

44.153200 

31 

20.035823 

16.692063 

83 

53.644301 

44.691654 

32 

20.682140 

17.230517 

84 

54.290618 

45.230107 

33 

21.328457 

17.768971 

85 

54.936935 

45.768561 

34 

21.974774 

18.307424 

86 

55.583252 

46.307015 

35 

22.621091 

18.845878 

87 

66.229569 

46.845468 

36 

23.267408 

19.384332 

88 

66.875885 

47.383922 

37 

23.913725 

19.922785 

89 

57.522202 

47.922376 

38 

24.560041 

20.461239 

90 

58.168519 

48.460829 

39 

25.206^58 

20.999693 

91 

58.814836 

48.999283 

40 

25.852675 

21.538146 

92 

69.461153 

49.537737 

41 

26.498992 

22.076600 

93 

60.107470 

50.076190 

42 

27.145309 

22.615054 

94 

60.753787 

50.614644 

43 

27.791626 

23.153507 

95 

61.400104 

51.153098 

44 

28.437943 

23.691961 

96 

62.046420 

51.691551 

46 

29.084260 

24.230415 

97 

62.692737 

52.23000« 

46 

29.7.30576 

24.768868 

98 

63.339054 

52.7684.59 

47 

30.376893 

25.307322 

99 

63.985371 

53.306912 

48 

31.023210 

25.845776 

100 

64.631688 

53.845366 

49 

31.669527 

26.384229 

101 

65.278005 

54.383820 

60 

32.^15844 

26.922683 

102 

65.924322 

54.922273 

51 

32.962161 

27.461137 

103 

66.570639 

55.460727 

52 

33.608478 

27.999590 

104 

67.216956 

55.999181 

264 


WEIGHTS  AND  MEASURES. 


TABIiE  OF  I>IS€HARO£S  (Continaed). 


Cub.  ft. 
per  sec. 

Millions  of 

Millions  of 

Cub.  ft. 

Millions  of 

Millions  of 

U.  S.  gallons 

Imperial  gallons 

per  sec. 

U.  S.  gallons 

Imperial  gallon! 

per  24  hours. 

per  24  hours. 

per  24  hours. 

per  24  hours. 

i05 

67.863272 

56.537634 

167 

107.934919 

89.921761 

106 

68.509589 

57.076088 

168 

108.581236 

90.460215 

107 

69.155906 

57.614542 

169 

109.227553 

90.998669 

108 

69.802223 

58.152995 

170 

109.873870 

91.537122 

109 

70.448540 

58.691449 

171 

110.520186 

92.075576 

110 

71.094857 

59.229903 

172 

111.166503 

92.614030 

111 

71.741174 

59.768356 

173 

111.812820 

93.152483 

112 

72.387491 

60.306810 

174 

112.459137 

93.690937 

113 

73.033807 

60.845264 

175 

113.105454 

94.229391 

114. 

73.680124 

61.383717 

176 

113.751771 

94.767844 

115 

74.326441 

61.922171 

177 

114.398088 

95.306298 

116 

74.972758 

62.460625 

178 

115.044405 

95.844751 

117 

75.619075 

62.999078 

179 

115.690722 

96.383205 

118 

76.265392 

63.537532 

180 

116.337038 

96.921659 

119 

76.911709 

64.075986 

181 

116.983355 

97.460112 

120 

77.558026 

64.614439 

182 

117.629672 

97.998566 

121 

78.204342 

65.152893 

183 

118.275989 

98.537020 

122 

78.850659 

65.691347 

184 

118.922306 

99.075473 

123 

79.496976 

66.229800 

185 

119.568623 

99.613927 

124 

80.143293 

66.768254 

186 

120.214940 

•  100.152381 

125 

80,789610 

67.306708 

187 

120.861257 

100.690834 

126 

81.435927 

67.845161 

188 

121.507573 

101.229288 

127 

82.082244 

68.383615 

189 

122.153890 

101.767742 

128 

82.728561 

68.922068 

190 

122.800207 

102.306195 

129 

83.374878 

69.460522 

191 

123.446524 

102.844649 

130 

84.021194 

69.998976 

192 

124.092841 

103.383103 

131 

84.667511 

70.537429 

193 

124.739158 

103.921556 

132 

85.313828 

71.075883 

194 

125.885475 

104.460010 

133 

85.960145 

71.614337 

195 

126.031792 

105.098464 

134 

86.606462 

72.152790 

196 

126.678108 

105.536917 

135 

•   87.252779 

72.691244 

197 

127.324425 

106.075371 

136 

87.899096 

73.229698 

198 

127.970742 

106.613825 

137 

88.545413 

73.768151 

199 

128.617059 

107.152278 

138 

89.191729 

74.306605 

200 

129.263376 

107.690732 

139 

89.838046 

74.845059 

201 

129.909693 

108.229186 

140 

90.484363 

75.383512 

202 

130.556010 

108.767639 

141 

91.130680 

75.921966 

203 

131.202327 

109.306093 

142 

91.776997 

76.460420 

204 

131.848644 

109.844547 

143 

92.423314 

76.998873 

205 

132.494960 

110.383000 

144 

93.069631 

77.537327 

206 

133.141277 

110.921454 

145 

93.715948 

78.075781 

207 

133.787594 

111.459908 

146 

94.362264 

78.614234 

208 

134.433911 

111.998361 

147 

95.008581 

79.152688 

209 

135.080228 

112.536815 

148 

95.654898 

79.691142 

210 

135.726545 

113.075269 

149 

96.301215 

80.229595 

211 

136.372862 

113.613722 

150 

96.947532 

80.768049 

212 

137.019179 

114.152176 

151 

97.593849 

81.306503 

213 

137.665495 

114.690630 

1521 

98.240166 

81.844956 

214 

138.311812 

115.229083 

153 

98.886483 

82.383410 

215 

138.958129 

115.767537 

154 

99.532800 

82.921864 

216 

139.604446 

116.305991 

155 

100.179116 

83.460317 

217 

140.250763 

116.844444 

156 

100.825433 

83.998771 

218 

140.897080 

117.382898 

157 

101.471750 

84.537225 

219 

141.543397 

117.921352 

158 

102.118067 

85.075678 

220 

142.189714 

118.459805 

159 

102.764384 

85.614132 

221 

142.836030 

118.998259 

160 

103.410701 

86.152586 

222 

143.482347 

119.536713 

361 

104.057018 

86.691039 

223 

144.128664 

120.075166 

162 

104.703335 

87.229493 

224- 

144.774981 

120.613620 

163 

105.349651 

87.767947 

225 

145.421298 

121.152074 

164 

105.995968 

88.306400 

226 

146.067615 

121.690527 

165 

106.642285 

88.844854 

227 

146.713932 

122  ?'?8981 

166 

107.288602 

89.383308 

228 

147.360249 

122.767434 

TIME. 


265 


TABI.C 

OF  ]>IS€IIARO£S  (Continued). 

Cub.  ft 

Millions  of 

Millions  of 

Cub.  ft. 

Millions  of 

Millions  of 

per  sec. 

U.  S.  gallons 

Ijpperial  gallons 

per  sec. 

U.  S.  gallons 

Imperial  galloni 

per  24  hours. 

per  24  hours. 

per  24  hours. 

per  24  hours. 

229 

148.006566 

123.305888 

240 

155.116051 

129.228878 

230 

148.652882 

123.844342 

241 

155.762368 

129.767332 

231 

149.299199 

124.382795 

242     . 

156.408685 

130.305786 

232 

149.945516 

124.921249 

243 

157.055002 

130.844239 

233 

150.591833 

125.459703 

244 

157.701519 

131.382693 

234 

151.238150 

125.998156 

245 

158.347636 

131.921147 

235 

151.884467 

126.536610 

246 

158.993952 

132.459600 

236 

152.530784 

127.075064 

247 

159.640269 

132.998054 

237 

153.177101 

127.613517 

248 

160.286586 

133.536508 

238 

153.823417 

128.151971 

249 

160.932903 

134.074961 

239 

154.469734 

128.690425 

250 

161.579220 

134.613415 

.TIME. 

60  seconds,*!  marked  s,  =  1  minute 
60  minutes,!        '*       m,  =  1  hour  =  3600  seconds 
24  hours,  "        h,  =  1  day    =  1440  minutes  =  86400  secondB 

7  days,  "        d,  =  1  week  =  168  hours       =  10080  minutes 


Arc        Time 
1°  =  4  minutes 
1'  =  4  seconds 
I'l  =  0.066...  second 


Time  Arc 

24  hours    =  360° 
1  hour      =    15° 
1  minute  =      0°  15' 
1  second  =     0°  0'  15" 


Methods  of  reckoning'  time.  Astronomers  distinguish  between  mean 
solar  time,  true  or  apparent  solar  time,  and  sidereal  time. 

At  a  standard  meridian  (see  page  267)  mean  solar  time  is  the  same  as 
ordinary  clock  time.  At  any  point  not  on  a  standard  meridian,  standard  time 
is  the  local  mean  solar  time  of  the  meridian  adopted  as  standard  for  such  point ; 
and  local  time  is  =  time  at  a  standard  meridian  phis  correction  for  longitude 
from  that  meridian  if  the  place  is  east  of  the  meridian,  and  vice  versa.  For  the 
amount  of  such  correction,  see  second  table  above.  A  true  or  apparent 
solar  day  is  the  interval  of  time  between  two  successive  culminations  of 
the  sun,  i.e.,  between  two  successive  transits  or  passages  of  the  sun  across  the 
meridian  of  the  same  point  on  the  earth  ;  but,  since  these  intervals  are  unequal, 
they  do  not  correspond  with  the  uniform  movement  of  clock  time.  A  fictitious 
or  imaginary  sun,  called  the  "mean  sun,"  is  therefore  supposed  to  move  along 
the  equator  in  such  a  way  that  the  interval  between  its  culminations  is  con- 
stant. This  interval  is  called. a  day,  or  mean  solar  day,  and  is  the  average  of  the 
lengths  of  all  the  apparent  solar  days  in  a  year.  Apparent  and  mean  time 
agree  at  four  points  in  the  year,  viz.,  about  the  middle  of  April  and  of  June, 
September  1  and  December  24.  The  sun  is  sometimes  behind  and  sometimes 
in  advance  of  the  mean  sun,  and  is  called  "  slow  "  or  "  fast  "  accordingly.  The 
sun  is  "  slow  "  in  winter,  the  maximum  being  about  February  11,  when  it  passes 
any  standard  meridian,  or  "  souths  "  (making  apparent  noon),  about  14m,  288, 
after  noon  by  a  correct  clock.  The  sun  is  "  fast,"  or  in  advance  of  the  clock,  in 
May  and  in  the  fall,  with  a  maximum,  about  November  2,  of  about  16m,  20s. 

The  difference  between  apparent  and  mean  time  is  called  the  equation  of 
time.  It  can  be  obtained  from  the  Nautical  Almanac,  or,  approximately,  by 
taking  the  mean  between  the  times  of  sunrise  and  sunset,  as  given  in  ordinary 
almanacs. 

As  solar  time  is  measured  by  the  apparent  daily  motion  of  the  sun,  so  sidereal 
time  is  measured  by  that  of  the  fixed  stars,  or,  more  strictly  speaking,  by  the 
motion  of  the  vernal  equinox  which  is  the  point  where  the  sun  crosses  the 
equator  in  the  spring. 

*  The  second  was  formerly  divided  into  60  equal  parts  called  thirds  (marked 
'") ;  but  it  is  now  divided  decimally. 

t  The  old  and  confusing  practice  of  designating  minutes,  seconds  and  thirds 
of  time  (see  footnote  *)  as  ',  "  and  '" ,  is  no  longer  in  vogue.  Days,  hours,  min- 
utes and  seconds  are  now  designated  by  d,  h,  m,  and  s,  respectively,  thus:  2d, 
20h,  48m,  55.43  s.,  and  the  symbols  '  and  "  designate  minutes  and  seconds  of  arc. 


266  TIME. 

A  sidereal  day  is  the  interval  of  time  between  two  successive  passages  of 
the  vernal  equinox  (or,  practically,  of  any  star)  past  the  meridian  of  a  given 

f>oint  on  the  earth.    It  is,  practically,  the  time  required  for  one  complete  revo- 
ution  of  the  earth  on  its  axis,  relatively  to  the  stars. 

The  length  of  the  sideral  day  is  23  h,  56  m,  4.09  s,  of  mean  solar  time,  or  3  m, 
55.91  s  of  mean  solar  time  less  than  the  mean  solar  day  of  24  hours.  lu  other 
words,  a  star  will,  on  any  night,  appear  to  set  3  m,  55.91  s  earlier  by  a  correct 
clock  than  it  did  on  the  preceding  night.  Hence,  substantially,  the  number  of 
sidereal  days  in  a  year  is  greater  by  1  than  the  number  of  solar  days. 

The  sidereal  day,  like  the  solar  day,  is  divided  into  24  hours.  These  hours 
are,  of  course,  shorter  than  those  of  the  solar  day  in  the  same  proportion  as  the 
sidereal  day  is  shorter  than  the  solar  day.  They  are  counted  from  0  to  24,  com- 
mencing with  sidereal  noon,  or  the  instant  when  the  vernal  equinox  passes  the 
upper  meridian. 

Tlie  civil  day  (=  24  hours  of  clock  or  mean  solar  time)  commences  at  mid- 
night; and  tlie  astronomical  solar  day  at  noon  on  the  civil  day  of  the 
same  date.  Thus,  on  a  standard  meridian,  Thursday,  May  9,  2  a.  m.  civil  time, 
is  Wednesday,  May  8,  14  h,  astronomical  time ;  but  Thursday,  May  9,  2  P.  M., 
civil  time,  is  Thursday,  May  9,  2  h,  astronomical  time. 

The  civil  month  is  the  ordinary  and. arbitrary  month  of  the  calendar, 
varying  in  length  from  28  to  81  mean  solar  days. 

A  sidereal  month  is  the  time  required  for  the  moon  to  perform  an  entire 
revolution  with  reference  to  the  stars.  Its  mean  length,  in  mean  solar  time,  is 
about  27  d,  7  h,  43  m,  12  s. 

A  lunation,  or  synodic  month  is  the  time  from  new  moon  to  new 
moon.    Its  mean  length  is  about  29  d,  12  h,  44  m,  3  s. 

The  tropical  or  natural  year  is  the  time  during  which  the  earth 
describes  the  circuit  from  either  equinox  to  the  same  again.  Its  mean  length, 
in  mean  solar  time,  is  now  about  365  d,  5  h,  48  m,  49  s. 

The  sidereal  year  is  the  time  during  which  the  earth  describes  its  orbit 
with  reference  to  the  stars."  Its  mean  length,  in  mean  solar  time,  is  about  365 
d,  6  h,  9  m,  10  s. 

The  civil  year  is  that  arbitrary  or  conventional  and  variable  division  of 
time  comprised  between  the  1st  of  .January  and  the  31st  of  the  following  Decem- 
ber, both  inclusive.  It  contains  ordinarily  365  mean  solar  days  of  24  hours,  but 
each  year  whose  number  is  divisible  by  4  contains  366  days,  and  is  called  a  leap 
year,  except  that  those  years  whose  numbers  end  in  00  and  are  not  multiples 
of  400  are  not  leap  years. 

To  regulate  a  watch  by  the  stars.  The  author,  after  having  regu- 
lated his  chronometer  for  a  year  by  this  method  only,  differed  but  a  few  seconds 
from  the  actual  time  as  deduced  from  careful  solar  observations.  Select  a 
window,  facing  west  if  possihle,  and  commanding  a  view  of  a  roof-crest  or  other 
fixed  horizontal  line,  preferably  about  40°  above  the  horizon,  in  order  to  avoid 
disturbance  due  to  refraction,  and  distant  say  50  feet  or  more.  Note  the 
time  when  any  bright  fixed  star  (not  a  planet)  passes  the  Vange  formed  between 
the  roof,  etc.,  and  any  fixed  horizontal  line  about  the  window  frame,  as  a  pin 
fixed  in  either  jamb.  The  sight  in  the  window,  and  the  watch,  must  be  illumi- 
nated. The  star  will  pass  the  range  3  m.  55.91  s.  earlier  on  each  succeeding 
evening.  Those  stars  which  are  nearest  the  equator  appear  to  move  the  fastest* 
and  are  therefore  best  suited  to  the  purpose.  If  the  first  observation  of  a  given 
star  be  made  as  late  as  midnight,  that  same  star  will  answer  for  about  three 
months,  until  at  last  it  will  begin  to  pass  the  range  in  daylight.  Before  this 
happens,  transfer  the  time  to  another  star  which  sets  later.  By  thus  tabulating, 
throughout  the  year,  about  half  a  dozen  stars  which  follow  each  other  at 
nearly  equal  intervals  of  time,  we  may  provide  a  standard  by  means  of  which 
correct  clock  time  may  be  ascertained  on  any  clear  night.  Experimenting  in 
this  way  with  two  of  the  best  chronometers,  the  author  found  that  their 
i^aies  varied,  at  times,  as  much  as  from  three  to  eight  seconds  per  day. 

An  average  man  takes  two  steps  (one  right,  one  left)  per  second. 
Hence,  march  music  usually  takes  one  second  per  measure  (or  "  bar").  Modern 
cratches  usually  tick  five  times,  and  clocks  either  one,  two,  or  four  times, 
per  second. 


STANDARD   RAILWAY   TIME.  267 

STAKDARn  RAH.WAY  TIME,  ADOPTED  1883. 

The  following  arrangement  of  standard  time  was  recommended  by  the  General 
and  Southern  Time  Conventions  of  the  railroads  of  the  United  States  and  Canada, 
held  respectively  in  St.  Louis,  Mo.,  and  New  York  city,  April,  1883,  and  in  Chicago, 
111.,  and  New  York  city,  in  October,  1883,  and  went  into  effect  on  most  of  the  rail- 
roads of  the  United  States  and  Canada,  November  18th,  1883.  Most  of  the  principal 
cities  of  the  United  States  have  made  their  respective  local  times  to  correspond  with 
it.  This  system  was  proposed  by  Mr.  W.  F.  Allen,  Secretary  of  the  Time  Conven- 
tions, and  its  adoption  was  largely  due  to  his  efforts.  We  are  indebted  to  Mr.  Allen 
for  documents  from  which  the  following  has  been  condensed.  Five  standards  of  timet 
or  five  "  times,"  have  been  adopted  for  the  United  States  and  Canada.  These  are, 
respectively,  the  mean  times  of  the  60th,  75th,  90th,  105th,  and  120th  meridians  west 
of  Greenwich,  England.  As  each  of  these  meridians,  in  the  above  order,  is  15°  west 
of  its  predecessor,  its  time  is  one  hour  slower.  Thus,  when  it  is  noon  on  the  90th 
meridian,  it  is  1  p.  m.  on  the  75th,  and  11  a.  m.  on  the  106th.  The  following  gives 
the  name  adopted  for  the  standai'd  time  of  each  meridian,  and  the  conventional 
color  adopted,  and  uniformly  adhered  to,  by  Mr.  Allen,  for  the  purpose  of  designat- 
ing it  and  its  time,  &c,  on  the  maps  published  under  his  auspices: 


Longitude  west 
from  Greenwich. 

Name  of 
Standard  Time. 

Conventional 
color. 

60° 

900 
105° 
120° 

Intercolonial. 

Eastern. 

Central. 

Mountain. 

Pacific. 

Brown. 

Red. 

Blue. 

Green. 

Yellow. 

Theoretically,  each  meridian  may  be  said  to  give  the  time  for  a  strip  of  country 
15°  wide,  running  north  and  south,  and  having  the  meridian  for  its  center.  Thus 
the  meridian  on  which  the  change  of  time  between  two  standard  meridians  is  sup- 
posed to  take  place,  lies  half-way  between  them.  But  it  would,  of  course,  not  bo 
practicable  for  the  railroads  to  use  an  imaginary  line  in  passing  from  one  time 
standard  to  another.  The  changes  are  made  at  prominent  stations  forming  the  ter- 
mini of  two  or  more  lines;  or,  as  in  the  case' of  the  long  Pacific  roads,  at  the  ends 
of  divisions.  As  far  as  practicable,  points  at  which  changes  of  time  had  previously 
been  made,  were  selected  as  the  changing  points  under  the  new  system.  Detroit, 
Mich.,  Pittsburgh,  Pa.,  Wheeling  and  Parkersburg,  W.  Ya.,  and  Augusta,  Ga.,  al- 
though not  situated  upon  the  same  meridian,  are  points  of  change  between  eastern 
and  central  standard  times.  A  train  arriving  at  Pittsburgh  from  the  east  at  noon, 
and  leaving  for  the  west  10  minutes  after  its  arrival,  leaves  (by  the  figures  shown 
upon  its  time-table,  and  by  the  watches  of  its  train. hands)  not  at  10  minutes  after 
12,  but  at  10  minutes  after  11.  • 

The  necessity  for  making  the  changes  of  time  at  principal  points,  instead  of  on  a 
true  meridian  line,  necessitates  also  so^ie  "overlapping"  of  the  times,  or  of  their 
colors  on  the  map.  Thus,  most  of  the  roads  between  Buffalo  and  Detroit,  on  the 
north  aide  of  Lake  Erie,  run  by  "eastern,"  or  " red,"  time;  while  those  on  the  south 
side  of  the  Lake,  between  Buffalo  and  Toledo,  immediately  opposite  to  and  directly 
south  of  them,  run  by  "  central  "  or  "  blue  "  time. 

If  the  changes  of  time  were  made  at  the  meridians  midway  between  the  standard 
ones,  it  would  not  be  necessary  for  any  town  to  change  its  time  more  than  30  min- 
utes. As  it  is,  somewhat  greater  changes  had  to  be  made  at  a  few  points.  Thus, 
standard  time  at  Detroit  is  32  minutes  ahead,  and  at  Savannah  36  minutes  back,  of 
mean  local  time. 

In  most  cases  the  necessary  change  was  made  npon  the  railways  by  simply  setting 
clocks  and  watches  ahead  or  back  the  necessary  number  of  minutes,  and  without 
making  any  change  in  time-tables. 

Halifax,  and  a  few  adjacent  cities,  use  the  time  of  the  60th  meridian,  that  being 
the  nearest  one  to  them ;  but  the  railroads  in  the  same  district  have  adopted  the 
75th  meridian,  or  eastern,  time;  so  that,  for  railroad  purposes,  intercolonial  time 
has  never  come  into  force. 

In  1873  there  were  71  time  standards  in  use  on  the  railroads  of  the  United  States 
and  Canada.  At  the  time  of  the  adoption  of  the  present  system  this  number  had 
been  reduced,  by  consolidation  of  roads,  &c,  to  53.  By  its  adoption,  the  number  be- 
came 5,  or,  practically,  4,  owing  to  the  adoption  of  eastern  time  by  the  intercolonial 
roads,  as  already  explained. 


268 


DIALS. 


DIALLING. 


To  make  a  borizontal  Sun-dial, 

Draw  a  line  a  h ;  and  at  right  angles  to  it,  draw  66.  From  any  convenient  point,  aa  c, 
in  a  by  draw  the  perp  c  o.  Make  the  angle  c  a  o  equal  to  the  lat  of  the  place ;  also 
the  angle  c  o  e  equal  to  the  same  ;  join  o  e.  Make  e  n  equal  to  o  e;  and  from  n  as  a 
center,  with  the  rad  en,  describe  a  quadrant  e  5;  and  div  it  into  6  equal  parts.  Draw  « 
y,  parallel  to  6,  6;  and 

from  n,  through   the  5  ^  DIAL 

points  on  the  quadrant, 
draw  lines  n  t,  n  i,  &c, 
terminating  in  e  y.  From 
a  draw  lines  a  5,  a  4,  &c, 
passing  through  t,  i,  &c. 
From  any  convenient 
point,  as  c,  describe  an 
arc  rmh,a,9&  kind  of  fin- 
ish or  border  to  half  the 
dial.  All  the  lines  may 
now  be  effaced,  except 
the  hour  lines  a  6,  a  5, 
a  4,  Ac,  to  a  12,  or  a  ^ ; 
unless,  as  is  generally  " 
the  case,  the  dial  is  to 
be  divided  to  quarters 
of  an  hour  at  least.  In 
this  case  each  of  the 
divisions  on  the  quad- 
rant e  s,  must  be  subdivided  into  4  equal  parts;  and  lines  drawn  from  n,  through 
the  points  of  subdivision,  terminating  in  ey.  The  quarter-hour  lines  must  be  drawn 
from  a,  as  were  the  hour  lines.  Subdivisions  of  5  min  may  be  made  in  the  same 
way ;  but  these,  as  well  as  single  min,  may  usually  be  laid  off  around  the  border,  by 
eye.  About  8  or  10  times  the  size  of  our  Fig  will  be  a  convenient  one  for  an  ordi- 
nary dial.  To  draw  the  other  half  of  the  Fig,  make  a  d  equal  to  the  intended  thick- 
ness of  the  gnomon,  or  style,  of  the  dial ;  and  draw  d  12,  parallel,  and  equal  to  a  12 ;  and 
drawthearc^j^rM;,  precisely  similar  to  the  arc  rm^.  Between  a;  and  w,  on  thearcx^w, 
space  off  divisions  equal  to  those  on  the  arc  rmh',  and  number  them  for  the  hours, 
as  in  the  Fig.  The  style  F,  of  metal  or  stone,  (wood  is  too  liable  to  warp,)  will  be 
triangular ;  its  thickness  must  throughout  be  equal  to  ad  or  hw;  its  base  must 
cover  the  space  adhw;  its  point  will  be  at  ad;  and  its  perp  height  h  m,  over  h w, 
must  be  such  that  lines  vd,ua,  drawn  from  its  top,  down  to  a  and  d,  will  make  the 
angles  uah^vdw.  each  equal  to  the  lat  of  the  place.  Its  thickness,  if  of  metal,  may 
conveniently  be  from  3^  to  1^  inch ;  or  if  of  stone,  an  inch  or  two,  or  more,  according 
to  the  size  of  the  dial.  Usually,  for  neatness  of  appearance,  the  back  huvw  of  the 
style  is  hollowed  inward.  The  upper  edges,  ua,  vd,  which  cast  the  shadows,  must 
be  sharp  and  straight.  The  dial  must  be  fixed  in  place  hor,  or  perfectly  level ;  ah 
and  dw  must  be  placed  truly  north  and  south ;  ad  being  south, and  hw  north.  The 
dial  gives  only  sun  or  solar  time ;  but  clock  time  can  be  found  by  means  of  the  "  fast 
or  slow  of  the  sun,"  as  given  by  all  almanacs.  If  by  the  almanac  the  sun  is  5  min, 
Ac,  fast,  the  dial  will  be  the  same ;  and  the  clock  or  watch,  to  be  correct,  must  be  S 
uin  slower  than  it ;  and  vice  versa. 

To  make  a  Tertical  Sun-Dial. 

Proceed  as  directed  above, except  that  the  angles  cao  and  coe  on  the  drawing, 
and  the  angle  uah  or  vdw  of  the  style,  must  be  equal  to  the  co-latitude  (=  dif- 
ference between  the  latitude  and  90°)  of  the  place,  and  the  hours  must  be  num- 
bered the  opposite  way  from  those  in  the  above  figure  ;  i  e,  from  htoy  number 
12,  11,  10,  9,  8,  7 ;  and  from  w  to  g  number  12, 1,  2,  3,  4,  5.  The  dial  plate  must  be 
placed  vertically,  in  the  position  shown  in  the  figure,  facing  exactly  south,  and 
with  ah  and  dvt  vertical. 


BOARD   MEASURE. 


269 


BOARD  MEASUEE. 


Remarlc  on  following'  table.    The  table  extends  to  12  ins  by  24  ins,  but 

it  Is  easy  to  find  for  greater  sizes  ;  thus,  for  example,  the  board  measure  in  a  piece  of  19  by  2'2,  -n-ill 
be  twice  that  of  a  piece  of  19  by  11,  or  17.42  X  2  =  34.84  ft  board  meas  ;  or  that  of  19)4  ^7  22,  will  b« 
that  of  1034  by  22  added  to  that  of  9  by  22,  or  18.79  -j-  16.50  =  35.29.  A  foot  of  board  meas  is  equal  t« 
I  foot  square  and  1  inch  thick,  or  to  144  cub  ins.    Hence  1  cub  ft  =  12  ft  board  meas. 


pj 

Feet  of  Board  Measure  contained  in  one  running  foot  of  Scantlings 

_d  . 

"^a 

of  different  dimensioas 

.     (Original.) 

|5 

1000  ft  board 

measure  = 

=  83Kcubft. 

il 

•d  o 

THICKNESS  IN  INCHES. 

13  O 

^H 

1 

IK 

1^ 

IH 

2 

2H 

2^ 

2% 

3 

Ft.  Bd.M. 

FtBd-M. 

Ft.  Bd.M. 

FtBd.M. 

Ft.Bd.M. 

Ft.  Bd.M. 

Ft.  Bd.M. 

Ft  Bd.M. 

FtBd.M. 

H 

.0208 

.0260 

.0313 

.0365 

.0417 

.0469 

.0521 

.0573 

.0625 

1. 

^ 

.0417 

.0521 

.0625 

.0729 

.0833 

.0938 

.1042 

.1146 

.1250 

.0625 

.0781 

.0938 

.1094 

.1250 

.1406 

.1563 

.1719 

.1875 

1. 

.0833 

.1042 

.1250 

.1458 

.1667 

.1875 

.2083 

.2292 

.2500 

.1042 

.1302 

.1563 

.1823 

.2083 

.2344 

.2604 

.2865 

.3125 

2^ 

iz 

.1250 

.1563 

.1875 

.2188 

.2500 

.2813 

.3125 

.3438 

.3750 

a^ 

.1458 

.1823 

.2187 

.2552 

.2917 

.3281 

.3646 

.4010 

.4375 

3. 

.1667 

.2083 

.2500 

.29ir 

.3333 

.3750 

.4166 

.4583 

.5000 

.1875 

.2344 

.2813 

.3281 

.3750 

.4219 

.4688 

.5156 

.5625 

1 

.2083 

.2604 

.3125 

.3646 

.4167 

.4688 

.5208 

.5729 

.6250 

a^ 

.2292 

.2865 

.3438 

.4010 

.4583 

.5156 

.5729 

.6302 

.6875 

8. 

.2500 

.3125 

.3750 

.4375 

.5000 

.5625 

.6250 

.6875 

.7500 

.2708 

.3385 

.4063 

.4739 

.5416 

.6094 

.6771 

.7448 

.8125 

■  4. 

M 

.2917 

.3646 

.4375 

.5104 

.5833 

.6563 

.7292 

.8021 

.8750 

r/ 

.3125 

.3906 

.4689 

.5469 

.6250 

.7031 

.7813 

.8594 

.9375 

4. 

.3333 

.4167 

.5000 

.5833 

.6667 

.7500 

.8333 

.9167 

1.000 

.3542 

.4427 

.5312 

.6198 

.7083 

.7969 

.8854 

.9740 

1.063 

5" 

IZ 

.3750 

.4688 

.5625 

.6563 

.7500 

.8438 

.9375 

1.031 

1.125 

H 

.3958 

.4948 

.5938 

.6927 

.7917 

.8906 

.9896 

1.086 

1.188 

5. 

.4167 

.5208 

.6250 

.7292 

.8333 

.9375 

1.042 

1.146 

1.250 

.4375 

.5469 

.6563 

.7656 

.8750 

.9844 

1.094 

1.203 

1.313 

g 

^ 

.4583 

.5729 

.6875 

.8020 

.9167 

1.031 

1.146 

1.260 

1.375 

^ 

.1792 

.5990 

.7188 

,8385 

.9583 

1.078 

1.198 

1.318 

1.438 

H 

t. 

.5000 

.6250 

,7500 

.8750 

1.000 

1.125 

1.250 

1.375 

1.500 

6 

y< 

.5208 

.6510 

.7813 

.9115 

1.042 

1.172 

1.302 

1.432 

1.563 

li 

.5417 

.6771 

.8125 

.9479 

1.083 

1.219 

1.354 

1.490 

1.625 

H 

.5625 

.7031 

.8438 

.9844 

1.125 

1.266 

1.406 

1.547 

1.688 

7. 

.5833 

.7292 

.8750 

1.021 

1.167 

1.312 

1.458 

1.604 

1.750 

7.* 

.6042 

.7552 

.9063 

1.057 

1.208 

1.359 

1.510 

1.661 

1.813 

}4 

^ 

.6250 

.7813 

.9375 

1.094 

1.250 

1.406 

1.563 

1.719 

1.875 

^ 

.6458 

.8073 

.9688 

1.130 

1.292 

1.453 

1.615 

1.776 

1.938 

H 

8. 

.6667 

1.000 

1.167 

1.333 

1.500 

1.667 

1.833 

2.000 

8. 

.6875 

.8594 

1.031 

1.203 

1.375 

1.547 

1.719 

1.891 

2.063 

9.* 

\/ 

.7083 

.8854 

1.063 

1.240 

1.417 

1.594 

1.771 

1.948 

2.125 

x/ 

.7292 

.9114 

1.094 

1.276 

1.458 

1.641 

1.823 

2.005 

2.188 

9. 

.7500 

.9375 

1.125  - 

1.313 

1.500 

1.688 

1.875 

2.062 

2.250 

.7708 

.9635 

1.156 

1.349 

1.542 

1.734 

1.927 

2.120 

2.313 

14 

.7917 

.9895 

1.188 

1.385 

1.583 

1.781 

1.979 

2.177 

2.375 

s^ 

.8125 

1.016 

1.219 

1.422 

1.625 

1.828 

2.031 

2.234 

2.438 

10. 

10. 

.8333 

1.042 

1.250 

1.458 

1.667 

1.875 

2.083 

2.292 

2.500 

.8542 

1.068 

1.281 

1.495 

1.708 

1.922 

2.135 

2.349 

2.563 

iz 

.8750 

1.094 

1.313 

1.531 

1.750 

1.969 

2.188 

2.406 

2.625 

y} 

&£ 

.8958 

1.120 

1.344 

1.568 

1.792 

2.016 

2.240 

2.463 

2.688 

x/ 

11. 

.9167 

1.146 

1.375 

1.604 

1.833 

2.063 

2.292 

2.521 

2.750 

11. 

.9375 

1.172 

1.406 

1.641 

1.875 

2.109 

2.344 

2.578 

2.813 

g 

IZ 

.9583 

1.198 

1.438 

1.67T 

1.917 

2.156 

2.396 

2.635 

2.875 

az 

.9792 

1.224 

1.469 

1.714 

1958 

2.203 

2.448 

2.693 

2.938 

% 

12. 

1.000 

1.250 

1.500 

1.750 

2.000 

2.250 

2.500 

2.750 

3.000 

12. 

H 

1.042 

1.302 

1.563 

1.823 

2.083 

2.344 

2.604 

2.865 

3.125 

H 

13 

1.083 

1.354 

1.625 

1.896 

2.167 

2.438 

2.708 

2.979 

3.250 

13 

H 

1.125 

1.406 

1.688 

1.969 

2.250 

2.531 

2.813 

3.094 

3.375 

}4 

u. 

1.167 

1.458 

1.750 

2.042 

2.333 

2.625 

2.917 

3.208 

3.500 

14. 

H 

1.208 

1.510 

1.813 

2.115 

2.417 

2.719 

3.021 

3.322 

3.625 

% 

15. 

1.250 

1.563 

1.875 

2.188 

2.500 

2.813 

3.125 

3.438 

3.750 

15 

^ 

1.292 

1.615 

1.938 

2.260 

2.583 

2.906 

3.229 

3.552 

3.875 

H 

16. 

1.333 

1.667 

2.000 

2.333 

2.667 

3.000 

3.333 

3.667 

4.000 

16. 

H 

1.375 

1.719 

2.063 

2.406 

2.750 

3.094 

3.438 

3.781 

4.125 

U 

17. 

1.417 

1.771 

2.125 

2.479 

2.833 

3.188 

3.542 

3.896 

4.250 

17 

14 

1.458 

1.823 

2.187 

2.552 

2.917 

3.281 

3.646 

4.010 

4.375 

\i 

18 

1.500 

1.875 

2.250 

2.625 

S.OOO 

3.375 

3.750 

4.125 

4.500 

18. 

19. 

1.583 

1.979 

2.375 

2.771 

3.167 

3.563 

3.958 

4.354 

4.750 

19. 

20. 

1.667 

2.083 

2.500 

2.917 

3.333 

3.750 

4.167 

4.583 

5.000 

20. 

21. 

1.750 

2.188 

2.625 

3.063 

3.500 

3.936 

4.375 

4.812 

5.250 

21. 

22. 

1.833 

2.292 

2.750 

3.208 

3.667 

4.125 

4.583 

5.042 

5.500 

22. 

23. 

1.917 

2.396 

2.875 

3.354 

3.833 

4.313 

4.792 

5.270 

5.750 

23. 

2i. 

2.000 

2.500 

3.000 

3.500 

4.000 

4.500 

6.000 

5.500 

6.000 

24. 

270 


BOARD    MEASURE. 


Table  of  Board  Measure  — (Continued.) 


0    . 

Feet  of  Board  Mea. 

ure  contained  in  one  runnine  foot  of  Scantlines 

5^ 

il 

of  different  dimensions.     (Original.) 

^1 

■ 

ga 

THICKNESS  IN  INCHES. 

%- 

3Ji 

3K 

8« 

4 

4M 

4>^ 

4% 

5 

5>i 

FtBd.M. 

Ft.Bd.M. 

Ft.Bd.M. 

Ft.Bd.M. 

Ft.Bd.M. 

Ft.Bd.M. 

Ft.Bd.M. 

Ft.Bd.M 

Ft.Bd.M. 

"■ 

g 

.0677 

.0729 

.0781 

.0833 

.0885 

.0938 

.0990 

.1042 

.1094 

34 

.1354 

.1457 

.1562 

.1667 

.1770 

.1875 

.1979 

.2083 

.2188 

3^ 

H 
1. 

.2031 

.2187 

.2344 

.2500 

.2656 

.2813 

.2969 

.3125 

.3281 

% 

.2708 

.2917 

.3125 

.3333 

.3542 

.3750 

.3958 

.4167 

.4375 

1. 

g 

.3385 

.3646 

.3906 

.4167 

.4427 

.4688 

.4948 

.5208 

.5469 

M 

.4063 

.4375 

.4688 

.5000 

.5313 

.5825 

.5938 

.6250 

.6563 

>i 

H 

.4740 

.5104 

.5469 

.5833 

.6198 

.6563 

.6927 

.7292 

.7656 

H 

2. 

.5417 

.5833 

.6250 

.6667 

.7083 

.7500 

.7917 

•8333 

.8750 

a. 

.6094 

.6563 

.7031 

.7500 

.7969 

.8438 

.8906 

.9375 

.9844 

Ya, 

.6771 

.7292 

.7813 

.8333 

.8854 

.9375 

.9896 

1.042 

1.094 

yi 

•/ 

.7448 

.8021 

,8594 

.9167 

.9740 

1.031 

1.089 

1.146 

1.203 

H 

sf* 

.8125 

.8750 

.9375 

1.000 

1.062 

1.125 

1.188 

1.250 

1.313 

3. 

H 

.8802 

.9479 

1.016 

1.083 

1.151 

1.219 

1.286 

1.354 

1.422 

34 

H 

.9479 

1.021 

1.094 

1.167 

1.240 

1.313 

1.385 

1.458 

1.531 

3i 

H 

1.016 

1.094 

1.172 

1.250 

1.327 

1.406 

1.484 

1.563 

1.641 

j^ 

4. 

1.083 

1.167 

1.250 

1.333 

1.416 

1.500 

1.588 

1.667 

1.750 

4. 

1.151 

1.240 

1.328 

1.417 

1.504 

1.594 

1.682 

1.771 

1.859 

^ 

14 

1.219 

1.313 

1.406 

1.500 

1.593 

1.688 

1.781 

1.875 

1.969 

yi 

% 

1.286 

1.384 

1.484 

1.583 

1.681 

1.781 

1.880 

1.979 

2.078 

% 

i. 

1.354 

1.457 

1.566 

1.666 

1.770 

1.875 

1.979 

2.083 

2.188 

5. 

Yi. 

•1.422 

1.530 

1.644 

1.750 

1.858 

1.969 

2.078 

2.188 

2.297 

% 

^ 

1.490 

1.603 

1.722 

1.833 

1.947 

2.063 

2.177 

2.292 

2.406 

%  ' 

1.557 

1.676 

1.800 

1.917 

2.035 

2.156 

2.276 

2.396 

2.516 

%. 

1.625 

1.750 

1.875 

2.000 

2.125 

2.250 

2.375 

2.500 

2.625 

6.* 

-u 

1.69S 

1.823 

1.953 

2.083 

2.214 

2..344 

2.474 

2.604 

2.7.^54 

^ 

rz 

1.760 

1.896 

2.031 

2.167 

2.302 

2.438 

2.573 

2.708 

2.843 

y*. 

1.828 

1.969 

2.109 

2.250 

2.391 

2.531 

2.672 

2.813 

2.953 

% 

7. 

1.896 

2.042 

2.188 

2.333 

2.479 

2.625 

2.771 

2.917 

3.063 

7. 

1.964 

2.115 

2.266 

2.416 

2.568 

2  719 

2.870 

3.021 

3.172 

M 

^ 

2.031 

2.187 

2.344 

2.500 

2  656 

2  813 

2.969 

3.125 

3.281 

3^ 

2.099 

2.260 

2.422 

2.583 

2.745 

2.906 

3.068 

8.229 

3.891 

8. 

2.167 

2.333 

2.500 

2.667 

2.833 

3.000 

3.167 

3.333 

3.500 

8. 

M 

2.234 

2.406 

2.578 

2.750 

2.922 

3.094 

3.266 

8.438 

3.609 

^ 

>^ 

2.302 

2.479 

2  656 

2.833 

3.010 

3.188 

3.365 

8.542 

3.718 

^ 

2.370 

2.552 

2.734 

2.916 

3.099 

3.281 

3.464 

3.646 

3.828 

5i 

9. 

2.  t38 

2.625 

2.813 

3.000 

3.187 

3.375 

3.563 

3.750 

3.938 

9. 

M 

2.505 

2.698 

2.891 

3.083 

3.276 

3.469 

3.661 

3.854 

4.047 

^ 

^ 

2.573 

2.771 

2.909 

3.167 

3.365 

3.563 

3.760 

3.958 

4.156 

?i 

2.641 

2.844 

3.047 

3.250 

3.453 

3.656 

3.859 

4.063 

4.266 

^ 

10. 

2.708 

2.917 

3.125 

3.333 

3.542 

3.750 

3.958 

t.l67 

4.375 

10. 

^ 

2.776 

2.990 

3.203 

3.416 

3.630 

3.844 

4.057    . 

4.271 

4.484 

34 

2.844 

3.063 

3.281 

3.500 

3.719 

3.938 

4.156 

4.375 

4.594 

H 

9i 

2.911 

3.135 

3.359 

3.583 

3.807 

4.031 

4.255 

4.479 

4.703 

H 

11. 

2.979 

3.208 

3.488 

3.666 

3.896 

4.125 

4.354 

4.588 

4.813 

11. 

S 

3.047 

8.281 

3.516 

3.750 

3.984 

4.219 

4.453 

4.688 

4.922 

^ 

3.115 

8.354 

3.594 

3.833 

4.073 

4.313 

4.552 

4.792 

5.031 

J< 

?i 

3.182 

3.427 

3.672 

3.916 

4.161 

4.406 

4.651 

4.896 

5.141 

y* 

IJ. 

3.250 

3.500 

3.750 

4.000 

4.250 

4.500 

4.750 

5.000 

5.250 

12. 

>^ 

3.385 

3.646 

3.906 

4.167 

4.427 

4.688 

4.948 

5.208 

5.469 

Ji 

18. 

3..521 

3.792 

4.063 

4.333 

4.604 

4.875 

5.146 

5.417 

5.688 

13. 

^ 

3.656 

3.938 

4.219 

4.500 

4.781 

5.063 

5.344 

5.625 

5.906 

yi 

14. 

3.792 

4.083 

4.375 

4.667 

4.958 

5.250 

5.542 

5.833 

6.125 

14. 

H 

3.927 

4.229 

4.531 

4.833 

5.135 

5.438 

5.740 

6.042 

6.344 

« 

15. 

4.063 

4.375 

4.688 

5.000 

5.313 

5.625 

5.938 

6.250 

6.563 

15. 

^ 

4.198 

4.521 

4.844 

5.166 

5.490 

5.813 

6.135 

6.458 

6.781 

Ji 

16. 

4.. 333 

4.667 

5.000 

5.333 

5.667 

6.000 

6.333 

6.667 

7.000 

1«. 

% 

4.469 

4.813 

5.156 

5.500 

5.844 

6.188 

6.531 

6.875 

7.219 

X 

IT. 

4.604 

4.958 

5.313 

5.667 

6.021 

6.375 

6.729 

7.083 

7.488 

IT. 

)^ 

4.740 

5.104 

5.469 

5.833 

6.198 

6.563 

6.927 

7.292 

7.656 

J« 

18. 

4.875 

5.250 

5.625 

6.000 

6.375 

6.750 

7.125 

7.500 

7.875 

18. 

19. 

5.146 

5.542 

5.938 

6.333 

6.729 

7.125 

7.521 

7.917 

8.318 

19. 

20. 

5.417 

5.838 

6.250 

6.667 

7.083 

7.500 

7.917 

8.333 

8.750 

20. 

21. 

5.688 

6.125 

6.563 

7.000 

7.438 

7.875 

8.313 

8.750 

9.188 

21. 

22. 

5.958 

6.417 

6.875 

7.333 

7.792 

8.250 

8.708 

9.167 

9.625 

22. 

23. 

6.229 

6.708 

7.188 

7.667 

8.145 

8.626 

9.104 

9.583 

10.06 

23. 

Si. 

C.500 

7.000 

7.500 

8.000 

8.500 

9.000 

9.600 

10.00 

10.60 

ai. 

BOARD   MEASURE. 


271 


Table  of  Board  HCeasnre— -(Contintied.) 


.s« 

Feet  of  Board  Measure  contained  in  one  running  foot  of  Scantlings            | 

H 

-  * 

of  different  dimensions 

(Original.) 

j 

A^ 

P 

2rt 

THICKNESS    IN 

INCHES. 

53^ 

^H 

6 

63i 

63i 

^Vi 

T 

1H 

7K 

Ft.Bd.M. 

Ft.Bd.M. 

Ft.Bd.M. 

Ft.Bd.M. 

FtBd.M. 

Ft.Bd.M. 

FkBd.M. 

Ft.Bd.M. 

FtBd.M. 

34 

.1146 

.1198 

.1250 

.1302 

.1354 

.1406 

.1458 

.1510 

.1563 

H 

1^ 

.2292 

.2396 

.2500 

.2604 

.2708 

.2813 

.2917 

.3021 

,3125 

H 

i^ 

.3438 

.  .3594 

.3750 

.3906 

.4063 

.4219 

.4375 

.4531 

.4688 

h 

1. 

.4583 

.4792 

.5000 

.5208 

.5417 

.5625 

.5«33 

.6042 

.6250 

1. 

M 

.5729 

.5990 

.6250 

.6510 

.6771 

.7031 

.7292 

.7552 

.7813 

g 

8 

.6875 

.7188 

.7500 

.7812 

.8125 

.8438 

.8750 

.9062 

.9375 

^ 

.8021 

.8385 

.8750 

.9115 

.9479 

.9844 

1.020 

1.057 

1.094 

H 

X. 

.9167 

.9583 

1.000 

1.042 

1.083 

1.125 

1.167 

1.208 

1.250 

2. 

1.031 

1.078 

1.125 

1.172 

1.219 

1.266 

1.313 

1.359 

1.406 

1^ 

1.146 

1.198 

1.250 

1.302 

1.354 

1.406 

1.458 

1.510 

1.563 

}i 

a^ 

1.260 

1.318 

1.375 

1.432 

1.490 

1.547 

1.604 

1.661 

1.719 

H 

8. 

1.375 

1.438 

1.500 

1.562 

1.625 

1.688 

1.750 

1.813 

1.875 

8. 

^ 

1.490 

1.557 

1.625 

1.693 

1.760 

1.828 

1.896 

1.964 

2.031 

H 

^ 

1.604 

1.677 

1.750 

1.823 

1.896 

1.969 

2.042 

2.115 

2.188 

H 

H 

1.719     1 

1.797 

1.875 

1.953 

2.031 

2.109 

2.188 

2.266 

2.344 

H 

4. 

1.833     , 

1.917 

2.000 

2.083 

2.167 

2.250 

2.333 

2.417 

2  500 

4. 

34 

1.948     1 

2.036 

2.125 

2.214 

2.302 

2.391 

2.479 

2.568 

2.656 

^ 

3^ 

2.063 

2.156 

2.250 

2.344 

2.438 

2.531 

2.625 

2.719 

2.813 

H 

2.177 

2.276 

2.375 

2.474 

2.573 

2.672 

2.771 

2.870 

2.969 

H 

5. 

2.292 

2.396 

2.500 

2.604 

2.708 

2.813 

2.917 

3.021 

3.125 

5. 

34 

2.406 

2.516 

2.625 

2.734 

2  844 

2.953 

3.063 

3.172 

3.281 

H 

3^ 

2.521 

2.635 

2.750 

2.865 

2.979 

3.094 

3.208 

3.323 

3.438 

« 

^ 

2.635 

2.755 

2.875 

2.995 

3.115 

3.234 

3.354 

3.474 

3.594 

H 

6 

2.750 

2.875 

3.000 

3.125 

3.250 

3.375 

,  3.500 

3.625 

3.750 

6. 

34 

2.865 

2.995 

3.125 

3.255 

3.385 

3.516 

3.646 

3.776 

3.906 

14 

2.979 

3.115 

3.250 

3.385 

3.521 

3.656 

3.792 

3.927 

4.063 

H 

3.094 

3.234 

3.375 

3.516 

3.656 

3.797 

3.938 

4.078 

4.219 

^ 

7. 

3.208 

3.354 

3.500 

3.646 

3.792 

3.938 

4.083 

4.229 

4.375 

7. 

34 

3.323 

3.474 

3.625 

3.776 

3.927 

4.078 

4.229 

4.380 

4.531 

g 

3.438 

3.594 

3.750 

3.906 

4.063 

4.219 

4.375 

4.531 

4.688 

M 

3.552 

3.714 

3.875 

4.03G 

4.198 

4.359 

4.521 

4.682 

4.844 

^ 

8. 

3.667 

3.833 

4.000 

4.167 

4.333 

4.500 

4.667 

4.833 

5.000 

8. 

3.781 

3.953 

4.125 

4.297 

4.469 

4.641 

4.813 

4.984 

5.156 

1^ 

34 

3.896 

4.073 

4.250  . 

4.427 

4.604 

4.781 

4.957 

5.135 

5.313 

^ 

a^ 

4.010 

4.193 

4.375 

4.557 

4.740 

4.922 

5.103 

5.286 

5.469 

^ 

9. 

4.125 

4.313 

4.500 

4.687 

4.875 

5.063 

5.249 

5.438 

5.625 

9. 

34 

4.240 

4.432 

4.625 

4.818 

5.010 

5.203 

5.395 

5.589 

5.781 

^ 

3I 

4.354 

4.552 

4.750 

4.948 

5.146 

5.344 

5.541 

5.740 

5.938 

h 

S 

4.469 

4.672 

4.875 

5.078 

5.281 

5.484 

5.687 

5.891 

6.094 

H 

10. 

4.583 

4.792 

5.000 

5.208 

5.417 

5.625 

5.833 

6.042 

6.250 

10. 

H 

4.698 

4.911 

5.125 

5.339 

5.552 

5.766 

5.979 

6.193 

6.406 

^ 

H 

4.813 

5.031 

5.250 

5.469 

5.688 

5.906 

6.125 

6.344 

6.5C3 

^ 

H 

4.927 

5.151 

5.375 

5.599 

5.823 

6.047 

6.271 

6.495 

6.719 

^ 

11. 

5.042 

5.271 

5  500 

5.729 

5.958 

6.188 

6.417 

6.646 

6.876 

11. 

34 

5.156 

5.391 

5.625 

5.859 

6.094 

6.328 

6.563 

6.797 

7.031 

g 

S 

5.271 

5.510 

5.750 

5  990 

6.229 

6.469 

6.708 

6.948 

7.188 

H 

5.385 

5.630 

5.875 

6.120 

6.365 

6.609 

6.854 

7.099 

7.344 

H 

12. 

5.500 

5.750 

6.000 

6.250 

6.500 

6.750 

7.000 

7.250 

7.500 

12. 

^ 

5.729 

5.990 

6.250 

6.510 

6.771 

7.031 

7.292 

7.552 

7.813 

H 

13 

5.958 

6.229 

6.500 

6.771 

7.042 

7.313 

7.583 

7.854 

8.125 

13. 

3>i 

6.188 

6.469 

6.750 

7.031 

7.313 

7.594 

7.875 

8.156 

8.-;  08 

H 

U 

6.417 

6.708 

7.000 

7.292 

7.583 

7.875 

8.167 

8.458 

8.750 

14. 

3i 

6.646 

6.948 

7.250 

7.552 

7.854 

8.156 

8.458 

8.760 

9.063 

H 

15 

6.875 

7.188 

7.500 

7.812 

8.125 

8.438 

8.750 

9.063 

9.375 

15. 

3^ 

7.104 

7.427 

7.750 

8.073 

8.396 

8.719 

9.042 

9.365 

9.688 

3i 

16. 

7.333 

7.667 

8.000 

8.333 

8.667 

9.000 

9..S33 

9.667 

10.00 

16. 

}i 

7.563 

7.906 

8.250 

8.594 

8.938 

9.281 

9.625 

9.969 

io.;'.i 

H 

17. 

7.792 

8.146 

8.500 

8.854 

9.208 

9.563 

9.917 

10.27 

10.6;} 

17. 

yi 

8.021 

8.385 

8.750 

9.115 

9.479 

9.844 

10.21 

10.57 

10.94 

H 

18. 

8.250 

8.625 

9.000 

9.375 

9.750 

10.13 

10.50 

10.88 

11.25 

18. 

19. 

8.708 

9.104 

9.500 

9.896 

10.29 

10.69 

11.08 

11.48 

11.88 

19. 

20. 

9.167 

9.583 

10.00 

10.42 

10.83 

11.25 

11.67 

12.08 

12.50 

20. 

21. 

9.625 

10.06 

10.50 

10.94 

11.38 

1K81 

12.25 

12.69 

13.13 

21. 

32. 

10.08 

10.54 

11.00 

11.46 

11.92 

12.38 

12.83 

13.29 

13.75 

22. 

23. 

10.54 

11.02 

11.50 

11.98 

12.46 

12.94 

13.42 

13.90 

14.38 

2». 

34. 

11.00 

11.50 

12.00 

12.50 

13.00 

13.50 

14.00 

14.50 

15.00 

24. 

272 


BOARD    MEASURE. 


Table  of  Board  Measure  —  (Continued.) 


a- 

Feet  of  Board  Measure  contained  in  one  running  foot  of  Scantlings 

fi 

of  dififerent  dimensions 

.     (Original.) 

•^  • 

5^ 

TI 

IICKN] 

ESS  IN 

INCHES. 

1               1 

m 

8 

8M 

8>^ 

m 

9 

9M 

9J^ 

m 

Ft.Bd.M. 

Ft.Bd.M. 

FtBd.M. 

Ft.Bd.M. 

Ft.Bd.M. 

Ft.Bd.M. 

FtBd.M. 

FtBd.M. 

Ft.Bd.M. 

M 

.1615 

.1667 

.1719 

.1771 

.1823 

.1875 

.1927 

.1979 

.2031 

Va. 

K 

.3229 

.3333 

.3438 

.3542 

.3646 

.3750 

.3854 

.3958 

.4063 

M 

?i 

.4844 

.5000 

.5156 

.5313 

.5467 

.5625 

.5781 

.5938 

.6094 

H 

1. 

.6458 

.6667 

.6875 

.7083 

.7292 

.7500 

.7708 

.7917 

.8125 

1. 

K 

.8073 

.8333 

.8594 

.8854 

.9115 

.9375 

.9633 

.9896 

1.016 

M 

.9688 

1.000 

1.031 

1.063 

1.094 

1.125 

1.156 

1.188 

1.219 

>i 

?i 

1.130 

1.167 

1.203 

1.240 

r.276 

1.313 

1.349 

1.385 

1.422 

y^ 

2. 

1.292 

1.333 

1.375 

1.417 

1.458 

1.500 

1.542 

1.583 

1.625 

2. 

H 

1.453 

1.500 

1.547 

1.594 

1.641 

1.688 

1.734 

1.781 

1.828 

^ 

^ 

1.615 

1.667 

1.719 

1.771 

1.822 

1.875 

1.927 

1.979 

2.031 

M 

?i 

1.776 

1.833 

1.891 

1.948 

2.005 

2.063 

2.120 

2.177 

2.234 

^ 

8. 

1.938 

2.000 

2.063 

2.125 

2.188 

2.250 

2.313 

2.375 

2.438 

8. 

K 

2.099 

2.167 

2.234 

2.302 

2.370 

2.438 

2.505 

2.573 

2.641 

/4 

}4 

2.260 

2.333 

2.406 

2.479 

2.552 

2.625 

2.698 

2.771 

2.844 

^ 

% 

2.422 

2.500 

2.578 

2.656 

2.734 

2.813 

2.891 

2.969 

3.047 

^ 

4. 

2.583 

2.667 

2.750 

2.833 

2.917 

3.000 

3.083 

3.167 

3.250 

4. 

V 

2.745 

2.833 

2.922 

3.010 

3.099 

3.188 

3.276 

3.365 

8.453 

yi 

j^ 

2.906 

3.000 

3.094 

3.188 

3.281 

3.375 

3.469 

3.563 

3.656 

H 

?i 

3.068 

3.167 

3.266 

3.365 

3.464 

3.563 

3.661 

3.760 

3.859 

H 

5. 

3.229 

3.333 

3.438 

3.542 

3.646 

3.750 

3.854 

3.958 

4.063 

5. 

yi 

3.391 

3.500 

3.609 

3.719 

3.836 

3.938 

4.047 

4.156 

4.266 

Ji 

/^ 

3.552 

3.667 

3.781 

3.896 

4.010 

4.125 

4.240 

4.354 

4.469 

H 

% 

3.714 

3.833 

3.953 

4.073 

4.193 

4.313 

4.432 

4.552 

4.672 

H 

B. 

3.875 

4.000 

4.125 

4.250 

4.375 

4.500 

4.625 

4.750 

4.875 

6. 

M 

4.036 

4.167 

4.297 

4.427 

4.557 

4.688 

4.818 

4.948 

5.078 

34 

J^ 

4.198 

4.333 

4.469 

4.604 

4.740 

4.875 

5.010 

5.146 

5.281 

}4 

?^ 

4.359 

4.500 

4.641 

4.781 

4.922 

5.063 

5.203 

5.344 

5.484 

H 

7. 

4.521 

4.667 

4.813 

4.958 

5.104 

5.250 

5.360 

5.542 

5.688 

7. 

1/ 

4.682 

4.8.33 

4.984 

5.135 

5.286 

5.438 

5.590 

5.740 

5.891 

yi 

iz 

4.844 

5.000 

5.156 

5.313 

5.469 

6.625 

5.782 

5.938 

6.094 

1^ 

fi 

5.005 

5.167 

5.328 

5.490 

5.651 

5.813 

5.975 

6.135 

6.297 

t£ 

8. 

5.167 

5.333 

5.500 

5.667 

5.833 

6.000 

6.167 

6.333 

6.500 

8. 

>i 

5.328 

5.500 

5.672 

5.844 

6.016 

6.188 

6.359 

6.531 

6.703 

Ji 

M 

5.490 

5.667 

5.844 

6.021 

6.198 

6.375 

6.552 

6.729 

6.906 

^ 

?i 

5.^)51 

5.833 

6.016 

6.198 

6.380 

6.563 

6.745 

6.927 

7.109 

^ 

9 

5.813 

6.000 

6.188 

6.375 

6  563 

6.750 

6.938 

7.125 

7.318 

9. 

M 

5.974 

6.167 

6.359 

6.552 

6.745 

6.938 

7.130 

7.323 

7.516 

1^ 

s 

6.135 

6.3.33 

6.531 

6.729 

6.927 

7.125 

7.323 

7.521 

7.719 

^ 

H 

6.297 

6.500 

6.703 

6.906 

7.109 

7.313 

7.516 

7.719 

7.922 

% 

10. 

6.458 

6.667 

6.875 

7.083 

7.292 

7.500 

7.708 

7.917 

8.125 

10. 

H 

6.620 

6.833 

7.047 

7.260 

7.474 

7.688 

7.901 

8.115 

8.328 

y^ 

^ 

6.781 

7.000 

7.219 

7.438 

7.656 

7.675 

8.094 

8.313 

8.531 

^ 

^ 

6.943 

7.167 

]1& 

7.615 

7.839 

8.063 

8.286 

8.510 

8.734 

^ 

11. 

7.104 

.  7..S33 

7.792 

8.021 

8.250 

8.479 

8.708 

8.938 

11. 

M 

7.266 

7.500 

7.735 

7.969 

8.203 

8.438 

8.672 

8.906 

9.141 

^ 

^ 

7.427 

7.667 

7.906 

8.146 

8.386 

8.625 

8.865 

9.104 

9.344 

^ 

^ 

7.589 

7.833 

8.078 

8.323 

8.568 

8.813 

9.057 

9.302 

9.547 

^ 

12. 

7.750 

8.000 

8.250 

8.500 

8.750 

9.000 

9.250 

9.500 

9.750 

12. 

H 

8.073 

8.333 

8.594 

8.854 

9.115 

9.375 

9.635 

9.896 

10.16 

H 

13. 

8.396 

8.666 

8.938 

9.208 

9.479 

9.750 

10.02 

10.29 

10.56 

IS. 

^ 

8.719 

9.000 

9.281 

9.563 

9.844 

10.13 

10.41 

10.69 

10.97 

H 

U. 

9.042 

9..S33 

9.625 

9.917 

10.21 

10.50 

10.79 

11.08 

11.38 

14. 

^ 

9.365 

9.666 

9.969 

10.27 

10.57 

10.88 

11.18 

11.48 

11.78 

H 

15. 

9.688 

10.000 

10.31 

10.63 

10.94 

11.25 

11.56 

11.88 

12.19 

15. 

^ 

10.01 

10.33 

10.66 

10.98 

11.30 

11.63 

11.95 

12.27 

12.59 

H 

1«. 

10.33 

10.67 

11.00 

11.33 

11.67 

12.00 

12.33 

12.67 

13.00 

16. 

3^ 

10.66 

11.00 

11.34 

11.69 

12.03 

12.38 

12.72 

13.06 

13.41 

H 

17. 

10.98 

11.33 

11.69 

12.04 

12.40 

12.75 

13.10 

13.46 

13.81 

IT. 

% 

11.30 

11.66 

12.03 

12.40 

12.76 

13.13 

13.49 

13.85 

14.22 

H 

18. 

11.63 

12.00 

12.38 

12.75 

13.13 

13.50 

13.88 

14.25 

14.63 

18. 

19. 

12.27 

12.67 

13.06 

13.46 

13.85 

14.25 

14.65 

15.04 

15.44 

19. 

20. 

12.92 

13.33 

13.75 

14.17 

14.58 

15.00 

15.42 

15.83 

16.25 

20. 

21. 

13.56 

14.00 

14.44 

14.88 

15.31 

15.75 

16.19 

16.63 

17.06 

21. 

22. 

14.21 

14.66 

15.13 

15.58 

16.04 

16.50 

16.96 

17.42 

17.88 

22. 

23. 

14.85 

15.33 

15.81 

16.29 

16.77 

17.25 

17.73 

18.21 

18.69 

28. 

24. 

15.50 

16.00 

16.50 

17.00 

17.50 

18.00 

18.50 

19.00 

19.50 

14. 

BOARD   MEASURE. 


273 


Table  of  Board  Measure— (Continued.) 


•s. 

Feet  of  Board  Measure  contained  in  one  running  foot  of  Scantlings 

3i 

of  dififerent  dimensioas.    (Original.) 

•So 

55 

TE 

[icKNEss  m 

INCH 

ES. 

10 

103^ 

10>^ 

10?^ 

11 

nn 

11^ 

UH 

12 

Ft  Bd.M. 

Ft.  Bd.M. 

Ft.  Bd.M. 

Ft.Bd.M. 

Ft.  Bd.M. 

Ft.Bd.M. 

FtB<LM. 

Ft.  Bd.M 

Ft.  Bd.M. 

34 

.2083 

.2135 

.2188 

.2240 

.2292 

.2344 

.2396 

.2448 

.2500 

}4 

.4167 

.4271 

.4375 

.4479 

.4583 

.4688 

.4792 

.4896 

.5000 

H 

.6250 

.6406 

.6563 

.6719 

.6875 

7031 

.7188 

.7344 

.7500 

1. 

.8333 

.8542 

.8750 

.8958 

.9167 

.9375 

.9583 

.0702 

1.000 

H 

1.042 

1.068 

1.094 

1.120 

1.146 

1.172 

1.198 

1.224 

1.250 

H 

1.250 

1.821 

1.313 

1.344 

1.375 

1.406 

1.438 

1.469 

1.500 

H 

1.458 

1.495 

1.531 

1.568 

1.604 

1.641 

1.677 

1.714 

1.750 

2. 

2. 

1.667 

1.708 

1.750 

1.792 

1.833 

1.875 

1.917 

1.958 

2.000 

H 

1.875 

1.922 

1.969 

2.016 

2.063 

2.109 

2.156 

2.203 

2.250 

^ 

H 

2.083 

2.135 

2.188 

2.240 

2.292 

2.344 

2.396 

2.448 

2.500 

H 

2.292 

2.349 

2.406 

2.464 

2.521 

2.578 

2.635 

2.693 

2.750 

3^ 

3. 

2.500 

2.563 

2.625 

2.688 

2.750 

2.813 

2.875 

2.938 

3.000 

2.708 

2.776 

2.844 

2.911 

2.979 

3.047 

3.115 

3.182 

3.250 

1^ 

2.917 

2.990 

3.063 

3.135 

3.208 

3.281 

3.354 

3.427 

3.500 

w 

9i 

3.125 

3.203 

3.281 

3.359 

3.438 

3.516 

3.594 

3.672 

3.750 

ay 

4. 

3.333 

3.417 

3.500 

3.583 

3.667 

3.750 

3.833 

3.917 

4.000 

4. 

3.542 

3.630 

3.719 

3.807 

3.896 

3.984 

4.073 

4.161 

4.250 

iz 

3.750 

3.844 

3.938 

4.031 

4.125 

4.219 

4.313 

4.406, 

4.500 

34  ■ 

a^ 

3.958 

4.057 

4.156 

4.255 

4.354 

4.453 

4.552 

4.651 

4.750 

H 

5. 

4.167 

4.271 

4.375 

4.479 

4.583 

4.688 

4.791 

4.896 

5.000 

5 

>^ 

4.375 

4.484 

4.594 

4.703 

4.813 

4.922 

5.031 

5.141 

5.250 

M 

4.583 

4.698 

4.813 

4.927 

5.042 

5.156 

5.270 

5.385 

5.500 

34 

H 

4.792 

4.911 

5.031 

5.151 

5.271 

5.391 

5.510 

5.6.30 

5.750 

H 

6. 

5.000 

5.125 

5.250 

5.375 

5.500 

5.625 

5.750 

5.875 

6.000 

6. 

^ 

5.208 

5.339 

5.469 

5.599 

5.729 

5.859 

5.990 

6.120 

6.250 

H 

5.417 

5.552 

5.688 

5.823 

5.958 

6.094 

6.229 

6.365 

6.500 

\y 

?^ 

5.625 

5.766 

5.906 

6.047 

6.188 

6.328 

6.469 

6.609 

6.750 

H. 

7. 

5.833 

5.979 

6.125 

6.271 

6.417  • 

6.563 

6.708 

6.854 

7.000 

7. 

6.042 

6.193 

6J544 

6.495 

6.646 

6.797 

6.948 

7.099 

7.250 

^ 

J^ 

6.250 

6.406 

8.563 

6.719 

6.875 

7.031 

7.188 

7.344 

7.500 

% 

p.  458 

6  620 

6.781 

6.943 

7.104 

7.266 

7.427 

7.589 

7.750 

K 

8. 

6.667 

6.833 

7.000 

7.167 

7.333 

7.500 

7.667 

7.833 

8.000 

8. 

^ 

6.875 

1.047 

7.219 

7.391 

7.563 

7.734 

7.906 

8.078 

8.250 

H, 

^ 

7.083 

7.260 

7.438 

7.615 

7.792 

7.969 

8.146 

8-323 

8.500 

H 

H 

7.292 

7.474 

7.656 

7.839 

8.021 

8.203 

8.385 

8.568 

8.750 

% 

9. 

7.500 

7.688 

7.876 

8.06:i 

8.250 

8.438 

8.625 

8.813 

9.000 

9.* 

>4 

7.708 

7.901 

8.094 

8.286 

8.479 

8.672 

8.865 

9.057 

9.250 

>^ 

7.917 

8.115 

8.313 

8.510 

8.709 

8.906 

9.104 

9.302 

9.500 

^ 

H 

8.125 

8.323 

8.531 

8.734 

8.839 

9.141 

9.344 

9.547 

9.750 

^ 

10. 

8.333 

8.542 

8.750 

8.958 

9.167 

9.375 

9.583 

9.792 

10.00 

10. 

34 

8.542 

8.755 

8.969 

9.182 

9.396 

9.609 

9.823 

10.04 

10.25 

3i 

3^ 

8.750 

8.969 

9.188 

9.*06 

9.625 

9.844 

10.06 

10.28 

10.50 

^ 

?i 

8.958 

9.182 

9.406 

9.630 

9.854 

10.08 

10.30 

10.53 

10.75 

H 

11. 

9.167 

9.396 

9.625 

9.854 

10.08 

10.31 

10.54 

10.77 

11.00 

11. 

^ 

9.375 

9.609 

9.844 

10  08 

10.31 

10.55 

10.78 

11.02 

11.25 

H 

M 

9.583 

9.823 

10.06 

10.30 

10.54 

10.78 

11.02 

11.26 

11.50 

H 

9i 

9.792 

10.04 

10.28 

10.53 

10.77 

11.02 

11.26 

11.51 

11.75 

H 

12. 

10.00 

10.25 

10.50 

10.75 

11.00 

11.25 

11.50 

11.75 

12.00 

12. 

14 

10.42 

10.68 

10.94 

11.20 

11.46 

11.72 

11.98 

12.24 

12.50 

H 

13. 

10.83 

11.10 

11. .38 

11.65 

11.92 

12.19 

12.46 

12.73 

13.00 

13 

3^ 

11.25 

11.53 

11.81 

12.09 

12.38 

12.66 

12.94 

13.22 

13.50 

H 

U. 

11.67 

11.96 

12.25 

12.54 

12.83 

13.13 

13.42 

13.71 

14.00 

14 

J^ 

12.08 

12.39 

12.69 

12.99 

13.29 

13.59 

13.90 

14.20 

14.50 

^ 

15. 

12.50 

12.81 

13.13 

13.44 

13.75 

14.06 

14.38 

14.69 

15.00 

15 

^ 

12.92 

13.24 

13.56 

13.89 

14.21 

14.53 

14.85 

15.18 

15.50 

}i 

16. 

13.33 

13.67 

14.00 

14.33 

14.67 

15.00 

15.33 

15.67 

16.00 

16. 

3^ 

13.75 

14.09 

14.44 

14.78 

15.13 

15.47 

15.81 

16.16 

16.50 

H 

17. 

14.17 

14.52 

14.88 

15.23 

15.58 

15.94 

16.29 

16.65 

17.00 

17. 

34 

14.58 

14.95 

15.31 

15.77 

16.04 

16.41 

16.77 

17.14 

17.50 

H 

re. 

15.00 

15.38 

15.75 

16.13 

16.50 

16.88 

17.25 

17.63 

18.00 

18. 

19. 

15.83 

16.23 

16.63 

17.02 

17.42 

17.81 

18.21 

18.60 

19.00 

19. 

20. 

16.67 

17.08 

17.50 

17.92 

18.33 

18.75 

19.17 

19.58 

20.00 

20. 

21. 

17.50 

17.94 

18.38 

18.81 

19.25 

19.69 

20.13 

20.56 

21.00 

21. 

22. 

18.33 

18.79 

19.25 

19.71 

20.17 

20.63 

21.08 

21.54 

22.00 

22. 

23. 

19.17 

19.65 

20.13 

20.60 

21.08 

21.56 

22.04 

22.52 

23.00 

28. 

a. 

30.00 

20.50. 

ai.oo 

21.50 

22.00 

22.50 

23.00 

23.50 

24.00 

M- 

18 


274 


LAND  SURVEYING. 


LAND  SURVEYING. 


In  surveying  a  tract  of  ground,  the  sides  which  compose  its  outline  are  desig» 
nated  by  numbers  in  the  order  in  which  they  occur.  That  end  of  each  side  whicn 
first  presents  itself  in  the  course  of  the  survey,  may  be  called  its  w«z?- end;  and  the 
other  its /ar  end.  The  number  of  each  side  is  placed  at  its  far  end.  Thus,  in'Fig.  1, 
the  survey  being  supposed  to  commence  at  the  corner  6,  and  to  follow  the'direo- 
tion  of  the  arrows,  the  first  side  is  6, 1 ;  and  its  number  is  placed  at  its  far  end  at  1; 
and  so  of  the  rest.  Let  NS  be  a  meridian  line,  that,  is,  a  north  and  south  line; 
and  EW  an  east  and  west  line.    Then  in  any  side  which  runs  northwardly. 


wliether  northeast,  as  side  2;  or  northwest,  as  sides  5  and  1 ;  or  due  north;  the 
distance  in  a  due  north  direction  between  its  near  end  and  its  far  end,  is  called 
its  northing;  thus,  a  1  is  the  northing  of  side  1;  16  the  northing  of  side  2;  4c 
of  side  5.  In  like  manner,  if  any  side  runs  in  a  southwardly  direction,  whether 
southeastwardly,  as  side  3;  or  southwestwardly,  as  sides  4  and  6 ;  or  due  south ; 
the  corresponding  distance  in  a  due  south  direction  between  its  near  end  and  its 
far  end,  is  called  its  southing;  thus,  d3  is  the  southing  of  side  3;  3e  of  side  4; 
/6  of  side  6.  Both  northings  and  southings  are  included  in  the  general  term 
Difference  of  Latitude  of  a  side ;  or,  more  commonly  but  erroneously,  its  latitude. 
The  distance  due  east, or  due  west,  between  the  near  and  the  far  end  of  any  side, 
is  in  like  manner  called  the  easting,  or  westing,  of  that  side,  as  the  case  may  be; 
thus,  6  a  is  the  westing  of  side  1 ;  5/  of  side  6;  c5  of  side  5;  «4  of  side  4;  and 
6  2  is  the  easting  of  side  2 ;  2d  of  side  3.  Both  eastings  and  westings  are  included 
in  the  general  term  Departure  of  a  side;  implying  that  the  side  departs  so  far 
from  a  north  or  south  direction.  We  may  say  that  a  side  norths,  wests,  southeast*, 
&c.  We  shall  call  the  northings,  southings,  &c.  the  Ns,  Ss,  £s,  and  Ws ;  the  lati- 
tudes, lats ;  and  the  departures,  deps. 

Perfect  accuracy  is  unattainable  in  any  operation  involving  the  measure- 
ments of  angles  and  distances.*  That  work  is  accurate  enough,  which  cannot 
be  made  more  so  without  an  expenditure  more  than  commensurate  with  the 
object  to  be  gained.  There  is  no  great  difficulty  in  confining  the  uncertainty 
within  about  one-half  per  cent,  of  the  content,  and  this  probably  never  pre- 
vents a  transfer  in  farm  transactions.  But  errors  always  become  apparent  when 
we  come  to  work  out  the  field  notes ;  and  since  the  map  or  plot  of  the  survey,  and 
the  calculations  for  ascertaining  the  content,  should  be  consistent  within  them-* 
selves,  we  do  what  is  usually  called  correcting  the  errors,  but  what  in  fact  is  simply 
humoring  them  in,  no  matter  how  scientific  the  process  may  appear.  We  distrib- 
ute them  all  around  the  survey.  Two  methods  are  used  for  this  purpose,  both 
based  upon  precisely  the  same  principle;  one  by  means  of  drawing;  the  other, 
more  exact  but  much  more  troublesome,  by  calculation.  The  graphic  method,  in 
the  hands  of  a  correct  draftsman,  is  sufficiently  exact  for  all  ordinary  purposes. 
Add  all  the  sides  in  feet  together;  and  divide  the  sum  by  their  number,  for  the 
average  length.  Divide  this  average  by  8;  the  quotient  will  be  the  proper  scale 
in  feet  per  inch.  In  other  words,  take  about  8  ins.  to  represent  an  average  side. 
We  shall  take  it  for  granted  that  an  engineer  does  not  consider  it  accurate  work  to 

*  A  100  ft.  chain  may  vary  its  length  5  feet  per  mile,  between  winter  and  summer,  by  mere 
change  of  temperature ;  and  this  alone  will  make  a  difference  of  about  1  acie  in  533.  The  atu. 
dent  should  practise  plotting  from  perfectUr  accurate  data  ;  as  from  the  example  in  table,  p.  281,  or 


lAND   SURVEYING. 


275 


ntaanre  his  angles  to  the  nearest  quarter  of  a  degree,  •which  is  the  usual  praetice  among  land-survey 
ors.  They  can,  by  means  of  the  engineer's  transit,  now  in  universal  use  on  our  public  works,  be  readily 
measured  within  a  minute  or  two ;  and  being  thus  much  more  accurate  than  the  compass  courses, 
(which  cannot  be  read  off  so  closely,  and  which  are  moreover  subject  to  many  sources  of  error,)  they 
serve  to  correct  the  latter  in  the  office.  The  noting  of  the  courses,  however,  should  not  be  confined  t« 
the  nearest, quarters  of  a  degree,  but  should  be  read  as  closely  as  the  observer  can  guess  at  the  minutes. 
The  back  courses  also  should  be  taken  at  every  corner,  as  an  additional  check,  and  for  the  detection 
of  local  attraction.  It  is 
well  in  taking  the  com- 
pass bearings,  to  adopt 
as  a  rule,  always  to  point 
the  north  of  the  compass- 
box  toward  the  object 
whose  bearing  is  to  be 
taken,  and  to  read  ofiF 
from  the  north  end  of  the 
needle.  A  person  who 
uses  indifferently  the  N 
and  the  S  of  the  box,  and 
of  the  needle,  will  be  very 
liable  to  make  mistakes. 
It  is  best  to  measure  the 
least  angle  (shown  by 
dotted  arcs,  Fig  2,)  at  the 
corners ;  whether  it  be 
exterior,  as  that  at  corner 
5;  or  interior,  as  all  the 
others ;  because  it  is  al- 
ways less  than  180°;  so 
that  there  is  less  danger 
of  reading  it  off  incor- 
rectly, than  if  it  exceeded 
180° ;  taking  it  for  grant- 

ed  that  the  transit  instrument  is  graduated  from  the  same  zero  to  180°  each  way ;  If  it  is  graduated 
from  zero  to  360°  the  precaution  is  useless.  When  the  small  angle  is  exterior,  subtract  it  from  360* 
for  the  interior  one. 

Supposing  the  field  work  to  be  finished,  and  that  we  require  a  plot  from  which  the  contents  may 
be  obtained  mechanically,  by  dividing  it  into  triangles,  (the  bases  and  heights  of  which  may  be 
measured  by  scale,  and  their  areas  calculated  one  by  one,)  a  protraction  of  it  may  be  made  at  once 
from  the  field  notes,  either  by  using  the  angles,  or  by  first  correcting  the  bearings  by  means  of  the 
angles,  and  then  using  them.  The  last  is  tlie  best,  because  in  the  first  the  protractor  must  be  moved 
to  each  angle  ;  whereas  in  the  last  it  will  remain  stationary  while  all  the  bearings  are  being  pricked 
off.  Every  movement  of  it  increases  the  liability  to  errors'.  The  manner  of  correcting  the  be»ring3 
is  explained  on  the  next  page. 

In  either  case  the  protracted  plot  will  certainly  not  close  precisely ;  not  only  in  consequence  of  errors  in 
the  field  work,  but  also  in  the  protracting  itself.  Thus  the  last  side,  No  6,  Fig  2,  instead  of  closing  in  at 
corner  6,  will  end  somewhere  else,  say,  for  instance,  at  t;  the  dist  t  6  being  the  closing  error,  which, 
however,  as  represented  in  Fig  2,  is  more  than  ten  times  as  great,  proportionally  to  the  size  of  the 
survey,  as  would  be  allowable  in  practice.  Now  to  humor-in  this  error,  rule  through  every  corner 
a  short  line  parallel  to  «6;  and,  in  all  cases,  in  the  direction  from  « (wherever  it  may  be)  to  the 
•tarting  point  6.  Add  all  the  sides  together ;  and  measure  «  6  by  the  scale  of  the  plot.  Then  befiM* 
mine  at  corner  1,  at  the  far  end  of  side  1,  say,  as  the 


6-^.'^ 


Fig.  2. 


Sum  of  all 
the  sides 


Total  closing 
error  t  6 


Bidel 


Error 
for  side  1. 


Lay  «£f  this  error  from  1  to  a. 
Sum  of  all 
the  sides 


Then  at  corner  2,  say,  as  the 

Total  closing        ,  ,  Sum  of  ,  Error 

error  <  6  •  •       sides  1  and  2         •         for  side  2. 

"Which  error  lay  off  from  2  to  &  ;  and  so  at  each  of  the  corners ;  always  using,  as  the  third  term,  th« 
«um  of  the  sides  between  the  starting  point  and  the  given  corner.  Finally,  join  the  points  o,  b,  c, 
d,  e,  6;  and  the  plot  is  finished. 

The  correction  has  evidently  changed  the  length  of  every  side;  lengthening  some  and  shortening 
others.  It  has  also  changed  the  angles.  The  new  lengths  and  angles  may  with  tolerable  accuracy 
ba  found  by  means  of  the  scale  and  protractor ;  and  be  marked  on  the  plot  instead  of  the  old  ones. 

from  those  to  be  found  in  books  on  surveying.  This  is  the  only  way  in  which  he  can  learn  what  i« 
meant  by  accurate  work.  His  semicircularprotractor  should  be  about  9  to  12  ins  in  diam  and  gradu- 
ated to  10  min.  His  straight  edge  and  triangle  should  be  of  metal;  we  prefer  German  silver,  which 
does  not  rust  as  steel  does  ;  and  they  should  be  made  with  scrupulous  accuracy  by  a  skilful  instru- 
ment-maker. A  very  fine  needle,  with  a  sealing-wax  head,  should  be  used  for  pricking  off  dists  and 
angles ;  it  must  be  held  vertically  ;  and  the  eye  of  the  draftsman  must  be  directly  over  it.  The  lead 
pencil  should  be  hard  (Faber's  No.  4  is  good  for  protracting),  and  must  be  kept  to  a  sharp  point  by 
rubbing  on  a  fine  file,  after  using  a  knife  for  removing  th«  wood.  The  scale  should  be  at  least  as  long 
as  the  longest  side  of  the  plot,  and  should  be  made  at  the  edge  of  a  strip  of  the  same  paper  as  the  plot 
is  drawn  on.  This  will  obviate  to  a  considerable  extent,  errors  arising  from  contraction  and  expan- 
sion. Unfortunately,  a  sheet  of  paper  does  not  contract  and  expand  in  the  same  proportion  length- 
«rise  and  crosswise,  thus  preventing  the  paper  scale  from  being  a  perfect  corrective.  In  plots  of  com- 
mon farm  surveys,  &c,  however,  the  errors  from  this  source  may  be  neglected.  For  such  plots  as  majf 
*e  protracted,  divided,  and  computed  within  a  time  too  shorttoadmit  of  appreciable  change,  theordi- 
iary  scales  of  wood,  ivory  or  metal  may  be  used ;  but  satisfactory  accuracy  cannot  be  obtained  with 
;hem  on  plots  requiring  several  davs.  if  the  air  be  meanwhile  alternately  moist  and  dry,  or  subject  to 
considerable  variations  in  temperature.  What  is  called  parchment  paper  is  worse  in  this  respect  thiu» 
good  ordinary  drawing-paper. 

With  the  foregoing  precautions  we  may  worli  from  a  drawing,  with  as  much  accuracy  as  is  uauaUy 
attained  in  the  field  work. 


276 


LAND  SURVEYING. 


When  the  plot  baa  many  sides,  tbis  calculating  the  error  for  each  ol  them  becomes  tedious ;  aa4 
since,  in  a  well-performed  survey  and  protraction,  the  entire  error  will  be  but  a  very  small  quantity, 
fit  should  not  exceed  about  x^tt  P**"'  °^  '^^  periphery,)  it  may  usually  be  divided  among  the  sides  by 
merely  placing  about  J4,  ^,  and  5i  of  it  at  corners  about  J4»  /^»  "■'id  %  ^^7  around  the  plot ;  and  at 

intermediate  corners  propor- 
tion it  by  eye.  Or  calculatioa 
may  be  avoided  entirely  by 
drawing  a  line  a  6  of  a  length 
equal  to  the  united  lengths 
of  all  the  sides ;  dividing  it 
Into  distances  a,  1 ;  1,  2  ;  &c,  equal  to  the  respective  sides.  Make  6  c  equal  to  the  entire  closing  error ; 
join  a  c  ;  and  draw  1 ,  1' ;  2,  2' ,  &c,  which  will  give  the  error  at  each  corner. 

When  the  plot  is  thus  completed,  it  may  be  divided  by  fine  pencil  lines  into  triangles,  whose 
bases  and  heights  may  be  measured  by  the  scale,  in  order  to  compute  the  contents.  With  care  in 
both  the  survey  and  the  drawing,  the  error  should  not  exceed  about  ^4^7  P^^'  °^  *'^®  ^'""^  *''^'=^'  ^* 
least  two  distinct  sets  of  triangles  should  be  drawn  and  computed,  as  a  guard  against  mistakes  ;  and  if 
the  two  sets  differ  in  calculated  contents  more  than  about  j^-^  part,  they  have  not  been  as  carefully 
prepared  as  they  should  have  been.  The  closing  error  due  to  imperfect  field-work,  may  be  accurately 
•alculated,  as  we  shall  show,  and  laid  down  on  the  paper  before  beginning  the  plot ;  thus  furnishing 
a  perfect  test  of  the  accuracy  of  the  protraction  work,  which,  if  correctly  done,  will  not  close  at  the 
point  of  beginning,  but  at  the  point  which  indicates  the  error.  But  this  caiculation  of  the  error,  by 
a  little  additional  trouble,  furnishes  data  also  for  dividing  it  by  calculation  among  the  difiT  sides ; 
besides  the  means  of  drawing  the  plot  correctly  at  once,  without  the  use  of  a  protractor  ;  thus  ena- 
bling us  to  make  the  subsequent  measurements  and  computations  of  the  triangles  with  more  cer- 
tainty. 

We  shall  now  describe  this  process,  but  would  recommend  that  even  when  it  is  employed,  and 
especially  in  complicated  surveys,  a  rough  plot  should  first  be  made  and  corrected,  by  the  first  of  the 
two  mechanical  methods  already  alluded  to.  It  will  prove  to  be  of  great  service  in  using  the  method 
by  calculation,  inasmuch  as  it  furnishes  an  eye  check  to  vexatious  mistakes  which  are  otherwise  apt 
to  occur  ;  for,  although  the  principles  involved  are  extremely  simple,  and  easily  remembered  when 
once  understood,  yet  the  continual  changes  in  the  directions  of  the  sides  will,  without  great  care, 
cause  us  to  use  Ns  instead  of  Ss ;  Es  instead  of  Ws,  &c. 

We  suppose,  then,  that  such  a  rough  plot  has  been  prepared,  and  that  the  angles,  bearings,  and 
distances,  as  taken  from  the  field  book,  are  figured  upon  it  in  lead  pencil. 

Add  together  the  interior  angles  formed  at  all  the  corners  :  call  their  sum  a.  Mult  the  number  of 
sides  by  180° ;  from  the  prod  subtract  360°  :  if  the  remainder  is  equal  to  the  sum  a,  it  is  a  proof  that 
the  angles  have  been  correctly  measured.*  This,  however,  will  rarely  if  ever  occur;  there  will 
always  be  some  discrepancy  ;  but  if  the  field  work  has  been  performed  with  moderate  care,  this  will 
not  exceed  about  two  min  for  each  angle.  In  this  case  div  it  in  equal  parts  among  all  the  angles, 
adding  or  subtracting,  as  the  case  may  be,  unless  it  amounts  to  less  than  a  min  to  each  angle,  when 
it  may  be  entirely  disregarded  in  common  farm  surveys.  The  corrected  angles  may  then  be  marked 
on  the  plot  in  ink,  and  the  pencilled  figures  erased.  We  will  suppose  the  corrected  ones  to  be  a« 
shown  in  Fig  3. 

Next,  by  means  of  these 
corrected  angles,  correct  the 
bearings  also,  thus.  Fig  3  ; 
Select  some  side  (the  longer 
the  better)  from  the  two  ends 
of  which  the  bearing  and  the 
reverse  bearing  agreed  ;  thus 
showing  that  that  bearing 
was  probably  not  influenced 
by  local  attraction.  Let  '?ide 
2"be  the  one  so  selected  ;  as- 
sume its  bearing,  N  75° 32'  E, 
as  taken  on  the  ground,  to  be 
correct;  through  either  end 
of  it,  as  at  its  far  end  2,  draw 
the  short  meridian  line ;  par- 
allel to  which  draw  others 
through  every  corner.  Now, 
having  the  bearing  of  side  2, 
N  75°  .32'  E,  and  requiring 
that  of  side  3,  it  is  plain  that 
the  reverse  bearing  from  cor- 
ner 2  is  S  75°  32'  W;  and 
that  therefore  the  angle  1,  2, 
m,  is  75°  32'.  Therefore,  if  we 
take  75°  32'  from  the  entire 
corrected  angle  1,  2,  3,  or  144° 
57',  the  rem  69°  25'  will  be 
the  angle  m  23  ;  consequently 
the  bearing  of  side  3  must  be 
8  69°  25'  E.  For  finding  the  bearing  of  side  4,  we  now  have  the  angle  23  a  of  the  reverse  bearing  of 
side  3  also  equal  to  69°  25' ;  and  if  we  add  this  to  the  entire  corrected  angle  234,  or  to69°  32',  we  have 
the  angle  a  34  =69°  25' +69°  32' =  138°  57';  which  taken  from  180°,  leaves  the  angle  6  34  —  410  3'; 
consequently  the  bearing  of  side  4  must  be  S  41°  3'  W.  For  the  bearing  of  side  5  we  now  have  the 
angle  34  c  =  41°  3',  which  taken  from  the  corrected  angle  345,  or  120°  43',  leaves  the  angle  c  45  =-"9° 
40'  •  consequently  the  bearing  of  side  5  must  be  N  79°  40'  W.  At  corner  5,  for  the  bearing  of  side  8, 
we  have  the  angle  45  d  =  79°  40',  which  taken  from  133°  10',  leaves  the  angle  d  56  =  53° 30'  ;  consi*- 
quently  the  bearing  of  side  6  must  be  S  53°  30'  W.     And  so  with  each  of  the   sides,  nothing  but 


Fig.  3. 


«  Because  in  every  straight-lined  figure  the  sum  of  all  its  interior  angles  is  equal  to  twice  as  vanj 
fight  angles  as  the  figure  has  sides,  minus  4  right  angles,  or  360°., 


LAND   SUBVEYING. 


277 


earcfnl  observation  la  necessary  to  see  hoir  the  several  angles  are  to  be  employed  at  each  oomer. 
Eules  are  sometimes  given  for  this  purpose,  but  unless  frequently  used,  they  are  soon  forgotten. 
The  plot  mechauically  prepared  obviates  the  necessity  for  such  rules,  inasmuch  as  the  principle  of 
proceeding  thereby  becomes  merely  a  matter  of  sight,  and  tends  greatly  to  prevent  error  from  using 
the  wrong  bearings  ;  while  the  protractor  will  at  once  detect  any  serious  mistakes  as  to  the  angles, 
and  thus  prevent  their  being  carried  farther  along.  After  having  obtained  all  the  corrected  bearings, 
they  may  be  figured  on  the  plot  instead  of  those  taken  in  the  field.  They  will,  however,  require  a 
still  further  correction  after  a  while,  since  they  will  beafiFected  by  the  adjustment  of  the  closing  error. 
We  now  proceed  to  calculate  the  closing  error  <  6  of  Fig  i,  which  is  done  on  the  principle  that  in  a 
correct  survey  the  northings  will  be  equal  to  the  southings,  and  the  eastings  to  the  westings.  Pre- 
pare a  table  of  7  columns,  as  below,  and  in  the  first  3  cols  place  the  numbers  of  the  sides,  and  their  oor. 
xected  courses ;  also  the  dists  or  lengths  of  the  sides,  as  measured  on  the  rough  plot,  ifsuchaooQ 
has  been  prepared;  but  if  not,  then  as  measured  on  the  ground.    Let  them  be  as  follows  : 


Side. 

Bearing. 

Dist.  Ft. 

Latitudes. 

Departures. 

N. 

S. 

E. 

W. 

1 
2 
3 

4 
6 

« 

N  16°  40' W 
N  75°  32' E 
S  69°  25' E 
S  41°    3' W 
N  790  40'  W 
g  530  30'  W 

1060 
1202 
1110 
850 
802 
705 

1015.5 
300.3 

143.9 

390.2 
641. 

419.3 

1163.9 
1039.2 

304. 

558.2 

789. 
566.7 

1459.7 
1450.5 

1450.5 

Error  in 
Lat. 

2203.1 

Error  in 
Dep. 

2217.9 
2203.1 

9.2 

14.8 

Kow,  by  means  of  tne  Table  of  Sines,  etc.,  find  the  N,  S,  E,  W,  of  the  several  sides,  and  place 
them  in  the  corresponding  four  columns.  Thus,  for  side  1,  which  is  1060  feet  long,  with  bearing 
N  16°  40'  W  ;  cos  16°  40'  =  O.itoSO  ;  sin  le-^  40'  =  0.2868. 

Here  N  =  1060  x  0.9580  =  1015.5;  and  W  =  1060  X  0.2868  =  304.  Proceed 
thus  with  all.  Add  up  the  four  cols ;  find  the  diff  between  the  N  and  S  cols ;  and  also  between 
the  E  and  W  ones.  In  this  instance  we  find  that  the  Ns  are  9.2  feet  greater  than  the  Ss;  and  that 
the  Ws  are  14.8  ft  greater  than  the  Es;  in  other  words,  there  is  a  closing  error  which  would  cause  a 
eorrect  protraction  of  our  first  three  cols,  to  terminate  9.2  feet  too  far  north  of  the  starting  point ;  and 
14.8  feet  too  far  west  of  it.  So  that  by  placing  this  error  upon  the  paper  before  beginning  to  protract, 
We  should  have  a  test  for  the  accuracy  of  the  protracting  work ;  but,  as  before  remarked,  a  little  more 
trouble  will  now  enable  us  to  di  v  the  error  proportionally  among  all  the  Ns,  Ss,  Es,  and  Ws,  and  thereby 
give  ns  data  for  drawing  the  plot  correctly  at  once;  without  using  a  protractor  at  all. 

To  divide  the  errors,  prepare  a  table  precisely  the  same  as  the  foregoing,  except  that  the  hor  spaces 
are  farther  apart ;  and  that  the  addings-up  of  the  old  N,  S,  E,  W  columns  are  omitted.  The  additioai 
here  noticed  are  made  subsequently. 

The  new  table  is  on  the  next  page. 

Remark.  The  bearing:  and  the  reverse  bearing:  from  the  two  ends 
of  a  line  will  not  read  preciat-ly  the  same  angle ;  and  the  difference  varies  with  the 
latitude  and  with  the  length  of  the  line,  but  not  in  the  same  proportion  with  either. 
It  is,  however,  generally  too  small  to  be  detected  by  the  needle,  being,  according  to 
Gummere,  only  three  quarters  of  a  minute  in  a  line  one  mile  long  in  lat  40°.  In 
higher  lats  it  is  more,  and  in  lower  ones  less.  It  is  caused  by  the  fact  that  meridians 
or  north  and  south  lines  are  not  truly  parallel  to  each  other ;  but  would  if  extended 
meet  at  the  poles. 

Sence  the  only  bearing:  that  can  be  run  in  a  straig:ht  line, 

with  strict  accuracy,  is  a  true  N  and  S  one ;  except  on  the  very  equator,  where  alone  a  due  E  and  W 

one  will  also  be  straight.    But  a  true  curved  E  and  W  line  may  be  found 

anywhere  with  sufficient  accuracy  for  the  surveyor's  purposes  thus.  Having  first  by  means  of  the  N 
staV  p  284,  or  otherwise  got  a  true  N  and  S  bearing  at  the  starting  point,  lay  off  from  it  90o,  for  a  true 
E  and  W  bearing  at  that  point.  This  E  and  W  bearing  will  be  tangent  to  the  true  E  aud  W  curve. 
Run  this  tangent  carefully  ;  and  at  intervals  (say  at  the  end  of  each  mile)  lay  off  from  it  (towards 
the  N  if  in  N  lat,  or  vice  versa)  an  offset  whose  length  in  feet  is  equal  to  the  proper  one  from  the 
following  table,  multiplied  by  the  square  of  the  distance  in  mUes  from  the  starting  point.  These 
offsets  will  mark  points  in  the  true  E  and  W  curve. 

l,atitude  :N^  or  IS. 

350    40°    450    50°    550    60=    65» 


6° 


10° 


150 


20° 


250 


30° 


Offsets  in  ft  one  mile  from  starting^  point. 

.058        .118        .179        .243        .311        .385        .467         .559        .667  .795  .952         1.15  1.43 

Or,  any  offset  in  ft  =  .6666  X  Total  Dist  in  miles2  x  Nat  Tang  of  Lat. 
A  rhumb  line  is  any  one  that  crosses  a  meridian  obliquely,  that  is,  i« 

neither  due  N  and  S,  nor  E  and  W. 


278 


LAND   SUKVEYING. 


Side. 

Bearing. 

Dist.  Ft. 

Latitudes. 

Departures. 

N. 

S. 

E. 

W. 

1 

N  16°  40'  W 
N  75°  32'  E 
S  69°  25' E 
S  41°    3'  W 

N  79°  40'  W 
S  53°  30'  W 

1060 
1202 
1110 
850 

802 

705 

1015.5 
1.7 

399.2 
1.8 

1163.9 
3.1 

304.0 

2.7 

2 

1013.8... 

300.3 
1.9 

...    301.3 

3 

298.4 

143.9 
1.3 

...  1167.0 

1039.2 
2.9 

4 

392    ... 

641.0 
1.3 

...  1042.1 

558.2 
2.2 

6 

642.3... 

419.3 
1.1 

...    556.0^ 

789.0 
2.1 

6 

142.6... 

...    786.9 

566.7 
1.8 

420.4... 

564.9 

5729 
Sum  of 
Sides. 

1454.8 
Cor'd  Ns. 

1454.7 
Cor'd  Ss. 

2209.1 
Cor'd  Es. 

2209.1 
Cor'd  Ws. 

Now  -we  have  already  found  by  the  old  table  that  the  Ns  and  the  Ws  are  too  long ;  consequently 
they  must  be  shortened  ;  while  the  Ss,  and  Ea,  must  be  lengthened ;  all  in  the  following  proportioas : 
As  the 

Sum  of  all     ,     Any  given     ,,     Total  err  of    ,     Err  of  lat,  or  dep, 
the  sides      •  side         •  •      lat  or  dep      •        of  given  side. 

Thus,  commencing  with  the  lat  of  side  1,  we  have,  as 

Sum  of  all  the  sides.     .     Side  1.     ,  .     Total  lat  err.     .    Lat  err  of  side  1. 
5729  •        1060       •  •  9.2  •  1.7 

Now  as  the  lat  of  side  1  is  north,  it  must  be  shortened ;  hence  it  become8  =  1015.5  —  1.7^10ia.8,  M 
figured  out  in  the  new  table.    Again  we  have  for  the  departure  of  side  1, 

Sum  of  all  the  sides.    ,    Side  1.     ,  ,    Total  dep  err.     ,    Dep  err  of  side  1. 
5729  •       1060       •  •  14.8  •  2.7 

Now  as  the  dep  of  side  1  is  west,  it  must  be  shortened ;  hence  it  becomes  304  —  2.7  =  301 .3  as  figured 
out  in  the  new  table.  ' 

Proceedinft  thus  with  each 
side,  we  obtain  all  the  corrected 
lats  and  deps  as  shown  in  the 
new  table ;  where  they  are  con- 
nected with  their  respective 
Bides  by  dotted  lines;  but  ia 
practice  it  is  better  to  cross  out 
the  original  ones  when  the  cal- 
culation is  finished  and  proved. 
If  we  now  add  up  the  4  cols  of 
corrected  N,  S,  E,W,  we  find  that 
the  Ns  =:  the  Ss ;  and  the  E8= 
the  Ws ;  thus  proving  that  the 
work  is  right.     There  is,  it  is 

5/        \  /  true,  a  discrepancy  of  .1  of  a  ft 

I -Y jl5_- —  y  between  the  Ns,  and  the  Ss ;  but 

this  is  owing  to  our  carrying 
out  the  corrections  to  only  one 
decimal  place ;  and  is  too  small 
to  be  regarded.  Discrepancies 
of  3  or  4  tenths  of  a  foot  will 
sometimes  occur  from  this 
cause;  but  may  be  neglected. 
The  corrected  lats  and  deps 
must  evidently  change  the 
bearing  and  distance  of  every 
•ide;  but  without  knowing  either  of  these,  we  ean  now  plot  the  survey  by  means  of  the  corrected 


Fig.  4. 


LAND   SURVEYING. 


279 


lats  and  deps  alone.    The  principal  is  self-evident,  explaining  itself.    First  draw  a  meridian  line 
N  8,  Fig.  4;  and  upon  it  fix  on  a  point  1,  to  represent  the  extreme  west*  comer  of  the  survey. 

Then  from  the  point  1,  prick  off  by  scale,  northward,  the  dist  1,  2'=the  corrected  northing  298.4 
of  side  2,  taken  from  the  last  table ;  from  2'  southward  prick  off  the  dist  2',  3',  the  corrected  south- 
ing 392  of  side  3  ;  from  3'  southward  prick  otf  3',  4',=southing  642.3  of  side  4  ;  from  4'  northward 
prick  off  4',  5'=northing  142.6  of  side  5;  from  5'  prick  off  southward  5',  G'^southing  of  side  6t. 
Then  from  the  points  2^,  3',  4/,  i/,  &,  draw  indefinite  lines  due  eastward,  or  at  right  angles  to  the 
neridian  line.  Make  by  scale,  2',  2  =  corrected  departure  of  side  2;  and  join  1,  2.  Make  3',  3  =  dep 
of  side  24-depof  side  3;  and  join  2,  3;  make  4',  4=3',  3  — depof  side  4;  and  join  3,  4;  make  5',  5 
=4',  4  — aep  of  side  5;  andjoin4,5;  make  6',  6=5',  5  — dep  of  side  6;  andjoin5,64  Finally  join 
S,  1 ;  and  the  plot  is  complete.  If  scrupulous  accuracy  is  not  requited,  the  contents  may  be  found  by 
the  mechanical  method  of  triangles;  the  bearings,  by  the  protractor;  and  the  lengths  of  the  sides 
by  the  scale  ;  all  with  an  approximation  sufficient  for  ordinary  purposes ;  and  perhaps  quite  as  close 
M  by  the  method  by  calculation,  when,  as  is  customary,  the  bearings  are  taken  only  to  the  nearest 

faarter  of  a  degree.  We  have  already  said  that  with  a  scale  of  feet  per  inch= ' 

1  o 

'the  error  of  area  need  not  exceed  the  '2'7)"5"th  part. 

But  if  it  is  required  to  calculate  the  area  of  the  corrected  survey  with  rigorous  exactness,  it  may 
be  done    on    the    following 
principle,  (see  Fig  5.)     If  a     ai 
meridian  line  N  S  be  sup-     IN 

posed  to  be  drawn  through      a  g 

the  extreme  west  corner  1  of    ^  — 

a  survey;  and  lines  (called 
niddle  dittancea)  drawn  (as 
the  dotted  ones  in  the  Fig) 
•it  right  angles  to  said  me- 
ridian, from  the  center  of 
each  side  of  the  survey; 
♦,hen  if  each  of  the  middle 
lists  of  such  sides  as  have 
aorthings,  be  mult  by  the 
jorrected  northing  of  its  cor- 
responding side ;  and  if  each 
9f  the  middle  dists  of  such 
tides  as  have  southings,  be 
mult  by  the  corrected  south- 
ing of  its  corresponding 
side ;  if  we  add  all  the  north 
prods  into  one  sum ;  and  all 
the  south  prods  into  another 
lum ;  and  subtract  the  least 
•f  these  sums  from  the  great- 
Mt(  the  rem  will  be  the  area 


Fifir.5. 


*  The  extreme  east  corner  would  answer  as  well,  with  a  slight  change  in  the  sabsequeat  «p«r* 
atious,  as  will  become  evident. 

t  Instead  of  pricking  off  these  northings  and  southings  in  succession, /Vom  each  other,  It  will  be 
more  correct  in  practice  to  prepare  first  a  table  showing  how  far  each  of  the  points  2', 3',  &o,  is  north 
or  south  from  1.  This  being  done,  the  points  can  be  pricked  off  north  or  south  from  1,  without  mov- 
ing the  scale  each  time;  and  of  course  with  greater  accuracy.  Such  a  table  is  readily  formed.  Rule 
it  as  below ;  and  in  the  first  three  columns  place  the  numbers  of  the  sides  (starting  with  side  2  from 
point  1 ;)  and  their  respective  corrected  northings  and  southings.  The  formation  of  the  4th  and  5th 
colt  by  means  of  the  3d  and  4th  ones,  explains  itself.  Its  accuracy  is  proved  by  the  final  result 
being  0. 




Dist  Nor 

Sft-om  Point  1.1 

Side. 

N.  lat. 

8.  lat. 

N. 

S. 

298.4 

298.4 

392. 

9e.s 

642.3 

735.9 

142.6 

693.3 

420.4 

1013.7 

1013.8 

000.0 

000.0 

J  A  similar  table  should  be  prepared  beforehand  for  the  dists  of  the  points  2,  8,  4,  Ac,  east  from  the 
xheridian  line.  It  is  done  in  the  same  manner,  but  requires  one  col  less,  as  all  the  dists  are  on  th« 
•am*  Bide  of  the  mer  line.    Thus,  starting  from  point  1,  with  side  2: 


Side. 

E.  dep. 

W.  dep. 

Dist  east  from 
meridian  line. 

* 

1167.0 
1042.1 

656.0 
786.9 
564.9 
301.3 

1167.0 
2209.1 
1653.1 
866.S 
301.8 
000.0 

This  wwrk  likewise  proves  itself  by  the  final  result  being  0. 


280 


LAND  SURVEYING. 


•r  the  BQrrey.*  The  corrected  northings  and  southings  we  have  already  found ;  as  also  the  eastings 
and  westings.  The  middle  dists  are  found  by  means  of  the  latter,  by  employing  their  halves ;  adding 
half  eastings,  and  subtracting  half  westings.  Thus  it  is  evident  that  the  middle  dist  2'  of  side  2,  is 
equal  to  half  the  easting  of  side  2.  To  this  add  the  other  half  easting  of  side  2,  and  a  half  easting 
of  side  3 ;  and  the  sum  is  plainly  equal  to  the  middle  dist  3'  of  side  3.  To  this  add  the  other  half 
easting  of  side  3,  and  subtract  a  half  westing  of  side  4.  for  the  middle  dist  4'  of  side  4.  From  this 
subtract  the  other  half  westing  of  side  4,  and  a  half  westing  of  side  5,  for  the  middle  dist  5'  of  sidt 
5;  and  so  on.     The  actual  calculation  may  be  made  thus : 

Half  easting  of  side  2  = =    583.5  B  =  mid  dist  of  iide  2. 

2  583.5  E 


1042.1      1167.0  E 
Half  easting  of  side  3  =  =    521.0  E 


1688.0  E  =  mid  dist  of  side  8. 
521.0  £ 


556  2209.0  E 

Half  westing  of  side  4  =  —    =    278.0  W 


1931.0  E  =  mid  dist  of  side  4. 
278.0  W 


786.9        1653.0  E 
Half  westing  of  side  5  =  =     393.5  W 


1259.5  E  =  mid  dist  of  «ide  6. 
393.5  W 


Half  westing  of  side  6  =  ■ 


301.3 
Half  westing  of  side  1  = = 


583.6  E  = 
282.4  W 


301.2  E 
150.6  W 


mid  dist  of  aide  6. 


150.6  E  =  mid  dist  of  side  1. 


The  work  always  proves  itself  by  the  last  two  results  being  equal. 

Next  make  a  table  like  the  following,  in  the  first  4  cols  of  which  place  the  numbers  of  the  sldee, 
the  middle  dists,  the  northings,  and  southings.  Mult  each  middle  dist  by  its  corresponding  northing 
or  southing,  and  place  the  products  in  their  proper  col.    Add  up  each  col ;  subtract  the  least  from  the 


Side. 


Middle  dist. 


150.6 
583.5 

1688 

1931 

1259.5 
583.6 


Northing. 


1013.8 

298.4 


142.6 


Southing. 


392 
642.3 


North  prod.    South  prod. 


152678 
174116 


1240281 
S45345 


2147322 
506399 


43560)1640923(37.67  Acret. 


*  Proof.  To  illustrate  the  principle  upon  which  this 
rule  is  based,  let  ab,  be,  and  c a,  Fig  6,  represent  in 
order  the  3  sides  of  the  triangular  plot  of  a  survey,  with 
a  meridian  line  d/ drawn  through  the  extreme  west  cor- 
ner, a.  Let  lines  b  d  and  cf  be  drawn  from  each  corner, 
perp  to  the  meridian  line ;  also  from  the  middle  of  each 
side  draw  lines  we,  mn,  so,  also  perp  to  meridian ;  and 
representing  the  middle  dists  of  the  sides.  Then  since 
the  sides  are  regarded  in  the  order  ab,  be,  ca,  it  is 
plain  that  a  d  represents  the  northing  of  the  side  a  b ; 
fa  the  northing  of  ca;  and  df  the  southing  of  6c. 
Now  if  we  mult  the  northing  ad  oT  the  side  ab,  by  its 
mid  dist  ew,  the  prod  is  the  area  of  the  triangle  abd. 
In  like  manner  the  northing  fa  of  the  side  c  a,  mult  by 
its  mid  dist  s  o,  gives  the  area  of  the  triangle  a  cf.  Again, 
the  southing  df  of  the  side  6  c,  mult  by  its  mid  dist  w»n, 
gives  the  area  of  the  entire  Gg  db  cfd.  If  from  this 
area  we  subtract  the  areas  of  the  two  triangles  abd, 
and  a  cf,  the  rem  is  evidently  the  area  of  the  plot  a  6  e. 
So  with  any  other  plot,  however  oomplicata«U 


liAND  SURVEYING. 


281 


greatest.    The  rem  will  be  the  area  of  the  survey  in  sq  ft ;  which,  div  by  43560,  (the  number  of  sq  ft 
u  an  acre,)  will  be  the  area  in  acres ;  in  this  instance,  37.67  ac. 

It  now  remains  only  to  calculate  the  corrected  bearings  and  lengths  of  the  sides  of  the  survey,  all 
of  which  are  necessarily  changed  by  the  adoption  of  the  corrected  lats  and  deps.  To  find  the  bearing 
of  any  side,  div  its  departure  (E  or  W)  by  its  lat  (N  or  S) ;  in  the  table  of  nat  tang,  find  the  quot ; 

the  angle  opposite  it  is  the  reqd  angle  of  bearing.     Thus,  for  the  course  of  side  1,  we  have  • '- 

'  1013.8  N 

=  .2972— nat  tang ;  opposite  which  in  the  table  is  the  reqd  angle,  16°  33' ;  the  bearing,  therefore,  ie 
K  163  33'  w. 

Again  :  for  the  dist  or  length  of  any  side,  from  the  table  of  nat  cosines  take  the  cos  opposite  to 
the  angle  of  the  corrected  bearing ;  divide  the  corrected  lat  (N  or  S)  of  the  side  by  the  cos.    Taos 
for  the  dist  of  side  1,  we  find  opposite  16°  33',  the  cos  .9586.    And 
Lat.        Cos. 
1013.8  -i-  .9586  =  1057.6  the  reqd  dist. 
The  following  table  contains  all  the  corrections  of  the  foregoing  survey ;  consequeutly,  if  the  bear. 


Side. 

Bearing. 

Dist.  Ft. 

1 
2 
3 

4 
5 
6 

N  16°  33'  W 
N  75°  39'  E 
S  69°  23'  E 
S  40°  53'  W 
N  79044' W 
S  530  21'  W 

1057.6 
1204.0 
1113.3 
849.6 
800.1 
704.3 

lags  and  dists  are  correctly  plotted,  they  will  close  perfectly.  The  young  assistant  is  advised  ta 
practise  doing  this,  as  well  as  dividing  the  plot  into  triangles,  and  computing  the  content.  In  this 
manner  he  will  soon  learn  what  degree  of  care  is  necessary  to  insure  accurate  results. 

The  following    hints    may  often  be  of  serrloe. 
Ist.  Avoid  taking  bearings  and 
4Ust3  along  a  circuitous  bound-  a 

ary  line  like  a  6  c,  Fig  7;  but  run  *. —----._-_. •* i* .....*X 

the  straight  line  a  c;   and   at  •  ^  '  ' 

right  angles  to  it,  measure  oflf- 
eets  to  the  crooked  line.  3d. 
Vishing  to  survey  a  straight 
Jine  from  a  to  c,  but  being  una- 
ble to  direct  the  instrument 
precisely  toward  c,  on  account 
•f  intervening  woods,  or  other 
•bstacles ;  first  run  a  trial  line, 
as  a  m,  as  nearly  in  the  proper 
direction  as  can  be  guessed  at. 
Measure  m  c,  and  say,  as  o  m  is  to  t»  c,  so  Is  100  ft  to  ?  Lay  off  o  o  equal  to  100  ft,  and  o  a  equal 
to  ?  ;  and  run  the  final  line  aa  c.  Or.  if  m  c  is  quite  small,  calculate  offsets  like  o  «  for  every  100  ft 
along  a  w»,  and  thus  avoid  the  necessity  for  running  a  second  line.  8d.  When  c  is  visible  from  a,  but 
the  intervening  ground  difficult  to  measure  along,  on  account  of  marshes,  &c.  extend  the  side  y  a 
to  good  ground  at  t:  then,  making  the  angle  y  t  d  equal  to  y  a  c,  run  the  line  t  n  to  that  point  d  at 
which  the  angle  n  <i  c  is  found  by  trial  to  be  equal  to  the  angle  a  t  d.  It  will  rarely  be  necessary  to 
make  more  than  one  trial  for  this  point  d ;  for,  suppose  it  to  be  made  at  x,  see  where  it  strikes  a  c  at 
»;  measure  t  c,  and  continue  from  x,  making  x  d=ic.  4th.  In  case  of  a  very  irregular  piece  of 
land,  or  a  lake.  Fig  8,  surround  it  by  straight  lines.  Survey  these,  and  at  right  angles  to  them« 
flteasure  offsets  to  the  crooked  boundary.   6tb.  Surveying  a  straight  line  from  w  toward  jr,  Fig  9l 


Fig.S. 


an  obstacle,  o,  is  net.  To  pass  It,  lay  off  a  right  angle  to  t  u;  measure  any  t  u ;  make  «  «  «  = 
.90°;  meaaure  «  v;  make  u  v  i  =  90°;  make  v  i  =  t  u;  make  v  i  y  =  90°.  Then  is  ti~uv;  and 
ly  is  in  the  straight  line.  Or,  with  less  trouble,  at  g  make  (  g  a— OOP;  measure  any  g  a;  make 
g  a  «=r60o ;  and  a  3  =  g  a:  make  ae  i  =  60°.  Then  is  gs  =  ^a  or  as;  and  i  s,  continued  toward 
y,  is  in  the  straight  line.  6th.  Being  between  two  objects,  m  and  n,  and  wishing  to  place  myself  in 
range  with  them,  I  lay  a  straight  rod  c  6  on  the  ground,  and  point  It  to  one  of  the  objects  m  ;  them 
going  to  the  end  c,  I  find  that  it  does  not  point  to  the  othT  object.  By  successive  trials,  I  find  the 
position  e  d  in  which  it  points  to  both  objects,  and  eonseq^^  ^ntly  is  in  range  with  ihem. 


282  CHAINING. 

CHAININO. 

Ctiains.  Engineers  have  abandoned  the  Gunter's  chain  of  66  ft,  divided 
into  100  links  of  7.92  ins  each.  They  now  use  a  chain  of  100  ft,  with  100  links 
of  1  ft  each,  and  calculate  areas  in  sq  ft,  the  number  of  which,  divided  by 
43,560,  reduces  to  acres  and  decimals,  instead  of  to  acres,  roods,  and  perches. 
Gunter's  chain  is  used  on  U.  S.  Government  land  surveys. 

Chains  are  commonly  made  of  iron  or  steel  wire.  Each  link  is  bent  at  each 
of  its  ends,  to  form  an  eye,  by  which  it  is  connected  with  the  adjacent  links, 
either  directly,  as  in  the  Grumman  patent  chain,  or,  more  commonly,  by  from 
1  to  3  small  wire  links.  The  wear  of  these  links  is  a  fruitful  source  of  inaccuracy, 
inasmuch  as  even  a  very  slight  wear  of  each  link  considerably  increases  the 
length  of  the  chain.  Hence,  chains  should  be  compared  with  some  standard, 
such  as  a  target  rod,  every  few  days  while  in  use.  For  transportation,  the 
lengths  are  folded  on  each  other,  making  a  compact  and  sheaf-like  bundle. 

Tapes.  With  improved  facilities  for  the  manufacture  of  steel  tape,  the  chain 
is  going  out  of  use.  The  tape,  being  much  lighter,  requires  much  less  pull,  and, 
as  there  are  no  links  to  wear,  its  length  is  much  more  nearly  constant  than  that 
of  the  chain.  It  is  replacing,  to  some  extent,  the  base-measuring  rod  for 
accurate  geodetic  work.  Steel  tapes  are  made  in  continuous  lengths  up  to  500, 
600,  and  even  1000  ft,  but  those  of  100  ft  are  the  most  commonly  used.  Very 
long  tapes  are  liable  to  breakage  in  handling.  Even  the  shorter  lengths,  unless 
handled  carefully,  are  apt  to  kink  and  break.  Breaks  are  diflficult  to  mend,  and 
the  repaired  joint  is  seldom  satisfactory ;  whereas  a  kink  in  a  wire  chain  seldom 
involves  more  than  a  temporary  change  of  length.  Being  run  over  by  a  car  or 
wagon  will  often  kink  steel  tapes  very  badly,  if  it  does  not  break  them.*  How- 
ever, the  lightness,  neatness,  and  reliability  of  the  tape  offset  these  disadvan- 
tages, which,  indeed,  the  surveyor  soon  learns  to  overcome. 

Tapes  for  general  field  work  are  usually  narrow  (from  0.10  to  0.25  in)  and 
thick  (from  0.013  to  0.025  in),f  and  are  graduated  by  means  of  small  brass 
and  copper  rivets,  spaced,  in  general,  5  ft  apart,  1  ft  apart  in  the  10  ft  at  each 
end,  and  0.1  ft  apart  in  the  ft  at  each  end.    They  are  usually  mounted  on  reels. 

Tapes  for  city  work  are  wider  (from  0.25  to  0.5  in)  and  thinner  (from  0.007  to 
0.010  in)t  and  are  graduated  (usually  to  0.01  ft)  throughout  their  length  by 
means  of  lines  and  numerals  etched  on  the  steel. 

Pins  are  ordinarily  of  wire,  pointed  at  the  lower  end,  and  bent  to  a  ring  at 
the  upper  end.  They  can  be  forced  into  almost  any  ground  that  is  not  exceed- 
ingly stony.  A  steel  ring,  like  a  large  key  ring,  is  often  used  for  carrying  the 
pins.  Each  pin  should  have  a  strip  of  bright  red  flannel  tied  to  its  top,  in  order 
that  it  may  be  readily  found,  among  the  grass,  etc.,  by  the  rear  chainman. 

Corrections  for  Sag  and  i^tretcti.  The  following  diagram  X  (see  p. 
283)     gives  the  correction  for  a  steel  tape  weighing  0.75  fb  per  100  ft.f  • 

*The  Nichols  Engineering  &  Contracting  Co.,  Chicago,  guarantees  that  its 
tapes  will  not  be  injured  by  being  run  over  by  wagons. 

fThe  sizes  of  tapes,  as  made  by  different  manufacturers,  vary  greatly.  In 
applying  the  corrections,  therefore,  the  width  and  thickness  of  the  tape  to  be 
used  should  be  carefully  measured,  and  its  weight  per  ft  computed. 

X  Deduced  from  diagrams  constructed  by  Mr.  J.  O.  Clarke,  Proceedings  Engi- 
neers' Club  of  Philadelphia,  April,  1901,  Vol.  XVIII,  No.  2.  from  the  formula : 

PS 
Stretch  =  ^ 

where 

P  =  pull  on  tape,  in  lbs. 

S  =  span  of  tape,  in  ins. 

E  =  modulus  of  elasticity  for  steel  =  27,500,000  lbs  per  sq  in. 

A  =  area  of  cross-section  of  tape  weighing  0.75  &  per  100  ft. 
=  0.0022  square  ins, 
and  from  the  equation  of  the  parabola,  according  to  which 

shortening  by  sag  =  ^jpg 

where  W  =  weight  of  tape  in  pounds  per  foot. 

Except  for  very  light  pulls,  this  last  formula  gives  practically  the  same  results 
as  the  equation  of  the  catenary,  which  is  absolutely  correct,  but  much  more 
eumbersome. 


CHAINING. 

jL 

J«, 

-\-1.0 

K 

1 1 

\ 

\ 

V 

1 

\ 

\ 

\ 

\ 

^+0.5 

\ 

\ 

Vi 

\ 

\ 

V 

%^ 

r^^ 

\ 

t 

\ 

s. 

\' 

-^2 

81 

\^. 

V 

\ 

1 

<?. 

i; 

V 

\r' 

^ 

N 

L 

s 

<^o^ 

N 

* 

« 

\ 

r- 

> 

s 

\ 

v 

\ 

• 

N^. 

^ 

f" 

, 

1   ^ 

\ 

N 

*s 

N 

s 

N 

^ 

>h 

< 

"J^ 

h 

•x 

1*^ 

*v 

sv. 

^ 

^e 

1 

*" 

;:; 

b; 

^ 

^ 

^ 

^ 

> 

L, 

1; 

=n 

=5 

^ 

p^ 

^ 

^ 

"^ 

r-\ 

"*■ 

"^ 

■is 

=55 

aas 

..^ 

*■ 

--- 

"-0.5 

"" 

— 

— 

5  10  15  20  23 

Pull  in  LJts.  on  Steel  Tape  Weighing  0.75  Lh.  per  ±00  Ft. 


The  diagram  shows  that  a  span  of  100  ft  of  tape  weighing  0.75  lb  per  100  ft, 
requires  a  pull  of  11.2  lbs  to  reduce  the  correction  to  zero  ;  a  span  of  50  ft,  7.3  lbs, 
etc.  At  these  tensions  (which  are  called  normal  pulls),  the  opposite  effects  01 
sag  and  of  stretch  are  equal.  At  higher  tensions,  the  lengthening  due  to  stretch 
exceeds  the  shortening  due  to  sag,  and  vice  versa. 

Tapes  of  otber  weights  require  pulls  proportional  to  their  weights  or 
to  their  areas  of  cross-section.     Thus,  a  tape,  of  any  length,  weighing  1  lb 

p6r  100  ft,  would  require,  for  any  given  correction,  a  pull  of  a;  =  -=r-  2/  =  13^  y, 

where  y  =  the  pull  for  the  same  correction  on  the  standard  tape,  weighing  0.75 
ib  per  100  ft. 

Conversely  ;  given  a  pull  of  10  lbs  on  a  50  ft  span  of  a  tape  weighing  0.6  lb  per 
100  ft ;  required  the  correction.     To  produce  the  same  error  in  the  tape  weighing 

0.75  lb  per  100  ft  would  require  a  pull  of  y  =  10  X  7;:^  =  12.5  lbs.    Referring  to 

60 
the  diagram  at  12.5  lbs  on  the  curve  for  a  50  ft  span,  we  find  correction  =  — 0.16. 
This  is  the  proper  correction  for  either  the  heavier  tape  with  12.5  lbs  or  for  the 
lighter  tape  with  10  lbs  pull. 

Corrections  for  temperature.  Tapes  are  usually  graduated  so  as  to 
be  of  standard  length  at  62°  Fahr.  For  ordinary  steel  tape,  the  correction  for 
temperature  is  about  0.0000065  ft  per  ft  per  degree  Fahr. 

Corrections  for  temperature  are  uncertain,  since  the  temperature  of  the  tape 
cannot  be  determined  with  any  accuracy.  Measurements  requiring  great 
accuracy  should  therefore  be  made  in  cloudy  weather,  or  at  night,  and  the  tape 
and  the' thermometer  should  be  kept  off  the  ground. 

When  measuring  over  sloping*  ground,  in  ordinary  work,  the  chain  or 
tape  should  be  held  as  nearly  horizontal  as  possible,  transferring  the  position  of 
the  raised  end  to  the  ground  by  means  of  a  plumb  line.  Where  the  ground  is 
steep,  it  becomes  necessary  to  use  a  short  length  of  tape,  as  the  down-hill  chain- 
man  could  not  otherwise  hold  his  end  high  enough ;  or  the  tape  may  be  held 
parallel  with  the  slope,  and  the  distance  corrected  by  the  following  formulas : 

H  TT 

-5-  =  cos  A  ;    H  =  S.  cos  A ;    S  = r  =  H.  sec  A ; 

S  '  '  cos  A 

R 

=5:  =  tan  A ;    R  =  H.  tan  A ; 


o^  =  sin  A ;     R  = 


S.  sin  A. 


284 


LOCATION   OF   THE    MERIDIAN. 


liOCATION  OF  THE  MERIDIAN. 
By  means  of  circnmpolar  stars. 

(1)  Seen  from  a  point  O  (Figs.  1  and  2)  on  the  earth,  a  circumpolar  star  e 
(star  near  the  pole  P)  appears  to  describe  daily*  and  counterclockwise  a 
small  circle,  euwl,  about  the  pole.  The  angle  P  O  e,  P  O  w,  etc.,  subtended 
by  the  radius  Pe,  P  w,  etc.,  of  this  circle,  or  the  apparent  distance  of  the 
star  from  the  pole,  is  called  Its  polar  distance.  The  polar  distances  of 
stars  vary  slightly  from  year  to  year.  See  Table  3.  They  vary  slightly  also 
during  each  year.  In  the  case  of  Polaris  this  latter  variation  amounts  to 
about  50  seconds  of  arc. 

(2)  The  altitude  of  the  pole  is  the  angle  N  O  P  of  the  pole's  elevation 
above  the  horizon  N  E  S  W,  and  is  =  the  latitude  of  the  point  of  obser- 


Fig.  1. 


Fig.  2. 


vation.  Declination  =  angular  distance  north  or  south  from  the  celestial 
equator.  Thus,  declination  of  pole  =  90°.  Declination  of  any  star  =  90°— its 
polar  distance. 

(3)  Let  Z  e  H  be  an  arc  of  a  vertical  circlet  passing  through  a  circumpolar 
Btar,  e,  and  let  H  be  the  point  where  this  arc  meets  the  horizon  N  E  S  W. 
Then  the  angle  N  Z  H  at  the  zenith  Z,  or  N  O  H  at  the  point  0  of  observa- 
tion, between  the  plane  N  Z  O  of  the  meridian  and  the  plane  H  Z  O  of  the 
star's  vertical  circle  (or  the  arc  N  H),  is  called  the  azimutlit  of  the  star. 
If  this  angle  N  O  H  be  laid  oflf  from  O  H,  on  the  ground,  the  line  O  N  will  be 
in  the  plane  of  the  meridian  N  Z  S,  or  will  be  a  nortli-and-soutli 
line.ll 

(4)  When  a  star  is  on  the  meridian  Z  N  of  the  observer,  above  or  below 
the  pole  P,  as  at  u  or  I,  it  is  said  to  be  at  its  upper  or  loi¥er  culmina- 
tion, respectively.  Its  azimuth  is  then  =  0,  the  line  O  H  coinciding  with 
the  meridian  line  O  N. 

(5)  When  the  star  has  reached  its  greatest  distance  east  or  west  from  the 
pole,  as  at  e  or  w,  it  is  said  to  be  at  its  eastern  or  western  elonga- 
tion.J 


*  In  23  h.  56.1  m. 

t  A  great  circle  is  that  section  of  the  surface  of  a  sphere  which  is  formed 
by  a  plane  passing  through  the  center  of  the  sphere.  A  vertical  circle  is  a 
great  circle  passing  through  the  zenith  Z. 

X  Astronomers  usually  reckon  azimuth  from  the  south  point  around 
through  the  west,  north,  and  east  points,  to  south  again  ;  but  for  our  pur- 
pose it  is  evidently  much  more  convenient  to  reckon  it  from  the  north 
point,  and  either  to  the  east  or  to  the  west,  as  the  case  may  be. 

II  The  point  N,on  the  horizon,  is  called  the  north  point,  and  must  not 
be  confounded  with  the  north  pole  P. 

§  As  seen  from  the  equator,  a  star,  at  either  elongation,  is,  like  the  pole 
itself,  on  the  horizon;  and  the  two  lines  Pe,  Viv,  joining  it  with  the  pole, 
form  a  single  straight  line  perpendicular  to  the  meridian,  and  lying  in  the 


LOCATION    OF    THE    MERIDIAN. 


285 


(6)  The  honr  ang-le  of  any  star,  at  any  given  moment,  is  the  time 
which  has  elapsed  since  it  was  in  upper  culmination.* 

(7)  Evidently  the  azimuth  of  a  star  is  continually  changing.  In  cir- 
cumpolar  stars  it  varies  from  0°  to  maximum  (at  elongation)  and  back  to 
0°  twice  daily,  as  the  star  appears  to  revolve  about  the  pole  ;  but  when  the 
star  is  near  either  elongation  the  change  in  azimuth  takes  place  so  slowly 
that,  for  some  minutes,  it  is  scarcely  perceptible,  the  star  appearing  to 
travel  vertically. 

(8)  Given  the  polar  distance  of  a  star  and  the  latitude  of  the  point  of 
observation,  the  azimutb  of  the  star,  at  elong-ation,  may  be  found 
by  the  formula.f 

c,.        ^     .       .1,    ^  ^  sine  of  polar  distance  of  star 

Sine  of  azimuth  of  star  = ; .   ..f   , ^ t-^ — -^—r- r^ — 

cosine  of  latitude  of  point  of  observation 

or  see  (11)  and  Table  3. 

(9)  The  following  circumpolar  stars  are  of  service  in  connection  with 
observations  for  determining  the  meridian.    See  Fig.  3. 

Constellation  Letter  Called 

Ursa  minor  (Little  bear)  a  (alpha)  Polaris 

Ursa  major  (Great  bear)  e  (epsilon)  Alioth 

"         "      (     "         "    )  ^(zeta)  Mizar 

Cassiopeia  6  (delta)  Delta  % 


poi*^^' 


(10)  Polaris,  or  the  north  star,  is  fortunately  placed  for  the  determi- 
nation of  the  meridian,  its  polar  distance  being  only  about  lVi°.  See  Table 
3.  Fig.  3  shows  the  circumpolar  stars  as  they  appear  about  midnight  in 
July ;  inverted,  as  in  January ;  with  the  left  side  uppermost,  as  in  April ; 
and,  with  the  right  side  uppermost,  as  in  October.  || 


horizon.  The  azimuth  of  the  star  is  then  =  its  polar  distance.  But  in 
other  latitudes  P  c  and  P  w  form  acute  angles  with  the  meridian,  as  shown, 
and  these  angles  decrease,  and  the  azimuth  of  the  star  at  elongation  in- 
creases, as  the  latitude  increases. 

*  In  lat.  40°  N.,  the  hour  angle,  ZPc  =  ZPty,  of  Polaris,  at  elongation,  is 
=  5  h.  55  m.  of  solar  time.    Caution.     It  will  be  noticed  that,  except  for 
an  observer  at  the  equator,  the  elongations  do  not  occur  at  90°  from  the 
meridian. 
t  In  the  spherical  triangle  Z  P  e,  we  have  : 

sin  e  Z  P  ^  sin  P  e 
sin  Z  e  P  "     sin  P  Z 
But,  since  Z  e  P  =  90°,  sin  Z  e  P  =  1.    Also,  sin  P  Z  =  cos  (90°  —  P  Z),  and 
c  Z  P  ^  azimuth  of  c. 

sin  P  e   _  sin  polar  distance  P  O  e 

cos  latitude 


Hence,  sin  azimuth  of  e  -- 


sin  P  Z 

t  5  Cassiopeia  is  here  called  Delta,  for  brevity. 

|l  Polaris  is  easily  found  by  means  of  the  two  well-known  stars 

called  the  "  pointers  "  in  "  the  dipper,"  Fig.  3,  which  forms  the  hinder 


286 


LOCATION    OF   THE    MERIDIAN. 


(11)  Table  3  gives  the  polar  distances  of  Polaris  and  their  log  sines  for 
January  1  in  each  third  year  from  1900  to  1930  inclusive,  the  log  cosines 
of  each  fifth  degree  of  latitude  from  20°  to  50°,  and  the  corresponding 
azimuths  of  Polaris  at  elongation.  Intermediate  values  may  be  taken  by 
interpolation.* 

(12)  By  observation  of  Polaris  at  elongation.  This  method 
has  the  convenience,  that  at  and  near  elongation  the  star  appears  to  travel 
vertically  for  some  minutes,  its  azimuth,  during  that  time,  remaining 
practically  constant ;  but  during  certain  parts  of  the  year  (see  Table  1),  the 
elongations  of  Polaris  take  place  in  daylight;  so  that  this  method  cannot 
then  be  used.  |  See  (18),  (19),  (22).  Nor  can  it  be  used  at  any  time  in  places 
south  of  about  4°  N.  lat.,  because  there  Polaris  is  not  visible. 

(13)  The  approximate  times  of  elongation  of  Polaris  for  certain  dates, 
in  1900,  are  given  in  Table  1,  with  instructions  for  finding  the  times  for 
other  dates.  Or,  watch  Polaris  in  connection  with  any  of  those  stars  which 
are  nearly  in  line  with  it  and  the  pole,  as  Delta,  Mizar,  and  Alioth.  See 
Fig.  3.  The  time  of  elongation  is  approximated,  with  sufficient  closeness 
for  the  determination  of  the  azimuth,  by  the  cessation  of  apparent  hori- 
zontal motion  during  the  observation. 

(14)  From  fifteen  to  thirty  minutes  before  the  time  of  elongation,  have 
the  transit,  see  (21),  set  up  and  carefully  centered  over  a  stake  previously 
driven  and  marked  with  a  center  point.  The  transit  must  be  in  adjust- 
ment, especially  in  regard  to  the  second  adjustment,  p.  294,  or  that  of  the 
horizontal  axis,  by  which  the  line  of  collimation  is  made  to  describe  a  ver- 
tical plane  when  the  transit  is  leveled  and  the  telescope  is  swung  upward 
or  downward. 

(15)  Means  must  be  provided  for  illuminating  the  cross-hairs  of  the  tran- 
sit. J  This  maybe  done  by  means  of  a  bull's 
eye,  or  a  dark  lantern,  so  held  as  not  to  throw 
its  light  into  the  eye  of  the  observer ;  or,  better, 
by  means  of  a  piece  of  tin  plate,  cut  and  per- 
forated as  in  Fig.  4,  bent  at  an  angle  of  45°,  as 
in  Fig.  5,  and  painted  white  on  the  surface 
next  to  the  telescope.  The  ring,  formed  by 
bending  the  long  strip,  is  placed  around  tlie 
object  end  of  the  telescope.  A  light,  screened 
from  the  view  of  the  observer,  is  then  held, 
at  one  side  of  the  instrument,  in  such  a  way 
that  its'  rays,  falling  upon  the  oblique  and 
whitened  surface  of  the  tin  plate,  are  reflected  directly  into  the  telescope. 

(16)  Bring  the  vertical  hair  to  cut  Polaris,  and,  by  means  of  the  tangent 
screw,  follow  the  star  as  it  appears  to  move,  to  the  right  if  approaching  east- 
ern elongation,  and  vice  versa,  keeping  the  hair  upon  the  star,  as  nearly  as* 
may  be.  As  elongation  is  approached,  the  star  will  appear  to  move  more 
and  more  slowly.  AVhen  it  appears  to  travel  vertically  along  the  hair,  it 
has  practically  reached  elongation,  and  the  vertical  plane  of  the  transit, 
with  the  vertical  hair  cutting  the  star,  is  in  the  plane  of  the  star's  vertical  circle. 
Depress  the  telescope,  and  fix  a  point  in  the  line  of  sight,  preferably  300 
feet  or  more  distant  from  the  transit.f  Immediately  reverse  the  transit, 
(swinging  it  horizontally  through  an  arc  of  180°),  sight  to  the  star  again, 


Fig.  4. 


Fig.  5. 


portion  of  the  "  great  bear  "  (Ursa  major),  a  line  drawn  through  these  two 
stars  passing  near  Polaris.  As  the  stars  in  the  handle  of  the  dipper  form 
the  tail  of  the  great  bear,  as  shown  on  celestial  maps,  so  Polaris  and  the 
stars  near  it  form  the  tail  of  the  little  bear  (Ursa  minor.)  Polaris  is  also 
nearly  midway  and  in  line  between  Delta  and  Mizar,  Polaris  forms,  with 
three  other  and  less  brilliant  stars,  a  quite  symmetrical  cross,  with  Polaris 
at  the  end  of  the  right  arm.  In  Fig.  3  this  cross  is  inverted.  Its  height  is 
about  5°,  or  =  the  distance  between  the  pointers. 

*  Part  of  a  table  computed  by  the  Surveying  Class  of  1882-3,  School  of 
Engineering,  Vanderbilt  University,  Nashville,  Tenn.,  and  published  by 
Prof  Olin  H.  Landreth. 

t  The  stake  must  be  illuminated.  This  may  be  done  by  throwing  light 
upon  that  side  of  the  stake  which  faces  the  transit,  or,  better,  by  holding  a 
sheet  of  white  paper  behind  the  stake,  with  a  lantern  behind  the  paper.  In 
the  latter  case,  the  cross-hairs  of  the  transit,  as  well  as  the  stake,  and  the 
knife-blade  or  pencil-point  with  which  the  assistant  marks  it,  show  out 
dark  against  the  illuminated  surface  of  the  paper. 

:j:  See  Note,  pnge  290. 


LOCATION    OF    THE    MERIDIAN.  287 

again  depress,  and,  if  the  line  of  sight  then  coincides  perfectly  with  the 
mark  first  set,  both  are  in  the  plane  of  the  star's  vertical  circle.  If  not, 
note  where  the  line  of  sight  does  strike,  and  make  a  third  mark,  midway 
between  the  two.  The  line  of  sight,  when  directed  to  this  third  mark,  is  in 
the  req^uired  plane,  from  which  the  azimuth,  found  as  in  (8),  has  yet  to  be 
laid  ofl  to  the  meridian,  to  the  left  from  eastern  elongation,  and  vice  versa. 

(17)  To  avoid  driving  the  distant  stake  and  marking  it  during  the  night, 
a  fixed  target  at  any  convenient  point  may  be  used,  and  the  horizontal 
angle  formed  between  the  line  of  sight  to  the  star  and  that  to  the  target 
merely  noted,  for  use  in  ascertaining  and  laying  off  the  azimuth  of  the 
target. 

(18)  By  observation  of  Polaris  at  ciilminatioii.  Owing  to, 
its  greater  difficulty,  this  method  will  generally  be  used  only  when  that 
by  elongation  is  impracticable.  It  consists  in  watching  Polaris  in  connec- 
tion with  another  circumpolar  star  (such  asMizar*or  Delta)  until  Polaris  is 
seen  in  the  same  vertical  plane  with  such  star,  and  then  waiting  a  short  and 
known  time  T,  as  follows,!  until  Polaris  reaches  culmination,  where- 
upon Polaris  is  sighted  and  the  line  of  collimation  is  in  the  meridian.  At 
their  upper  culminations,  Mizar  and  Delta  are  too  near  the  zenith  to  be 
conveniently  observed  at  latitudes  north  of  about  25°  and  30°  respectively. 
At  their  lower  culminations  they  are  too  near  the  horizon  to  be  used  to 
advantage  at  places  much  below  about  38°  of  N.  latitude.  In  general, 
Delta  is  conveniently  observed  at  lower  culmination  from  February  to 
August,  and  Mizar  during  the  rest  of  the  year. 

Mizar  Delta 

T=  T  = 

In  1900 2.6   mins       3.4   mins 

In  1910 6.5   mins        7.2   mins 

Mean  annual  increase,  1900-1910 .     0.39  min         0.38  min 

(19)  By  observation  of  Polaris  at  any  point  in  its  patli. 
Table  1  gives  the  mean  solar  times  of  upper  culmination  of  Polaris  on  the 
1st  of  each  month  in  1900,  and  directions  for  ascertaining  the  times  on  other 
dates  ;  and  Table  2  gives  the  azimuths  of  Polaris  corresponding  to  different 
values  of  its  hour  angle  in  civil  or.  mean  solar  time,  for  different  latitudes 
from  30°  to  50°,  and  for  the  years  1901  and  1906.  For  hour  angles  and  lati- 
tudes intermediate  of  those  in  the  table,  the  azimuths  may  be  taken  by 
interpolation.    See  Caution  and  formula,  p.  290. 

(20)  The  local  time  %  of  observation  must  be  accurately  known,  and  the 
time  of  the  preceding  upper  culmination  (as  obtained  from*Table  1)  deducted 
from  it.  The  difference  is  the  hour  angle.  If  the  hour  angle,  thus  found, 
is  11  h.  58  m.  or  less,  the  star  is  west  of  the  meridian.  If  it  is  greater  than 
11  h.  58  m.,  the  star  is  east  of  the  meridian.  In  that  case  deduct  the  hour 
angle  from  23  h.  56  m.  and  enter  the  table  with  the  remainder  as  the  hour 
angle.    See  Fig.  1. 

(31)  Where  great  accuracy  is  not  required,  Polaris  may  be  observed  by 
means  of  a  plumb-line  and  sight.  A  brick,  stone,  or  other  heavy  object 
will  answer  perfectly  as  a  plumb-bob.  It  should  hang  in  a  pall  of  water. 
A  compass  sight,  or  any  other  device  with  an  accurately  straight  slit  about 
1//16  inch  wide,  may  be  used.  The  sight  must  remain  always  perfectly  verti- 
cal, but  must  be  adiustable  horizontally  for  a  few  feet  east  and  west.  The 
plumb-line  and  sight  should  be  at  least  15  feet  apart,  and  so  placed  that  the 
star  and  plumb-line  can  be  seen  together  through  the  sights  throughout  the 
observation.  The  plumb-line  must  be  illuminated.  It  is  well  to  arrange 
all  these  matters  on  an  evening  preceding  that  of  the  observation.  When 
the  star  reaches  elongation,  the  sight  must  be  fastened  in  range  with  the 
plumb-line  and  the  star.  From  the  line  thus  obtained,  lay  off  the  azimuth ; 
to  the  west  for  eastern  elongation,  and  vice  versa. 

(33)  By  any  star  at  eqnal  altitudes.  This  method,  applicable 
to  south  as  well  as  to  north  latitudes,  consists  in  observing  a  star  when  it 
is  at  any  two  equal  altitudes,  E.  and  W.  of  the  meridian,  thus  locating,  on  the 
horizon,  two  points  of  equal  and  opposite  azimuth.  The  meridian  will 
be  midway  between  the  two  points. 

*  Mizar  will  be  recognized  by  the  small  star  Alcor,  close  to  it. 

t  Deduced  from  values  calculated  in  astronomical  time  (p.  266)  by  the 
U.  S.  Coast  and  Geodetic  Survey. 

X  liOcal  time  agrees  with  standard  time  (p.  267)  on  the  standard 
meridians  only.  For  other  points  add  to  standard  time  4  minutes  for  each 
degree  of  longitude  east  of  a  standard  meridian,  and  vice  versa. 


288 


LOCATION    OF   THE    MERIDIAN. 


(33)  By  equal  shadows  from  the  son.  Fig.  6.  Approximate. 
At  the  solstices  (about  June  21  and  December  21)  the  path  abed  traversed 
before  and  after  noon,  by  the  end  of 
the  solar  shadow  O  a,  etc.,  of  a  verti- 
cal object  O,  or  by  the  shadow  oi  a 
knot  tied  in  a  plumb-line  suspended 
over  O,  will  intersect  a  circular  arc 
a  N  d,  described  about  O,  at  equal  dis- 
tances, am,  md,  from  the  meridian 
O  N.  The  observations  should  be 
made  within  two  hours  before  and 
after  noon.  At  the  vernal  equinox 
(March  21)  the  line  thus  located  will 
then  be  west,  and  at  the  autumnal 
equinox  (Sept.  21)  east,  of  the  merid-  ,.         .  .      .^ 

ian,  by  less  than  2U  minutes  of  arc.  For  intermediate  dates  the  error  is 
nearly  proportional  to  the  time  elapsed.  It  is  well  to  draw  several  arcs 
of  difierent  radii,  O  a,  O  6,  etc.,  note  two  points  where  the  path  of  the  shadow 
intersects  each  arc,  and  take  the  mean  of  all  the  results.  A  small  piece  of 
tin  plate,  with  a  hole  pierced  through  it,  may  be  placed  with  the  hole 
vertically  over  O  ;  and  the  bright  spot,  formed  by  the  light  shining  through 
the  hole,  used  in  place  of  the  end  of  the  shadow. 

Table  1. 

Approximate  civil  times  of  elongation  and  cnlmination 
of  Polaris  in  lat.  40°  N.,  long.  90°  W.  from  Greenwich,  on  the  first  of  each 
month,  in  1900. 

The  times  given  in  this  table  are  mean  solar  or  local  times. 

P.  M.  times  (between  noon  and  midnight)  are  printed  in  bold-face. 

In  latitude  25^,  W.  elongations  occur  later  and  E.  earlier )  ,     nearlv  o  mins 

In  latitude  50°,  W.  "  "     earlier  and  E.  later  P^  ^^^^^^^  ^'^^• 

The  correction  for  longitude  amounts  to  scarcely  a  minute  of  time  in  any 
part  of  the  United  States. 

For  other  days  of  the  month,  deduct  5.94  min.  for  each  succeeding  day. 

In  general,  the  times  are  a  little  later  each  year.  In  1904  ihey  will  beabout 
5^  minutes  later,  but  in  1905,  only  about  3  minutes  later,  than  in  1900.  This 
discrepancy  is  due  to  the  occurrence  of  leap-year  in  1904. 

Inasmuch  as  this  table  serves  chiefly  to  put  the  observer  on  guard,  and  as 
he  should  be  at  his  post  from  15  to  30  minutes  in  advance  of  these  times, 
the  gradual  increase  in  the  times  is  of  little  consequence.  The  position  of 
the  star  at  elongation  is  determined  by  observation. 

At  culmination,  where  the  change  in  azimuth  is  most  rapid,  an  error  in 
time  of  2  minutes  in  observing  Polaris  involves  an  error  of  about  1  minute 
of  azimuth. 


At  elongation, 

an  error  in  time  of' 

20  minutes 

10  minutes 

5  minutes 

1  minute 


will  make  an  error  in  azimuth  of 
less  than  30      seconds 
less  than   6  " 

less  than   2  " 

about         0.06  second 


In  all  latitudes,  each  culmination  follows  the  preceding  one  bv  11  h.  58  m. 
mean  solar  time.  The  times  between  successive  elongations  vary  with  the 
latitudes.  . 

(E,  eastern  :  W,  western.)     1900. 

May  1.  E.  Jun.  1.  E 

4.50  A.  M.  2.49  A.  M. 

Nov.l.W.  Dec.l.W. 

4.33  A.  M.  2.35  A.  M. 


Elong>ations. 


Jan.  1.  W. 
12.31  A.  M. 
July  1.   E. 


Feb.  1.  W. 
10..30  P.  M. 

Aug.  1.    E. 


12.51  A.  M.     10.46  P.  M 


Mar.  1.   W. 

8.40  P.  M. 

Sept.  1.  E. 

8.45  P.  M. 


Apr.  1.  W. 
6.38  P.  M. 

Oct.  1.  E  . 
6.47  P.  M. 


Culminations.    (U,  upper ; 

Jan.  1.  U.  Feb.  1.  L.  Mar.  1.  L.  Apr.  1.  L. 

6.38  P.  M.  4.38  A.  M.  2.47  A.  M.  12.45  A.  M. 

L.  Aug.  1.  U.  Sept.  1.  U.  Oct.  1.   U. 

.  M.  4.45  A.  M.  2.43  A.  M.  12.46  A.  M. 


July  1. 
6.44  P. 


L,  lower.)     1900. 

May  1.  L. 
10.43  P.  M 

Nov.  1.  U. 
10.40  P.  M. 


June  1.  L. 
8.43  P.  M. 

Dec.  1.  U. 
8  42  P.  M. 


LOCATION    OF   THE    MERIDIAN. 


289 


Table  3. 

AZIMUTHS  OF  POLARIS. 


Hour 

A.ugle 

Azimuth  for  latitude 

Hour 

Angle 

Azimuth  for  latitude 

Mean  solar  time 

Meau  solar  time 

1901 

1906 

30° 

35° 

40° 

0     / 

450 

0     / 

50° 

1901 

1906 

30° 

0     f 

35° 

0     / 

40° 

0     / 

45° 
0     / 

50° 

h  m 

h  m 

o     / 

0     / 

0     / 

h  m 

h  m 

0     f 

0    4 

0    4 

0    2 

0    20    210    2 

0     2 

6  57 

6  27 

1  21 

1  26 

1  32 

1  39 

1  49 

0    8 

0    9 

.0    3 

0     30    410    4 

0     4 

7  12 

6  52 

1  20 

1  24 

1  80 

1  37 

1  47 

0  13 

0  13 

0     5 

0     5,0    50     6 

0     6 

7  25 

7     9 

1  18 

1  22 

1  28 

1  35 

1  44 

0  17 

0  17 

0     6 

0     7[0     70     8 

0     8 

7  38 

7  25 

1   16 

1  20 

1  26 

1  83 

1  42 

0  21 

0  21 

0     8 

0     8  0    9  0  10 

0  11 

7  4^ 

7  38 

1  14 

1  19 

1  24 

1  31 

1  39 

0  25 

0  26 

0     9 

0  10  0  11|0  12 

0  13 

7  57 

7  47 

1  13 

1  17 

1  22 

1  29 

1  37 

0  29 

0  30 

0  11 

0  12  0  12!0  14 

0  15 

8    5 

7  57 

1  11 

1  15 

1  20 

1  27 

1  35 

0  33. 

0  34 

0  12 

0  13  0  14  0  16 

0  17 

8  13 

8     6 

1   10 

1  14 

1  18 

1  25 

1  33 

0  38 

0  38 

0  14 

0  15  0  16  0  18 

0  19 

8  20 

8  13 

1     8 

1  12 

1  17 

1  23 

1  31 

0  42 

0  43 

0  15 

0  17,0  18  0  20 

0  21 

8  27 

.  8  21 

1     7 

1  11 

1  15 

1  21 

1  29 

0  46 

0  47 

0  17 

0  no  19  0  21 

0  23 

8  34 

8  28 

1     5 

1     9 

1   13 

1  19 

1  27 

0  50 

0  51 

0  19 

0  20,0  21  0  23 

0  25 

8  40 

8  35 

1     3 

1     7 

1  12 

1  18 

1  25 

0  54 

0  55 

0  20 

0  22  0  23  0  25 

0  27 

8  47 

8  42 

1     2 

1     6 

1  10 

1  16 

1  23 

0  59 

1     0 

0  22 

0  23  0  25  0  27 

0  29 

8  53 

8  48 

1     0 

1     4 

1     8 

1  14 

1  21 

1     3 

1     4 

0  23 

0  25  0  26  0  29 

0  32 

8  58 

8  53 

0  59 

1     3 

1     7 

1  12 

1  19 

1    8 

1    9 

0  2.-, 

0  26  0  28  0  31 

0  34 

9     4 

8  69 

0  58 

1     1 

1     5 

1  10 

1  17 

1  13 

1  15 

0  27 

0  29  0  31  0  34 

0  37 

9     9 

9     5 

0  56 

1     0 

1     3 

1     8 

I  15 

1  19 

1  20 

0  29 

0  31  ;0  33  0  36 

0  39 

9  15 

9  11 

0  55 

0  58 

1     2 

i  7 

1  13 

1  24 

1  25 

0  31 

0  33' 0  35  0  38 

0  42 

9  20 

9  16 

0  53 

0  56 

1     0 

1    5 

1  11 

1  29" 

1  31 

0  32 

0  35  0  37  0  40 

0  44 

9  25 

9  22 

0  51 

0  54 

0  58 

1     3 

1    9 

1  34 

1  37 

0  34 

0  37  0  39  0  43 

0  47 

9  31 

9  27 

0  50 

0  53 

0  56 

1     1 

1    7 

1  40 

1  42 

0  36 

0  39  0  41  0  45 

0  49 

9  36 

9  33 

0  49 

0  52 

0  55 

0  59 

1    5 

1  45 

1  47 

0  38 

0  41  0  43  0  47 

0  51 

9  41 

9  38 

0  47 

0  50 

0  53 

0  57 

1    2 

1  50 

1  53 

0  39 

0  42  0  45  0  49 

0  54 

9  47 

9  44 

0  45 

0  48 

0  51 

0  55 

1    0 

1  56 

1  58 

0  41 

0  44  0  47  0  51 

0  56 

9  52 

9  49 

0  44 

0  46 

0  49 

0  63 

0  58 

2     1 

2    3 

0  43 

0  46  0  49  0  53 

0  59 

9  57 

9  55 

0  42 

0  44 

0  47 

0  51 

0  56 

2    6 

2    9 

0  45 

0  48  0  51  0  55 

1     1 

10    2 

10     0 

0  40 

0  42 

0  45 

0  49 

0  54 

2  11 

2  14 

0  46 

0  50  0  53  0  57 

1     3 

10     8 

10     5 

0  39 

0  41 

0  43 

0  47 

0  51 

2  17 

2  20 

0  48 

0  51  0  54  0  59 

1    5 

10  13 

10  11 

0  37 

0  39 

0  41 

0  45 

0  49 

2  22 

2  25 

0  50 

0  53  0  56  1     1 

1     8 

10  18 

10  16 

0  35 

0  37 

0  40 

0  43 

0  47 

2  27 

2  31 

0  51 

0  54  0  58  1     3 

1  10 

10  24 

10  21 

0  33 

0  35 

0  38 

0  41 

0  44 

2  33 

2  36 

0  53 

0  56  1     0  15 

1  12 

10  29 

10  27 

0  32 

0  34 

0  36 

0  39 

0  42 

2  38 

2  42 

0  54  0  5811     2  17 

1  14 

10  34 

10  33 

0  30 

0  32 

0  34 

0  37 

0  40 

2  43 

2  47 

0  56,0  59  1     3  1     9 

1  16 

10  39 

10  38 

0  28 

0  29 

0  31 

0  34 

0  37 

2  49 

2  53 

0  57  1     ill     5  1  11 

1  18 

10  45 

10  43 

0  26 

0  27 

0  '19 

0  32 

0  35 

2  54 

2  59 

0  59,1     3;i     7  1  13 

1  20 

10  50 

10  49 

0  24 

0  25 

0  27 

0  29 

0  32 

3    0 

3    5 

1     Oil     4'1     8  1  15 

1  22 

10  55 

10  54 

0  23 

0  24 

0  25 

0  27 

0  30 

3    5 

3  10 

I     2I1     6  1  10  1  16 

1  24 

10  59 

10  58 

0  21 

0  22 

0  24 

0  26 

0  28 

3  11 

3  16 

1     3  1     7;i  12^1  18 

1  26 

11     4 

11     3 

0  20 

0  21 

0  22 

0  24 

0  26 

3  18 

3  23 

1     5|l     91  13  1  20 

1  28 

11     8 

11     7 

0  18 

0  19  0  20 

0  22 

0  25 

3  24 

3  30 

1     6  1  10  1  15  1  22 

1  30 

11   12 

11   11 

0  17 

0  18  0  19 

0  20 

0  22 

3  31 

3  37 

1     8^1  12:1  17|1  24 

1  32 

11   16 

11    15 

0  15 

9  16  0  17 

0  18 

0  20 

3  38 

3  45 

1     9  1   14|l  19'l  26 

1  34 

11  20 

11   20 

0  14 

0  14 

0  15  0  16 

0  18 

3  45 

3  52- 

1  11 

1  15  1  2011  27 

1  36 

11  25 

11   24 

0  12 

0  13 

0  14  0  15 

0  16 

3  53 

4     1 

1  12 

1  17  1  22jl  29 

1  38 

11  29 

11  28 

0  11 

0  11 

0  12  0  13 

0  14 

4    1 

4  11 

1  14 

1  1811  241  31 

1  40 

11  33 

11  32 

0     9 

0     9  0  10  0  11 

0  12 

4  10 

4  20 

1  15 

1  20 jl  25  1  33 

1  42 

11  37 

11  37 

0     8 

0     80     80    9 

0  10 

4  20 

4  33 

1  17 

1  22  1  27,1  35 

1  44 

11  41 

11  41 

0     6 

0     60     70     7 

0    8 

4  33 

4  49 

1  19 

1  24  1  29:1  37 

1  47 

11  45 

11  45 

0     5 

0    5  0    50    5 

0    6 

4  46 

5     6 

1  20 

1  25|l  3111  39 

1  49 

11  50 

11  49 

0     3 

0     30     30    4 

0    4 

5    1 

5  31 

I  22 

1  27! 1  33  1  41 

1  51 

11  54 

11  54 

0     1 

0     1,0    20    2 

0    2 

When  the  star  is  near  elongation  (hour  angles  hetween  5  h  and  7  h).  a  con- 
siderable change  in  hour  angle  corresponds  to  but  a  small  change  in  azimuth. 
At    such    times  it  will  usually  be   better  to  use  the  method   employing  the 
elongation. 
19 


290 


LOCATION    OF    THE    MERIDIAN. 


Table  3. 

POLARIS.    POLAR  DISTANCES,  AND  AZIMUTH  AT  ELONGATION. 


Azimuth  at  Elongation,  in  Latitude 

»H 

Polar 
Dist.  of 
Polaris 

Log  sin 
pol  dist. 

$ 

ao° 

35° 

30° 

35° 

40° 

45° 

50° 

o   /     // 

O        f 

o      / 

o      / 

o     r 

O        f 

O        f 

o      / 

1900 

1  13  33 

8.33  027 

1   18.3 

1   21.1 

1    24.9 

1    29.8 

1   36.1 

1    44.1 

1    54.4 

1903 

1  12  37 

8.32  472 

1    17.3 

1   20.1 

1    23.8 

1    28.7 

1   34.8 

1    42.7 

1   53.0 

1906 

1  11  41 

8.31  910 

1    16.3 

1    19.1 

1    22.8 

1    27.6 

1    33.6 

1    41.4 

1   51.5 

1909 

1  10  45 

8.31  341 

1    15.3 

1    18.1 

1    21.7 

1    26.4 

1.  32.3 

1    40.1 

1   50.1 

1912 

1     9  49 

8.30  765 

1    14.3 

1    17.0 

1    20.6 

1   25.2 

1    81.1 

1    38.7 

1   48.6 

1915 

1     8  53 

8.30181 

1    13.3 

1    16.0 

1    19.5 

1    24.1 

1    29.9 

1    37.5 

1   47.2 

1918 

1     7  58 

8.29  594 

1    12.3 

1    15.0 

1    18.5 

1    23.0 

1    28.7 

1    36.1 

1   45.7 

1921 

1    7    2 

8.28  999 

1    11.4 

1    14.0 

1     17.4 

1   21.9 

1    27.5 

1    34.8 

1   44.3 

1924 

1    6    7 

8.28  401 

1    10.4 

1    13.0 

1    16.3 

1    20.7 

1   26.3 

1    33.5 

1    42.9 

1927 

1    5  12 

8.27  794 

1     9.4 

1    11.9 

1    15.3 

1    19.6 

1    25.1 

1   32.2 

1   41.4 

1930 

1     4  16 

8.27  169 

1     8.4 

1   10.9 

1    14.2 

1    18.5 

1   23.9 

1    30.9 

1    40.0 

Log  cos 

lat 

9.97  299 

9.95  728 

9.93  753 

9:91  337 

9.88  425 

9.84  949 

9.80  807 

Owing  to  changes  in  the  position  of  Polaris  during  the  year,  the  positioni 
given  in  the  table  may  at  times  be  in  error  by  as  much  as  a  minute.  Thi 
error  is  greater  in  the  higher  latitudes. 

Having  the  north  polar  distance,  /),  of  a  star,  and  the  latitude,  L,  of  th( 
point  of  observation,  we  have,  declination  of  star  =  5  =  90°  —  p ;  and  th( 
aKiiiiiith,  a,  of  the  star,  corresponding  to  any  hour  angle,  h,  may  b< 
found  by  the  following  formulas : 

rr,       i.r       cot  p       t&uS  COS  M .  tau  h 

Tan  M  = r:  = r.    Then       Tan  a  = — -z—. . 

cosh        cosh  cos(L  — M) 

The  declinations,  5,  of  Polaris  are  given  in  the  U.  S.  Ephemeris  or  Nautica! 
Almanac.  From  these  the  polar  distances  may  be  obtained  more  accuratelj 
than  from  our  Table  3. 

Caution.  When  it  is  desired  to  determine  the  meridian  within  out 
minute  of  arc,  it  is  well  to  use  more  than  one  method  and  compare  th€ 
results.  For  example,  observe  Polaris  both  E.  and  W.  of  the  meridian,  anc 
a  star  at  equal  altitudes  south  of  the  zenith. 

Note. — If  Polari.s  be  found  during  twilight,  in  the  morning  or  evening,  obser 
vations  of  it  may  be  made  without  artificial  illumiuation  of  the  cross-hairs 
For  times  of  elongation,  see  Table  L 


Conversion  of  Arc  into  Time,  and  vice  versa. 


Arc         Time 
1°  =  4  minutes 
1'  =  4  seconds 
V  =  0.066...  second 


Time  Arc 

24  hours    =  360° 
Ihour      =    15° 
1  minute  =      0°  15' 
1  second  =     0°  O'  15" 


THE   engineer's   TRANSIT.  291 

THE  ENGINEER'S  TRANSIT. 


292 


THE    ENGINEER  S   TRANSIT. 


The  details  of  the  transit,  like  those  of  the  level,  are  differently  arranged  by 
diff  makers,  and  to  suit  particular  purposes.  We  describe  it  in  its  modern  form, 
as  made  by  Heller  and  Brightly,  of  Philada.  Without  the  long-  bubble-tube 
F  F,  Fig  1,  under  the  telescope,  and  the  graduated  arc  g,  it  is  their  plain 
transit.  With  these  appendages,  or  rather  with  a  graduated  circle  in  place  of 
the  arc,  it  becomes  virtually  a  Complete  Theodolite. 

B  D  D,  Fig  2,  is  the  tripod-head.  The  screw-threads  at  v  receive  the  screw 
of  a  wooden  tripod-head-cover  when  the  instrument  is  out  of  use.  S  S  A  is  the 
lower  parallel  plate.  After  the  transit  has  been  set  very  nearly  over  the 
center  of  a  stake,  the  shifting^-plate^  c^c^  cc,  enables  us,  by  slightly  loosening 
the  levelling'-screwis  K,  to  shift  the  upper  parts  horizontally  a  trijfle,  and 
thus  bring  the  plumb-bob  exactly  over  the  center  with  less  trouble  than  by  the 
elder  method  of  pushing  one  or  two  of  the  legs  further  into  the  ground,  or  spread- 
ing them  more  or  less.  The  screws,  K,  are  then  tightened,  thereby  pushing  up- 
ward the  upi>er  parallel  plate  mmmxx,  and  with  it  the  half-ball  &,  thus 
pressing  c  b  tightly  up  against  the  under  side  of  S.    The  plumb-line 


through  tlie  vert  hole  in  b.  Screw-caps.  /,  g,  protect  the  levelling-screws  from 
dust  &c.  The  feet,  %  of  the  screws,  work  in  loose  sockets,./,  made  flat  at  bottom, 
to  preserve  S  from  being  indented.  The  parts  thus  far  described  are  generally 
left  attached  to  the  legs  at  all  times.    Fig  1  shows  the  method  of  attachment. 

To  set  the  upper  parts  upon  the  parallel  plates.  Place  the 
lower  end  of  U  U  in  y  x,  holding  the  instrument  so  that  the  three  blocks  on  m  wi 
(of  which  the  one  shown  at  F  is  movable)  may  enter  the  three  corresponding 


THE   engineer's   TRANSIT.  293 

recesses  in  a,  thus  allowing  a  to  bear  fully  on  m,  upon  which  the  upper  paite 
then  rest.  (The  inner  end  of  the  spring-catch,  I,  in  the  meantime  enters  a  groove 
around  U,  just  below  a,  and  prevents  the  upper  parts  from  falling  off,  if  the  in- 
strument  is  now  carried  over  the  shoulder.)  Revolve  the  upper  parts  horizontallj 
a  trifle,  in  either  direction,  until  they  are  stopped  by  the  striking  of  a  small  lug 
on  a  against  one  of  the  blocks  F.  The  recesses  in  a  are  now  clear  of  the  blocks. 
Tighten  q,  thereby  pushing  inward  the  movable  block  F,  which  clamps  the 
bevelled  flange  a  between  it  and  the  two  fixed  blocks  on  m  m,  and  confines  the 
spindle  U  to  the  fixed  parallel  plates.  It  remains  so  clamped  while  the  instrument 
is  being  used. 

To  remove  the  upper  parts  from  tlie  parallel  plates.  Loosen 
g,  bring  the  recesses  in  a  opposite  the  blocks  F,  Hold  back  I,  and  lift  the  upper 
parts,  which  are  then  held  together  by  the  broad  head  of  the  screw  inserted  into 
the  foot  of  the  spindle  w. 

T  T  is  the  outer  revolving  spindle,  east  in  one  with  the  support- 
ing-plate Z  Z,  to  which  is  fastened  the  graduated  limb  O  O.  The  limb 
extends  beyond  the  compass-box,  and  thus  admits  of  larger  graduations  than 
would  otherwise  be  obtainable,  w  wi%  the  inner  revolving  spindle.  At 
it«  top  it  has  a  broad  flange,  to  which  is  fastened  the  vernier  pl-ate  P.  To  the 
latter  are  fastened  the  compass-box  C,  the  two  bubble-tubes  M  M,  the  standards 
V  V,  supporting  the  telescope,  &c.  Each  bubble-tube  is  supported  and  adjusted 
by  four  capstan-head  nuts,  two  at  each  end.  The  bent  strip,  curving  over  the 
tube,  protects  the  glass  from  accidental  blows  in  swinging  the  telescope. 

Control  of  motions  of  graduated  limb  O  O  and  vernier 
plate  P. — The  tangent-screw  G  and  a  spiral  spring  (not  shown)  opposite  to  it 
are  fixed  to  the  graduated  limb  0  0,  and  hold  between  them  a  projection  y  from 
the  loose  collar  t,  which  is  thus  confined  to  the  limb  and  made  to  travel  with  it. 
The  clamp-screw  H  passes  through  the  collar  t  and  presses  against  the  small  lug 
shown  at  its  inner  end.  When  H  is  tiglitened,  this  lug  is  pressed  against  the 
fixed  spindle  U  U,  to  whicii  the  graduated  limb  is  thus  made  fast.  A  slow  mo- 
tion may,  liowever,  still  be  given  to  the  limb  by  means  of  the  tangent-screw  G. 

The  motion  of  the  vernier  plate  P  over  the  graduated  limb  O  O  is  similarly 
governed  by  the  tangent-screw  b  and  its  spiral  spring  (not  shown),  fixed  to  the 
vernier  plate  P,  and  the  clamp-screw  e,  which  passes  through  the  collar  z,  and 
presses  against  the  small  lug  shown  at  its  inner  end.  In  Heller  and  Brightly's 
instruments,  the  screw  b  is  provided  with  means  for  taking  up  its  "wear,"  or 
"  lost-motion." 

There  are  two  verniers.  One  is  shown  at  p.  Fig  1.  Both  may  be  read,  and 
their  mean  taken,  when  great  accuracy  is  required.  Ivory  reflectors,  c,  facilitate 
their  reading.  Before  the  instrument  is  moved  from  one  place  to  another,  the 
eompass-needle,  k,  Fig  2,  should  always  be  pressed  up  against  the  glass  cover 
of  the  compass-box  by  means  of  the  upright  milled-head  screw  seen  on  the  ver- 
nier-plate in  Fig  1,  just  to  the  right  of  the  nearest  standard.  The  pivot-point  is 
thus  protected  from  injury.. 

R,  Fig  1,  is  a  ring  with  a  clamp  (the  latter  not  shown)  for  holding  the  telescope 
in  any  required  position.  It  is  best  to  let  the  e2/e-end,E,  of  the  telescope  revolve 
downward,  as  otherwise  the  shade  on  O,  if  in  use,  may  fall  otf.  The  tangent-screw, 
d,  moves  a  vert  arm  attached  to  R,  and  is  thus  used  for  slightly  changing  the 
elevation  of  the  telescope.  In  the  arm  is  a  slit  like  that  seen  in  the  vernier-arm 
I.  By  means  of  the  screw  D,  the  movable  vernier-arm  Y  may  be  clamped  at 
any  desired  point  on  the  vertical  limb  g.  When  0°  of  the  vernier  is  placed  at 
30°  on  the  arc  g,  and  the  index  of  the  opposite  arm  is  placed  over  a  small  notch 
on  the  horizontal  brace  (not  seen  in  our  figs)  of  the  standards,  the  two  slit^  will 
be  opposite  each  other,  and  may  be  used  for  laying  off  offsets,  &c,  at  right-angles 
to  the  line  of  sight. 

One  end,  R,  of  the  telescope  axis  rests  in  a  movable  box,  under  which  is  a  screw. 
By  means  of  the  screw,  the  box  may  be  raised  or  lowered,  and  the  axis  thus  ad- 
justed for  very  slight  derangements  of  the  standards.  For  E,  B,  0,  and  A,  see 
Level,  p  306.    «  is  a  dAist-guard  for  the  object-slide. 

Stadia  Hairs.  Immediately  behind  the  capstan-screw,  p,  Fig  1,  is  seen  a 
smaller  one.  This  and  a  similar  o'ne  on  the  opposite  side  of  the  telescope,  work 
in  a  ring  inside  the  telescope,  and  hold  the  ring  in  position.  Across  the  ring  are 
stretched  two  additional  horizontal  hairs,  called  stadia  hairs,  placed  at  such  a 
distance  apart,  vertically,  that  they  will  subtend  say  10  divisions  of  a  graduated  rod 
placed  100  ft  from  the  instrumeut,  16  divisions  at  150  ft,  &c.  They  are  thus  used  for 
measuring  hor  and  sloping  distances. 

The  long  bubble- tube.  B'  F,  Fig  1,  enables  us  to  use  the  transit  as  a  level, 
aJthough  it  is  not  so  well  adanVid  as  the  latter  to  this  purpose. 


294  THE  engineer's  transit. 

To  adjust  a  plain  Transit* 

When  either  a  level  or  a  transit  is  purcliased,  it  is  a  good  precaution  (but  on* 
which  the  writer  has  never  seen  alluded  to)  to  first  screw  the  object-glass  firmly  horn* 
to  its  place ;  and  then  make  a  short  continuous  scratch  upon  the  ring  of  the  glass,  and 
upon  its  slide;  so  as  to  be  able  to  see  at  any  time  when  at  work,  that  the  glass  is 
always  in  the  same  position  with  regard  to  the  slide.  For  if,  after  all  the  adjustments 
are  completed,  the  position  of  the  glass  should  become  changed,  (as  it  is  apt  to  be  if 
unscrewed,  and  afterward  not  screwed  up  to  the  same  precise  spot,)  the  adjustments 
may  thereby  become  materially  derauged ;  especially  if  the  object-glass  is  eccentric, 
or  not  truly  ground,  which  is  often  the  case.  Such  scratches  should  be  prepared  bj 
the  maker.  In  making  adjustments,  as  well  as  when  using  a  transit  or  level,  be 
careful  that  the  eye-glass  and  object-glass  are  so  drawn  out  that  there  shall  be  no 
parallax.  The  eye-glass  must  first  be  drawn  out  so  as  to  obtain  perfect  distinctnegs 
of  the  cross-hairs ;  it  must  not  be  disturbed  afterward ;  but  the  object-glass  must 
be  moved  for  different  distances. 

First,  to  ascertain  tliat  tlie  bnbble-tubes,  M  m,  are  placed 
parallel  to  the  vernier-plate,  and  that  therefore  when  both  bubbles  are  in 
the  centers  of  their  tubes  the  axis  of  the  inst  is  vert.  By  means  of  the  four  levelling- 
screws,  K,  bring  both  bubbles  to  the  centers  of  their  tubes  in  one  position  of  the 
inst ;  then  turn  the  upper  parts  of  the  inst  half-way  round.  If  the  bubbles  do  not 
remain  in  the  center,  correct  half  the  error  by  means  of  the  two  capstan-nuts 
r  r ;  and  the  other  half  by  the  levelling-screws  K.  Repeat  the  trial  until  both 
bubbles  remain  in  the  center  while  the  inst  is  being  turned  entirely  around  on 
its  spindle. 

Second,  to  see  that  the  standards  have  suffered  no  derang^e- 
ment ;  that  is,  that  they  are  of  equal  height  and  perpendicular  to  the  vernier- 
plate,  as  they  always  are  when  they  leave  the  maker's  hands.  Level  the  inst 
perfectly;  then  direct  the  intersection  of  the  hairs  to  some  point  of  a  high  object 
(as  the  top  of  a  steeple)  near  by;  clamp  the  inst  by  means  of  screws  H  and  e, 
and  lower  the  telescope  until  the  intersection  strikes  some  point  of  a  low  object. 
(If  there  is  none  such  drive  a  stake  or  chain-pin,  &c,  in  the  line.)  Then  un- 
clamp  either  H  or  e,  and  turn  the  upper  parts  of  the  inst  half-way  round  ;  fix  the 
intersection  again  upon  the  high  point;  clamp;  lower  the  telescope  to  the  low 
point.  If  the  intersection  still  strikes  the  low  point,  the  standards  are  in  order. 
If  notj  correct  one-half  of  the  difference  by  means  of  the  adjusting-block  and 
screw  at  the  end,  R,  of  the  telescope  axis,  Fig.  1,  and  repeat  the  trial  de  novo, 
resetting  the  stake  or  chain-pin  at  each  trial.  If  the  inst  has  no  adjusting- block 
for  the  axis,  it  should  be  returned  to  the  maker  for  correction  of  any  derange- 
ment of  the  standards. 

A  transit  may  be  used  for  running  straight  lines,  even  if  the  standards  become 
Slightly  bent,  by  the  process  described  at  the  end  of  the  fourth  adjustment. 

Third,  to  see  that  the  cross-hairs  are  truly  vert  and  hor 
M^hen  the  inst  is  level.  When  the  telescope  inverts,  the  cross-hairs  are 
nearer  the  eye-end  than  when  it  shows  objects  erect.  The  maker  takes  care  to  place 
the  cross-hairs  at  right-angles  to  each  other  in  their  ring,  or  diaphragm  ;  and  gene- 
rally he  so  places  the  ring  in  the  telescope,  that  when  levelled,  they  shall  be  vert 
and  hor.  Sometimes,  however,  this  is  neglected ;  or  the  ring  may  by  accident  be- 
come turned  a  little.  To  be  certain  that  one  hair  is  vert,  (in  which  case  the  other 
must,  by  construction,  be  hor,)  after  having  adjusted  the  bubble-tubes,  level  the  in- 
strument  carefully,  and  take  sight  with  the  telescope  at  a  plumb-line,  or  other  vert- 
straight  edge.  If  the  vert  hair  coincides  with  this  object, 
it  is,  so  far, in  adjustment;  but  if  not,  then  loosen  slightlf 
only  tioo  adjacent  screws  of  the  four,  p  p  i  i.  Fig  1 ;  and 
with  a  knife,  key,  or  other  small  instrument,  tap  very 
gently  against  the  screw-heads,  so  as  to  turn  the  ring  a 
little  in  the  telescope;  i)ersevering  until  the  hair  be* 
comes  truly  vertical.  When  this  is  done,  tighten  th« 
screws.  In  the  absence  of  a  plumb-line,  or  vert  straight 
edge,  sight  the  cross-hair  at  a  very  small  distinct 
point;  and  see  if  the  hair  still  cuts  that  point,  when 
the  telescope  is  raised  or  lowered  by  revolving  it  on 
its  axis. 
The  mode  of  performing  the  foregoing  will  be  readily 
nnderstood  from  this  Fig,  which  represents  a  section  across  the  top  part  of  the  tele- 
iscope,  and  at  the  cross-hairs.  The  hair-ring,  or  diaphragm,  a;  vert  hair,  v;  tele- 
scope tube,  g  ;  ring  outside  of  telescope  tube,  d;  &  is  one  of  the  four  capstan-headed 
screws  which  hold  the  hair-ring,  a,  in  its  place,  and  also  serve  to  adjust  it.  The 
lower. ends  of  these  screws  work  in  the  tnickness  of  the  hair-ring;  so  that  wheci 
they  are  loosened  somewhat,  they  do  not  lose  their  hold  on  the  ring.     Small  loose 


THE  engineer's  TRANSIT. 


295 


^m 


washers,  c,  are  placed  under  the  heads  h  cf  the  screws.  A  space  y  y  is  left  around 
each  screw  where  it  passes  through  the  telescope  tube,  to  allow  the  screws  and  ring 
together  to  be  moved  a  little  sideways  when  the  screws  h  are  slightly  loosened. 

Fourth,  to  see  tliat  ttie  vertical  hair  is  in  the  line  of  colli- 
mation.  Plant  the  tripod  firmly  upon  the  ground,  as  at  a.  Level  the  inst ; 
clamp  it;  and  direct  the  vert  hair  by  means  of  tangent-screw  G  (figs.  1  and  2) 
upon  some  convenient  object  b\  or  if  there  is  none  such,  drive  a  thin  stake,  or  a 
chain-pin.    Then  revolving  the  telescope  vert  on  its  axis,  ^q, 

observe  some  object,  as  c,  where  the  vert  hair  now  strikes;   t.  a       ^^""^ 

or  if  there  is  none,  place  a  second  pin.     Unclamp  the  instru-  , •^rrrl ,o 

ment  by  the  clamp-screw  H;  and  turn  the  whole  upper  •  ^^-^ 

part  of  it  around  until  the  vert  hair  again  strikes  ^- -.^Fifi".  4, 
Clamp  again;  and  again  revolve  the  telescope  vert  on  its 
axis.  If  the  vert  hair  now  strikes  c,  as  it  did  before,  it  shows  that  c  is  really 
at  0 ;  and  that  b,  a,  c,  are  in  the  .same  straight  lute ;  and  therefore  this  adjustment 
is  in  order.  If  not,  observe  where  it  does  strike,  say  at  m,  (the  dist  a  m  being 
taken  equal  to  a  c,)  and  place  a  pin  there  also.  Measure  m  c ;  and  place  a  pin 
at  t',  in  the  line  m.  c,  making  m  v  =  one-fourth  of  m  c.  Also  put  a  pin  at  o,  half- 
way between  m  and  c,  or  in  range  with  a  and  b.  By  means  of  the  two  hor 
screws  that  move  the  ring  carrying  the  cross-hairs,  adjust  the  vert  hair  until  it 
cuts  V.  Now  repeat  the  entire  operation  ;  and  persevere  until  the  telescope,  after 
being  directed  to  6,  shall  strike  the  same  object  o,  both  times,  when  revolved  on 
its  axis.  See  whether  the  movement  of  the  ring  in  this  4th  adjustment  has  dis- 
turbed the  verticality  of  the  hair.  If  it  has,  repeat  the  3d  adjustment.  Then  re- 
peat the  4th,  if  necessary  ;  and  so  on  until  both  adjustments  are  found  to  be  riyht 
at  the  same  time.  'I'hus  a  straight  line  may  be  run,  even  if  the  hairs  are  out  of 
adjustment ;  but  with  somewhat  more  trouble.  For  at  each  station,  as  at  a,  two 
back-sights,  and  two  foresights,  a  c  and  a  m,  may  be  taken,  as  when  making  the 
adjustment ;  and  the  point  o,  half-way  between  c  and  ni,  will  be  in  the  straight  line. 
The  inst  may  then  be  moved  to  o,  and  the  two  back-sights  be  taken  to  a  ;  and  so  on. 

Angles  measured  by  the  transit,  whether  vert  or  hor,  will  evidently  not  be 
affected  by  the  hairs  being  out  of  adjustment,  provided  either  that  the  vert 
hair  is  truly  vert,  or  that  we  use  the  intersection  of  the  hairs  when  measuring. 

The  foreg-oing  are  all  the  adjustments  needed,  unless  the  tran- 
sit is  requirea  for  levelling,  in  which  case  the  folio wiug  one  must  be  attended  to : 


Big.  5 


To  adjust  the  long  bubble* tnbe,  F  F,  Fig.  1,  we  first  place  the  line 
of  sight  of  the  telescope  hor,  and  then  make  the  bubble-tube  hor,  so  that  the 
two  are  parallel.  Drive  two  pegs,  a  and  b  Fig.  5,  with  their  tops  at  precisely 
the  same  level  (see  Rem.  p.  296)  and  at  least  about  100  ft.  apart ;  300  or  more 
will  be  better.  Plant  the  inst  firmly,  in  range  with  them,  as  at  c,  making  h  c 
an  aliquot  part  of  a  b.  and  as  short  as  will  permit  focusing  on  a  rod  at  b.  The 
inst  need  not  be  leveled.  Suppose  the  line  of  sight,  to  cut  e  and  d.  Take  the 
readings  b  e  and  a  d.  Their  diff  is  be  —  ad  =  an  —  ad  =  dn;  and  a  h  :  a  c  :  : 
dn:  d  s]  s  being  the  height  of  the  target  at  a  when  the  readings  (a  s,  b  o)  on  the 

two  stakes   are  equal,     a  s  =  a  d-\-d  s=a  d  -\ — •    If  the  reading  on  a 

exceeds  that  on  6  (as  when  the  line  of  sight  is  vfg)  the  diff  of  readings  is  =  a  ^r  — 


b /=  ag  —  ai  =  gi;  and  as  =  ag  —  gs  =  ag  — 


giXac 
ab 


Sight  to  s,  bring  the 


bubble  to  the  cen  of  its  tube  by  means  of  the  two  small  nuts  n  n  at  one  end  of  the 
tube.  Fig.  1,  and  assume  that  the  telescope  and  tube  are  parallel.*    The  zeros  of 

*  This  neglects  a  small  error  due  to  the  curvature  of  the  earth ;  for  a  hor  line  at  t;  is  t;  h,  tan- 
gential to  the  curved  (or  "  level")  surface  of  still  water  at  v,  whereas  u  s  is  tangential  to  water  surf 
at  a  point  midway  between  a  and  h.  Hence  if  the  telescope  at  v  points  to  a  it  will  not  be  parallel  to 
the  level  bubble-tube.  To  allow  for  this,  and  for  the  refraction  by  the  air,  which  diminishes  the 
error,  raise  the  target  on  o  to  a  point  A  above  ».  h  s  =  .0000000205  X  square  of  a  c  in  ft  ;  but  whea 
•  c  is  660  ft,  ha\s  only  aboufc  one  tenth  of  an  inch  and  barely  covers  the  apparent  thickness  of  the 
w«M-hair  in  the  tele.scoi>» 


j:> 


.rrii 


296  THE  engineer's  transit. 

the  vert  circle,  and  of  its  vernier,  may  now  be  adjusted,  if  they  require  it,  by 
loosening  the  vernier  screws  and  then  moving  the  vernier  until  the  two  coin- 
cide. ,  ^    .  ,      , 

Rem.  If  no  level  is  at  hand  for  levelling  the  two  pegs  a  and  6,  it  may  be  done 
by  the  transit  itself,  thus :  Carefully  level  the  two  short  bubbU-s,  by  means  of  the 
levelling-screws  K.  Drive  a  peg  m,  from  100  to  300  feet  froni  the  instrument  o. 
Then  placing  a  target-rod  on  m,  clamp  the  target  tight  at  whatever  height,  as  ««, 
the  hor  hair  happens  to  cut  it ;  it  being  of  no  im- 
portance whether  the  telescoi>e  is  level  or  not; 
although  it  might  as  well  be  as  nearly  so  as  can 
conveniently  be  guessed  at.    Clamp  the  telescope 

-Q- ■ H  in  its  position   by  the  clamp-ring  R,  Fig.  1.    Re- 

^  volve    the    Inst    a  considerable  way   round ;    say 
^S-  0«  nearly  or  quite  half  way.    Place  another  peg  71, 

at  precisely  the  same  dist  from  the  instrument  that  m  is ;  and  continue  to  drive  it  un- 
til the  hor  hair  cuts  the  target  placed  on  it,  and  still  kept  clamped  to  the  rod,  at  the 
same  height  as  when  it  was  on  m.  When  this  is  done,  the  tops  of  the  two  pegs  are 
on  a  level  with  each  other,  and  are  ready  to  be  used  as  before  directed. 

When  a  transit  is  intended  to  be  used  for  surveying  farms,  &c,  or  for  retracing 
lines  of  old  surveys,  it  is  very  useful  to  set  the  compass  so  as  to  allow  for  the  "va- 
riation "   during    the    interval    between    the    two    surveys.       For  this  purpose  a 
"  variation- vernier  "  is  added  to  such  transits;  and  also  to  the  compass. 
When  the  graduations  of  a  transit  are  figured,  or  numbered,  so  as  to  read  both 
10  0  10 

ways  from  zero,  thus,     1 1  t  1 1  i  i  i  1  ii  i  i  I  i  i  i  1 1  i  i  1 1  I  i  i  i        the  vernier  also  is  made 

double;  that  is,  it  also  is  graduated  and  numbered  from  its  zero  both  ways.  In  this 
case,  if  the  angle  is  measured  from  zero  toward  the  right  hand,  the  reading  must  be 
made  from  the  right  hand  half  of  the  vernier  ;  and  vice  versa.  If  the  figuring  is 
single,  or  only  in  one  direction,  from  zero  to  360°,  then  only  the  single  vernier  is 
necessary,  as  the  angles  are  then  measured  only  in  the  direction  that  the  figuring 
counts.  Engineers  differ  in  their  preferences  for  various  manners  of  figuring  the 
graduations.  The  writer  prefers  from  zero  each  way  to  180°,  with  two  double  ver- 
niers. 

To  replace  cross-hairs  in  a  level,  or  transit.  Take  out  the  tube 
from  the  eye  end  of  the  telescope.  Looking  in,  notice  which  side  of  the  cross- 
hair diaphragm  is  turned  toward  the  eye  end.  Then  loosen  the  four  screws  which 
hold  the  diaphragm,  so  as  to  let  the  latter  fall  out  of  the  telescope.  Fasten  on  new 
hairs  with  beeswax,  varnish,  glue,  or  gum-arabic  water,  &c.  This  requires  care. 
Then,  to  return  the  diaphragm  to  its  place,  press  firmly  into  one  of  the  screw-holes 
on  the  circiimf  of  the  diaphragm  itself,  the  end  of  a  piece  of  stick,  long  enough  to 
reach  easily  into  the  telescope  as  far  as  to  where  the  diaphragm  belongs.  By  this 
stick,  as  a  handle,  insert  the  diaphragm  edgewise  to  its  place  in  the  telescope,  and  hold 
it  there  until  two  opposite  screws  are  put  in  place  and  screwed.  Then  draw  the  stick 
out  of  the  hole  in  the  diaphragm ;  and  with  it  turn  the  diaphragm  until  the  same 
side  presents  itself  toward  the  eye  end  as  before ;  then  put  in  the  other  two  screws. 

The  so-called  cross  hairs  are  actually  spider-web,  so  fine  as  to  be  barely  visible  to 
the  naked  eye.  Holler  &  Brightly  use  very  fine  platina  wire,  which  is  much  better. 
Human  hair  is  entirely  too  coarse. 

To  replace  a  spirit-level,  or  bubble-g-lass.  Detach  the  level  from 
the  instrument;  draw  off  its  sliding  ends;  push  out  the  broken  glass  vial,  and  the 
cement  which  held  it ;  insert  the  new  one,  with  the  proper  side  up  (the  upper  side 
is  always  marked  with  a  file  by  the  maker);  wrapping  some  paper  around  it?  ends, 
if  it  fits  loosely.  Finally,  put  a  little  putty,  or  )nelted  beeswax  over  the  ends  of  tlio 
vial,  to  secure  it  against  moving  in  its  tube. 

In  purchasing  instruments,  especially  when  they  are  to  be  used  far  from  a  maker, 
it  is  advisable  to  provide  extras  of  such  parts  as  may  be  easily  broken  or  lost;  such 
as  glass  compass-covers,  and  needles;  adjusting  pins;  level  vials;  magnifiers,  Ac. 


Tbeodolite  adjustments  are  performed  like  those  of  the  level  and  transit. 

let.  That  of  the  cross-hairs ;  the  same  as  in  the  level. 

2d.  The  long  bubble-tube  of  the  telescope ;  also  as  in  the  level. 

3d.  The  two  short  bubble-tubes ;  as  in  the  transit. 

4th  The  vernier  of  the  vert  limb ;  as  in  the  transit  with  a  vert  circle. 

5th.  To  see  that  the  vert  hair  travels  vertically  ;  as  in  the  fourth  adjustment 
of  the  transit.  In  some  theodolites,  no  adjustment  is  provided  for  this;  but  in 
large  ones  it  is  provided  for  by  screws  under  the  feet  of  the  standards. 

6om«time§  a  second  telescope  is  added ;  it  is  p'aced  below  the  hor  lin^b,  and  is 


THE  BOX  OR  POCKET  SEXTANT. 


297 


called  a  watcher.  It  has  its  own  clamp,  and  tangent-screw.  Its  use  is  to  ascertain 
whether  the  zero  of  that  limb  has  moved  during  the  measurement  of  hor  angles. 
When,  previously  to  beginning  the  measurement,  the  zero  and  upper  telescope  are 
directed  toward  the  first  object,  point  the  lower  telescope  to  any  small  distant 
object,  and  then  clamp  it.  During  the  subsequent  measurement,  look  through  it, 
from  time  to  time,  to  be  sure  that  it  still  strikes  that  object ;  thus  proving  that  Tom 
slipping  has  occurred. 


THE  BOX  OE  POCKET  SEXTANT. 


The  portability  of  the  pocket  sextant,  and  the  fact  that  it  reads  to  single  minutes, 
render  it  at  times  very  useful  to  the  engineer.  By  it,  angles  can  be  measured  while 
in  a  boat,  or  on  horseback:  and  in  many  situations  which  preclude  the  use  of  a 
transit.  It  is  useful  for  obtaining  latitudes,  by  aid  of  an  artificial  horizon.  When 
closed,  it  resembles  a  cylindrical  brass  box,  about  3  inches  in  diameter,  and  1]/^ 
inches  deep.  This  box  is  in  two  parts; 
by  unscrewing  which,  then  inverting 
one  part,  and  then  screwing  them  to- 
gether again,  the  lower  part  becomes  a 
handle  for  holding  the  instrument. 
Looking  down  upon  its  top  when  thus 
arranged,  we  see,  as  in  this  figure,  a 
movable  arm  I  C,  called  the  iiKlex, 
which  turns  on  a  center  at  C,  and  car- 
ries the  vernier  V  at  its  other  end.  Gr 
Gr  is  the  graduated  arc  or  limb.  It 
actually  subtends  about  73°,  but  is  di- 
vided into  about  146°.  Its  zero  is  at 
one  end.  Its  graduations  are  not  shown 
in  the  Fig. 

Attached  to  the  index  is  a  small  mov- 
able lens,  (not  shown  in  the  figure,) 
likewise  revolving  around  C,  for  read- 
ing the  fine  divisions  of  the  limb.  When 
measuring  an  angle,  the  index  is  moved 
by  turning  the  milled-head  P  of  a 
pinion,  which  works  in  a  rack  placed  within  the  box.  The  eye  is  applied  to  a  cir- 
cular hole  at  the  side  of  the  box,  near  A.  A  small  telescope,  about  3  inches  long, 
accompanies  the  instrument",  but  may  generally  be  dispensed  with.  When  so,  the 
eyehole  at  A  should  be  partially  closed  by  a  slide  which  has  a  very  small  eye-hole 
in  it;  and  which  is  moved  by  the  pin  h,  moving  in  the  curved  slot.  Another  slide, 
at  the  side  of  the  box,  carries  a  dark  glass  for  covering  the  eye-hole  when  observing 
the  sun.  When  the  telescope  is  used,  it  is  fastened  on  by  the  milled-head  screw  T. 
The  top  part  shown  in  our  figure,  can  be  separated  from  the  cylindrical  part,  by 
removing  3  or  4  small  screws  around  its  edge ;  and  the  interior  can  then  be  exam- 
ined, and  cleaned  if  necessary.  Like  nautical,  and  other  sextants,  this  one  has 
two  principal  glasses,  both  of  them  mirrors.  One,  the  liidex-g-lass,  is  attached 
to  the  underside  of  the  index,  at  C ;  its  upper  edge  being  indicated  by  tlie 
two  dotted  lines.  The  other,  the  liorlzoii-g-lass,  (because,  when  meas- 
uring the  vert  angles  of  celestial  bodies,  it  is  directed  toward  the  horizon,)  is  also 
within  the  box;  the  position  of  its  upper  edge  being  shown  by  the  dotted  lines  at 
R.  The  horizon-glass  is  silvered  only  half-way  down ;  so  that  one  of  the  observed 
objects  may  be  seen  directly  through  its  lower  half,  while  the  image  of  the  other 
object  is  seen  in  the  upper  half,  reflected  from  the  index-glass.  That  the  instrument 
may  be  in  adjustment,  ready  for  use,  these  two  glasses  must  be  at  right  angles  to  th« 
plane  of  the  instrument ;  that  is,  to  the  under  side  of  the  top  of  the,  box,  to  which  they 
are  attached;  and  must  also  be  parallel  to  each  other,  when  the  zeros  of  the  vernier 
and  of  the  limb  coincide.  The  index-glass  is  already  permanently  fixed  by  the 
maker,  and  requires  no  other  adjustment.  But  the  horizon-glass  has  two  adjust- 
ments, which  are  made  by  a  key  like  that  of  a  watch,  and  having  a  milled-head  K. 
It  is  screwed  into  the  top  of  the  box,  so  as  to  be  always  at  hand  for  use.  When 
needed,  it  is  unscrewed.    This  key  fits  upon  two  small  square-heads,  (like  that  for 


298 


THE   COMPASS. 


winding  a  watch;)  one  of  which  is  shown  at  S;  while  the  other  is  near  it,  but  on  the 
SIDE  of  the  box.  These  squares  are  the  heads  of  two  small  screws.  If  the 
horizon  glass  H  should,  a^;  in  this  sketch,  (where  it  is  shown  endwise,)  not  be  at 
right  angles  to  the  top  U  ij  of  the  box,  it  is  brought  right  by  turning  the  square- 
bead  S  of  the  screw  S  T ;  and  if,  after  being  so  far  rectified,  it  still  is  not  parallel  to 
the  index-glass  when  the  zeros  coincide,  it  is  moved 
a  little  backward  or  forward  by  the  square  head 
at  the  side. 

To  adjust  a  box  sextant,  bring  the  two 
zeros  to  coincide  precisely ;  then  look  through  the 
eye-hole,  and  the  lower  or  unsilvered  part  of  the 
horizon-glass,  at  some  distant  object.  If  the  instru- 
ment is  in  adjustment,  the  object  thus  seen  directly, 
will  coincide  precisely  with  its  reflected  image, 
seen  at  the  same  time,  at  the  same  spot.  But  if  it 
is  not  in  adjustment,  the  two  will  appear  separated 
either  hor  or  vert,  or  both,  thus,  *  * ;  in  which  case 
apply  the  key  K  to  the  square-head  S ;  and  by  turning  it  slightly  in  whichever  direc- 
tion may  be  necessary,  still  looking  at  the  object  and  its  image,  bring  the  two  into  a  hor 
position,  or  on  a  level  with  each  other,  thus,  *  *.  Then  applj  the  key  to  the  square- 
head in  the  side  of  tlie  box;  and  by  turning  it  slightly,  bring  the  two  to  coincide 
perfectly.     The  instrument  is  then  adjusted. 

In  some  instruments,  the  hor  glass  has  a  hinge  at  v,  to  allow  it  play  while  being 
adjusted  by  the  single  screw  ST;  but  others  dispense  with  this  hinge,  and  use  ttoo 
pcrews  like  S  on  top  of  the  box,  in  addition  to  the  one  in  the  side. 

If  a  sextant  is  used  for  measuring  vert  angles  by  means  of  an  artificial 
horizon,  the  actual  altitude  will  be  but  one-half  of  that  read  off  on  the 
limb ;  because  we  then  read  at  once  both  the  actual  and  the  reflected  angle.  The 
great  objection  to  the  sextant  for  engineering  purposes,  is  that  it  does  not  measure 
angles  horizontally,  as  the  transit  does;  unless  when  the  observer,  and  the  two  ob- 
jects happen  to  be  in  the  same  hor  plane. 
Thus  an  observer  with  a  sextant  at  A,  if 
measuring  the  angle  subtended  by  the 
mountain-peaks  B  and  C,  must  hold  the 
graduated  plane  of  the  sextant  in  the 
plane  of  A  B  C ;  and  must  actually  meas- 
ure the  angle  BAG;  whereas  what  he 
wants  is  the  hor  angle  n Am.  This  is 
greater  than  BAG,  because  the  dists  A n 
and  A  m  are  shorter  than  A  B  and  A  C. 
The  transit  gives  the  hor  angle  nAm,  be- 
<;ause  its  graduated  plane  is  first  fixed  hor  by  the  le veiling-screws :  and  the  subse- 
quent measurement  of  the  angle  is  not  affected  by  his  directing  merely  the  line  of 
sight  upward,  to  any  extent,  in  order  to  fix  it  upon  B  and  G.  For  more  on  this  sub- 
ject ;  and  for  a  method  of  partially  obviating  this  objection  to  the  sextant,  see  the 
note  to  Example  2,  Gase  4,  of  "  Trigonometry." 

The  nautical  sextant,  used  on  ships,  is  constructed  on  the  same  principle 
as  the  box  sextant ;  and  its  adjustments  are  very  similar.  In  it,  also,  the  index- 
glass  is  permanently  fixed  by  the  maker;  and  the  horizon-glass  has  the  two  adjust- 
ments of  the  box  sextant.  It  also  has  its  dark  glasses  for  looking  at  the  sun ;  and 
*  small  sight-hole,  to  be  used  when  the  telescope  is  dispensed  with. 


THE  COMPASS. 


To  adjust  a  Compass. 

The  first  adjustment  is  that  of  the  bubbles.  Plant  firmly  ;  and  level  the 
Instrument,  in  any  position ;  that  is,  bring  the  bubbles  to  the  centers  of  their  tubes. 
Then  turn  the  instrument  half-way  round.  If  the  bubbles  then  remain  at  the  cen- 
ters, they  are  in  adjustment;  but  if  not,  correct  one-half  the  difif  in  each  bubble, 
by  means  of  the  adjusting-screws  of  the  tubes.  Level  the  instrumemt  again;  turn 
it  half  round ;  and  if  the  bubbles  still  do  not  remain  at  the  center,  the  adjusting- 
•crews  must  be  again  mored  a  little,  so  as  to  rectify  half  the  remaining  diflf.   Gener- 


THE   COMPASS.  299 

ally  several  trials  must  bo  thus  made,  until  the  bubbles  will  remain  at  the  cente 
while  the  compass  is  being  turned  entirely  around. 

Second  adjust/iiient.  Level  the  compass,  and  then  see  that  the  needle  ii 
hor ;  and  if  not,  make  it  so  by  means  of  the  small  piece  of  wire  which  is  wrapped 
around  it ;  sliding  the  wire  toward  the  high  end.  A  needle  thus  horizontally  ad- 
Justed  at  one  place,  will  not  remain  so  if  removed  far  north  or  south  from  that  place. 
If  carried  to  the  north,  tlie  north  end  will  dip  down ;  and  if  to  the  south,  the  south 
end  will  do  so.    The  sliding  wire  is  intended  to  counteract  this. 

Third  adjustment.  This  is  always  fixed  right  at  first  by  the  maker;  that 
is,  the  sights,  or  slits  for  sighting  through,  are  placed  at  right  angles  to  the  compass 
plate  ;  so  that  when  the  latter  is  levelled  by  the  bubbles,  the  sights 
are  vert.  To  test  whether  they  are  so,  hang  up  a  plumb-line  ;  and 
having  levelled  the  compass,  take  sight  at  the  line,  and  see  if  the 
slits  coincide  with  it.  If  one  or  both  slits  should  prove  to  be 
out  of  plumb,  as  shown  to  an  exaggerated  extent  in  this  sketch, 
it  should  be  unscrewed  from  the  compass,  and  a  portion  of  its  foot 

on  the  high  side  be  filed  or  ground  off,  as  per  the  dotted  line ;  or       ^^^_^ 

aa  a  temporary  expedient,  a  small  wedge  may  be  placed  under  the   ^  j 

low  side,  so  as  to  raise  it. 

Fourth  adjustment,  to  straighten  the  needle,  if  it  should  become  bent. 
The  compass  being  levelled,  and  the  needle  hor,  and  loose  on  its  pivot,  see  whether 
its  two  ends  continue  to  point  to  exactly  opposite  graduations,  (that  is,  graduations 
180°  apart;)  while  the  compass  is  turned  completely  around.  If  it  does,  the  needle 
is  straight ;  and  its  pin  is  in  the  center  of  the  graduated  circle  ;  but  if  it  does  not, 
then  one  or  both  of  these  require  adjusting.  First  level  the  compass.  Then  turn  it 
until  some  graduation  (say  90°)  comes  precisely  to  the  north  end  of  the  needle.  If 
the  south  end  dues  not  then  point  precisely  to  the  opposite  90°  division,  lilt  off  the 
needle,  and  bend  the  pivot-point  until  it  does;  remembering  that  every  time  said 
point  is  bent,  the  compass  must  be  turned  a  hairsbreadth  so  as  to  keep  the  north  end 
of  the  needle  at  its  90°  mark.  Then  turn  the  compass  half-way  round,  or  until  the 
opposite  90°  mark  comes  precisely  to  the  north  end  of  the  needle.  Make  a  fine  pen- 
cil mark  where  the  soiith  end  of  the  needle  now  points.  Then  take  off  the  needle, 
and  bend  it  until  its  south  end  points  half-way  between  its  90°  mark  and  the  pencil 
mark,  while  its  north  end  is  kept  at  90°  by  moving  the  compass  round  a  hairsbreadth. 
The  needle  will  then  be  straight,  and  must  not  be  altered  in  making  the  following, 
adjustment,  although  it  will  not  yet  cut  opposite  degrees. 

Fifth  adjustment,  of  the  pivot-pin.  After  being  certain  that  the  needle  is 
straight,  turn  the  compass  around  until  a  part  is  arrived  at  where  the  two  ends  of  the 
needle  happen  to  cut  opposite  degrees.  Then  turn  the  compass  quarter  way  around, 
or  through  90°.  If  the  needle  then  cuts  opposite  degrees,  the  pivot-point  is  already 
in  adjustment ;  but  if  the  needle  does  not  so  cut,  bend  the  pivot-point  until  it  does. 
Repeat,  if  necessary,  until  the  needle  cuts  opposite  degrees  while  being  turned  entirely 
around. 

Oftre  and  nicety  of  observation  are  necessary  in  making  these  adjustments  properly ; 
because  the  entire  error  to  be  rectified  is,  in  itself,  a  minute  quantity ;  and  the  novice 
is  very  apt  to  increase  his  trouble  by  not  knowing  how  to  use  his  mag'nifier, 
when  looking  at  the  end  of  the  needle  and  the  corresponding  graduations.  The  mag- 
nilier  must  always  be  held  with  its  center  directly  over  the  point  to  be  examined ;  antl 
it  must  be  held  parallel  to  the  graduated  circle.  Otherwise  annoying  errors  of 
several  minutes  wTLl  be  made  in  a  single  observation ;  and  the  accumulation  of  two 
or  three  such  errors,  arising  from  a  cause  unknown  to  him,  may  compel  him  to 
abandon  the  adjustments  in  despair.  This  suggestion  applies  also  to  the  reading  of 
angles  taken  by  the  transit,  &c  ;  although  the  errors  are  not  then  likely  to  be  so 
great  as  in  the  case  of  the  compass.  In  purchasing  a  magnifier  for  a  compass,  see 
that  no  part  of  it,  as  hinges,  or  rivets,  are  made  of  iron ;  for  such  would  change  the 
direction  of  the  needle. 

If  the  sight-slits  of  a  compass  are  not  fixed  by  the  maker  in  line  with  the  two 
opposite  zeros,  the  engineer  cannot  remedy  the  defect.  This  can  be  ascertained  by 
passing  a  piece  of  fine  thread  through  the  slits,  and  observing  whether  it  stands 
precisely  over  the  zeros. 


300 


THE    COMPASS. 


100  200  300  400  500  GOO       6  100  200  300  400  500  '   ^ 


THE  COMPASS. 


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302  CONTOUR  LINES. 

United  States,  by  Henry  Gannett,  in  17th  Annual  Report  of  U.  S.  Geological 
Survey,  1895-6. 

Electricity,  either  atmospheric,  or  excited  by  rubbing  the  glass  cover  of 
the  compass  box,  sometimes  gives  trouble.  It  may  be  removed  by  touching  the 
glass  with  the  moist  tongue  or  finger. 

I>£]»IAOXi:TIZATIO]!ir. 

The  needle,  if  of  soft  metal,  sometimes  loses  part  of  its  magnetism,  and  consequently 
does  not  work  well.  It  may  be  restored  by  simply  drawing  the  north  pole  of  a 
common  magnet  (either  straight  or  horseshoe)  about  a  dozen  times,  from  the  center 
to  the  end  of  the  south  half  of  tlie  needle ;  and  the  south  pole,  in  the  same  way,  along 
the  north  half;  pressing  the  magnet  gently  upon  the  needle.  After  each  stroke, 
remove  the  magnet  several  inches  from  the  needle,  while  bringing  it  back  to  the 
center  for  making  another  stroke.  Each  half  of  the  needle  in  turn,  while  being  thus 
operated  on,  should  be  held  flat  upon  a  smooth  hard  surface.  Sluggish  action  of  the 
needle  is,  however,  more  generally  produced  by  the  dulling  or  other  injury  of  the 
point  of  the  pivot.  Remagnetizing  will  throw  the  needle  out  of  balance ;  which  must 
be  counteracted  by  the  sliding  wire. 

In  order  to  prevent  mistakes  by  reading*  sometimes  from  one  end, 
B,nd  sometimes  from  the  other  end  of  the  needle,  it  is  best  to  always  point  the  N  of 
the  compass- box  toward  the  object  whose  bearing  is  to  be  taken ;  and  to  read  oflf 
from  the  noith  end  of  the  needle.    This  is  also  more  accurate. 


CONTOUE  LINES. 


A  CONTOUR  LINE  is  a  curved  hor  one,  every  point  in  which  represents  the  same  level; 
thus  each  of  the  contour  lines  88c,  9ic,  94c,  «S;c,  Fig  1,  indicates  that  every  point  in 
the  ground  through  which  it  is  traced  is  at  the  same  level ;  and  that  that  level  or 
height  is  everywhere  88,  91 ,  or  94  ft  above  a  certain  other  level  or  height  called 
datum:  to  which  all  others  are  referred. 

Frequently  the  level  of  the  starting  point  of  a  survey  is  taken  as  being  0,  or  zero, 
or  datum ;  and  if  we  are  sure  of  meeting  with  no  points  lower  than  it,  this  answers 
every  purpose.  But  if  there  is  a  probability  of  many  lower  points,  it  is  better  to 
assume  the  starting  point  to  be  so  far  above  a  certain  supposed  datum,  that  none  of 
these  lower  points  shall  become  minus  quantities,  or  below  said  supposed  datum  or 
zero.  The  only  object  in  this  is  to  avoid  the  liability  to  error  which  arises  when 
some  of  the  levels  are  +,  or  plus  ;  and  some  — ,  or  minus.  Hence  we  may  assume 
the  level  of  the  starting  point  to  be  10,  100,  1000,  &c,  ft  above  datum,  according  to 
circumstances. 

The  vert  dists  between  each  two  contour  lines  are  supposed  to  be  equal ;  and  in 
railroad  surveys  through  well-known  districts,  where  the  engineer  knows  that  his 
actual  line  of  survey  will  not  require  to  be  much  changed,  the  dist  may  be  1  or  2  ft 
only  ;  and  the  lines  need  not  be  laid  down  for  widths  greater  than  100  or  200  ft  on 
each  side  of  his  center-stakes.  But  in  regions  of  which  the  topography  is  compara- 
tively unknown ;  and  where  consequently  unexpected  obstacles  may  occur  which 
require  the  line  to  be  materially  changed  for  a  considerable  dist  back,  the  observa- 
tions should  extend  to  greater  widths ;  and  for  expedition  the  vertical  dists  apart 
may  be  increased  to  3,  5,  or  even  10  ft,  depending  on  the  character  of  the  country, 
&c.  Also,  when  a  survey  is  made  for  a  topographical  map  of  a  State,  or  of  a  county, 
vert  dists  of  5  or  10  ft  will  generally  suffice. 

Let  the  line  A  B,  Fig  1,  starting  from  O,  represent  three  stations  (S  1,  S  2,  S  3,)  of 
the  center  line  of  a  railroad  survey ;  and  let  the  numbers  100,  103,  101,  104,  along 
that  line  denote  the  heights  at  the  stakes  above  datum,  as  determined  by  levelling. 
Then  the  use  of  the  contour  lines  is  to  show  in  the  office  what  would  be  the  effect 
of  changing  the  surveyed  center  line  A  B,  by  moving  any  part  of  it  to  the  right  oi 


CONTOUR   LINES. 


303 


left  hand.*  Thus,  if  it  should  be  moved  100  ft  to  the  left,  the  starting  point  0  would 
be  on  ground  about  6  ft  higher  than  at  present ;  inasmuch  as  its  level  would  then 
be  about  106  ft  above  datum,  instead  of  100.  Station  1  would  be  about  7  ft  higher, 
or  110  ft  instead  of  103.  Station  2  would  be  about  7  ft  higher,  or  108  ft  instead  oi 
101.     If  the  line  b?i  thrown  to  the  right,  it  will  plainly  be  on  lower  ground. 

The  field  observations  for  contour  lines  are  sometimes  made  with  the  spirit-level; 
but  more  frequently  oy  a  slope-man,  with  a  straight  12-ft  graduated  rod,  and  a  slope 
Instrument,  or  clinometer.    At  each  station  he  lays  his  rod  upon  the  ground,  as 


Fig.  1. 

nearly  Sii  right  angles  to  the  center  line  A  B  as  he  can  judge  by  eye;  and  placing 
the  slope  instrument  upon  it,  he  takes  the  angle  of  the  slope  of  the  ground  to  the 
nearest  %ofsi  degree.  He  also  observes  how  far  beyond  the  rod  the  slope  continues 
the  same ;  and  with  the  rod  he  measures  the  dist.  Then  laying  down  the  rod  at  that 
point  also,  he  takes  the  next  slope,  and  measures  its  length ;  and  so  on  as  far  as  may 
be  judged  necessary.  His  notes  are  entered  in  his  field-book  as  shown  in  Fig  2 ;  the 
angles  of  the  slopes  being  written  above  the  lines,  and  their  lengths  below ;  and 
should  be  accompanied  by  such  remarks  as  the  locality  suggests ;  such  as  woods, 
rocks,>  mari»h.  sand,  field,  garden,  across  small  run,  j&c,  &c. 

*  In  thus  using  the  words  right  and  left  we  are  supposed  to  have  our  backs  turned  to  the  starting 

point  of  the  survey.  In  a  river,  the  ri^bt  bank  or  shore  is  that  which 
IS  on  the  right  hand  as  we  descend  it,  that  is,  in  speaking  of  its  right  or  lefl 

bank,  we  are  supposed  to  have  our  backs  turned  towards  its  head,  or  origin ;  and  so  with  a  survey 


304 


CONTOUR    LINES. 


It  is  not  absolutely  necessary  to  represent  the  slopes  roughly  in  the  field-book,  ai 


in  Fig  2 ;  for  by  using  the  sign  +  to  signify 
the  slopes  may  be  writ- 
ten in  a  straight  line, 
as  in  Fig  2i^. 

The  notes  having  been 
taken,  the  preparation 
of  the  contour  lines  by 
means  of  them,  is  of 
course  office-work ;  and 
is  usually  done  at  the 
*!ame  time  as  the  draw- 
ing of  the  map,  &c.  The 
field  observations  at  each 
station  are  then  sepa- 
rately drawn  by  protrac- 
tor and  scale,  as  shown 
in  Fig  3  for  the  starting 


up;"  — "down;"  and  cz  "level," 


point  0.  The  scale  should  not  be  less  than  about  j^  inch  to  a  ft,  if  anything  lik« 
accuracy  is  aimed  at.  Suppose  that  at  said  station  the  slopes  to  the  right,  taken  iu 
their  order,  are,  as  in  Fig  2,  15°,  4°.  and  26° ;  aiid  those  to  the  left,  20°,  10°,  and  16° ; 
and  their  lengths  as  in  the  same  Fig.  Draw  a  hor  line  ho,  Fig  3;  and  consider  the 
center  of  it  to  be  the  station-stake.  From  this  point  as  a  center,  lay  off  these  angles 
with  a  protractor,  as  shown  on  the  arcs  in  Fig  3.  Then  beginning  say  on  the  right 
hand,  with  a  parallel  ruler  draw  the  first  dist  a  c,  at  its  proper  slope  of  15° ;  and  of 
its  proper  length,  45  ft,  by  scale.  Then  the  same  with  c  y  and  y  t.  Do  the  same  with 
those  on  the  left  hand.  We  then  have  a  cross-section  of  the  ground  at  Sta  0.  Then 
on  the  map,  as  in  Fig  1,  draw  a  line  as  in  n,  or  h  w,  at  right  angles  to  the  line  of  road, 
and  passing  through  ths  station-stake.  On  this  line  lay  down  the  hor  dists  ad,ds,sVf 
ae,eg,  g  k,  marking  them  with  a  small  star,  as  is  done  and  lettered  in  Fig  1,  at  Sta  0. 

When  extreme  accuracy  is  pretended  to,  these  hor  dists  must  be  found  by  measure 
on  Fig  3;  but  as  a  general  rule  it  will  be  near  enough,  when  the  slopes  do  not  ex- 
ceed 10°  to  assume  them  to  be  the  same  as  the  sloping  dists  measured  in  the  field. 
Next  ascertaiQ  how  high  each  of  the  points  cytlniis  above  datum.  Thus,  measure 
by  scale  the  vert  dist  d  c.  Suppose  it  is  found  to  be  5  ft ;  or  in  other  words,  that  c 
is  5  ft  below  station-stake  0.  Then  since  the  level  at  stake  0  is  100  ft  above  datum, 
that  at  c  must  be  5  ft  less,  or  100  —  5  =  95  ft  above  datum ;  which  may  be  marked  in . 
light  lead-pencil  figures  on  the  map,  as  at  d,  Fig  1.  Next  for  the  point  y,  suppose 
we  find  s  2/  to  be  11  ft,  or  y  to  be  11  ft  below  stake  0 ;  then  its  height  above  datum 
must  be  100  — 11  =  89;  Avliich  also  write  in  pencil,  as  at  s.  Proceed  in  the  same 
way  with  t.  Next  going  to  the  left  hand  of  the  station-stake,  we  find  eZ  to  be  say 
2  ft;  but  I  is  above  the  level  of  the  station-stake,  therefore  its  height  abov»  datum  it 


h-^ 


Fig.  8. 


100  -f-  2  ==  102  ft,  as  figured  at  e  on  the  map.  Let  ng  be  5  ft;  then  is  n,  100  -f-  5  =. 
105  ft  above  datum,  as  marked  at  g ;  and  so  on  at  each  station.  When  this  has  been 
done  at  several  stations,  we  may  draw  iu  the  contour  lines  of  that  portion  by  hand 
thus :  Suppose  they  are  to  represent  vert  heights  of  3  ft.  Beginning  at  Station  O 
(of  which  the  height  above  datum  is  100  ft)  to  lay  dow^n  a  contour  line  103  ft  above 
datum,  we  see  at  once  that  the  height  of  103  ft  must  be  at  t,  or  at  3/^  the  dist  from  e 
to  g.  Make  a  light  lead-pencil  dot  at  t;  and  then  go  to  the  next  Station  1.  Here 
we  see  that  the  heiglit  of  103  ft  coincides  with  the  station-stake  itself;  place  a  dot 
there,  and  go  to  Sta  2.    The  level  at  this  stake  is  101 ;  therefore  the  contour  for  103 


CONTOUR  LIKES. 


305 


ft  must  evidently  be  2  ft  higher,  or  at  i,.%  of  the  dist  from  Sta  2  to  +104;  thereior* 
make  a  dot  at  i.  Then  go  to  Sta  3.  Here  the  level  being  lOi  above  datum,  the  con- 
tour of  103  must  be  at  y,  or  -5-  of  the  dist  from  Sta  3  to  +99 ;  put  a  dot  at  y.  Finally 
draw  by  hand  a  curving  line  through  t,  SI,  i,  and  y  5  and  the  contour  line  of  103  ft 
ig  done.  All  the  others  are  prepared  in  the  same  way,  one  by  one.  The  level  of  each 
must  be  figured  upon  it  at  short  intervals  along  the  map,  as  at  103  c,  106  c,  &c. 

Or,  instead  of  first  placing  the  +  points  on  the  map,  to  denote  the  slope  dists  actu- 
ally measured  upon  the  ground,  we  may  at  once,  and  with  less  trouble,  find  and  show 
those  only  which  represent  the  points  t,  S 1,  t,  y,  &c,  of  the  contours  themselves. 
Thus,  say  that  at  any  given  station-stake.  Fig  4,  the  level  is  104 ;  that  the  cross-sec- 
tion  c  s  of  the  ground  has  been  prepared  as  before  ;  and  that  we  want  the  hor  diiii 
from  the  stake,  to  contour  lines  for  94,  97, 100  ft,  &c,  3  ft  apart  vert. 


Draw  a  vert  line  v  I,  through  the  station-stake,  and  on  it  by  scale  mark  levels 
of  94,  97,  100,  &c.  ft.  This  is  readily  done,  inasmuch  as  we  have  the  level  104  of 
the  stake  already  given.  Through  these  levels  draw  the  hor  lines  «,  ft,  m,  n,  Ac. 
to  the  ground-slopes.  Then  these  lines,  measured  by  the  scale,  plainly  give  the 
required  dists. 

When  the  ground  is  very  irregular  transversely,  the  cross-sections  must  be 
taken  in  the  field  nearer  together  than  100  ft.  The  preparation  of  contour  lines 
will  be  greatly  facilitated  by  the  use  of  paper  ruled  into  small  squares  of  not  less 
than  about  ^V  inch  to  a  side,  for  drawing  the  cross-sections  upon. 

When  the  ground  is  very  steep,  it  is  usual  to  shade  such  portions  of  the  map  to 
represent  hill-side.  The  closer  together  the  contours  come,  the  steeper  of  course 
is  the  ground  between  thera  ;  and  the  shading  should  be  proportionally  darker 
at  such  portions.     But  for  ivorking  maps  it  is  best  to  omit  the  shading. 

In  surveys  of  wide  districts,  the  transit  instrument  with  a  graduated  vertical 
circle  or  arc,  g,  p.  291,  is  used  for  measuring  the  angles  of  slope,  instead  of. 
the  common  slope-instrument. 

In  many  cases,  notes  similar  to  the  following  will  serve  the  purpose  of  contour 
lines  on  railroad  surveys. 

Sta  60 —  3.  1  R.    +  2.  1  L. 

61 +  2.  2  R.    —1.3  L. 

62 =  1.  R.     +  4.  2  L. 


63.. 


"Which  means  that  at  station  60,  the  slope  of  the  ground  on  the  rieht,  as  nearly  as  he  can  judge  by 
eye,  or  by  his  hand-level,  is  about  3  ft  downward,  for  I  chain,  or  100  ft ;  and  on  the  left,  about  2  ft 
upward  in  1  chain.  A/  61 ,  2  ft  up,  in  2  chains  to  the  right;  and  1  ft  down  in  3  chains  to  the  left. 
At  62,  level  for  1  chain  to  the  right;  and  ascending  4  ft  in  2  chains  to  the  left.  At  68,  the  same  as  at 
62.  At  some  spots  it  will  be  well  to  add  a  sketch  of  a  cross-section,  like  Fig  2;  only,  instead  of  the 
angles,  use  ft  of  rise  or  fall,  to  indicate  the  slopes,  as  judged  by  eye,  or  by  a  hand-level.  By  this 
method,  the  result  at  every  stntion  will  be  somewhat  in  error;  but  these  small  errors  will  balance 
each  other  so  nearly  that  the  total  may  be  regarded  as  sufficiently  correct  for  all  the  purposes  of  a 
preliminary  estimate  of  the  cost  of  a  road.  When  the  final  stakes  for  guiding  the  workmen  are 
placed,  the  slopes  should  be  carefully  taken,  \n  order  to  calculate  the  quauiity  of  excavation  accu- 
rately for  payment. 

20 


306 


TH£  LJBV£IL. 


THE  LEVEL. 


Although  the  levels  of  different  makers  vary  somewhat  in  their  details,  still  theil 
principal  parts  will  be  understood  from  the  following  figure.  The  telescope  T  T 
rests  upon  two  supports  YY,  called  Ys ;  out  of  which  it  can  be  lifted,  first  removing 
the  pins  s  s  which  confine  the  semicircular  clips  e  e,  and  then  opening  the  clips. 
The  pins  should  be  tied  to  the  Ys,  by  pieces  of  string,  to  prevent  their  being  lost. 
The  slide  of  the  object-glass  O,  is  moved  backward  or  forward  by  a  rack  and  pinion, 
by  means  of  the  milled  head  A.  The  slide  of  the  eye-glass  E,  is  moved  in  the  same 
way  by  the  milled  head  e.  A  cylindrical  tube  of  brass,  called  a  shade,  is  usually 
furnished  with  each  level.  It  is  intended  to  be  slid  on  to  the  object-end  O  of  the 
telescope,  to  prevent  the  glare  of  the  sun  upon  the  object-glass,  when  the  sun  is 
low.  At  B  is  an  outer  ring  encircling  the  telescope,  and  carrying  4  small  capstan- 
headed  screws ;  two  of  which,  »j9,  are  at  top  and  bottom;  while  the  other  two, 
of  which  i  is  one,  are  at  the  sides,  and  at  right  angles  top  p.  Inside  of  this  outer 
ring  is  another,  inside  of  the  telescope,  and  which  has  stretched  across  it  two 
spider-webs,  usually  called  the  cross-hairs.  These  are  much  finer  than  they  ap- 
pear to  be,  being  considerably  magnified.  They  are  at  right  angles  to  each  other ; 
and,  in  levelling,  one  is  kept  vert,  and  the  other  hor.    They  are  liable  at  times  to  b« 


thrown  out  of  this  position  by  a  partial  revolution  of  the  telescope,  when  carrying 
the  level,  or  when  setting  the  tripod  down  suddenly  upon  the  ground ;  but  since,  in 
levelling,  the  intersection  of  the  hairs  is  directed  to  the  target-rod,  this  derangement 
does  not  affect  the  accuracy  of  the  work.  Still  it  is  well  to  keep  them  nearly  vert 
and  hor,  by  keeping  the  bubble-tube  D  D  as  nearly  directly  over  the  bar  V  F  as  can 
be  judged  by  eye.  This  euables  the  leveller  to  see  that  the  rod-man  holds  his  rod 
nearly  vert,  which  is  absolutely  essential  for  correct  levelling.  If  perfect  verticality 
is  desired,  as  is  sometimes  the  case,  when  staking  out  work,  it  may  be  obtained  (if 
the  instrument  is  in  perfect  adjustment,  and  levelled)  by  sighting  at  a  plumb-line,  or 
other  vert  object,  and  then  turning  the  telescope  a  little  in  its  Ys,so  as  to  bring  the 
hair  to  correspond.  When  this  is  done,  a  short  continuous  scratch  may  be  made  on 
the  telescope  and  Y,  to  save  that  troncle  in  future.  Heller  &  Brightly,  however, 
provide  their  levels  with  a  small  projection  inside  of  the  Ys,  and  a  corresponding 
stop  on  the  telescope,  the  contact  of  which  insures  the  verticality  of  the  hair. 
Should  the  hairs  be  broken  by  accident,  they  may  be  replaced  as  directed  here- 
after. 

The  small  holes  around  the  heads  of  the  4  small  capstan-screwsp,  i,  just  referred  to, 
are  for  admitting  the  end  of  a  small  steel  pin.  or  lerer,  for  turning  them.  If  first 
the  upper  screw  p  be  loosened,  and  then  the  lower  one  tightened,  the  interior  ring 
will  be  lowered,  and  the  horizontal  hair  with  it.    But  on  looking  through  the  tele- 


THE  LEVEL.  307 

scope  they  will  appear  to  be  raised.  If  first  the  lower  one  be  loosened,  and  the  upper 
one  tightened,  the  hor  hair  will  be  actually  raised,  but  apparently  lowered.  This  is 
because  the  glasses  in  the  eye-piece  E  reverse  the  apparent  position  of  objects  inside 
cf  the  telescope ;  which  effect  is  obviated,  as  regards  exterior  objects,  by  means  of 
the  object-glass  0.  This  must  be  remembered  when  adjusting  the  cross-hairs  ;  for  if  a 
hair  appears  to  strike  too  high,  it  must  be  raised  still  higher;  if  it  appears  to  be 
already  too  far  to  the  right  or  left,  it  must  be  actually  moved  still  more  in  the  same 
direction. 

This  remark,  however,  does  not  apply  to  telescopes  which  make  objects  appear 
inverted. 

There  is  no  danger  of  injuring  the  hairs  by  these  motions,  inasmuch  as  the  four 
screws  act  against  the  ring  only,  and  do  not  come  in  contact  with  the  hairs  them- 
selves. 

Under  the  telescope  is  the  bubble-tube  D  D.  One  end  of  this  tube  can  be  raised  or 
lowered  slightly  by  means  of  the  two  capstan-headed  nuts  n  n,  one  of  which  must 
be  loosened  before  the  other  is  tightened.  On  top  of  the  bubble-tube  are  scratchei 
for  showing  when  the  bubble  is  central  in  the  tube.  Frequently  these  scratches,  or 
marks,  are  made  on  a  strip  of  brass  placed  above  the  tube,  as  in  our  fig.  There  are 
several  of  them,  to  allow  for  the  lengthening  or  shortening  of  the  bubble  by  changes 
of  temperatuie.  At  the  other  end  of  the  bubble-tube  are  two  small  capstan-screws, 
placed  on  opposite  sides  horizontally.  The  circular  head  of  one  of  them  is  shown 
near  t.  By  means  of  these  two  screws,  that  end  of  the  tube  can  be  slightly  moved 
hor,  or  to  right  or  left.  Under  the  bubble-tube  is  the  bar  V  ¥ ;  at  one  end  of  which, 
as  at  V,are  two  large  capstan-nuts  w  w,  which  operate  upon  a  stout  interior  screw 
which  forms  a  prolongation  of  the  Y.  The  holes  in  these  nuts  are  larger  than  the 
others,  as  they  require  a  larger  lever  for  turning  them.  If  the  lower  nut  is  loosened 
and  the  upper  one  tightened,  the  Y  above  is  raised ;  and  that  end  of  the  telescope 
becomes  farther  removed  from  the  bar;  and  vice  versa.  Some  makers  place  a  similar 
screw  and  nuts  under  both  Ys  ;  while  others  dispense  with  the  nuts  entirely,  and 
substitute  beneath  one  end  of  the  bar  a  large  circular  milled  head,  to  be  turned  by 
the  fingers.  This,  however,  is  exposed  to  accidental  alteration,  which  should  be 
Avoided. 

When  the  portions  above  m  are  put  upon  m,  and  fastened  by  the  screw  Y,  all 
the  upper  part  may  be  swung  round  hor,  in  either  direction,  Dy  loosening  the 
clamp-screi¥  H ;  or  such  motion  may  be  prevented  by  tightening  thatserew. 
It  frequently  happens,  after  the  telescope  has  been  sighted  very  nearly  upon  an 
object,  and  then  clamped  by  H,  that  we  wish  to  bring  the  cross-hairs  to  coincide 
more  precisely  with  the  object  than  we  can  readily  do  by  turning  the  telescope  bp 
hand;  and  in  this  case  we  use  the  tan^eiit-stcrenr  b,  by  means  of  which  a 
Blight  but  steady  motion  may  be  given  after  the  instrument  is  clamped.  For 
fuller  remarks  on  the  clamp  and  tangent-screws,  see  "Transit." 

The  parallel  plates  m  and  S  are  operated  by  four  levellin^-screws; 
three  of  which  are  seen  in  the  figure,  at  K  K.  The  screws  work  in  sockets  R ; 
■which,  as  well  as  the  screws,  extend  above  the  upper  plate.  When  the  instrument 
is  placed  on  the  ground  for  levelling,  it  is  well  to  set  it  so  that  the  lower  parallel 
plate  S  shall  be  as  nearly  horizontal  as  can  be  roughly  judged  by  eye ;  in  order 
to  avoid  much  turning  of  the  levelling  screws  K  K  in  making  the  upper  plate 
m  hor.  The  lower  plate  S,  and  the  brass  parts  below  it,  are  together  called  the 
tripod-bead  ;  and,  in  connection  with  three  wooden  legs  Q  Q  Q,  constitute 
the  tripod.  In  the  figure  are  seen  the  heads  of  wing-nuts  J  which  confine  the 
legs  to  the  tripod-head.  Under  the  center  of  the  tripod-head  should  always  be 
placed  a  small  ring,  from  which  a  plumb-bob  may  be  suspended.  This  is  not 
needed  in  ordinary  levelling,  but  becomes  useful  when  ranging  center-stakes,  &c. 

To  adjust  a  liCvel. 

This  is  a  quite  simple  operation,  but  requires  a  little  patience.  Be  careful  to  avoid 
ttraining  any  of  the  screws.  The  large  Y  nuts  ww  sometimes  require  some  force  to 
start  them ;  but  it  should  be  applied  by  pressure,  and  not  by  blows.  Before  begin- 
ning to  adjust,  attend  to  the  object-glass,  as  directed  in  the  first  sentence  under  "To 
adjust  a  plain  transit." 

Three  adjustments  are  necessary;  and  must  he  made  in  the  following  order: 

First,  that  of  tlie  cross-hairs ;  to  secure  that  their  intersection  shall 
continue  to  strike  the  same  point  of  a  distant  object,  while  the  telescope  is  being 
turned  round  a  complete  revolution  in  its  Ys.  This  is  called  adjusting  the  line 
of  collimatton,  or  sometimes,  the  line  of  sight;  but  it  is  not  strictly  the  line 
of  sight  until  all  the  adjustments  are  finished;  for  until  then,  the  line  of  collimation 
will  not  serve  for  taking  levelling  sights.      If  cross-hairs  break,  see  t»  296. 

Second,  that  of  the  bubble-tube  D  D,  to  place  it  parallel  to  th^  line 


308  THE  LEVEL. 

of  eollimation.  previously  adjusted ;  so  that  when  the  bubble  stands  at  the  centre  of 
Its  tube,  indicating  that  it  is  level,  we  know  that  our  sight  through  the  telescope  ii 
hor.    To  replace  broken  bubC>le  tube,  see  p  296. 

Tbird,  that  ©f  the  Ys,  by  which  the  telescope  and  bubble-tube  are  supported; 

so  that  the  bubble-tube,  and  line  of  sight,  shall  be  perp  to  the  vert  axis  of  the  instru- 
ment; so  fis  to  remain  hor  while  the  telescope  is  pointed  to  objects  in  difif  directions, 
as  when  taking  baiCk  and  fore  sights. 

To  make  tlie  first  adjustment,  or  that  of  the  cross-hairs,  plant  the 
ivi^od.  firmly  upon  the  ground.  In  this  adjustment  it  is  not  necessary  to  level  the 
instrument.  Open  the  clips  of  the  Ys ;  unclamp ;  draw  out  the  eye-glass  E,  until 
the  cro88-ha;irs  are  seen  perfectly  clear ;  sight  the  telescope  toward  some  clear  dis- 
tant point  of  an  object ;  or  still  better,  toward  some  straight  line,  whether  vert  or 
Dot.  Move  the  object-glass  0,  by  means  of  the  milled  head  A,  so  that  the  object  shall 
be  clearly  seen,  i¥ithout  parallax,  that  is,  without  any  apparent  dancing 
about  of  the  cross-hairs,  if  the  eye  is  moved  a  little  up  or  down  or  sideways.  To 
secure  this,  the  object-glass  alone  is  moved  to  suit  different  distances ;  the  eye-giass 
is  not  to  be  changed  after  it  is  once  properly  fixed  upon  the  cross-hairs.  The  neglect 
of  parallax  is  a  source  of  frequent  errors  in  levelling.  Clamp ;  and,  by  means  of  the 
tangent-screw  b,  bring  either  one  of  the  cross-hairs  to  coincide  precisely  with  the 
object.  •  Then  gently,  and  without  jarring,  revolve  the  telescope  half-way  round  in 
its  Ys.  When  this  is  done,  if  the  hair  still  coincides  precisely  with  the  object,  it  is 
in  adjustment ;  and  we  proceed  to  try  the  other  hair.  But  if  it  does  not  coincide, 
then  by  means  of  the  4  screws  jo,  i,  move  the  ring  which  carries  the  hairs,  so  as  to 
rectify,  as  nearly  as  can  be  judged  by  eye,  only  one-half  of  the  error;  remembering 
that  the  ring  must  be  moved  in  the  direction  opposite  to  what  appears  to  be  the 
right  one ;  unless  the  telescope  is  an  inverting  one.  Then  turn  the  telescope  back 
again  to  its  former  position ;  and  again  by  the  tangent-screw  bring  the  cross-hair  to 
coincide  with  the  object.  Then  again  turn  the  telescope  half-way  round  as  before. 
The  hair  Mill  now  be  found  to  be  more  nearly  in  its  right  place,  but,  in  all  probabil- 
ity, not  precisely  so ;  inasmuch  as  it  is  diliicult  to  estimate  one-half  the  error  accu- 
rately by  eye.  Therefore  a  little  more  alteration  of  the  ring  must  be  made ;  and  it 
may  be  necessary  to  repeat  the  operation  several  times,  before  the  adjustment  is 
perfect.  Afterward  treat  the  other  hair  in  precisely  the  same  manner.  When  both 
are  adjusted,  their  intersection  will  strike  the  same  precise  spot  while  the  telescope 
is  being  turned  entirely  round  in  its  Ys.  This  must  be  tried  before  the  adjustment 
can  be  pronounced  perfect;  because  at  times  the  adjustment  of  the  second  hair, 
slightly  deranges  that  of  the  ;first  one ;  especially  if  both  were  much  out  in  the  be- 
ginning. 

To  make  the  second  adjustment,  or  to  place  the  bubble-tube  parallel 
to  the  line  of  eollimation.  This  consists  of  two  dis- 
tinct adjustments,  one  vert,  and  one  hor.  The  first 
of  these  is  effected  by  means  of  the  two  nuts  n  n  on 
the  vert  screw  at  one  end  of  the  tube  ;  and  the  second 
by  the  two  hor  screws  at  the  other  end,  t,  of  the  tube. 
Looking  at  the  bubble-tube  endwise,  from  t  in  the 
foregoing  Fig,  its  two  hor  adjusting-screws  1 1  are 
seen  as  in  this  sketch.  The  larger  capstan-headed 
nut  below,  has  nothing  to  do  with  the  adjustments; 
it  merely  holds  the  end  of  the  tube  in  its  place. 

To  make  the  vert  adjustment  of  the  bubble-tiibe,  by  means  of  the  two  nuts  nn.  Place 
the  telescope  over  a  diagonal  pair  of  the  levelling-screws  K  K ;  and  clamp  it  there. 
Open  the  clips  of  the  Ys ;  and  by  means  of  the  levelling-screws  bring  the  bubble  to 
the  center  of  its  tube.  Lift  the  telescope  gently  out  of  the  Ys,  turn  it  end  for  end,  and 
put  it  back  again  in  its  reversed  position.  This  being  done,  if  the  bubble  still  remains 
at  the  center  of  its  tube,  this  adjustment  is  in  order  ;  but  if  it  moves  toward  one  end, 
that  end  is  too  high,  and  must  be  lowered;  or  else  the  other  end  must  be  raised. 
First,  correct  half  the  error  by  means  of  the  levelling-screws  K  K,  and  then  the  re- 
maining half  by  means  of  the  tM'o  small  capstan-headed  nuts  n  n.  To  raise  the  end 
w,  first  loosen  the  upper  nut  and  then  tighten  the  lower  one ;  to  do  which,  turn  each 
nut  so  that  the  near  side  moves  toward  your  right.  To  lower  it,  first  loosen  the  lowet 
nut,  then  tighten  the  upper  one,  moving  the  near  side  of  each  nut  toward  your  left 
Having  thus  brought  the  bubble  to  the  middle  again,  again  lift  the  telescope  out  of 
its  Ys ;  turn  it  end  for  end,  and  replace  it.  The  bubble  will  now  settle  nearer  th« 
center  than  it  did  before,  but  will  probably  require  still  further  adjustment.  If  so, 
correct  half  the  remaining  error  by  the  levelling-screws,  and  half  by  the  nuts,  as  be- 
fore ;  and  so  continue  to  repeat  the  operation  until  the  bubble  remains  at  the  centei 
in  both  positions.    For  another  method,  see  "  To  adjust  the  long  bubble-tube,"  p  295. 

Horizontal  adjustment  of  bubble-tube ;  to  see  that  its  axis  is  in  the  same  plane 
with  that  of  the  telescope,  as  it  usually  is  in  new  instruments.    It  is  not  easily  de^ 


THE   LEVEL.  309 

ranged,  except  by  blows.  Have  the  bubble-tube,  as  nearly  as  may  be,  directly  under 
the  telescope,  or  over  the  center  of  the  bar  V  F.  Bring  the  telescope  over  two  of  the 
levelling-screws  K  K ;  clamp  it  there ;  center  the  bubble  with  said  screws ;'  turn  the 
telescope  in  its  Ys,  say  about  ^^  inch,  bringing  the  bubble-tube  out  from  over  the 
center  of  the  bar,  first  on  one  side,  then  on  the  other.  If  the  bubble  stays  centered 
while  so  swung  out,  this  adjustment  is  correct.  If  it  runs  toward  opposite  ends  of  its 
tube  when  swung  out  on  opposite  sides  of  the  center,  move  the  end  t  of  the  tube  by 
the  two  horizontal  screws  1 1  until  the  bubble  stays  centered  when  the  tube  is  swung 
out  on  either  side.  If  the  bubble  runs  toward  the  same  end  of  its  tube  on  both  sideSy 
the  tube  is  not  truly  cylindrical,  but  slightly  conical,*  so  that  if  the  telescope  is 
turned  in  its  Ys  the  bubble  will  leave  the  center,  even  when  the  horizontal  adjust- 
ment is  correct.  It  is  known  to  be  correct,  in  such  tubes,  if  the  bubble  runs  the  same 
distance  from  the  center  when  swung  out  the  same  distance  on  each  side. 

Having  made  the  horizontal  adjustment,  turn  the  telescope  back  in  its  Ys  until  the 
bubble-tube  is  over  the  bar.  Repeat  the  vertical  adjustment  (p  308),  which  may  have 
become  deranged  in  making  this  horizontal  one.  Persevere  until  both  adjustments 
are  found  to  be  correct  at  the  same  time. 

To  make  the  third  ad  justinent,  or  to  adjust  the  heights  of  the  Ys,  m 
fts  to  make  the  line  of  collimation  parallel  to  the  bar  V  F,  or  perp  to  the  vert  axis 
of  the  instrument.  The  other  adjustments  being  made,  fasten  down  the  clips  of  the 
Ys.  Make  the  instrument  nearly  level  by  means  of  all  four  of  the  levelling-screws 
K.  Place  the  telescope  over  two  of  the  levelling-screws  which  stand  diagonally; 
and  leave  it  there  undamped.  Then  bring  the  bubble  to  the  center  of  its  tube,  by 
the  two  levelling-screws.  Swing  the  upper  part  of  the  instrument  half-way  around, 
BO  that  the  telescope  shall  again  stand  over  the  same  two  screws;  but  end  for  end. 
This  done,  if  the  bubble  leaves  the  center,  bring  it  half-way  back  by  the  large  cap- 
stan nuts  w,  w;  and  the  other  half  by  the  two  levelling-screws.  Remember  that  to 
raise  the  Y,  and  the  end  of  the  bubble  over  w,  w,  the  lower  w  must  be  loosened ;  and 
the  upper  one  tightened  ;  and  vice  versa.  Now  place  the  telescope  over  the  other 
diagonal  pair  of  levelling-screws:  and  repeat  the  whole  operation  with  them.  Hav- 
ing completed  it,  again  try  with  the  first  pair;  and  so  keep  on  until  the  bubble  re- 
mains at  the  center  of  its  tube,  in  every  position  of  the  telescope. 

Correct  levelling  may  be  performed  even  if  all  the  foregoing  adjustments  are 
out  of  order ;  provided  each  fore-sight  be  taken  at  precisely  the  same  distance  from 
the  instrument  as  the  back-sight,  is.  But  a  good  leveller  will  keep  his  instrument  always 
in  adjustment;  and  will  test  the  adjustments  at  least  once  a  day  when  at  work.  As 
much,  however,  depends  upon  the  rodman,  or  target-man,  as  upon  the  leveller.  A  rod- 
man  who  is  careless  about  holding  the  rod  vert,  or  about  reading  the  sights  correctly, 
■hould  be  discharged  without  mercy. 

The  levelling-screws  in  many  instruments  become  very  hard  to  turn  if  dirty.  Clean 
with  water  and  a  tooth-brush.    Use  no  oil  on  field  instruments. 

Forms  for  level  note-books.  When  the  distance  is  short,  so  as  not  to 
require  two  sets  of  books,  the  following  is  perhaps  as  good  as  any. 

UlU'I^tl's.IsSitJ  ""f-  heve..|Grade.|  Cut.  |  Fill.  | 
But  on  public  works  generally  the  original  field-books  have  only  the  first  five  cols. 
After  the  grades  have  been  determined  by  means  of  the  profile  drawn  from  these, 
the  results  are  placed  in  another  book,  which  has  only  the  first  col  and  the  last  four. 
In  both  cases,  the  right-hand  page  is  reserved  for  memoranda.  The  writer  considers 
it  best,  both  with  the  level  and  with  the  transit,  to  consider  the  term  "  Station  "  to 
a^ply  to  the  whole  dist  between  two  consecutive  stakes ;  and  that  its  number  shall 
be  that  written  on  the  last  stake.  Thus,  with  the  transit.  Station  6  means  the  dist 
from  stake  5  to  stake  6 ;  that  it  has  a  bearing  or  course  of  so  and  so ;  and  its  length 
is  so  and  so.  And  with  the  level.  Station  6  also  means  the  dist  from  stake  5  to  stake 
t) ;  the  back-sight  for  that  dist  being  taken  at  stake  5,  and  the  fore-sight  on  stake 
6 ;  and  that  the  level,  grade,  cut,  or  fill  is  that  at  stake  6.  The  starting-point  of  the 
survey,  whether  a  stake,  or  any  thing  else,  we  call  and  mark  simply  0. 

*  This  defect  can  be  remedied  only  by  removing  the  tube  and  inserting  a  correctly- 
shaped  one,  and  this  is  best  done  by  an  instrument -maker ;  but  coiTect  work  can 
be  done  in^  spite  of  it,  thus:  Make  all  the  adjustments  as  nearly  correct  as  possible. 
Level  the  instrument.  By  turning  the  telescope  in  its  Ys,  make  the  vertical  hair 
coincide  with  a  plumb-line  or  other  vertical  line,  and  make  a  short  continuous  knife- 
scratch  on  the  collar  nearest  the  object-glass,  and  on  the  adjoining  Y.  Lift  the  tele- 
scope out  of  its  Ys,  turn  it  end  for  end,  replace  it  in  its  Ys ;  again  bring  the  upright 
hair  vertical,  and  make  on  the  other  Y  a  scratch  coinciding  with  that  on  the  collar. 
Then,  in  levelling  or  in  adjusting,  always  see  that  the  scratch  on  the  collar  coincides 
with  that  on  the  adjoining  Y  when  the  bubble-tube  is  under  the  telescope. 


310 


THE  HAND-LEVEL. 
THE    HAJ^I>-I.EVEI.. 

si 


This  very  useful  little  instrument,  as  arranj2;ed  by  Professor  Locke,  of  Cincinnati,  is 
but  about  five  or  six  inches  long.  Simply  holding  it  in  one  hand,  and  looking  through 
It  in  any  direction,  we  can  ascertain  at  once,  approximately,  what  objects  are  at  the 
same  level  with  the  eye.  E  is  the  eye  end :  and  0  the  object  end.  L  is  a  small 
level,  enclosed  in  a  kind  of  brass  boxing  t  g,  the  bottom  of  which  is  open,  with  a  cor- 
responding opening  under  it,  through  the  top  of  the  main  tube  E  0.  Immediately 
at  the  bottom  of  the  small  level  L,  is  a  cross-wire,  stretched  across  said  opening,  and 
carried  by  a  small  plate,  which,  for  adjusting  the  wire,  can  be  pushed  backward  a 
trifle  by  tightening  the  screw  t,  or  pushed  forward  by  a  small  spring  within  the  box- 
ing, near  g,  when  the  screw  t  is  loosened.  At  m  is  a  small  semicircular  mirror  a  a, 
silvered  on  the  back  m.  This  is  placed  at  an  angle  of  45°,  and  occupies  one-half  the 
width  of  the  tube  E  0.  Through  the  forementioned  openings,  the  images  of  the 
cross-wire  and  of  the  level-bubble  are  reflected  down  on  the  unsilvered  face  a  a  of 
the  mirror,  and  thence  to  the  eye,  as  shown  by  the  single  dotted  lines  c  and  w;  and 
when  the  instrument  is  adjusted,  and  held  level,  the  wire  will  appear  to  be  at  the 
center  of  the  bubble.  At  7e  is  one-half  of  a  plano-convex  lens,  at  the  inner  end  of  a 
short  tube  k  p,  which  may  be  moved  backward  or  forward  by  a  pin  n,  projecting 
through  a  short  slit  in  the  main  tube.  By  this  means  the  image  of  the  cross-wire  is 
rendered  distinct ;  and  the  half  lens  must  be  moved  until,  when  viewing  an  object, 
tbe  wire  shall  show  no  parallax;  but  appear  steady  against  the  object  when  the  ey* 
is  slightly  moved  up  or  down.  At  each  end  of  the  tube  E  0  is  a  circular  piece  of 
plain  glass  for  excluding  dust. 

To  adj  ust  tlie  band-level,  first  fix  two  precisely  level  marks,  say  from 
50  feet  to  100  yards  apart.  This  being  done,  rest  the  instrument  against  one  of  the 
level  marks,  and  take  sight  at  the  other.  If,  then,  the  wire  does  not  appear  to  be 
precisely  at  the  center  of  the  bubble,  move  it  slightly  backward  or  forwai  d,  as  the 
case  may  be,  by  the  screw  t,  until  it  does  so  appear. 
The  two  level  marks  may  be  fixed  by  means  of  the  ^,^ 

hand-level  itself,  even  if  it  is  entirely  out  of  adjust-   ^  rtrr-^ 

ment,  thus :    First,  by  the  pin  n  arrange  the  half  lens   "^  •'     JSZIZ____  TTl 

k,  so  as  to  show  the  wire  distinctly  and  without  paral-       0      ^  ~ /. 

lax.     Then  holding  the  level  steadily,  at  any  selected 
object,  as  a,  so  that  the  wire  appears  to  cut  the  center 

of  the  bubble,  see  where  it  cuts  any  other  convenient  object,  as  b.  Then  go  to  5, 
and  from  it,  in  like  manner,  sight  back  toward  a.  If  the  instrument  is  in  adjust- 
ment, the  wire  will  cut  a ;  but  if  not,  it  will  strike  either  above  it  or  below  it,  as  at  c. 
In  either  case,  make  a  mark  m,  half-way  between  c  and  a.  Then  b  and  m  will  be  the 
two  level  marks  required.  With  care,  these  adjustments,  when  once  made,  will 
remain  in  order  for  years.  The  instrument  generally  lias  a  small  ring  r,  for  hanging 
it  around  the  neck :  it  is  not  adapted  to  very  accurate  work,  but  admirably  so  for 
exploring  a  route.  The  height  of  a  bare  hill  can  be  found  by  begii^ning  at  the  foot, 
and  sighting  ahead  at  any  little  chance  object  which  the  cross-wire  may  strike,  as  a 
pebble,  twig,  &c;  then  going  forward,  stand  at  that  object,  and  fix  the  wire  on 
another  one  still  farther  on,  and  so  to  the  top.  At  each  observation  we  plainly  rise 
a  height  equal  to  that  of  the  eye,  say  5%  feet,  or  whatever  it  may  be.  Whether 
going  up  or  down  it,  if  the  hill  is  covered  with  grass,  bushes,  &c,  a  target  rod  must 
be  used  for  the  fore-sights ;  and  the  constant  height  of  the  eye  may  be  regarded  as 
the  back-sight  at  each  station.  An  attachment  may  be  made  for  screwing  the  level 
to  a  small  ball  and  socket  on  top  of  a  cane,  or  of  a  longer  stick,  for  occasional  u»e, 
when  rather  more  accuracy  is  desired. 


LEVELS. 


311 


To  adjust  a  builder's  plnmb- 
level,  t'b  d%  stand  it  upon  any  two  sup- 
ports m  and  n,  and  mark  where  the  plumb- 
line  cuts  at  o.  Then  reverse  it,  placing  the 
foot  t  upon  n,  and  d  upon  w,  and  mark  where 
the  line  now  cuts  at  c.  Half-way  between  o 
and  c  maKe  the  permanent  mark.  Wlienever 
the  line  cuts  this,  the  feet  t  and  d  are  on  a 
level. 


To  adjust  a  slope-instrament,  or  clinometer.  As  usually  made, 
the  bubble-tube  is  attached  to  the  movable  bar  by  a  screw  near  each  end,  and  the 
head  of  one  of  the  screws  conceals  a  small  slot  in  the  bar,  which  allows  a  slight  vert 
motion  to  the  screw  when  loose,  and  with  it  to  that  end  of  the  tube.  Therefore,  in 
order  to  adjust  the  bubble,  this  screw  is  first  loosened  a  little,  and  then  moved  up 
«r  down  a  trifle,  as  may  be  reqd.    It  is  then  tightened  again. 


312  liE YELLING    BY    THE    BAROMETER. 


liETEIililHrO  BY  THE  BAROMETER. 

1.  Many  circumstances  combine  to  render  the  results  of  this  kind  ofierellin?  un- 
reliable where  great  accuracy  is  required.  This  fact  was  most  conclusively  proved 
by  the  observations  made  by  Captain  T,  J.  Cram,  of  the  U.  S.  Coast  Survey.  See 
Report  of  U.  S.  C.  S.,  vol.  for  1854.  It  is  difficult  to  read  off  from  an  aneroid  (the 
kind  of  barom  generally  employed  for  engineering  purposes)  to  within  from  two  to 
five  or  six  ft,  depending  on  its  size.  The  moisture  or  dryness  of  the  air  affects  the 
results;  also  winds,  the  vicinity  of  mountains,  and  the  daily  atmospheric  tides, 
which  cause  incessant  and  irregular  fluctuations  in  the  barom.  A  barom  hanging 
quietly  in  a  room  will  often  vary  jq  of  an  inch  within  a  few  hours,  corresponding 
to  a  diff  of  elevation  of  nearly  100  ft.  No  formula  can  possibly  be  devised  that  shall 
embrace  these  sources  of  error.  The  variations  dependent  upon  temperature,  lati- 
tude, Ac,  are  in  some  measure  provided  for;  so  that  with  very  delicate  instruments,  a 
skilful  observer  may  measure  the  diff  of  altitude  of  two  points  close  together,  such 
as  the  bottom  and  top  of  a  steeple,  with  a  tolerable  confidence  that  he  is  within  two 
or  three  feet  of  the  truth.  But  if  as  short  an  interval  as  even  a  few  hours  elapses 
between  his  two  observations,  such  changes  may  occur  in  the  condition  of  the  atmo- 
sphere that  he  may  make  the  top  of  the  steeple  to  be  lower  than  its  bottom ;  or  at 
least,  cannot  feel  by  any  means  certain  that  he  is  not  ten  or  twenty  ft  in  error;  and 
this  may  occur  without  any  perceptible  change  in  the  atmosphere.  Whenever  prac- 
ticable, therefore,  there  should  be  a  person  at  each  station,  to  observe  at  both  points 
at  the  same  time.  Single  observations  at  points  many  miles  apart,  and  made  on  dif- 
ferent days,  and  in  different  states  of  the  atmosphere,  are  of  little  value.  In  such 
cases  the  mean  of  many  observations,  extending  over  several  days,  weeks,  or  months, 
and  made  when  the  air  is  apparently  undisturbed,  will  give  tolerable  approximations 
to  the  ttuth.  In  the  tropics  the  range  of  the  atmospheric  pres  is  much  less  than 
in  other  regions,  seldom  exceeding  ^  inch  at  any  one  spot ;  also  more  regular  in 
time,  and,  therefore,  less  productive  of  error.  Still,  the  barometer,  especially  either 
the  aneroid,  or  Bourdon's  mietallic,  may  be  rendered  highly  useful  to  the  civil  engi- 
neer, in  cases  where  great  accuracy  is  not  demanded.  By  hurrying  from  point  to 
point,  and  especially  by  repeating,  he  can  form  a  judgment  as  to  which  of  two  sum- 
mits is  the  lowest.  Or  a  careful  observer,  keeping  some  miles  ahead  of  a  surveying 
party,  may  materially  lessen  their  labors,  especially  in  a  rough  country,  by  select- 
ing the  general  route  for  them  in  advance.  The  accounts  of  the  agreement  within 
a  few  inches,  in  the  measurements  of  high  mountains,  by  diff  observers,  at  diff 
periods  ;  and  those  of  ascertaining  accurately  the  grades  of  a  railroad,  by  means  of 
an  aneroid,  while  riding  in  a  car,  will  be  believed  by  those  only  who  are  ignorant 
of  the  subject.    Such  results  can  happen  only  by  chance. 

When  possible,  the  observations  at  different  places  should  be  taken  at  the  same 
time  of  day,  as  some  check  upon  the  effects  of  the  daily  atmospheric  tides ;  and  in 
very  important  cases,  a  memorandum  should  be  made  of  the  year,  month,  day,  and 
hour,  as  well  as  of  the  state  of  the  weather,  direction  of  the  wind,  latitude  of  the 
place,  &c,  to  be  referred  to  an  expert,  if  necessary. 

Ttie  effects  of  latitude  are  not  included  in  any  of  our  formulas.  When 
reqd  they  may  be  found  in  the  table  page  314.  Several  other  corrections  must  be 
made  when  great  accuracy  is  aimed  at ;  but  they  require  extensive  tables. 

In  rapid  railroad  exploring,  however,  such  refinements  may  be  neglected,  inas- 
much as  no  approach  to  such  accuracy  is  to  be  expected ;  but  on  the  contrary,  errors 
oi  from  1  to  10  or  more  feet  in  100  of  height,  will  frequently  occur. 

As  a  very  rong-li  average  we  may  assume  that  the  barometer  falls  y^^  . 
inch  for  every  90  feet  that  we  ascend  above  the  level  of  the  sea,  up  to  1000  ft.    But 
in  fact  its  rate  of  fall  decreases  continually  as  we  rise ;  so  that  at  one  mile  high  it 
falls  ^Q  inch  for  about  106  ft  rise.    Table  2  shows  the  true  rate. 


LEVELLING   BY   THE   BAROMETER. 


313 


To  ascertain  the  difT  of  heigrht  between  two  points. 

KuLE  1,  Take  readings  of  the  barom  and  therm  (Fah)  in  the  shade  at  both 
Btations.  Add  together  the  two  readings  of  the  barom,  and  div  their  sum  by  2,  for 
their  mean ;  which  call  b.  Do  the  same  with  the  two  readings  of  the  thermom,  and 
call  the  mean  t.  Subtract  the  least  reading  of  the  barom  from  the  greatest ;  and  call 
the  diff  d.  Then  mult  together  this  diff  d;  the  number  from  the  next  Table  No.  1, 
opposite  t;  and  the  constant  number  30.    Div  the  prod  by  b.    Or 

Height  _  DiflF(<i)of  ^>  Tabular  number  opposite  ^  p  „^far.<-  -^n 
in  feet  ""     barom       ^       mean  (Q  of  thermom       ^   i^oDstant  du. 

mean  (b)  of  barom. 
Example.    Reading  of  the  barom  at  lower  station,  26.64  ins  ;  and  at  the  upper 
6ta  20.82  ins.    Thermom  at  lowest  sta,  70°;  at  upper  sta,  40°.     What  is  the  ditf  io 
height  of  the  two  stations  ?     Here, 

Barom,  26.64  Therm,  70^ 

20.82  "       40° 

Also,  

2)47.46  2)110 

23.73  mean  of  bar,  or  b.  55°  mean  of 

therm,  or  t. 
The  tabular  number  opposite  55°,  is  917.2. 
Bar.        Bar. 
Again,  26.64  —  20.82  =  6.82,  diff  of  bar ;  or  d.    Hence, 
d,      Tab  No.  Con. 
Height  _  5.82X917.2X30  _  160143.12  _  6748.5  ft;  answer, 
in  feet  23.73  (or  6)  23.73 

Then  correct  for  latitude,  if  more  accuracy  is  reqd,  by  rule  on  next  page. 
The  screw^  at  the  bach  of  an  aneroid  is  for  adjusting  the  index  by  a  stand- 
ard barom.  After  this  has  been  done  it  must  by  no  means  be  meddled  with.  In 
Bome  instruments  specially  made  to  order  with  that  intention,  this  screw  may  be 
lised  also  for  turning  the  index  back,  after  having  risen  to  an  elevation  so  great  that 
the  index  has  reached  the  extreme  limit  of  the  graduated  arc.  After  thus  turning 
it  back,  the  indications  of  the  index  at  greater  heights  must  be  added  to  that  at- 
tained when  it  was  turned  back. 

TABIi£  1.    For  Rale  1. 


Mean 

Meao 

Mean 

Mean 

of 

No. 

of 

No. 

of 

No. 

of 

No. 

Ther. 

Ther. 

Ther. 

Ther. 

0° 

801.1- 

30° 

864.4 

60° 

927.7 

90° 

991.0 

1 

803.2 

31 

866.5 

61 

929.8 

91 

993.1 

2 

805.3 

32 

868.6 

62 

931.9 

92 

995.2 

3 

807.4 

33 

870.7 

63 

934.0 

93 

997.3 

i 

809.5 

34 

872.8 

64 

9.36.1 

94 

999.4 

5 

8H.7 

35 

874.9 

65 

938.2 

95 

1001.6 

6 

813.8 

36 

877.0 

66 

940.3 

96 

1003.7 

7 

815.9 

37 

879.2 

67 

942.4 

97 

1005.8 

8 

818.0 

38 

881.3 

68 

944.5 

98 

1007.9 

9 

820.1 

39 

883.4 

69 

946.7 

99 

1010.0 

10 

822.2 

40 

885.4 

70 

948.8 

100 

1012.1 

11 

824.3 

41 

887.5 

71 

950.9 

101 

1014.2 

12 

826.4 

42 

889.6 

72 

953.0 

102 

1016.3 

13 

828.5 

43 

891.7 

73 

955.1 

103 

1018.4 

U 

830.6 

44 

893.8 

74 

957.2 

104 

1020.5 

15 

832.8 

45 

896.0 

75 

959.3 

105 

1022.7 

16 

834.9 

46 

898.1 

76 

961.4 

106 

1024.8 

17 

837.0 

47 

900.2 

77 

963.5 

107 

1026.9 

18 

839.1 

48 

902.3 

78 

965.6 

108 

1029.0 

19 

841.2 

49 

904.5 

79 

967.7 

109 

1031.1 

20 

843.3 

50 

906.6 

80 

969.9 

110 

1033.2 

21 

845.4 

51 

908.7 

81 

972.0 

111 

1035.3 

22 

847.5 

52 

910.8 

82 

974.  J 

112 

1037.4 

23 

849.6 

53 

913.0 

83 

976.2 

113 

1039.5 

24 

851.8 

54 

915.1 

84 

97^.3 

114 

1041.6 

25 

853.9 

55 

917.2 

86 

980.4 

115 

1043.8 

26 

856.0 

56 

919.3 

86 

982.6 

116 

1045.9 

27 

858.1 

57 

921.4 

87 

984.7 

117 

1048.0 

28 

860.2 

58 

923.5 

88 

986.8 

118 

1050.1 

29 

862.3 

59 

925.6 

89 

988.9 

119 

1052.2 

314 


LEVELLING   BY   THE   BAROMETER. 


Rule  2.  Belville's  short  approx  rule  is  the  one  best  adapted  to  rapid 
field  use,  namely,  add  together  the  two  readings  of  the  barom  only.  Also  find  the 
diff  between  said  two  readings ;  then,  as  the  Slim  of  the  two  reading's 
is  to  their  diff,  so  is  55000  feet  to  the  reqd  altitude. 

Correction  for  latitude  is  usually  omitted  where  great  accui-acy  is  not 
required.  To  apply  it,  first  find  the  altitude  by  the  rule,  aa  before.  Then  divide  it 
by  the  number  in  the  following  table  opposite  the  latitude  of  the  place.  (If  the  two 
.  places  are  in  different  latitudes,  use  their  mean.)  Add  the  quotient  to  the  altitude 
if  the  latitude  is  less  than  45°.  Subtract  it  if  the  latitude  is  more  than  45°.  No  cor- 
rection required  for  latitude  45°. 


Table  of  corrections 

for  latitude. 

Lat. 

Lat. 

Lat. 

Lat. 

Lat. 

Lat. 

0° 

352 

14° 

399 

280 

630 

420 

3367 

540 

1140 

680 

490 

2 

354 

16 

416 

30 

705 

44 

10101 

56 

941 

70 

460 

i 

356 

18 

436 

32 

804 

45 

00 

58 

804 

72 

436 

6 

360 

20 

460 

34 

941 

46 

10101 

60 

705 

74 

416 

8 

367 

22 

490 

36 

1140 

48 

3367 

62 

630 

76 

399 

10 

375 

24 

527 

38 

1458 

50 

2028 

64 

572 

78 

386 

12 

386 

26 

572 

40 

2028 

52 

1458 

66 

527 

80 

375 

LiCTelling^  by  Barometer;  or  by  the  boiling  point. 

Rule  3.  The  following  table.  No.  2,  enables  us  to  measure  heights  either  by  means 
of  boiling  water,  or  by  the  barom.  The  third  column  shows  the  approximate  alti- 
tude above  sea-level  corresponding  to  diff  heights,  or  readings  of  the  barom  ;  and  to 
the  diff  degrees  of  Fahrenheit's  thermom,at  which  water  boils  in  the  open  air.  Thus 
when  the  barom,  under  undisturbed  conditions  of  the  atmosphere,  stands  at  24.08 
inches,  or  when  pure  rain  or  distilled  water  boils  at  the  temp  of  201°  Fah  ;  the  place 
is  about  5764  ft  above  the  level  of  the  sea,  as  shown  by  the  table.  It  is  therefore 
very  eeisy  to  find  the  diff  of  altitude  of  two  places.  Thus  :  take  out  from  table  No  2, 
the  altitudes  opposite  to  the  two  boiling  temperatures  ;  or  to  the  two  barom  readings. 
Subtract  the  one  opposite  the  lower  reading,  from  that  opposite  the  upper  reading. 
The  rem  will  be  tlie  reqd  height,  as  a  rough  approximation.  To  correct  this,  add 
together  the  two  therm  readings ;  and  div  the  sum  by  2,  for  their  mean.  From  table 
for  temperature,  p  816,  take  out  the  number  opposite  this  mean.  Mult  the  ap- 
proximate height  just  found,  by  this  tabular  number.    Then  correct  for  lat  if  reqd. 

Ex.  The  same  as  preceding ;  namely,  barom  at  lower  sta,  26.64 ;  and  at  upper  sta, 
20.82.  Thermom  at  lower  sta,  70°  Fah ;  and  at  the  upper  one,  40°.  What  is  the  dilf 
of  height  of  the  two  stations  ? 

Alt. 

Here  the  tabular  altitudes  are,  for  20.82 9579 

and  for  26.64 3115 

6464  ft,  approx  height. 
70°  -f  40°      110° 
To  correct  this,  we  have  ^ =  — ^  =  55°  mean ;  and  in  table  p  316,  opp  to 

55°,  we  find  1.048.    Therefore  6464  X  1.048  =  6774  ft,  the  reqd  height. 

This  is  about  26  ft  more  than  by  Rule  1  ;  or  nearly  .4  of  a  ft  in  each  100  ft. 

At  70°  Fah,  pure  water  will  boil  at  1°  less  of  temp,  for  an  average  of  about  550  ft 
of  elevation  above  sea-level,  up  to  a  height  of  l^  a  mile.  At  the  height  of  1  mile,  1° 
of  boiling  temp  will  correspond  to  about  560  ft  of  elevation.  In  table  p  315  the 
mean  of  the  temps  at  the  two  stations  is  assumed  to  be  32°  Fah  ;  at  which  no  correc- 
tion for  temp  is  necessary  in  using  the  table;  hence  the  tabular  number  opposite 
32°,  in  table  p  316,  is  1. 

This  diff  produced  in  the  temp  of  the  boiling  point,  by  change  of  elevation,  must 
not  be  confounded  with  that  of  the  atmosphere,  due  to  the  same  cause.  The  air  be- 
comes cooler  as  we  ascend  above  sea-level,  at  the  rate  (very  roughly)  of  about  1°  Fah 
for  every  200  ft  near  sea-level,  to  350  ft  at  the  height  of  1  mile. 

The  follonring'  table,  Jfo.  2,  (so  far  as  it  relates  to  the  barom.)  was  de- 
duced  by  the  writer  from  the  standard  work  on  the  barom  by  Lieut.-Col.  R.  S.  Wil- 
liamson, U.  S.  army.* 

•  Published  bj  permissioo  of  Ooverament  la  1868  by  Van  Nostraud,  N.  Y. 


LEVELLING    BY    THE    BAROMETER,    ETC. 


315 


TABLE  3. 

Levelling:  by  Barometer;  or  by  the  boiling  point. 

Assumed  temp  in  the  shade  32°  Fah.     If  not  32°  mult  barom  alt  as  per  Table,  p  316 


Boil 

Altitude 

Boil 

Altitude 

Boil 

Altitude 

Boil 

Altitud* 

point 

Barom. 

above 

point 

Barom. 

above 

point 

Barom. 

above 

point 

Barom. 

above 

indeg 

aea  level 

in  deg 

sea  level 

in  deg 

sea  level 

indeg 

sea  level 

Fah. 

Ins. 

Feet. 

Fah. 

Ins. 

Feet. 

Fah. 

Ins. 

Feet. 

Fah. 

Ins. 

Feet. 

1840 

16.79 

15221 

.3 

19.66 

11083 

.6 

22.93 

7048 

.9 

2659 

3164 

.1 

16.83 

15159 

.4 

19.70 

11029 

.7 

22.98 

6991 

206 

26.64 

3115 

.2 

16.86 

15112 

.5 

19.74 

10976 

.8 

23.02 

6945 

.1 

26-69 

3066 

.3 

16.90 

15050 

.6 

19.78 

10923 

.9 

23.07 

6888 

26.75 

3007 

.4 

16.93 

15003 

.7 

19.82 

10870 

199 

23.11 

6843 

26.80 

2958 

.5 

16.97 

14941 

.8 

19.87. 

10804 

,1 

23.16 

6786 

26.86 

2899 

.6 

17.00 

14895 

.9 

19.92 

10738 

.2 

23.21 

6729 

26.91 

2850 

.7 

1704 

14833 

192 

19.96 

10685 

,3 

23.26 

6673 

26.97 

2792 

.8 

17.08 

14772 

.1 

20.00 

10633 

.4 

23.31 

6617 

27.02 

2743 

.9 

17.12 

14710 

.2 

20.05 

10567 

.5 

23.36 

6560 

27.08 

2685 

186 

17.16 

14649 

.3 

20.10 

10502 

.6 

23.40 

6516 

27.13 

2637 

.1 

17.20 

14588 

.4 

20.14 

10450 

.7 

23  45 

6460 

207 

27.18 

2589 

.2 

17.23 

14543 

.5 

20.18 

10398 

.8 

23.49 

6415 

27.23 

2540 

.3 

17.27 

14482 

.6 

20.22 

10346 

.9 

23.54 

6359 

27.29 

2483 

.4 

17.31 

14421 

.7 

20.27 

10281 

200 

23.59 

6304 

27.34 

2435 

.5 

17.35 

14361 

.8 

20.31 

10230 

.1 

23  64 

6248 

27.40 

2377 

.6 

17.38 

14315 

.9 

20.35 

10178 

.2 

23.69 

6193 

27.45 

2329 

.7 

17.42 

14255 

193 

20.39 

10127 

.3 

23.74 

6137 

27.51 

2272 

.8 

17.46 

14195 

.1 

20.43 

10075 

.4 

23.79 

6082 

27.56 

2224 

.9 

17.50 

14135 

,2 

20.48 

10011 

.5 

23  84 

6027 

27.62 

2167 

186 

17.54 

14075 

.3 

20.53 

9947 

.6 

23.89 

5972 

27.67 

2120 

.1 

17.58 

14015 

.4 

20.57 

9896 

.7 

23.94 

5917 

208 

27.73 

2063 

.2 

17.62 

13956 

.5 

20.61 

9845 

.8 

23.98 

5874 

27.78 

2016 

.8 

17.66 

13886 

.6 

20.65 

9794 

.9 

24.03 

5819 

27.84 

1959 

.4 

17.70 

13837 

.7 

20.69 

9743 

201 

24.08 

5764 

27.89 

1912 

.5 

17.74 

13778 

.8 

20.73 

9693 

.1 

24.13 

5710 

27.95 

1856 

.6 

17.78 

13718 

.9 

20.77 

9642 

.2 

24.18 

5656 

28.00 

1809 

.7 

17.82 

13660 

194 

20.82 

9579 

.3 

24.23 

5602 

28.06 

1753 

17.86 

13601 

.1 

20.87 

9516 

.4 

24.28 

5547 

28.11 

1706 

17.90 

13542 

.2 

20.91 

9466 

.5 

24.33 

5494 

28.17 

1650 

187 

17.93 

13498 

.3 

20.96 

9403 

.6 

24.38 

5440 

28.23 

1595 

17.97 

13440 

.4 

21.00 

9353 

.7 

24.43 

5386 

209 

28.29 

1539 

18.00 

13396 

.5 

21.05 

9291 

.8 

24.48 

5332 

28.35 

1483 

18.04 

13338 

.6 

21.09 

9241 

.9 

24.53 

5279 

28.40 

1437 

18.08 

13280 

.7 

21.14 

9179 

202 

24.58 

5225 

28.45 

1391 

18.12 

13222 

.8 

21.18 

9130 

.1 

24.63 

5172 

28.51 

1336 

18.16 

13164 

.9 

21.22 

9080 

.2 

24.68 

5119 

28.56 

1290 

18.20 

13106 

195 

21.26 

9031 

.3 

24.73 

5066 

28.62 

1235 

18.24 

13049 

.1 

21.31 

8969 

.4 

24,78 

5013 

28.67 

1189 

18.28 

12991 

.2 

21.35 

8920 

.5 

24.83 

4960 

28.73 

1134 

188 

18.32 

12934 

.3 

21.40 

8859 

.6 

24.88 

4907 

28.79 

1079 

18.36 

12877 

.4 

.   21.44 

8810 

.7 

24.93 

4855 

210 

28.85 

1025 

18.40 

12820 

.5 

21.49 

8749 

.8 

24.98 

4802 

28.91 

970 

18.44 

12763 

.6 

21.53 

8700 

.9 

25.03 

4750 

28.97 

916 

18.48 

12706 

.7 

21.58 

8639 

203 

25.08 

4697 

29.03 

862 

18.52 

12649 

.8 

21.62 

8590 

.1 

25.13 

4645 

29.09 

808 

18.56 

12593 

.9 

21.67 

8530 

,2 

25.18 

4593 

29.15 

754 

18.60 

12536 

196 

21.71 

8481 

.3 

25.23 

4541 

29.20 

709 

18.64 

12480 

.1 

21.76 

8421 

.4 

25.28 

4489 

29.25 

664 

18.68 

12424 

.2 

21.81 

8361 

.5 

25.33 

4437 

29.31 

610 

189 

18.72 

12367 

.3 

21.86 

8301 

.6 

25.38 

4:^86 

29.36 

565 

18.76 

12311 

.4 

21.90 

8253 

.7 

25.43 

4334 

211 

29.42 

512 

18.80 

12256 

.5 

21.95 

8193 

.8 

25.49 

4272 

29.48 

458 

18.84 

12200 

.6 

21.99 

8145 

.9 

25.54 

4221 

29,54 

405 

18.88 

12144 

.7 

22.04 

8086 

204 

25.59 

4169 

29.60 

352 

18.92 

12089 

.8 

22.08 

8038 

.1 

25.64 

4118 

29.65 

308 

18.96 

12033 

.9 

22.13 

7979 

.2 

25.70 

4057 

29.71 

255 

19.00 

11978 

197 

22.17 

7932 

.3 

25.76 

3996 

29.77 

202 

19.04 

11923 

.1 

22.22 

7873 

.4 

25.81 

3945 

29  83 

149 

19.08 

11868 

.2 

22.27 

7814 

.5 

25.86 

3894 

29,88 

105 

190 

19.13 

11799 

.8 

22.32 

7755 

.6 

25.91 

3844 

29.94 

52 

19.17 

11745 

.4 

22.36 

7708 

.7 

25.96 

3793 

212 

30.00 

sea  leT=0 

19.21 

11690 

.5 

22.41 

7649 

.8 

26.01 

3742 

B 

elow  sea 

level. 

19.25 

11635 

.6 

22.45 

7602 

.9 

26.06 

3692 

30.06 

—  52 

19.29 

11581 

.7 

22.50 

7544 

205 

26.11 

3642 

2 

30.12 

—104 

19.33 

11527 

.8 

22.54 

7498 

.1 

26.17 

3582 

3 

30.18 

—156 

19.37 

11472 

.9 

22.59 

7439 

.2 

26.22 

3532 

4 

30.24 

—209 

19.41 

11418 

198 

22.64 

7381 

.3 

26.28 

3472 

5 

30.30 

—261 

19.45 

1 1364 

.1 

22.69 

732,4 

.4 

26.33 

3422 

6 

30.35 

-304 

19.49 

11310 

.2 

22.74 

7266 

.5 

26.38 

3372 

7 

30.41 

-356 

Itl 

19.54 

11243 

.3 

22.79 

7208 

.6 

26.43 

3322 

8 

30.47 

—408 

19.58 

11190 

.4 

22.84 

7151 

.7 

26.48 

3273 

9 

30.53 

—459 

'.2    1     19.62 

11136 

.6 

22.89 

7001 

.8 

26.54 

3213 

213 

30.59 

—611 

316 


SOUND. 


Corrections  for  temperature;  to  be  used  in  connection  with 
Rule  3,  when  $?reater  accuracy  is  necessary.  Also  in  con^ 
nection  with  Table  2  when  the  temp  is  not  »2°. 


Mean 

Mean 

Mean 

Mean 

temp 

Mult 

temp 

Mult 

temp 

Mult 

temp 

Mult 

in  the 

by 

in  the 

by 

in  the 

by        in    the 

by 

shade. 

shade. 

shade. 

shade. 

Zero. 

.933 

28° 

.992 

56° 

1.050 

84° 

1.108 

2° 

.937 

30 

.996 

58 

1.054 

86 

1.112 

4 

.942 

32 

1.000 

60 

1.058 

88 

1.117 

6 

.946 

34 

1.004 

62 

1.062 

90 

1.121 

8 

.950 

36 

1.008 

64 

1.066 

92 

1.125 

10 

.954 

38 

1.012 

66 

1.071 

94 

1.129 

12 

.958 

40 

1.016  • 

68 

1.075 

96 

1.133 

14 

.962 

42 

1.020 

70 

1.079 

98 

1.138 

16 

.967 

44 

1.024 

72 

1.083 

100 

1.142 

18 

.971 

46 

1.028 

74 

1.087 

102 

1.146 

20 

.975 

48 

1.032 

76 

1.091 

104 

1.150 

22 

.979 

50 

1.036 

78 

1.096 

106 

1.164 

24 

.983 

52 

1.041 

80 

1.100 

108 

1.158 

26 

.987 

54 

1.046 

82 

1.104 

110 

1.163 

SOUND. 


■JOke  velocity  of  sound  in  quiet  open  air,  has  been  experimentally  deter- 
xnlned  to  be  very  approximately  1U90  feet  per  second,  when  the  temperature  is  at 
freezing  point,  or  3'^°  Fahaenheit.  For  every  degree  Fahrenheit  of  increase  of 
temperature,  the  velocity  increases  by  from  i^  foot  to  1}^  feet  per  second,  according 
to  different  authorities.  Takiug  the  increase  at  1  foot  per  second  for  each  degree 
(which  agrees  closely  with  theoretical  calculations),  we  have 
at   _    30°  Fahr  1030  feet  per  sec  =  0.1951  mile  per  sec  ==  1  mile  in  5.13  seconda. 


20° 

"     1040 

" 

<.' 

=  0.1970 

" 

" 

=  1    " 

5.08 

10° 

"     1050 

<' 

(( 

=  0.1989 

« 

" 

—=1     " 

5.03 

0 

"     1060 

'• 

" 

=  0.2008 

(I 

« 

=  1     " 

4.98 

10° 

«     1070 

<' 

'< 

=  0.2027 

" 

« 

=  1     " 

4.93 

20° 

"     1080 

« 

« 

=  0.2045 

" 

" 

=  1     " 

4.88 

32° 

"     1092 

(; 

'• 

=  0.2068 

" 

« 

=  1     " 

4.83 

40° 

"    1100 

" 

« 

=  0.2083 

" 

« 

==1     " 

4.80 

50° 

"    1110 

" 

« 

=  0.2102 

" 

<« 

=  1     " 

4.78 

60° 

''     1120 

" 

« 

=  0.2121 

" 

" 

=  1     " 

4.73 

70° 

«'     1130 

" 

« 

=  0.2140 

« 

«• 

=  1     " 

4.68 

80° 

«     1140 

" 

« 

=  0.2159 

" 

" 

=  1     " 

4.63 

90° 

"     1150 

« 

« 

=  0.2178 

« 

<* 

=  1     " 

4.59 

100° 

"    1160 

" 

i( 

=  0.2197 

" 

« 

=  1     " 

4.55 

110° 

"    1170 

« 

«« 

=  0.2216 

" 

« 

=  1     " 

4.51 

120° 

"    1180 

" 

« 

=  0.2235 

u 

« 

=  1    " 

4.47 

If  the  air  is  calm,  fog  or  rain  does  not  appreciably  affect  the  result ;  but  winds  do. 
Very  loud  sounds  appear  to  travel  somewhat  faster  than  low  ones.  The  watchword 
of  sentinels  has  been  heard  across  still  water,  on  a  calm  night,  10}^  miles;  and  a 
cannon  20  miles.  Separate  sounds,  at  intervals  of  ^^  of  a  second,  cannot  be  distin- 
guished, but  appear  to  be  connected.  The  distances  at  which  a  speaker  can  be 
understood,  in  front,  on  one  side,  and  behind  him.  are  about  as  4,  3,  and  1. 

Dr.  Charles  M.  Cresson  informs  the  writer  that,  by  repeated  trials,  he  found  that 
in  a  Philadelphia  gas  main  20  inches  diameter  and  16000  feet  long,  laid  and  covered 
in  the  earth,  but  empty  of  gas,  and  having  one  horizontal  bend  of  90°,  and  of  40  feet 
radius,  the  sound  of  a  pistol-shot  travelled  16000  feet  in  precisely  16  seconds,  or  1000 
feet  per  second.  The  arrival  of  the  sound  was  barely  audible;  but  was  rendered 
very  apparent  to  the  eye  by  its  blowing  off  a  diaphragm  of  tissue-paper  placed  over 
the  end  of  the  main. 

Two  boats  anchored  some  distance  apart  may  serve  as  a  base  line  for 
triangulating  objects  along  the  coast;  the  distance  between  them  being  first  found 
by  firing  guns  on  board  one  of  them. 

In  -water  the  velocity  is  about  4708  feet  per  second,  or  about  4  times  that 
in  air.  In  '«voods,  it  is  from  10  to  16  times ;  and  in  metals,  from  4  to  16  times 
greater  than  in  air,  according  to  some  authorities. 


HEAT. 


317 


Approximate  expansion  of  solids  by  heat:  and  tlieir  melt* 
ing^  points  by  Fahrenheit's  thermometer.  | 


Fire-brick.. 
Granite.... 


;  from 


J  fn 
i  to 

Glass  rod. 

Glass  tube 

"      crown 

"      plate 

Platina 

Marble,  granular,  white,  dry.. 

"  "  *'     moist. 

"  *    black,  compact 

Antimony    

Obst  iron 

Slate 

Steel 

"    blistered 

"    untempered 

"     tempered  yellow 

"    hardened 

"     annealed 

Iron,  rolled 

"    soft,  forged 

•*    wire 

Bismuth .... 

Gold,  annealed 

Copper average 

Sandstone^*. 

Brass average 

'•    wire 

Silver .... 

Tin average 

Lead average 

Pewter 

Zinc  (most  of  all  metalf) 

"White  pine 


For  1  degree. 


1  part  in 
365220 
187560 


221400 
214200 
211500 
209700 


173000 
128000 
405000 
166500 
16200011 
173000 
151200 
159840 
167400 
131400 


147600 
14994011 
147420 
146340 
129600 
123120 
104400 
103320 
97740 
94140 
95040 
87840 
63180 
78840 
61920 
440530 


^  inch  in 
3804  ft. 
1954 
2375 
2306 
2231 
2203 
2184 
2175 
1802 
1333 
4219 
1722 
1688 
1802 
1575 
1665 
1744 
1369 
1530 
1537 
1562 
1536 
1524 
1350 
1282 
1088 
1076 
1018 

981 

990 

915 

658 

821 
645 
4588 


For  180  degrees.* 


1  part  in 

%  inch  in 

2029 

21.14  ft. 

1042 

10.85 

1267 

13.20 

1230 

12.81 

1190 

12.40 

1175 

12.24 

1165 

12.13 

1160 

12.08 

961 

10.00 

711 

7.41 

2250 

23.44 

925 

9.63 

900 

9.38 

961 

10.00 

840 

8.75 

888 

9.25 

930 

9.69 

730 

7.60 

816 

8.50 

820 

8.54 

833 

8.68 

819 

8.53 

8K 

8.55 

720 

7.50 

684 

7.12 

580 

6.04 

574 

5.98 

543 

5.66 

523 

5.45 

528 

11 

488 

351 

3.66 

438 

4.56 

344 

3.58 

2447 

25.49 

Melting 

point 

in  Deg.t 


955 
1920  to 

2800 
2370  to 

2550 


3000  t* 
3500 

506 
2016 
2000 

1873 

1861 
444 
612 


Heat  of  a  common  wood-fire*  variously  estimated  from  800  to  1140  deg.    That  of  a  ohareoal 

eue  about  2200    ;  coal  about  2400°. 

Each  12°  to  15°  of  heat  produce  in  wrot  iron  an  expansion  equal  to 
that  produced  by  a  tension  of  about  1  ton  per  sq  inch  of  section ;  varying  with  the  quality  of  the  iron. 


*  By  adding  y^^f  P'*^'"'  *^  f^-  lengths  in  the  two  cola  under  180°,  ire  get  the  lengths  corresponding  t« 
a  number  of  degrees  -Jh  less  than  180° ;  or  to  1630.63  deg;  which  may  be  taken  as  about  the  extremet 
of  temp  in  the  colder  portions  of  the  United  States.  In  the  Middle  States  the  extremes  rarely  reach 
135°,  or  U  part  less  than  180°. 

No  dependence  whatever  is  to  be  placed  on  results  obtained  by  Wedgewood's  pyrometer. 

t  The  table  shows  that  the  contraction  and  expansion  of  stone  will  cause 

open  joints  in  winter  ;  and  crushing  of  the  mortar  in  summer,  at  the  ends  of  long  coping-stones. 

I  The  melting:  points  are  quite  uncertain.    We  give  the  mean  of 

the  best  authorities.  Assuming  that  with  a  change  of  temp  of  about  163°,  wrought  iron  will  alter  its 
length  1  part  in  916 ;  this  in  a  mile  amounts  to  5.764  ft,  or  about  5  ft  9J^  ins  ;  and  in  100  ft  to  .109  of  a 
foot;  or  lys  ins ;  so  that  a  ditf  of  5  ft,  or  more,  can  readily  result  from  measuring  a  mile  in  winter 
and  in  summer  with  the  same  chain  ;  and  a  25  ft  rail  will  change  its  length  full  %  of  an  inch. 

II  Hence  fvrought  escpands  by  heat  about  one  eleventh  part  more  than  cast;  whereas 
under  tension  within  elastic  limits  cast  stretches  twice  as  much  as  wrt. 


318 


THERMOMETERS. 


THEEMOMETERS. 


Below  about  35°  below  zero  of  Fah.  the  mercurial  thermometer  and  barometer  become  too  irregular  It 
be  depended  on.   Mercury  begins  to  freeze  at  abou t— 40°  Fp.h.  Below — 4(Ppure  alcohol  is  used. 

To  cbaiige  degrees  of  Fahrenheit  to  the  corresponding*  de* 
g'rees  of  Centigrade ;  take  a  Fah  reading  32°  lower  than  the  given  one;  mult 
this  lower  reading  by  5  ;  divide  the  prod  by  9.  Thus  :  -|-UO  Fah  =  (14  —32)  X  5  -r  9  =  —18  X  5  -r  9= 
— IQoCent.     Again,  —13°  Fah  =  (— 13  — 32)  X  5-r9  =  — 45  X  5-r9  =  — '25°  Cent. 

To  chang-e  Fah  toReau;  take  a  Fah  reading  32°  Zower  than  the  given 
one;  mult  this  lower  reading  by  4 ;  div  the  prod  by  9.  Thus:  +14°  Fah  =  (14— 32)  X  4-r9  =  — 18X 
4-r9=— SOReau.     Again,  —13°  Fah  =  (—13 —32)  X  4-r9  =  — 45  X  44-9  =  — 20°  Reau. 

To  change  €ent  to  Fah;  mult  the  Cent  reading  by  9;  divide  by  5.  Take  a 
Fah  reading  320  Uqher  than  the  quot.  Thus :  -fioo  Cent  =  (10  X  9  -j-  5)  +32  =  18  +32  =  -j-SQO  Fah. 
Again,  —20°  Cent  =  (—20  X  9  -j-  5)  -f  32  =  —4°  Fah. 

To  change  Cent  to  Iteau;  mult  by  4;  div  by  5.  Thus:  -t-10°Cent  =10X 
4-r5-=+80Reau.  Again,— 10oCent=— lOX  4t-5=:— 8°  R^au. 

To  change  Reau  to  Fah;  mult  by  9;  div  by  4.  Take  a  Fah  reading  32° 
higher.  Thus :  +16°  Reau  =  fie  X9  +  4)  4-  82=  36  +  32  =  +68°  Fah.  Again.  —8°  Reau  =  (-8  X  9  •r 
4)  4-32  =  —18  +32  =  +14"  Fah. 

To  change  Reau  to  €ent;  mult  by  5 ;  div  by  4.    Thus :  +8°  Reau  =  +  8° 

X5 +4= +10° Cent.  Again,— 8° Reau =—8X5+4= —10° Cent. 

TABIiF  1.  Fahrenheit  compared  with  Centigrade  and  R^an- 
mnr.    In  this  table  the  Cent  and  Reau  readings  are  given  to  the  nearest  decimal. 


F. 

c. 

R. 

F. 

c. 

R. 

F. 

c. 

R. 

F. 

c. 

R. 

F. 

c. 

R. 

o 

o 

o 

o 

o 

o 

o 

o 

o 

o 

o 

o 

o 

o 

0 

212 

100 

80.0 

158 

TO.O 

56.0 

104 

40  0 

32.0 

50 

10.0 

8.0 

—3 

-19.4 

-15.6 

211 

99.4 

79.6 

157 

69.4 

55.6 

103 

39.4 

31.6 

49 

9.4 

7.6 

—4 

-20.0 

—16.0 

210 

98.9 

79.1 

156 

68.9 

55.1 

102 

38.9 

31.1 

48 

8.9 

7.1 

—5 

-20.6 

—16.4 

209 

98.3 

78.7 

1.55 

68.3 

54.7 

101 

38  3 

30.7 

47 

8.3 

6.7 

-6 

-21.1 

—16.9 

208 

97.8 

78.2 

154 

67.8 

54.2 

100 

37.8 

30.2 

46 

7.8 

6.2 

—7 

-21.7 

-17.3 

207 

97.2 

77.a 

153 

67.2 

53.8 

99 

37.2 

29.8 

45 

7.2 

5.8 

—8 

—22.2 

-17.8 

206 

96.7 

77.3 

152 

66.7 

53.3 

98 

36.7 

29.3 

44 

6.7 

5.3 

—9 

-22.8 

—18.2 

205 

96.1 

76.9 

151 

66.1 

52.9 

97 

36.1 

28.9 

43 

.  6.1 

4.9 

-10 

-23.3 

-18.7 

204 

95.6 

76.4 

i.y) 

65.6 

52.4 

96 

35.6 

28.4 

42 

5.6 

4.4 

—11 

—23.9 

—19.1 

203 

95.0 

76.0 

U9 

65.0 

52.0 

95 

35.0 

28.0 

41  . 

5.0 

4.0 

—12 

-24.4 

-19.6 

202 

94.4 

75.6 

148 

64.4 

51.6 

94 

84.4 

27.6 

40 

4.4 

3.6 

—13 

-250 

-20.0 

201 

93.9 

75.1 

147 

63.9 

51.1 

93 

33.9 

27.1 

39 

3.9 

3,1 

—14 

-25.6 

-20.4 

200 

93.3 

74.7 

146 

63.3 

50.7 

92 

33.3 

26.7 

38 

8.3 

2.7 

-15 

—26.1 

-20.9 

199 

92.8 

74.2 

145 

62.8 

50.2 

91 

32.8 

26.2 

37 

2.8 

2.2 

—16 

-26.7 

-21.3 

198 

92.2 

73.8 

144 

62.2 

49.8 

90 

32.2 

25.8 

36 

2.'/ 

1.8 

—17 

-27.2 

—21.8 

197 

91.7 

73.3 

143 

61.7 

49.3 

89 

31.7 

25.3 

85 

1.7 

1.8 

—18 

—27.8 

-22.2 

196 

91.1 

72.9 

142 

61.1 

48.9 

88 

31.1 

24.9 

34 

1.1 

0.9 

—19 

—28.3 

-22.7 

195 

90.6 

72.4 

141 

60.6 

48.4 

87 

30.6 

24.4 

33 

0.6 

0.4 

—20 

-28.9 

—23.1 

194 

90.0 

72.0 

140 

60.0 

48.0 

86 

30.0 

24.0 

32 

0.0 

0.0 

—21 

—29.4 

—23.6 

193 

89.4 

71.6 

139 

59.4 

47.6 

85 

29.4 

23.6 

31 

—0.6 

-0.4 

—22 

—30.0 

—24.0 

192 

88.9 

71.1 

138 

58.9 

47.1 

84 

28.9 

23.1 

.SO 

— l.l 

—0.9 

—23 

-30.6 

—24.4 

191 

88.3 

70.7 

137 

58.3 

46.7 

83 

28.3 

22.7 

29 

—1.7 

—1.3 

—24 

-31.1 

-24.9 

190 

87.8 

70.2 

136 

57.8 

46.2 

82 

27.8 

22,2 

28 

—2.2 

—1.8 

—25 

-31.7 

—25.3 

189 

87.2 

69.8 

135 

57.2 

45.8 

81 

27.2 

21.8 

27 

—2.8 

—2.2 

—26 

—32.2 

-25.8 

188 

86.7 

69.3 

134 

56.7 

45.3 

80 

26.7 

21.3 

26 

—3.3 

—2.7 

—27 

—32.8 

—26.2 

187 

86.1 

68.9 

133 

56.1 

44.9 

79 

26.1 

20.9 

25 

—8.9 

—3.1 

—28 

-33.3 

-26.7 

186 

85.6 

68.4 

132 

55.6 

44.4 

78 

25.6 

20.4 

24 

—4,4 

—3.6 

—29 

—33.9 

-27.1 

185 

85.0 

68.0 

131 

55.0 

44.0 

77 

25.0 

20.0 

33 

-6,0 

—4.0 

—80 

-34.4 

-27.« 

184 

81.4 

67.6 

130 

54.4 

43.6 

76 

24.4 

19.6 

22 

-5.6 

—4.4 

—31 

—35.0 

—28.0 

18:^ 

83.9 

67.1 

129 

53.9 

43,1 

75 

23.9 

19.1 

21 

—6.1 

—4.9 

—32 

—85.6 

—28.4 

182 

83.3 

66.7 

128 

53.3 

42.7 

74 

23.3 

18.7 

20 

-6.7 

-5.3 

—33 

—36.1 

—28.9 

181 

82.8 

66.2 

127 

52.8 

42.2 

73 

22.8 

18.2 

19 

-7.2 

—5.8 

—34 

—36.7 

-29.3 

180 

82.2 

65.8 

126 

52.2 

41.8 

72 

22.2 

17.8 

18 

-7.8 

-6.2 

-35 

—87.2 

-29.8 

179 

81.7 

65.3 

125 

51.7 

41.3 

71 

21.7 

17.3 

17 

—8.3 

-6.7 

-36. 

-37.8 

-30.2 

178 

81.1 

64.9 

124 

51.1 

40.9 

70 

21.1 

16.9 

16 

—8.9 

—7.1 

-87 

-38.8 

—30.7 

177 

80.6 

64.4 

123 

50.6 

40.4 

69 

20.6 

16.4 

15 

—9.4 

-7.6 

-38 

—38.9 

—31.1 

176 
175 

80.0 
79.4 

64.0 
63.6 

122 
121 

50.0 
49.4 

40.0 
39.6 

16.0 
15.6 

14 
13 

—10-0 
—10.6 

1  -8.0 
—8.4 

—39 
—40 

-39.4 
-40.0 

—81.6 
-32.0 

67 

\%A 

174 

78.9 

63.1 

120 

48.9 

39.1 

66 

18.9 

15.1 

12 

—11.1 

—8.9 

—41 

-40.6 

—32.4 

173 

78.3 

62.7 

119 

48.3 

38.7 

65 

18.3 

14.7 

11 

—11.7 

\   —9.3 

-42 

—41.1 

-.S2.9 

172 

77.8 

62.2 

118 

47.8 

38.2 

64 

17.8 

14.2 

10 

—12.2 

—9.8 

-43 

—41,7 

— 33.S 

171 

77.2 

61.8 

117 

47.2 

37.8 

63 

17.2 

13,8 

9 

—12.8 

—10.2 

—44 

—42,2 

-33.8 

170 

76.7 

61.3 

116 

46.7 

37.3 

62 

16.7 

13.3 

8 

—13.3 

—10.7 

—45 

-42,8 

-34.2 

169 

76.1 

60.9 

115 

46.1 

36.9 

61 

16.1 

12.9 

7 

—13.9 

-11.1 

-46 

-43.3 

-34.7 

168 

75.6 

60.4 

114 

456 

36.4 

60 

15.6 

12.4 

6 

-14.4 

-11.6 

—47 

—43.9 

-35.1 

167 

75.0 

60.0 

113 

45.0 

36.0 

59 

15.0 

12,0 

5 

—15.0 

—12.0 

—48 

—44.4 

-36.6 

166 

74.4 

59.6 

112 

44.4 

35.6 

58 

14.4 

11.6 

4 

—15.6 

—12.4 

—49 

—45.0 

-36.0 

165 

73  9 

59.1 

111 

43.9 

35.1 

57 

13.9 

11.1 

3 

-16.1 

—12.9 

-50 

-45.6 

i— 36.4 

164 

73.3 

58.7 

110 

43.3 

34.7 

56 

13.3 

10.7 

2 

-16.7 

—18.3 

—51 

—46.1 

1—36.9 

163 

72.8 

58.2 

109 

42.8 

34.2 

55 

12.8 

10.2 

1 

—17.2 

-13.8 

—52 

—46.7 

-87.1 

162 

72.2 

57.8 

108 

42.2 

.33  8 

54 

12.2 

9.8 

0 

—17.8 

-14.2 

—53 

—47.2 

— ST.8 

'161 

71.7 

57.3 

107 

41.7 

33.3 

53 

11.7 

9.3 

—1 

-18.3 

—14.7 

—54 

-47.8 

-S8.I 

160 

71.1 

56.9 

106 

41.1 

32.9 

52 

11.1 

8.9 

—2 

—18.9 

j— 15.1 

-55 

— 4«.3 

— S«.1 

1S9 

70.6 

56.4 

105 

40.6 

32.4 

51 

10.6 

8.4 

1 

THERMOMETERS. 


319 


TAB1.E  2,    Centigrade  compared  with   FatireaiBieit  and 
Reaumur. 


c. 

F. 

R. 

c. 

F. 

R. 

c. 

F. 

R. 

€. 

F. 

R. 

0 

o 

o 

o 

o 

c 

o 

o 

o 

o 

o 

o 

Exact. 

Exact. 

Exact. 

Exact. 

Exact. 

Exact. 

Exact. 

Exact. 

(00 

212.0 

80.0 

62 

143.6 

49.6 

24 

75.2 

19.2 

—14 

6.8 

-11.2 

99 

210.2 

79.2 

61 

141.8 

48.8 

23 

73.4 

18.4 

-15 

5.0 

—12.0 

98 

208.4 

78.4 

60 

140.0 

48.0 

22 

71.6 

17.6 

—16 

3.2 

-12.8 

97 

206.6 

77.6 

59 

138.2 

47.2 

21 

69.8 

16.8 

-17 

1.4 

— 1S.6 

96 

204.8 

76.8 

58 

136.4 

46.4 

20 

68.0 

16.0 

-18 

-0.4 

-14.4 

95 

203.0 

76.0 

57 

134.6 

45.6 

19 

66.2 

15.2 

-19 

-2.2 

—15.2 

94 

201.2 

75.2 

56 

132.8 

44.8 

18 

64.4 

14-4 

—20 

-4.0 

—16.0 

93 

199.4 

74.4 

55 

131.0 

44.0 

17 

62.6 

13.6 

-21 

—5.8 

—16.8 

92 

197.6 

73.6 

54 

129.2 

43  2 

16 

60.8 

12.8 

—22 

—7.6 

-17.6 

91 

195.8 

72.8 

53 

127.4 

42.4 

15 

59.0 

12.0 

—23 

-9.4 

-18.4 

90 

194.0 

72.0 

52 

125.6 

41.6 

14 

57.2 

11.2 

—24 

-11.2 

—19.2 

89 

192.2 

71.2 

51 

123.8 

40.8 

13 

55.4 

10.4 

-25 

—13.0 

—20.0 

88 

190.4 

70.4 

50 

122.0 

40.0 

12 

53.6 

9.6 

—26 

—14.8 

-20.  g 

87 

188.6 

69.6 

49 

120.2 

39.2 

11 

51.8 

8.8 

-27 

-16.6 

—21.6 

86 

186.8 

68.8 

48 

118.4 

38.4 

10 

50.0 

8.0 

—28 

-18.4 

—22.4 

85 

185.0 

68.0 

47 

116.6 

37.6 

9 

48.2 

7.2 

—29 

—20.2 

—23.2 

84 

183.2 

67.2 

46 

114.8 

36.8 

8 

46.4 

6.4 

-30 

—22.0 

-24.0 

83 

181.4 

66.4 

45 

113.0 

.S6.0 

7 

44.6 

5.6 

—31 

—23.8 

-24.8 

82 

179.6 

65.6 

44 

111.2 

35.2 

6 

42.8 

4.8 

—32 

-25.6 

-25.6 

81 

177.8 

64.8 

43 

109.4 

34.4 

5 

41.0 

4.0 

-33 

—27.4 

—26.4 

80 

176.0 

64.0 

42 

107.6 

33.6 

4 

39.2 

3.2 

—34 

—29.2 

—27.2 

79 

174.2 

63. 2 

41 

105.8 

32.8 

3 

37.4 

2.4 

—35 

—31.0 

—28.0 

78 

172.4 

62.4 

40 

104.0 

32.0 

2 

35.6 

1.6 

—36 

—32.8 

—28.8 

77 

170.6 

61.6 

39 

102.2 

31.2 

1 

33.8 

0.8 

—37 

-34.6 

-29.6 

76 

168.8 

60.8 

38 

100.4 

30.4 

0 

32.0 

0.0 

-38 

—36.4 

-30.4 

75 

167,0 

60.0 

37 

98.6 

29.6 

-1 

30.2 

-0.8 

-39 

—38.2 

—31.2 

74 

165.2 

59.2 

36 

96.8 

28.8 

-2 

28.4 

-1.6 

—40 

^40.0 

—32.0 

7S 

163.4 

58.4 

35 

95.0 

28.0 

—3 

2«.6 

—2.4 

—41 

—41.8 

-32.8 

72 

161.6 

57.6 

34 

93.2 

27.2 

—4 

24.8 

—3.2 

—42 

-43.6 

-33.6 

71 

159.8 

56.8 

33 

91.4 

26.4 

—5 

23.0 

—4.0 

—  43 

-45.4 

—34.4 

70 

158.0 

56.0 

32 

89.6 

25.6 

-6 

21.2 

-4.8 

—44 

—47.2 

— S5.2 

69 

156.2 

55.2 

31 

87.8 

24.8 

—7 

19.4 

-5.6 

—45 

—49.0 

-36.0 

68 

154.4 

54.4 

30 

86.0 

24.0 

—8 

17.6 

-6.4 

—46 

-50.8 

-36.8 

67 

152.6 

53.6 

29 

84.2 

23.2 

-9 

15.8 

—7.2 

—47 

-52.6 

-37.6 

66 

150.8 

52  8 

28 

82.4 

22.4 

—10 

14.0 

-8.0 

—48 

-54.4 

—38.4 

65 

149.0 

52.0 

27 

80.6 

21.6 

-11 

12.2 

—8.8 

—49 

-56.2 

—39,2 

64 

147.2 

51.2 

26 

78.8 

20.8 

-12 

10.4 

—9.6 

-50 

-.78.0 

— 40.# 

68 

145.4 

50.4 

25 

77.0 

20.0 

—13 

8.6 

—10.4 

TABL.F   3.    Reaumur  compared  witli  Falirenbeit  and 
Centig-rade. 


R. 

F. 

c. 

R. 

F. 

€. 

R. 

F. 

c. 

K. 

F. 

c. 

o 

o 

o 

o 

o 

o 

o 

o 

o 

O 

o 

o 

Exact. 

Exact. 

Exact. 

Exact. 

Exact. 

Exact. 

Exact. 

Exact. 

80 

212.00 

100.00 

49 

142.25 

61.25 

19 

74.75 

23.75 

— li 

7.25 

—13.75 

79 

209.75 

98.75 

48 

HO.OO 

60.00 

18 

72.50 

22.50 

—12 

5.00 

—15,00 

78 

207.50 

97.50 

47 

137.75 

58.75 

17 

70.25 

21.25 

—13 

2.75 

—16.25 

77 

205.25 

96.25 

46 

135  50 

57.50 

16 

68.00 

20.00 

—14 

0,50 

—17,50 

76 

203.00 

95.00 

45 

133.25 

56.25 

15 

65.75 

18,75 

—15 

-1,75 

-18,75 

75 

200.75 

93.75 

44 

131.00 

55  00 

14 

63.50 

17,50 

—16 

-4.00 

—20.00 

74 

198.50 

92,50 

43 

128.75 

53.75 

13 

61.25 

16.25 

—17 

—6.25 

—21.25 

73 

196.25 

91,25 

42 

126.50 

52.50 

12 

59.00 

15.00 

-18 

-8.50 

-22,50 

72 

194.00 

90.00 

41 

124.25 

51.25 

11 

56.75 

13.75 

-19 

—10.75 

-!«}.75 

71 

191.75 

88.75 

40 

122.00 

50.00 

10 

54.50 

12,50 

-20 

—13.00 

-25.00 

70 

189.50 

87.50 

39 

119.75 

48.75 

9 

52.25 

11.25 

—21 

—15.25 

—26.25 

69 

187.25 

86.25 

38 

117.50 

47.50 

8 

50.00 

10,00 

—22 

—17.50 

—27.50 

68 

185.00 

85  00 

37 

115.25 

46.25 

7 

47.75 

8.75 

-23 

-19.75 

—28.75 

67 

182.75 

83.75 

36 

113.00 

45.00 

6 

45,50 

7.50 

—24 

—22.00 

—30.00 

66 

180.50 

82.50 

35 

110.75 

43.75 

5 

43.25 

6.25 

-25 

—24,25 

-31.25 

65 

178  25 

81.25 

34 

108.50 

42.50 

4 

41.00 

5.00 

-26 

—26,50 

-32.50 

64 

176.00 

8(t.0O 

33 

106.25 

41.25 

3 

38.75 

3.75 

-27 

—28.75 

—33.75 

63 

173.75 

78.75 

32 

104.00 

40.00 

2 

36,50 

2.50 

-^28 

—31.00 

—35.00 

62 

171.50 

77,50 

31 

101.75 

38.75 

1 

34,25 

1.25 

-29 

-33.25 

-36.25 

61 

169.25 

76.25 

30 

99.50 

37  50 

0 

32.00 

0.00 

-30 

—35.50 

—37.50 

60 

167.00 

75.00 

29 

97.25 

36.25 

—1 

•29.75 

-1.25 

-31 

—37.75 

-38.75 

59 

164.75 

73.75 

28 

95.00 

35.00 

—2 

27,50 

-2.50 

—32 

-40.00 

-40.00 

58 

162.50 

72.50 

27 

92.75 

33  75 

—3 

25,25 

—3.75 

-33 

—42.25 

-41.25 

57 

160.25 

71.25 

26 

90.50 

32  50 

—4 

23.00 

-5.00 

-34 

-44.50 

-42.50 

56 

158.00 

70.00 

25 

88.25 

31.25 

-5 

20.75 

-6.25 

-35 

-46.75 

-43.75 

55 

155.75 

68.75 

24 

86.00 

30.00 

—6 

18.50 

-7.50 

-36 

-49.00 

-45.00 

54 

153.50 

67.50 

23 

83.75 

28.75 

—7 

16.25 

—8.75 

--37 

-51.25 

—46.25 

53 

151.25 

66.25 

22 

81,50 

27.50 

-8 

14.00 

—10.00 

-38 

—53.50 

-47.50 

52 

149.00 

65.00 

21 

79,25 

26.25 

—9 

11.75 

—11,25 

-39 

-.55.75 

-48.75 

61 

146,75 

63.75 

20 

77,00 

25.00 

—10 

9.50 

—12.50 

-40 

—58.00 

-50.08 

50 

144,50 

62.50 

320  AIK. 

AIR-ATMOSPHERE. 

The  atmosphere  is  known  to  extend  to  at  least  45  miles 

above  the  earth.  It  is  a  mixture  of  about  79  measures  of  nitrogen  gas  and  21 
of  oxygen  gas;  or  about  77  nitrogen,  23  oxygen,  by  weight.  It  generally  con- 
tains, however,  a  trace  of  water,  and  of  carbonic  acid  and  carburetted  hydrogen 
gases,  and  still  less  ammonia. 

Density  of  air.  Under  *'  normal"  or  "standard"  conditions  (sea  level, 
lat45°,  barometer  760  mm  =  29.922  ins,  temperature  0°C  =  32°F)  dry  air 
weig'hs  1.292673  kilograms  per  cubic  meter  *  =  2.17888  flbs  avoir  per  cubic  yard. 
For  other  lats  and  elevations — 

R 

Density,  in  kg  per  cu  m,   =  1.292673  X    r^   ,  ^  ^  X   (1  —0.002837  cos  2  lat)  * 

Is,  -f-  z  a 
where  R  =  earth's  mean  radius  =  6,366,198  meters;   h  ==  elevation  above  sea 
level,  in  meters.     For  other  temperatures,  see  below. 

Under  normal  conditions,  but  with  0.04  parts  carbonic  acid  (C  Oj)  in  100  parts 
of  air,  density  =  1.293052  kg  per  cu  m.f  =  2.17952  lbs  avoir  per  cu  yd. J 

The  atmospheric  pressure,  at  any  given,  place,  may  vary  2  inches  or 
more  from  day  to  day.  Tne  average  pressure,  at  sea  level  |  varies  from 
about  745  to  770  millimeters  of  mercury  according  to  the  latitude  and  locality. 
760  millimeters*  is  generally  accepted  as  the  mean  atmospheric  pressure,  and 
called  an  atmosphere.  The  '^  metric  atmosphere,''  taken  arbitrarily 
at  1  kilogram  per  square  centimeter,  is  in  general  use  in  Continental  Europe. 
The  pressure  diminishes  as  the  altitude  increases,!  Therefore,  a  pump  in  a  high 
region  will  not  lift  water  to  as  great  a  height  as  in  a  low  one.  The  pressure  of 
air,  like  that  of  water,  is,  at  any  given  point,  equal  in  all  directions. 

It  is  often  stated  that  the  temperature  of  the  atmosphere  lowers  at 
the  rate  of  1°  Fah  for  each  300  feet  of  ascent  above  the  earth's  surface: 
but  this  is  liable  to  many  exceptions,  and  varies  much  with  local  causes.  Actual 
observation  in  balloons 'seems  to  show  that,  up  to  the  first  1000  feet,  1°  in  about 
200  feet  is  nearer  the  truth  ;  at  2000  feet,  1°  in  250  feet ;  at  4000  feet,  1°  in  300  feet ; 
and,  at  a  mile,  1°  in  350  feet. 

In  breathing*,  a  grown  person  at  rest  requires  from  0.25  to  0.35  of  a  cubic 
foot  of  air  per  minute;  which,  when  breathed,  vitiates  from  3.5  to  5  cubic  feet. 
When  walking,  or  hard  at  work,  he  breathes  and  vitiates  two  or  three  times  as 
much.  About  5  cubic  feet  of  fresh  air  per  person  per  minute  are  required  for  the 
perfect  ventilation  of  rooms  in  winter ;  8  in  summer.     Hospitals  40  to  80. 

Beneath  the  general  level  of  the  surface  of  the  earth,  in  temperate 
regions,  a  tolerably  uniform  temperature  of  about  50°  to  60°  Fah  exists  at 
the  depth  of  about  50  to  60  feet ;  and  increases  about  1°  for  each  additional  50  to 
60  feet;  all  subject,  however,  to  considerable  deviations  owing  to  many  local 
causes.  In  the  Rose  Bridge  Colliery,  England,  at  the  depth  of  2424  feet,  the 
temperature  of  the  coal  is  93.5°  Fah ;  and  at  the  bottom  of  a  boring  4169  feet 
deep,  near  Berlin,  the  temperature  is  119°. 

The  air  is  a  very  sloiv  conductor  of  heat^  hence  hollow  walls 
serve  to  retain  the  heat  in  dwellings;  besides  keeping  them  dry.  It  rushes 
into  a  vacuum  near  sea  level  with  a  velocity  of  about  1157  feet  per  second  ; 
or  13.8  miles  per  minute ;  or  about  as  fast  as  sound  ordinarily  travels  through 
quiet  air.    See  Sound. 

liike  all  other  elastic  fluids,  air  expands  equally  with 
equal  increases  of  temperature.  Every  increase  of  5°  Fah,  expands 
the  bulk  of  any  of  them  slightly  more  than  1  per  cent  of  that  which  it  has  at  0® 
Fah  ;  or  500°  about  doubles  its  bulk  at  zero.  The  bulk  of  any  of  them  diminishes 
inversely  in  proportion  to  the  total  pressure  to  which  it  is  subjected. 

This  holds  good  with  air  at  least  up  to  pressures  of  about  750  lbs  per  square 
inch,  or  50  times  its  natural  pressure  ;  the  air  in  this  case  occupying  one-fiftieth 
of  its  natural  bulk.  In  like  manner  the  bulk  will  increase  as  the  total  pressure 
is  diminished.    Substances  which  follow  these  laws,  are  said  to  be  perfectly 

*H.  V.  Regnault,  Memoires  de  1' Academic  Rovale  des  Sciences  de  I'lnstitut  de 
France,  Tome  XXI,  1847.  Translation  in  abstract,  Journal  of  Franklin  Insti- 
tute, Phila.,  June,  1848. 

fTravaux  et  Memoires  du  Bureau  International  desPoidset  Mesures,  Tomel, 
page  A  54.  Smithsonian  Meteorological  Tables,  1893,  published  in  Smithsoniam 
Miscellaneous  Collections,  Vol.  XXXV,  1897. 

t  See  Conversion  Tables. 

I  See  Leveling  by  the  Barometer. 


WIND. 


321 


elastiCo  Under  a  pressure  of  about  5}^  tons  per  square  inch,  air  would  become 
as  dense  as  water.  Since  the  air  at  the  surface  of  the  earth  is  pressed  14%  Bbs  per 
square  inch  by  the  atmosphere  above  it,  and  since  this  is  equal  to  the  weight  of  a 
column  of  water  1  inch  square  and  34  feet  high,  it  follows  that  at  the  depths  of 
34,  68,  102  feet,  &c,  below  water,  air  will  be  compressed  into  3^,  %,  %,  &c, 
of  its  bulk  at  the  surface. 

In  a  diviiig'-bell,  men,  after  some  experience,  can  readily  work  for  several 
hours  at  a  depth  of  51  feet,  or  under  a  pressure  of  2^  atmospheres ;  or  373^  lbs 
per  square  inch.  But  at  90  feet  deep,  or  under  3.64  atmospheres,  or  nearly  55 
fcs  per  square  inch,  they  can  work  for  but  about  an  hour,  without  serious  suffer- 
ing from  paralysis,  or  even  danger  of  death.  Still,  at  the  St.  Louis  bridge,  work 
was  done  at  a  depth  of  1103^  feet ;  pressure  63.7  lbs  per  square  inch. 

Tlie  dew^  point  is  that  temp  (varying)  at  which  the  air  deposits  its  vapor. 

Tlie  greatest  tieat  of  tlie  air  in  tlie  sun  probably  never  exeeeds 
145°  Fah ;  nor  the  greatest  cold  —  74°  at  niglit.  About  130°  above,  and  40°  below 
zero,  are  the  extremes  in  the  U.  S.  east  of  the  Mississippi;  and  65°  below  in  the 
N.  W. ;  all  at  common  ground  level.  It  is  stated,  however,  that  — 81°  has  t.een 
observed  in  N.  E.  Siberia;' and  +101°  Fah  in  the  shade  in  Paris;  and  +153°  in 
the  sun  at  Greenwich  Observatory,  both  in  July,  1881.  It  has  frequently  ex- 
ceeded +100°  Fah  in  the  shade  in  Philadelphia  during  recent  years. 


WIND. 

The  relation  between  tlie  velocity  of  wind,  and  its  press- 
ure against  an  obstacle  placed  either  at  right  angles  to  its  course,  or  inclined 
to  it,  has  not  been  weM  determined ;  and  still  less  so  its  pressure  against  curved 
surfaces.  The  pressure  against  a  large  surface  is  probably  proportionally  greater 
than  against  a  small  one.  It  is  generally  supposed  to  vary  nearly  as  the  squares 
of  the  velocities;  and  when  the  obstacle  is  at  right  angles  to  its  direction,  the 
pressure  in  Bbs  per  square  foot  of  exposed  surface  is  considered  to  be  equal  to 
the  square  of  the  velocity  in  miles  per  hour,  divided  by  200.  On  this  basis, 
which  is  probably  quite  defective,  the  following  table,  as  given  by  Smeaton,  is 
prepared. 


Vel.  in  Miles 

Vel.  in  Ft. 

Pres.  in  Lbs. 

Remarks. 

per  Hour. 

per  Sec. 

per  Sq.  Ft. 

1 

1.467 

.005 

Hardly  perceptible.               

Pleasant.                           ^^^ff 

2 

2.933 

.020 

3 

4.400 

.045 

c 

4 

5.867 

.080 

5 
10 

7.33 
14.67 

.125 
.5 

rk^/jh 

12}4 

18.33 

.781 

Fresh  breeze.                           O 

15 

22. 

1.125 

20 

29.33 

2. 

^  .  ,      .    ^                     The  pres  against 

25 

36.67 

3.125 

Brisk  wind.                 a     semicylindrical 

30 

44. 

4.5 

Strong  wind.               surface  acbnom 

40 

58.67 

8. 

High  wind.                  is  about  half  that 

50 

73.33 

12.5 

Storm.                          against     the      flat 

60 

88. 

'  18. 

Violent  storm.            surf  a 6  nm. 

80 

117.3 

32. 

Hurricane. 

100 

146.7 

50. 

Violent  hurricane,  uprooting  large  trees. 

Tredg-old  recommends  to  allow  40  lbs  per  sq  ft  of  roof  for  the 

pres  of  wind  against  it  ;  but  as  roofs  are  constructed  with  a  slope,  and  consequently  do  not  receive 
the  full  force  of  the  wind,  this  is  plainly  too  much.*  Moreover,  only  one-half  of  a  roof  is  usually  ex- 
posed, even  thus  partially,  to  the  wind.  Probably  the  force  in  suoh  cases  varies  approximately  as  the 
sines  of  the  angles  of  slopes.  According  to  observations  in  Liverpool,  in  1860,  a  wind  of  38  miles  per 
hour,  produced  a  pres  of  14  lbs  per  sq  ft  against  an  object  perp  to  it:  and  one  of  70  miles  per  hour, 
(the  severest  gale  on  record  at  that  city,)  42  lbs  per  sq  foot.  These  would  make  the  pres  per  sq  ft, 
more  nearly  equal  to  the  square  of  the  vel  in  miles  per  hour,  div  by  100;  or  nearly  twice  as  great  as 
given  in  Smeaton's  table.  We  should  ourselves  give  the  preference  to  the  Liverpool  observations.  A 
very  violent  gale  in  Scotland,  registered  by  an  excellent  anemometer,  or  wind-gauge,  45  R)s  per  sq 
ft.    It  is  stated  that  as  high  as  55  B)s  has  been  observed  at  Glasgow.     High  winds  often  lift  roofs. 

The  gauge  at  Girard  College,  Philada,  broke  under  a  strain  of  42  lbs  per  sq  ft;  a  tornado  passing 
at  the  moment,  within  34  mile. 

By  inversion  of  Smeaton's  rule,  if  the  force  in  lbs  per  sq  ft,  be  mult  by  200,  the  sq  rt  of  the  prod 
will  give  the  vel  in  miles  per  hour.    Smeaton's  rule  is  used  by  the  U.  S.  Signal  Service. 

*  The  writer  thinks  8  lbs  per  sq  foot  of  ordinary  double-sloping  rooft,  or  16  lbs  for  shed-roofs,  sufia- 
cient  allowance  for  pres  of  wind. 

21 


322 


RAIN   AND   SNOW. 


RAIN  AND  SNOW. 

Tlie  annnal  precipitation  "i"  at  any  given  place  varies  greatly  from 
year  to  year,  the  ratio  between  maximum  and  minimum  being  frequently  greater 
than  2:1.  Beware  of  averag'es.  In  estimating ^00^5,  take  the  maximum 
falls,  and  in  estimating  water  supply,  the  minimum,  not  only  per  annum,  but  for 
short  periods.  In  estimating  water  supply,  make  deductions  for  evaporation 
and  leakage. 

Maxima  and  minima  deduced  from  observations  covering  only  4  or  5  years  are 
apt  to  be  misleading.  Data  covering  even  10  or  more  years  may  just  miss  includ- 
ing a  very  severe  flood  or  drought.  Records  of  from  15  to  20  years  may  usually 
be  accepted  as  sufficient. 

Table  1.   Average  Precipitation*  in  the  United  States,  in  ins. 
(From  Bulletin  C  of  U.  S.  Department  of  Agriculture,  compiled  to  end  of  1891.) 


State.  Spr. 

Alabama 14.9 

Arizona 1.3 

Arkansas 14.3 

California 6.2 

Colorado 4.2 

Connecticut 11.1 

Delaware 10.2 

Dist.  Columbia.11.0 

Florida 10.2 

Georgia 12.4 

Idaho 4.4 

Illinois 10.2 

Indiana 11. 0 

Indian  T'y 10.6 

Iowa 8.3 

Kansas 8.9 

Kentucky 12.4 

Louisiana 13.7 

Maine 11.1 

Maryland 11.4 

Massachusetts.  ..11.6 

Michigan 7.9 

Minnesota 6.5 

Mississippi 14.9 

Missouri 10.0 


Sum. 

Aut. 

Win.  Ann'l 

13.8 

10.0 

14.9    53.6 

4.3 

2.2 

3.1    10.9 

12.5 

11.0 

12.8  50.6 

0.3 

3.5 

11.9   21.9 

5.5 

2.8 

2.3   14.8 

12.5 

11.7 

11.5  46.8 

11.0 

10.0 

9.6   40.8 

12.4 

9.4 

9.0  41.8 

21.4 

14.2 

9.1   54.9 

15.6 

10.7 

12.7  61.4 

2.1 

3.6 

7.0  17.1 

11.2 

9.0 

7.7  38.1 

11.7 

9.7 

10.3  42.7 

11.0 

8.9 

5.7  36.2 

12.4 

8.1 

4.1   32.9 

11.9 

67 

3.5   31.0 

12.5 

9.7 

11.8  46.4 

15.0 

10.8 

14.4  53.9 

10.5 

12.3 

11.1   45.0 

12.4 

10.7 

9.5  44.0 

11.4 

11.9 

11.7  46.6 

9.7 

9.2 

7.0  33.8 

10.8 

5.8 

8.1  26.2 

12.6 

10.1 

15.4  53.0 

12.4 

9.1 

6.5  38.0 

State.      .   .  Spr. 

Montana 4.2 

Nebraska 8.9 

Nevada 2.3 

N.  Hampshire.  9.8 

New  Jersey 11.7 

New  Mexico 1.4 

New  York 8.5 

N.  Carolina 12.9 

N.  Dakota 4.6 

Ohio 10.0 

Oregon 9.8 

Pennsylvania. ..10.3 
Rhode  Island  ...11.9 

S.  Carolina 9.8 

S.  Dakota 7.2 

Tennessee 13.5 

Texas 8.1 

Utah 3.4 

Vermont 9.2 

Virginia 10.9 

Washington 8.6 

W.  Virginia 10.9 

Wisconsin 7.8 

Wyoming 4.3 

United  States...  9.2 


Sum. 

4.9 
10.9 

0.8 
12.2 
13.3 

5.8 
10.4 
16.6 

8.0 
11.9 

2.7 
12.7 
10.7 
16.2 

9.7 
12.5 

8.6 

1.5 
12.2 
12.5 

3.9 
12.9 
11.6 

3.5 
10.3 


Aut.  Win.  Ann'l 

2.6  2.3  14.0 
4.9  2.2  26.9 
1.3  3.2  7.6 

11.4  10.7  44.1 
11.2  11.1  47.3 

3.5  2.0  12.7 

9.7  7.9  36.5 
12.0  12.2  53.7 

2.8  1.7  17.1 
9.0  9.1  40.0 

10.5  21.0  44.0 
10.0  9.5  42.5 
11.7  12.4  46.7 

e.7  9.7  45.4 

3.5  2.5  22.9 
10.2  14.5  50.7 

7.6  6.0  30.3 
3.5  10.6 
9.3  42.1 
9.7  42.6 

16.8  39.8 

9.0  10.0  42.8 

7.8  5.2  32.5 

2.2  1.6  11.6 

8.3  8.6  36.3 


2.2 
11.4 

9.5 
10.5 


At  Philadelphia,  in  1869,  during  which  occurred  the  greatest  drought  known 
there  for  at  least  50  years,  43.21  inches  fell ;  August  13,  1873,  7.3  inches  in  1  day; 
August,  1867,  15.8  inches  in  1  month;  July,  1842,  6  inches  in  2  hours;  9  inches 
per  month  not  more  than  7  or  8  times  in  25  years.  From  1825  to  1893,  greatest 
in  one  year,  61  inches,  in  1867  ;  least,  30  inches,  in  1825  and  1880.  At  Norristown, 
Pennsylvania,  in  1865,  the  writer  saw  evidence  that  at  least  9  inches  fell  within 

5  hours.  At  Genoa,  Italy,  on  one  occasion,  32  inches  fell  in  24  hours ;  at  Geneva, 
Switzerland,  6  inches  in  3  hours  ;  at  Marseilles,  France,  13 inches  in  14  hours; 
in  Chicago,  Sept.,  1878,  .97  inch  in  7  minutes. 

Near  l<ondon,  Kng^Ianci,  the  mean  total  fall  for  many  years  is  23  inches. 
On  one  occasion,  6  inches  fell  in  1}^  hottrs !  In  the  mountain  districts  of  the 
English  lakes,  the  fall  is  enormous ;  reaching  in  some  years  to  180  or  240  inches; 
or  from  15  to  20  feet !  while,  in  the  adjacent  neighborhood,  it  is  but  40  to  60 
inches.  At  Liverpool,  the  average  is  34  inches;  at  Edinburgh,  30:  Glasgow,  22; 
Ireland,  36;  Madras,  47;  Calcutta,  60;  maximum  for  16  vears,  82;  Delhi,  21; 
Gibraltar,  30 ;  Adelaide,  Australia,  23  ;  West  Indies,  36  to  96 ;  Rome,  39.  On  the 
Khassya  hills  north  of  Calcutta,  500  inches,  or  41  feet  8  inches,  have  fallen  in  the 

6  rainy  months  !  In  pther  mountainous  districts  of  India,  annual  falls  of  10  to 
20  feet  are  common, 

A  moderate  steady  rain,  continuing  24  hours,  will  yield  a  depth  of  about  an  inch. 

As  a  s^eneral  rule,  more  rain  falls  in  warm  than  in  cold 

eountries;  and  more  in  elevated  regions  than  in  low  ones.     Local  peculiar- 

*  Precipitation  includes  snow,  hail,  and  sleet,  melted.  Unmelted  snow  is 
estimated  at  10  inches  snow  =  1  inch  rain. 


RAIN   AND  SNOW. 


323 


ities,  however,  aoraetiraes  reverse  this ;  and  also  cause  great  differences  in  the 
amounts  in  places  quite  near  each  other;  as  in  the  English  lake  districts  just 
alluded  to.  It  is  sometimes  dit&cult  to  account  for  these  variations,  in  some 
lagoons  in  New  Granada,  South  America,  the  writer  has  known  three  or  four 
heavy  raina  to  occur  weekly  for  some  months,  during  which  not  a  drop  fell  on 
hills  about  1000  feet  high,  within  ten  miles'  distance,  and  within  full  sight.  At 
another  locality,  almost  a  dead-level  plain,  fully  three-quarters  of  the  rains  that 
fell  for  two  years,  at  a  spot  two  miles  from  his  residence,  occurred  in  the  morn- 
ing ;  while  those  which  fell  about  three  miles  from  it,  in  an  opposite  direction, 
were  in  the  afternoon. 

Tlie  relation  between  precipitation  and  stream-flow  is  greatly 
affected  by  the  existence  of  forests  or  crops,  by  the  slope  and  character  of  ground 
on  the  water-shed,  especially  as  to  rate  of  absorption,  by  the  season  of  the  year, 
the  frost  in  the  ground,  etc.  The  stream-flow  may  ordinarily  be  taken  as  vary- 
ing between  0.2  and  0.8  of  the  rainfall.  Streams  in  limestone  regions  frequently 
lose  a  very  large  proportion  of  their  flow  through  subterranean  caverns. 

Assuming  a  fall  of  2  feet  in  1  year  (==  76,379  cubic  feet  per  square  mile  per 
day),  that  half  the  rainfall  is  available  for  water  supply,  and  that  a  per  capita 
consumption  of  4  cubic  feet  (=  30  gallons)  per  day  is  sufficient,  one  square  mile 
will  supply  19,095  persons ;  or  a  square  of  38.25  feet  on  a  side  will  supply  one 
person. 

An  inch  of  rain  anionnt!>a  to  3630  cnMc  feet;  or  27155  U.  S. 
gallons;  or  101.3  tons  per  acre;  or  to  2323200  cubic  feet;  or  17378743  U.  S.  gal- 
lons ;  or  64821  tons  per  square  mile  at  623^  ft)s  per  cubic  foot. 

The  most  destructive  rains  are  usually  those  which  fall  upon  snow,  under 
which  the  ground  is  frozen,  so  as  not  to  absorb  water. 

Table  2.    Maximum  intensity  of  rainfall  for  periods  of  5. 10,  and 

60  minutes  at  Weather  Bureau  stations  equipped  with  self-registering 

gauges,  compiled  from  all  available  records  to  the  end  of  1896. 

(From  Bulletin  D  of  U.  S.  Department  of  Agriculture.) 


Stations. 

Rate  per  hour  for — 

Stations. 

Rate  per  hour  for — 

5min. 

lOmins. 

60mins. 

5min. 

lOmins. 

60mins. 

Ins. 
9.00 
8.40 
8.16 
7.80 
7.80 
7.50 
7.44 
7.20 
7.20 
6.72 
6.60 
6.60 
6.60 

Inches. 
6.00 
6.00 
4.86 
4.20 
6.60 
5.10 
7.08 
6.00 
4.92 
4.98 
6.00 
3.90 
4.80 

Inches. 
2.00 
1.30 
2.18 
1.25 
2.40 
1.78 
2.20 
2.15 
1.60 
1.68 
2.21 
1.60 
1.86 

Chicago 

Ins. 
6.60 
6.48 
6.00 
6.00 
5.76 
5.64 
5.46 
5.40 
5.40 
4.80 
4.56 
3.60 
3.60 

Inches. 
5.92 
5.58 
4.80 
4.20 
5.46 
3.66 
5.46 
4.80 
4.02 
3.84 
4.20 
3.30 
2.40 

Inches. 
1.60 

St.  Paul  ..  .. 

Galveston 

Omaha 

2.55 

New  Orleans 

1.55 

Milwaukee 

Kansas  City 

Washington 

Jacksonville    .. 

Dodge  City 

Norfolk 

1.34 
1  55 

Cleveland 

Atlanta  

1.12 
1  50 

Detroit 

Key  West 

Philadelphia... 

St.  Louis 

Cincinnati 

Denver 

2.25 

New  York  City.. 
Boston 

1.50 
2.25 

1.70 

Indianapolis 

1.18 

Duluth 

1.35 

The  w^ei^ht  of  freshly  fallen  snow^,  as  measured  by  the  author, 

Taries  from  about  5  to  12  ft)s  per  cubic  foot ;  apparently  depending  chiefly  upon 
the  degree  of  humidity  of  the  air  through  which  it  had  passed.  On  one  occasion, 
when  mingled  snow  and  hail  had  fallen  to  the  depth  of  6  inches,  he  found  its 
weight  to  be  31  ibs  per  cubic  foot.  It  was  very  dry  and  incoherent.  A  cubic  foot 
of  heavy  snow  may,  by  a  gentle  sprinkling  of  water,  be  converted  into  about 
half  a  cubic  foot  of  slush,  weigbing  20  lbs.;  which  will  not  slide  or  run  off 
from  a  shingled  roof  sloping  30°,  if  the  weather  is  cold.  A  cubic  block  of  snow 
saturated  with  water  until  it  weighed  45  lbs  per  cul)ic  foot,  just  slid  on  a  rough 
board  inclined  at  45°;  on  a  smoothly  planed  one  at  30°;  and  on  slate  at  18°;  all 
approximate.  A  prism  of  snow,  saturated  to  52  Tbs  per  cubic  foot,  one  inch 
square,  and  4  inches  high,  bore  a  weigrht  of  7  fibs  ;  which  at  first  compressed 
it  about  one-quarter  part  of  its  length.  European  engineers  consider  6  lbs  per 
square  foot  of  roof  to  be  sufficient  allowance  for  the  Aveig^ht  of  snow^ ; 


324  RAIN   AND  SNOW. 

and  8  !bs  for  the  pressure  of  wind ;  total,  14  lbs.  The  writer  thinks  that  in  the 
U.  S.  the  allowance  for  snow  should  not  be  taken  at  less  than  12  lbs ;  or  the  total 
for  snow  and  wind,  at  20  lbs.  There  is  no  danger  that  snow  on  a  roof  will 
become  saturated  to  the  extent  just  alluded  to ;  because  a  rain  that  would  supply 
the  necessary  quantity  of  water  would  also  by  its  violence  wash  away  the  snow ; 
but  we  entertain  no  doubt  whatever  that  the  united  pressures  from  snow  and 
wind,  in  our  Northern  States,  do  actually  at  times  reach,  and  even  surpass, 
20  fibs  per  square  foot  of  roof.  Tbe  limit  of 

perpetual  $iiiow  at  the  equator  is  at  the  height  of  about  16000  feet,  or  say 
3  miles  above  sea-level;  in  lat  45°  north  or  south,  it  is  about  half  that  height; 
while  near  the  poles  it  is  about  at  sea-level. 

Rain  Oau^es.  Plain  cylindrical  vessels  are  ill  adapted  to  service  as  rain 
gauges;  because  moderate  rains,  even  though  sufficient  to  yield  a  large  run-off 
from  a  moderate  area,  are  not  of  sufficient  depth  to  be  satisfactorily  measured 
unless  the  depth  be  exaggerated.  The  inaccuracy  of  measurement,  always  con- 
siderable, is  too  great  relatively  to  the  depth. 

In  its  simplest  and  most  usual  form,  the  gauge  (see  Fig.)  consists  essentially 
of  a  funnel,  A,  which  receives  the  rain  and  leads  it  into  a  measuring 
tube,  B,  of  smaller  cross-section.     The  funnel  should  have  a  vertical 


and  fairly  sharp  edge,  and,  in  order  to  minimize  the  loss  through        \A/^ 
evaporation,  it  should  fit  closely  over  the  tube,  and  its  lower  end  C^ 

should  be  of  small  diameter. 

The  depth  of  water  in  the  tube  is  ascertained  by  inserting,  to  the 
bottom  of  the  tube,  a  measuring  stick  of  some  unpolished  wood 
which  will  readily  show  to  what  depth  it  has  been  wet.  The  stick 
may  be  permanently  graduated,  or  it  may  be  compared  with  an  ordi- 
nary scale  at  each  observation.  The  tube  is  usually  of  such  diameter 
that  the  area  of  its  cross-section,  minus  that  of  the'stick,  is  one-tenth 
of  the  area  of  the  funnel  mouth.  The  depth  of  rainfall  is  then  one- 
tenth  of  the  depth  as  measured  by  the  stick. 


Dimensions  of  Standard  U.  S.  Weather  Bureau  Rain  Gauge.  Ins. 

A.  Receiver  or  funnel.  Diameter         8 

B.  Measuring  tube.  Height  20  ins.  "  2.53 
C  C.    Overflow  attachment  and  snow  gauge.                         "  8 

Such  gauges,  with  the  tubes  carefully  made  from  seamless  drawn  brass  tubing, 
cost  about  $5.00  each  ;  but  an  intelligent  and  careful  tinsmith,  given  the  dimen- 
sions accurately,  can  construct,  of  galvanized  iron,  for  about  $1.00  a  gauge  that 
will  answer  every  purpose  of  the  engineer. 

The  exposure  has  a  very  marked  effect  upon  the  results  obtained.  The 
funnel  should  be  elevated  about  3  ft,  in  order  to  prevent  rain  from  splashing  back 
into  it  from  the  ground  or  roof.  If  on  a  roof,  the  latter  should  be  fiat,  and  pref- 
erably 50  ft  wide  or  wider,  and  the  gauge  should  be  placed  as  far  as  possible 
from  the  edges  ;  else  the  air  currents,  produced  by  the  wind  striking  the  side  of 
the  building,  will  carry  some  of  the  rain  over  the  gauge.  No  objects  much  higher 
than  the  gauge  should'  be  near  it,  as  they  produce  variable  air  currents  which 
may  seriously  affect  its  indications. 

An  overflow  tank,  C,  should  be  provided,  for  cases  of  overfilling  the  tube. 

Water,  freezing  in  the  gauge,  may  burst  it,  or  force  the  bottom  off",  or  at  least 
so  deform  the  gauge  as  to  destroy  its  accuracy. 

To  measure  snow,  the  funnel  is  removed,  and  the  snow  is  collected  in 
the  overflow  attachment  or  other  cylindrical  vessel  deep  enough  to  prevent  the 
snow  from  being  blown  out,  and  the  cross-sectional  area  of  which  is  accurately 
known.  The  snow  is  then  melted,  either  by  allowing  it  to  stand  in  a  warm 
place,  or,  with  less  loss  through  evaporation,  by  adding  an  accurately  known 
quantity  of  luke-warm  water.  In  the  latter  case,  the  volume  of  the  added  water 
must  of  course  be  deducted  from  the  measurement. 

Sainfall  equivalent  of  snow.  Ten  inches  of  snow  are  usually  taken 
as  equivalent  to  1  in  of  rain  ;  but,  according  to  various  authorities,  the  equiva- 
lent may  vary  between  2>^  and  34;  i.  e.,  between  25  and  1.84  ft)s.  per  cubic  foot. 

Self-recording:  gauges,  of  which  several  forms  are  on  the  market,  are 
quite  expensive,  and,  even  when  purchased  from  regular  makers,  seldom  per- 
fectly reliable.  Gauges  using  a  small  tipping  bucket  register  inaccurately  in 
heavy  rains ;  those  using  a  float  are  limited  as  to  the  total  depth  which  they  can 
register ;  while  those  which  weigh  th«  rain,  if  exposed,  are  affected  by  wind. 


EAIN  AND  8NOW. 


825 


Table  3.    Details  of  Precipitation,  in  the  United  States. 

Records  from  1871  or  earlier,  to  1891  inclusive. 

(From  Bulletin  C  of  U.  S.  Department  of  Agriculture,  1894.) 


State. 


City. 


'  o  ^ 


Trace 

to 

0.25 

inch. 


Average  percentage 
days  with  rain.  * 


0.26 
to 

0.50 
in. 


0.51 

to 
1.00 

in. 


1.012.01 

to     to 

2.00  3.00 

ins.  ins. 


Max.  No. 
of  con- 
sec,  days 


-a  5 


Alabama 

Arizona 

Arizona 

California 

Colorado 

Connecticut 

Dis  of  Columbia, 

Florida 

Florida 

Georgia 

Georgia 

Illinois 

Illinois 

Indiana 

Iowa 

Iowa i. 

Kansas 

Louisiana 

Louisiana 

Maine 

Maryland 

Massachusetts... 

Michigan 

Michigan 

Minnesota 

Minnesota 

Mississippi 

Missouri 

Nebraska 

New  Jersey 

New  Mexico 

New  York 

New  York 

New  York 

North  Carolina.. 

Ohio 

Ohio 

Ohio 

Pennsylvania ... 
Pennsylvania ... 
South  Carolina.. 

Tennessee 

Tennessee 

Tennessee 

Texas 

Virginia 

Virginia 

Wisconsin 

Wyoming 


Mobile 

Yumaf 

Whipple  Barr'ks 
San  Francisco.... 

Denver 

New  London 

Washington 

Jacksonville 

Key  West 

Augusta 

Savannah 

Cairo 

Chicago 

Indianapolis 

Davenport 

Keokuk 

Leavenworth 

New  Orleans 

Shreveport 

Portland 

Baltimore 

Boston 

Detroit 

Marquette 

Duluth 

St.  Paul 

Vicksburg 

St.  Louis 

Omaha 

Atlantic  City  J... 

Fort  Wingate 

Buffalo 

New  York 

Oswego 

Wihuington 

Cincinnati 

Cleveland 

Toledo 

Philadelphia  §.... 

Pittsburg , 

Charleston , 

Knoxville , 

Memphis , 

Nashville , 

Galveston , 

Lynchburg , 

Norkfolk 

Milwaukee 

Cheyenne , 


40.1 
6.6 

15.7 
21.8 
28.7 
43.0 
40.8 
44.7 
38.8 
37.2 
36.4 
38.5 
44.0 
44.2 
40.8 
35.4 
31.9 
40.6 
35.9 
43.4 
41.2 
39.8 
46.7 
49.0 
44.5 
38.9 
36.4 
38.4 
34.1 
41.0 
14.6 
54.2 
39.6 
58.3 
36.2 
46.4 
52.2 
44.3 
43.6 
51.3 
34.7 
43.4 
39.8 
42.0 
34.6 
40.3 
42.0 
44.7 
30.5 


24.3 
5.6 

10.2 
14.2 
24.2 
28.1 
27.2 
29.8 
27.9 
22.9 
22.3 
25.0 
32.5 
30.3 
29.7 
24.5 
20.7 
24.9 
22.6 
29.8 
27.5 
25.8 
36.0 
39.0 
34.7 
30.3 
21.1 
26.4 
24.4 
27.5 
10.1 
41.3 
25.7 
46.5 
20.7 
32.5 
40.2 
33.9 
30.6 
38.3 
20.6 
26.8 
24.4 
26.4 
22.8 
26.2 
26.2 
34.0 
26.7 


7.3 
1.7 
4.2 
4.7 
6.5 
11.8 
4.2 
6.2 
8.2 
4.9 
8.6 
4.2 
5.6 
4.3 
5.2 
4.8 
3.6 
8.9 
6.9 
3 

4.5 
4.4 
3.9 
5.2 
3.1 
3.7 
5.4 
4.5 
5.0 
9.0 
4.5 
3.2 
6.2 
3.6 
8.0 
3.0 
3.6 
3.2 
5.2 
3.6 
8.3 
5.6 
8.9 
5.2 
7.9 
4.7 
4.6 
3.7 
1.9 


*  For  instance,  Alabama,  Mobile,  trace  to  0.25  inch,  24.3,  means  that  on  24.3  per 
cent,  of  the  days  embraced  within  the  20  years,  rain  fell  to  a  depth  of  from  a 
trace  to  0.25  inch. 

t  From  October  1875  only.    J  From  January  1874  only.    §  From  May  1872  only. 


326  WATER. 


WATER. 

Pure  water,  as  boiled  and  distilled,  is  composed  of  the  two  gases,  hydro, 
gen  and  oxygen;  in  the  proportions  of  2  measures  hydrogen  to  1  of  oxygen; 
or  1  weight  of  hydrogen  to  8  of  oxygen.  Ordinarily,  however,  it  contains  sev- 
eral foreign  ingredients,  as  carbonic  and  other  acids;  and  soluble  mineral,  or 
organic  substances.  When  it  contains  much  lime,  it  is  said  to  be  hard;  and  will 
not  make  a  good  lather  with  soap.  Tlie  air  in  its  ordinary  state  contains 
about  4  grains  of  water  per  cubic  foot. 

The  average  pressure  of  tlie  air  at  sea  level,  wiW  balance  a 
column  of  ivater  34  feet  high  ;  or  about  30  inches  of  mercury.  At  its  boil- 
ing point  of  212°  Fah,  its  bulk  is  about  one  twenty-third  greater  than  at  70°. 

Its  wei^lit  per  cubic  foot  is  taken  at  623^  lbs, or  1000  ounces  avoir;  but  623^ 
ft)s  would  be  nearer  the  truth,  as  per  table  below.  It  is  about  815  times  heavier 
than  air,  wlien  both  are  at  the  temperature  of  62°;  and  the  barometer  at  30 
inches.  With  barometer  at  30  inches  the  weight  of  perfectly  pure  water  is  as 
follows.    At  about  39°  it  has  its  maximum  density  of  62.425  lbs  per  cubic  foot. 


Temp,  Fah.  Lbs  per  Cub  Ft. 

32° 62.417 

40° 62.423 

50° 62.409 

60° 62.367 


Temp,  Fah.  Lbs  per  Cub  Ft. 

70° 62.302 

80° 62.218 

90°.. 62.119 

212°- .59.7 


Weight  of  sea  iprater  64.00  to  64.27  Rs  per  cubic  foot,  or  say  1.6  to  1.9  fts 

per  cubic  foot  more  than  fresh  water.     See  also  p  328. 

Water  has  its  maximum  density  when  its  temperature  is  a  little  above 
39°  Fah  ;  or  about  7°  above  the  freezing  point.  By  best  authorities  39.2°.  From 
about  39°  it  expands  either  by  cold,  or  by  heat.  When  the  temperature  of  32° 
reduces  it  to  ice,  its  weight  is  but  about  57.2  ft)s.  per  cubic  foot ;  and  its  specific 
gravity  about  .9175,  according  to  the  investigations  of  L.  Dufour.  Hence,  as 
ice,  it  has  expanded  one-twelfth  of  its  original  bulk  as  water ;  and  the  sudden 
expansive  force  exerted  at  the  moment  of  freezing,  is  sufficiently  great  to 
split  iron  water-pipes;  being  probably  not  less  than  30000  lbs  per  square  inch. 
Instances  have  occurred  of  its  splitting  cast  tubular  posts  of  iron  bridges,  and 
of  ordinary  buildings,  when  full  of  rain  water  from  exposure.  It  also  loosens 
and  throws  down  masses  of  rock,  through  the  joints  of  which  rain  or  spring 
water  has  found  its  way.  Retaining- walls  also  are  sometimes  overthrown,  or 
at  least  bulged,  by  the  freezing  of  water  which  has  settled  between  their  backs 
and  the  earth  filling  which  they  sustain  ;  and  walls  which  are  not  founded  at  a 
sufficient  depth,  are  often  lifted  upward  by  the  same  process. 

It  is  said  that  in  a  grlass  tube  }^  inch  in  diameter,  water  will  not 
freeze  until  the  temperature  is  reduced  to  23°;  and  in  tubes  of  less  than  -^ 
inch,  to  3°  or  4°.  Neither  will  it  freeze  until  considerably  colder  than  32°  in 
rapid  running  streams.  Anchor  ice,  sometimes  found  at  depths  as  great  as 
25  feet,  consists  of  an  aggregation  of  small  crystals  or  needles  of  ice  frozen  a*' 
the  surface  of  rapid  open  water;  and  probably  carried  below  by  the  force  of  tho 
stream.   It  does  not  form  under  frozen  water. 

Since  ice  floats  in  water;  and  a  floating  body  displaces  a  weight  of  the 
liquid  equal  to  its  own  weight,  it  follows  that  a  cubic  foot  of  floating  ice  weighing 
57.2  ibs,  must  displace  57.2  lbs  of  water.  But  57.2  Bbs  of  water,  one  foot  square,  is  11 
inches  deep:  therefore,  floating  ice  of  a  cubical  or  parallelopipedal  shape.  \v\\\ 
have  \}i  of  its  volume  under  water;  and  only  ^^  above ;  and  a  square  foot  of  ice 
of  any  thickness,  will  require  a  weight  equal  to  ^  of  its  own  weight  to  sink  it 
to  the  surface  of  the  water.  In  practice,  however,  this  must  be  regarded  merely 
as  a  close  approximation,  since  the  weight  of  ice  is  somewhat  affected  by  en- 
closed air-bubbles. 

Pure  water  is  usually  assumed  to  boil  at  212°  Fah  in  the  open  air,  at  the 
level  of  the  sea ;  the  barometer  being  at  30  inches ;  and  at  about  1°  less  for  every 
520  feet  above  sea  level,  for  heights  within  1  mile.  In  fact,  its  boiling  point 
varies  like  its  freezing  point,  with  its  purity,  the  density  of  the  air,  the  material 
of  the  vessel,  &c.  In  a  metallic  vessel,  it  may  boil  at  210°;  and  in  a  glass  one, 
at  from  212°  to  220° ;  and  it  is  stated  that  if  all  air  be  previously  extracted,  it 
requires  275°. 

It  evaporates  at  all  temperatures;  dissolves  more  substances  than  any 
other  agent:  and  has  a  greater  capacity  for  heat  than  any  other  known  substance. 

It  is  compressed  at  the  rate  of  about  one-21740th,  (or  about  y^  of  an 
inch  in  18^^  feet,)  by  each  atmosphere  or  pressure  of  15  ibs  per  square  inch. 
When  the  pressure  is  removed.  U«  pinaHoUy  restores  its  original  bulk. 


WATER.  327 

Effect  on  metals.  The  lime  contained  in  many  waters,  forms  deposits  in 
metallic  water-pipes^  and  in  channels  of  earthenware,  or  of  masonry  ;  especially 
if  the  current  be  slow.  Some  other  substances  do  the  same;  obstructing  the 
flow  of  the  water  to  such  an  extent,  that  it  is  always  expedient  to  use  pipes  of 
diameters  larger  than  would  otherwise  be  necessary.  The  lime  also  forms  very 
hard  incrustations  at  tlie  bottoms  of  boilers;  very  much  impair- 
ing their  efficiency;  and  rendering  them  more  liable  to  burst.  Such  water  is 
unfit  for  locomc^ives.  We  have  seen  it  stated  that  the  Southwestern  R  R  Co, 
England,  prevent  this  lime  deposit,  along  their  limestone  sections,  by  dissolving 
1  ounce  of  sal-ammoniac  to  90  gallons  of  water.  The  salt  of  sea  water  forms 
similar  deposits  in  boilers;  as  also  does  mud,  and  other  impurities. 

Water,  either  when  very  pure,  as  rain  water ;  or  when  it  contains  carbonic 
acid,  (as  most  water,  does,)  produces  carbonate  of  lead  in  lead 
pipes;  and  as  this  is  an  active  poison,  such  pipes  should  not  be  used  for  such 
wat^ers.  Tinned  lead  pipes  may  be  substituted  for  them.  If,  however,  sulphate 
of  lime  also  be  present,  as  is  very  frequently  the  case,  this  effect  is  not  always 
produced;  and  several  other  substances  usually  found  in  spring  and  river 
water,  also  diminish  it  to  a  greater  or  less  degree.  Fresb  i¥ater  corrodes 
^¥roug-bt  iron  more  rapidly  than  cast;  but  the  reverse  appears  to 
be  the  case  with  sea  ivater;  although  it  also  affects  wrought  iron  very 
quickly;  so  that  thick  flakes  may  be  detached  from  it  with  ease.  The  corrosion 
of  iron  or  steel  by  sea  water  increases  with  the  carbon.  Cast-iron  cannons 
from  a  vessel  which  had  been  sunk  in  the  fresh  water  of  the  Delaware  River 
for  more  than  40  years,  were  perfectly  free  from  rust.  Gen.  Pasley,  who  had 
examined  the  metals  found  in  the  ships  Royal  George,  and  Edgar,  the  first  of 
which  had  remained  sunk  in  the  sea  for  62  years,  and  the  last  for  133  years, 
"stated  that  the  cast  iron  had  generally  become  quite  soft;  and  in  some  cases 
resembled  plumbago.  Some  of  the  shot  when  exposed  to  the  air  became  hot; 
and  burst  into  many  pieces.  The  wrought  iron  was  not  so  much  injured, 
except  when  in  cmiiact  with  copper,  or  brass  gun-metal.  Neither  of  these  last  was 
much  afiected,  except  when  in  contact  with  iron.  Some  of  the  wrought  iron 
was  reworked  by  a  blacksmith,  and  pronounced  superior  to  modern  iron."  "Mr. 
Cottam  stated  that  some  of  the  guns  had  been  carefully  removed  in  their  soft 
state,  to  the  Tower  of  London ;  and  in  time  (within  4  years)  resumed  their  orig- 
inal hardness.  Brass  cannons  from  the  Mary  Rose,  which  had  been  sunk  in  the 
sea  for  292  years,  were  considerably  honeycombed  in  spots  only;  (perhaps  where 
iron  had  been  in  contact  with  them.)  The  old  cannons,  of  wrought-iron  bars 
hooped  together,  were  corroded  about  i^  inch  deep;  but  had  probably  been  pro- 
tected by  mud.  The  cast-iron  shot  became  redhot  on  exposure  to  the  air;  and 
fell  to  pieces  like  dry  clay!" 

"Unprotected  parts  of  cast-iron  sluice-valves,  on  the  sea  gates  of  the  Cale- 
donian canal,  were  converted  into  a  soft  plumbaginous  substance,  to  a  depth 
of  %  of  an  inch,  within  4  years;  but  where  they  had  been  coated  with  common 
Swedish  tar,  they  were  entirely  uninjured.  This  softening  effect  on  cast  iron 
appears  to  be  as  rapid  even  when  the  water  is  but  slightly  brackish ;  and  that 
only  at  intervals.  It  also  takes  place  on  cast  iron  imbedded  in  salt  earth.  Some 
water  pipes  thus  laid  near  the  Liverpool  docks,  at  the  expiration  of  20  years 
were  soft  enough  to  be  cut  by  a  knife;  while  the  same  kind,  on  higher  ground 
beyond  the  influence  of  the  sea  water,  were  as  good  as  new  at  the  end  of  50  years." 

Observation  has,  however,  shown  that  tbe  rapidity  of  this  action 
«lepends  mucb  on  tbe  quality  of  tbe  iron ;  that  which  is  dark- 
colored,  and  contains  much  carbon  mechanically  combined  with  it,  corrodes 
most  rapidly :  while  hard  white,  or  light-gray  castings  remain  secure  for  a  long 
time.  Some  cast-iron  sea-piles  of  this  character,  showed  no  deterioration  in  40 
years. 

Contact  witb  brass  or  copper  is  said  to  induce  a  galvanic  action 
which  greatly  hastens  decay  in  either  fresh  or  salt  water.  Some  muskets  were 
recovered  from  a  wreck  which  had  been  submerged  in  sea  water  for  70  years 
near  New  York.  The  brass  parts  were  in  perfect  condition  ;  but  the  iron  parts 
had  entirely  disappeared.  Oalvanizing'  (coating  with  zinc)  acts  as  a  pre- 
servative to  the  iron,  but  at  the  expense  of  the  zinc,  which  soon  disappears. 
The  iron  then  corrodes.  If  iron  be  well  heated,  and  then  coated  with  bot 
coal-tar,  it  will  resist  the  action  of  either  salt  or  freshwater  for  many  years. 
It  is  very  important  that  the  tar  be  perfectly  purified.  Such  a  coat- 

ing, or  one  of  paint,  will  not  prevent  barnacles  and  otber  sbells  from 
attaching  themselves  to  the  iron.  Asphaltum,  if  pure,  answers  as  well  as 
coal-tar. 

Copper  and  bronze  are  very  little  aflfected  by  sea  water. 

No  galvanic  action  has  been  detected  where  brass  feriiles  are  inserted  intt 
the  water-pipes  in  Philadelphia. 


328  TIDES. 

Tbe  most  prejudicial  exposure  for  iron,  as  well  as  for  wood,  is 
that  to  alternate  wet  and  dry.  At  some  dangerous  spots  in  Long  Island  Sound, 
it  has  been  the  practice  to  drive  round  bars  of  rolled  iron  about  4  inches  diam- 
eter, for  supporting  signals.  These  wear  away  most  rapidly  between  high  and 
low  water;  at  the  rate  of  about  an  inch  in  depth  in  20  years;  in  which  time  the 
4-inch  bar  becomes  reduced  to  a  2-inch  one,  along  that  portion  of  it.  Under 
fresh  water  especially,  or  under  ground,  a  thin  coating  of  coal-pitch  varnish, 
carefully  applied,  will  protect  iron,  such  as  water-pipes,  Ac,  for  a  long  time. 
See  page  655.  The  sulphuric  acid  contained  in  the  water  from  coal  minei 
corrodes  iron  pipes  rapidly.  In  the  fresb  water  of  canals,  iron  boate 
have  continued  in  service  from  20  to  40  years.  l¥oocl  remains  sound  for 
centuries  under  either  fresh  or  salt  water,  if  not  exposed  to  be  worn  away  by 
the  action  of  currents ;  or  to  be  destroyed  by  marine  insects. 

Sea  water  weig^ns  from  64  to  64.27  lbs  per  cubic  foot,  or  say  from  1-6  to 
1.9  R)s  per  cubic  foot  more  than  fresh  water,  varying  with  the  locality,  and  jiot 
appreciably  with  the  depth.  Theexcess,  over  the  weight  of  fresh  water,  is  chiefly 
common  salt.  At  64  ft)s  per  cubic  foot,  35  cubic  feet  weigh  2240  lbs.  Sea  ivater 
freeases  at  about  27°  Fahr.  The  ice  is  fresh;  but  (especially  at  low  tempera- 
tures) brine  may  be  entrapped  in  the  ice. 

A  teaspoonful  of  powdered  alum,  well  stirred  into  a  bucket  of  dirty  w^ater, 
will  generally  purify  it  sufficiently  within  a  few  hours  to  be  drinkable.  If  a 
bole  3  or  4  feet  deep  be  dug  in  the' sand  of  the  sea-shore,  the  infiltrating  water 
will  usually  be  sufficiently  fresh  for  washing  vi'ith  soap;  or  even  for  drinking. 
It  is  also  stated  that  water  may  be  preserved  sweet  for  many  years  by  placing 
in  the  containing  vessel  1  ounce  of  black  oxide  of  manganese'  for  each  gallon 
of  water. 

It  is  said  that  water  kept  in  zinc  tanks;  or  flowing  through  iron 
tubes  galvanized  inside,  rapidly  becomes  poisoned  by  soluble  salts  of  zinc 
formed  thereby;  and  it  is  recommended  to  coat  zinc  surfaces  with  asphalt 
varnish  to  prevent  this.  Yet,  in  the  city  of  Hartford,  Conn,  service  pipes  of 
iron,  galvanized  inside  and  out,  were  adopted  in  1855,  at  the  recommendation 
of  the  water  commissioners;  and  have  been  in  use  ever  since.  They'are  like- 
wise used  in  Philadelphia  and  other  cities  to  a  considerable  extent.  In  many 
hotels  and  other  buildings  in  Boston,  the  "Seamless  Drawn  Brass  Tube"  of  the 
American  Tube  Works  at  Boston,  has  for  many  years  been  in  use  for  service 
pipe;  and  has  given  great  satisfaction.  II  is  stated  that  the  softest  water  may 
be  kept  in  brass  vessels  for  years  without  any  deleterious  result. 

The  action  of  lead  upon  some  waters  (even  pure  ones)  is  highly  poison- 
ous. The  subject,  however,  is  a  complicated  one.  An  injurious  ingredient  may 
be  attended  by  another  which  neutralizes  its  action.  Organic  matter,  whether 
vegetable  or  animal,  is  injurious.  Carbonic  acid,  when  not  in  excess,  is  harm- 
less. 

Ice  may  be  so  impure  that  its  water  is  dangerous  to  drink. 

The  popular  notion  that  hot  water  freezes  more  quickly 
than  cold,  with  air  at  the  same  temperature,  is  erroneous. 

TIDES.  .     ^ 

The  tides  are  those  well-known  rises  and  falls  of  the  surface  of  the  sei. 
and  of  some  rivers,  caused  by  the  attraction  of  the  sun  and  moon.  There  are 
two  rises,  floods,  or  high  tides;  and  two  falls,  ebbs,  or  low  tides,  every  24  hours 
and  50  minutes  (a  lunar  day) ;  making  the  average  of  5  hours  123x^  minutes 
between  high  and  low  water.  These  intervals  are,,  however,  subject  to 
g-reat  variations;  as  are  also  the  heights  of  the  tides;  and  this  not  only 
at  different  places,  but  at  the  same  place.  These  irregularities  are  owing  to  the 
shape  of  the  coast  line,  the  depth  of  water,  winds,  and  other  causes.  Usually  at 
new  and  full  moon,  or  rather  a  day  or  two  after,  (or  twice  in  each  lunar  month, 
at  intervals  of  two  weeks,)  the  tides  rise  higher,  and  fall  lower  than  at  other 
times;  and  these  are  called  spring  tides.  Also,  one  or  two  days  after  the 
moon  is  in  her  quarters,  twice  in  a  lunar  month,  they  both  rise  and  fall  less  than 
at  other  times ;  and  are  then  called  neap  tides.  From  neap  to  spring  they 
rise  and  fall  more  daily ;  and  vice  versa.  The  time  of  hig-h  w^ater  at  any 
place,  is  generally  two  or  three  hours  after  the  moon  has  passed  over  either 
the  upper  or  lower  meridian;  and  is  called  the  establishment  of  that 
place;  because,  when  this  time  is  established,  the  time  of  high  water  on  any 
other  day  may  be  found  from  it  in  most  cases.  The  total  height  of  spring  tides 
is  genei'ally  from  1}/^  to  2  times  as  great  as  that  of  neaps.  The  great  tidal 
wave  is  merely  an  undulation,  unattended  by  any  current,  or  progressive  motion 
of  the  particles  of  water.  Each  successive  high  tide  occurs  about  24  minute* 
later  than  the  preceding  one;  aod  so  with  the  low  tides 


EVAPORATION   AND   LEAKAGE.  329 

EVAPOEATION,  FILTRATION,  AND  LEAKAGE. 

The  amount  of  evaporation  from  surfaces  of  water  exposed  to 

the  natural  effects  of  the  open  air,  is  of  course  greater  iu  summer  than  in  winter;  although  it  is  quit© 
perceptible  in  eren  the  coldest  weather.  It  is  greater  in  shallow  water  than  iu  deep,  inasmuch  as  the 
bottom  also  becomes  heated  by  the  sun.  It  is  greater  in  runuing,  than  in  standing  water;  on  much 
the  same  principle  that  it  is  greater  during  winds  than  calms.  It  is  probable  that  the  average  daily 
I088  from  a  reservoir  of  moderate  depth,  from  evaporation  alone,  throughout  the  3  warmer  month* 
of  the  year,  (June,  July,  August,)  rarely  exceeds  about  y^-g-  inch,  in  any  part  of  the  United  States.  Or 
JLj.  inch  during  the  9  colder  months ;  except  in  the  Southern  States.  These  two  averages  would  give 
a  daily  one  of  .15  inch  ;  or  a  total  annual  loss  of  55  ins,  or  4  ft  7  ins.     It  probably  is  3.5  to  4  ft. 

By  some  trials  by  the  writer,  in  the  tropics,  ponds  of  pure  water 

8  ft  deep,  in  a  stiff  retentive  clay,  and  fully  exposed  to  a  very  hot  sun  ail  day,  lost  during  the  dry  sea- 
son, precisely  2  ins  in  16  days  ;  or  ^  inch  per  day  ;  while  the  evaporation  from  a  glass  tumbler  wa« 
yi  inch  per  day.  The  air  in  that  region  is  highly  charged  with  moisture;  and  the  dews  are  heavy. 
Every  day  during  the  trial  the  thermometer  reached  from  115°  to  125°  in  the  sun. 

The  total  annual  evaporation  in  several  parts  of  England  and  Scotland  is  stated  to  average  from  22 
to  38  ins  ;  at  Paris,  34 ;  Boston,  Mass,  32  ;  many  places  in  the  U.  S.,  30  to  36  ins.  This  last  would  give 
a  daily  average  of  -^t^  inch  for  the  whole  year.  Such  statements,  however,  are  of  very  little  value, 
unless  accompanied  by  memoranda  of  the  circumstances  of  the  case ;  such  as  the  depth,  exposure, 
size  and  nature  of  the  vessel,  pond.  &c,  which  contains  the  water.  <fec.  Sometimes  the  total  annual 
evaporation  from  a  district  of  country  exceeds  the  rain  fall ;  and  vice  versa. 

On  canals,  reservoirs,  &c,  it  is  usual  to  combine  the  loss  by  evaporation, 
with  that  by  filtration.  The  last  is  that  which  soaks  into  the  earth ;  and  of  which  some  portion 
passes  entirely  through  the  banks,  (when  in  embankt;)  and  if  in  very  small  quantity,  may  be  dried 
up  by  the  sun  and  air  as  fast  as  it  reaches  the  outside;  so  as  not  to  exhibit  itself  as  water;  but  if  in 
greater  quantity,  it  becomes  apparent,  as  leakage. 

E.  H.  Gill.  C  E,  states  the  average  evaporation  and  filtra- 
tion on  the  San<ly  and  Beaver  canal.  Ohio,  (38  ft  wide  at  water  sur- 
face ;  26  ft  at  bottom ;  and  4  ft  deep,)  to  be  but  13  cub  ft  per  mile  per  minute,  in  a  dry  season.  Here 
the  exposed  water  surf  in  one  mile  is  200640  sq  ft;  and  in  order,  with  this  surf,  to  lose  13  cub  ft  per 
min,  or  18720  cub  ft  per  day  of  24  hours,  the  quantity  lost  must  be  oVoVto  ~ '^^^^  Tt,  =  lH  inch  in 
depth  per  day.  Moreover,  one  mile  of  the  canal  contains  675840  cub  ft ;  therefore,  the  number  of  days 
reqd  for  the  combined  evaporation  and  filtration  to  amount  to  as  much  as  all  the  water  in  the  canal,  is 
6J75  84J)  _  gg  ^^yg^     Observations  in  warm  weather  on  a  22-mile  reach  of  the  Chenango  canal,  N 

York,  (40;  28;  and  4  ft,)  gave  653^  cub  ft  per  mile  per  min  ;  or  5  times  as  much  as  in  the  preceding 
case.  This  rate  would  empty  the  canal  in  about  8  days.  Besides  this  there  was  an  excessive  leakage 
at  the  gates  of  a  lock,  (of  ouly  5?^  ft  lift,)  of  479  cub  ft  per  min,  22  cub  ft  per  mile  per  min  ;  and  at 
aqueducts,  and  waste-weirs,  others  amounting  to  19  cub  ft  per  mile  per  min.  The  leakage  at  other 
locks  with  lifts  of  8  ft,  or  less,  did  not  exceed  about  350  cub  ft  per  min.  at  each.  On  other  canals,  it 
has  been  found  to  be  from  50,  to  500  ft  per  min.  On  the  Chesapeake  and  Ohio  canal,  (where  50.  32, 
and  6  ft,)  Mr.  Fisk,  C  E,  estimated  the  lo.ss  by  evap  and  filtration  in  2  weeks  of  warm  weather,  to  be 
equal  to  all  the  water  in  the  canal.     ProfeSSOr   RaukinC   aSSUmCS   2   inS   per 

day,  for  leakage  of  canal  be<l,  and  evaporation,  on  English 

canals.  J.  B.  Jervls,  C  E,  estimated  the  loss  from  evap,  filtration,  and  leakage  through  lock- 
gates,  on  the  original  Erie  canal,  (40,  28,  and  4  ft,;  at  100  cub  ft  per  mile  per  min;  or  144000  cub  ft 
per  day.   The  water  surf  in  a  mile  is  211200  sq  ft ;  therefore,  the  daily  loss  would  be  equal  to  a  depth  of 

On  the  Delaware  division  of  the  Pennsylvania  canals,  when 

the  supply  is  temporarily  shut  off  from  any  long  reach,  the  water  falls  from  4  to  8  ins  per  day.  The 
filtration  will  of  course  be  much  greater  on  "embankts,  than  in  cuts.  In  some  of  our  canals,  the  depth 
at  high  embankts  becomes  quite  considerable  ;  the  earth,  from  motives  of  economy,  not  being  filled  in 
level  under  the  bottom  of  the  canal;  but  merely  left  to  form  its  own  natural  slopes.  At  one  spot  at 
least,  on  the  Ches  and  Ohio  canal,  where  one  side  is  a  natural  face  of  vertical  rock,  this  depth  is  40 
ft.  Such  depths  increase  the  leakage  very  greatly  ;  especially  when,  as  is  frequently  the  case,  the  em- 
bankts are  not  puddled;  and  the  practice  is  not  to  be  commended,  for  other  reasons  also. 

I'he  total  averag-e  loss  from  reservoirs  of  moderate  depths, 

in  case  the  earthen  dams  be  constructed  with  proper  care,  and  well  settled  by  time,  will  not  exceed 
about  from  )^  to  1  inch  per  day  ;  but  in  new  ones,  it  will  usually  be  considerably  greater. 

The  loss  from  ditches,  or  channels  of  small  area,  is  much 

greater  than  that  from  navigable  canals  ;  so  that  long  canal  feeders  usually  deliver  but  a  small  pro- 
portion of  the  water  which  enters  t,hem  at  their  heads. 


330  FORCE   IN   RIGID    BODIES. 


MECHANICS.    FOEOE  IN  EIGID  BODIES. 


In  the  following  pages  we  endeavor  to  make  clear  a  few  elementary  principles 
of  Mechanics.  The  opening  articles  are  devoted  chiefly  to  the  subject  of  matter  in 
motion;  for,  while  an  acquaintance  with  this  is  perhaps  not  absolutely  required  in 
obtaining  a  working  knowledge  of  those  principles  of  Statics  which  enter  so  largely 
into  the  computations  of  the  civil  engineer,  yet  it  must  be  an  important  aid  to  their 
intelligent  appreciation. 

Art.  1  (a).  Meclianics  may  toe  defined  as  that  branch  of  science  which 
treats  of  the  effects  of  force  upon  matter. 

This  broad  definition  of  the  word  "Mechanics"  includes  hydrostatics,  hydraulics, 
pneumatics,  etc.,  if  not  also  electricity,  optics,  acoustics,  and  indeed  all  branches  of 
physics;  but  we  shall  here  confine  ourselves  chiefly  to  the  consideration  of  tlie  action 
of  extraneous  forces  upon  bodies  supposed  to  be  rigid,  or  incapable  of  change  of  shape. 

(b)  Mechanics  is  divided  into  tw  >  branches,  namely: 

Kinematics;  or  the  study  of  the  motions  of  bodies,  without  reference  to  the 
causes  of  niutiou;  and 

Dynamics,  or  the  study  of  force  and  its  effects. 

The  latter  is  sub-divided  into 

Kinetics;  which  treats  of  the  relations  between  force  and  motion;  and 

iiilatics;  which  considers  those  special,  but  very  numerous,  cases,  where  equal 
and  opposite  forces  counteract  each  other  and  thus  destroy  each  other's  motions. 

Art.  3  (a).  Matter,  or  substance,  may  be  defined  as  whatever  occupies  space; 
as  meta',  stone,  wood,  water,  air,  steam,  gas,  etc. 

(to)  A  toody  is  any  portion  of  matter  which  is  either  more  or  less  completely 
separated  in  fact  from  all  other  matter,  or  which  we  take  into  consideration  by  itself 
and  as  if  it  were  so  separated.  Thus,  a  stone  is  a  body,  whether  it  be  fallinsr  through 
the  air  or  lying  detached  upon  the  ground,  or  built  up  into  a  wall.  Also,  the  wall  is 
a  body ;  or,  if  we  wish,  we  may  consider  any  portion  of  the  wall,  as  any  particular 
cubic  foot  or  inch  in  it,  as  a  body.  The  earth  and  the  other  planets  are  bodies,  and 
their  smallest  atoms  are  bodies. 

A  tram  of  cars  may  b©  regarded  as  a  body ;  as  may  also  each  car,  each  wheel  or 
axle  or  other  part  of  the  car,  each  passenger,  etc.,  etc. 

Similarly,  the  ocean  is  a  body,  or  we  may  take  as  a  body  any  portion  of  it  at  pleas- 
ure, such  as  a  cubic  foot,  a  certain  bay,  a  drop,  etc. 

(c)  But  in  what  follows  we  shall  (as  already  stated)  consider  chiefly  rigid  bodies; 
i.  e.,  bodies  which  undergo  no  change  in  shape,  such  as  by  being  crushed  or  stretched 
or  pulled  apart,  or  penetrated  by  another  body.  All  actual  bodies  are  of  course  more 
or  less  subject  to  some  such  changes  of  shape ;  t.  e.,  no  body  i«  in  fact  absolutely 
rigid;  but  we  may  properly,  for  convenience,  suppose  such  bodies  to  exist,  because 
many  bodies  are  so  nearly  rigid  that  under  oniinary  circumstances  they  undergo 
little  or  no  change  of  shape,  and  because  such  change  as  does  occur  may  be  con- 
sidered under  the  distinct  head  of  Strength  of  Materials. 

(d)  But  while  bodies  are  thus  to  be  regarded  as  incapable  of  change  of  form,  it  is 
squally  important  that  we  regard  them  as  susceptible  to  change  of  position  as  wholes. 
Thus,  they  may  be  upset  or  turned  around  horiKontally  or  in  any  other  direction,  or 
moved  along  in  any  straight  or  curved  line,  with  or  without  turning  around  a  point 
within  themselves.     In  short  they  are  capable  of  motion,  as  wholes. 


•   FORCE   IN    RIGID    BODIES.  331 

Art.  3  (a).  Motion  of  a  body  is  change  of  its  position  in  relation  to  another 
body  or  to  some  real  or  imaginary  point,  which  (for  conyenience)  we  regard  as  fixed, 
or  at  rest.  Thus,  while  a  stone  falls  fron.  a  roof  to  the  ground,  its  position,  relatively 
to  the  roof,  is  constantly  changing,  as  is  also  that  relatively  to  the  ground  and  that 
relatively  to  any  given  point  in  the  wall :  and  we  say  that  the  stone  is  in  motion  rela- 
tively to  either  of  tfiose  bodies,  or  to  any  point  in  them.  But  if  two  stones,  A  and  B, 
fall  from  the  roof  at  the  same  instant  and  reach  the  ground  at  the  same  (subsequent) 
instant,  we  say  that  although  each  moves,  relatively  to  roof  and  ground,  yet  they 
have  no  motion  relatively  to  each  other ;  or,  they  are  at  rest  relatively  to  each  other; 
for  their  position  in  regard  to  each  other  does  not  change ;  i.  e.,  in  whatever  direction 
and  at  whatever  distance  stone  A  may  be  from  stone  B  at  the  time  of  starting,  it 
remains  in  that  same  direction,  and  at  that  same  distance  from  B  during  the  whole 
time  of  the  fall.  Similarly,  the  roof,  the  wall  and  the  ground  are  at  rest  relatively 
to  each  other,  yet  th«y  are  in  motion  relatively  to  a  falling  stone.  They  are  also  in 
motion  relatively  to  the  tun.  owing  to  the  earth's  daily  rotation  about  its  axis,  and 
its  annual  movement  around  the  sun. 

(1>)  If  a  train-man  walks  toward  the  rear  along  the  top  of  a  freight  train  ju«t  ae 
fast  as  the  train  moves  forward,  he  is  in  motion  relatively  to  the  train;  but,  as  a 
whole,  he  is  at  rest  relatively  to  buildings,  etc.  near  by ;  for  a  spectator,  standing  at 
a  little  distance  from  the  track,  sees  him  continually  opposite  the  same  part  of  such 
building,  etc.  If  the  man  on  the  train  now  stops  walking,  he  comes  to  rest  relatively 
to  the  train,  but  at  the  same  time  comes  into  motion  relatively  to  the  surrounding 
buildings,  etc.,  for  the  spectator  sees  him  begin  to  move  along  with  the  train. 

(c)  Since  we  know  of  no  absolutely  fixed  point  in  space,  we  cannot  say,  of  any 
body,  what  its  absolute  motion  is.  Consequently,  we  do  not  know  of  such  a  thing  as 
absolute  red,  and  are  safe  in  saying  that  all  bodies  are  in  motion. 

Art.  4  (a).  The  velocity  of  a  moving  body  is  its  rate  of  motion.  A  body  (as  a 
railroad  train)  is  said  to  move  with  uniform  velocity ,  or  constant  velocity, 
when  the  distances  moved  over  in  equal  times  are  equal  to  each  other,  no  matter  how 
small  those  times  may  be  taken. 

(to)  The  velocity  Is  expressed  by  stating  the  distance  passed  over  during  some 
given  time,  or  which  would  be  passed  over  during  that  time  if  the  uniform  motion 
continued  so  long  Thus,  if  a  railroad  train,  moving  with  constant  velocity,  passes 
over  10  miles  in  half  an  hour,  we  may  say  that  its  velocity,  during  that  time,  is 
(t.  e.,  that  it  moves  at  the  rate  of)  20  miles  per  hour,  or  105,600  feet  per  hour,  or  1760 
feet  per  minute,  or  29%  feet  per  second.  Or,  we  may,  if  desirable,  say  that  it  moves 
at  the  rate  of  10  miles  in  half  an  hour,  or  8*^  feet  in  three  seconds,  etc.;  but  it  is 
generally  more  convenient  to  state  the  distance  passed  over  in  a  unit  of  time,  as  in 
one  day.  one  hour,  one  second,  etc. 

(c)  If,  of  two  trains,  A  and  K,  moving  with  constant  velocity, 

A  moves  10  miles  in  half  an  hour, 
B  moves  10  miles  in  quarter  of  an  hour, 
then  the  velocities  are, 

A,  20  niilps  per  hour, 

B,  40  iniles  per  hour. 

In  other  words,  the  velocity  of  a  body  (which  may  be  defined  as  the  distance  passed 
over  in  a  given  time)  is  inversely  as  the  time  required  to  pans  over  a  given  distance. 

(d)  By  unit  velocity  is  meant  that  velocity  whi«h,  by  common  consent,  is  taken 
as  equal  to  uniti/  or  one.  Where  English  measures  are  used,  the  unit  velocity  gen- 
erally adopted  in  the  study  of  Mechanics  is  1  foot  per  second. 

(e)  When  we  say  that  a  body  has  a  velocity  of  20  miles  per  hour,  or  10  feet  per 
second,  etc..  we  do  not  imply  that  it  will  necessarily  travel  20  miles,  or  10  feet,  etc. ; 
for  it  may  not  have  suflScient  time  for  that.  We  mean  merely  that  it  is  traveling  at 
the  rate  of  20  miles  per  hour,  or  10  feet  per  second,  etc. ;  so  that  if  it  continued  to  move 
at  that  same  rate  for  an  hour,  or  a  second,  etc.,  it  would  travel  20  miles,  or  10  feet.  etc. 

(f)  When  velocity  increases,  it  is  said  to  be  accelerated.  When  it  decreases. 
it  is  said  to  be  retarded.  If  the  acceleration  or  retardation  is  in  exact  proportion 
to  the  time  ;  that  is,  when  during  any  and  every  equal  interval  of  time,  the  same  degree 
of  change  takes  place,  it  is  uniformly  accelerated,  or  retarded.  When  otherwise,  the 
words  variable  and  variably  are  UHed. 

(g)  A  body  may  have,  at  the  same  time,  t-wo  or  more  Independent  veloci- 
ties requiring  to  be  considered.     For  instance,  a  ball  fired  vertically  upward  from  a 


332  FORCE   IN    RIGID   BODIES. 

gun,  and  then  falling  again  to  the  earth,  has,  during  the  whole  time  of  its  rise  and 
fell,  (Ist)  the  uniform  upward  velocity  with  which  it  leaves  the  muzzle,  and  (2nd)  the 
continually  accelerated  downward  velocity  given  to  it  by  gravity,  which  acts  upon  it 
during  the  whole  time.  Its  resultant  (or  apparent)  velocity  at  any  moment  is  the 
difftrence  between  these  two. 

Thus,  immediately  after  leaving  the  gun,  the  downward  velocity  given  by 
gravity  is  very  small,  and  the  resultant  velocity  is  therefore  upward  and  very 
nearly  equal  to  the  whole  upward  velocity  due  to  the  powder.  But  after  awhile 
the  downward  velocity  (by  constantly  increasing)  becomes  equal  to  the  upward 
velocity;  i.  e.,  their  difference,  or  the  resultant  velocity,  becomes  nothing ;  the  ball 
at  that  instant  stands  still ;  but  its  downward  velocity  continues  to  increase,  and 
immediately  becomes  a  little  greater  than  the  upward  velocity ;  then  greater  and 
greater,  until  the  ball  strikes  the  ground.     At  that  instant  its  resultant  velocity  is 

{the  downward  velocity  which  it  would  "j  (    the  uniform  upward 

have  acquired  by  falling  during  the       >  —  -<   velocity  given  by  the 
whole  time  of  its  rise  and  fall.  )  [  powder. 

We  have  here  neglected  the  resistance  of  the  air,  which  of  course  retards  both 
the  Bficent  and  the  descent  of  the  ball. 

(h)  As  a  further  illustration,  regard  a  6  n  c  as  a  raft  drifting  in  the  directioQ 
c  a  or  n  b.     A  man  on  the  raft  walks  with  uniform  velocity  from  corner  n  to 
corner  c  while  the  raft  drifts  (with  a  uniform  velocity  a 
little  greater  than  that  of  the  man)  through  the  distance  n  b.  /^^^ 

Therefore,  when  the  man  reaches  corner  c,  that  corner  has  \^i^^*\ 

moved  to  the  point  which,  when  he  started,  was  occupied  by  /"Iff^^^^ 

a.    The  man's  resultant  motion,  relatively  to  the  bed  of  the  /       ; 

river  or  to  a  point  on  shore,  has  therefore  been  w  <x.    His  /        •,     / 

motion  at  right  angles  to  n  a,  due  to  his  walking,  is  i  c,  but       C'^.-- -jt  /' 

that  due  to  the  drifting  of  the  raft  is  o  b.     These  two  are  "•-i'' 

equal  and  opposite.      Hence  his  resultant  motion  at  right  ii. 

angles  to  n  a  is  nothing  ;  he  does  not  move  from  the  line  n  a. 
His  walking  moves  him  through  a  distance  equal  to  n  i,  in  the  direction  n  a: 
and  the  drifting  through  a  distance  equal  to  i  a,  and  the  sum  of  these  two  is  n  a. 

(i)  All  the  motionB  which  we  see  given  to  bodies  are  but  changes  in  their  unknown 
absolute  motions.  For  convenience,  we  may  confine  our  attention  to  some  one  or 
more  of  these  changes,  neglecting  others. 

Thus,  in  the  case  of  the  ball  fired  upward  from  a  gun  (see  (g)  above)  we  may 
neglect  its  uniform  upward  motion  and  consider  only  its  constantly  accelerated 
downward  motion  under  the  action  of  gravity  ;  or,  as  is  more  usual,  w©  may  consider 
only  the  resultant  or  appa/rent  motion,  which  is  first  upward  and  then  downward.  In 
both  csusies  we  neglect  the  motions  of  the  ball  caused  by  the  several  motions  o* 
the  earth  in  space. 

Art.  5  (a).  Force,  ttie  cause  of  cliange  of  motion.  Suppose  » 
perfectly  smooth  ball  resting  upon  a  perfectly  hard,  frictionless  and  level  surface, 
and  suppose  the  resistance  of  the  air  to  be  removed.  In  order  to  merely  move  the 
ball  horizontally  (i.  e.,  to  set  it  in  motion — to  change  its  state  of  motion)  some  force 
must  act  upon  it.  Or,  if  such  a  ball  were  already  iu  motion,  we  could  not  retard 
or  hasten  it,  or  turn  it  from  its  path  without. exerting  force  upon  it.  For,  as  stated 
in  Ne^vtoii's  first  law  of  motion,  "every  body  continues  in  Its 
state  of  rest  or  of  motion  iu  a  straight  line,  except  iu  so  far  as  it  may  be  com- 
pelled by  impressed  forces  to  change  that  wtate."  On  the  other  hand,  if  a  force  acts 
upon  a  body,  the  motion  of  the  body  must  undergo  change. 

(b)  Force  1«  an  action  betMreen  tvro  bodies,  tending  eitber  to 
separate  tbeni  or  to  bring  tbem  closer  togetber.  For  instance,  when 
a  stone  falls  to  the  ground,  we  explain  the  fact  by  saying  that  a  force  (the  attraction 
of  gravitation)  tends  to  draw  the  earth  and  the  stone  together. 

Magnetic  and  electric  attraction,  and  the  cohesive  force  between  the  particles  of  a 
body,  are  other  instances  of  attractive  force. 

(c)  Force  applied  by  contact.  In  practice  we  apply  force  to  a  body  (B) 
by  causing  contact  between  it  and  another  body  (A)  which  has  a  tendency  to  motion 
toward  B.  A  repuUive  force  is  thus  called  into  action  between  the  two  bodies  (i» 
some  way  which  we  cannot  understand),  and  this  force  pushes  B  forward  (or  in  the 


FORCE   IN    RIGID    BODIEd.  333 

direction  of  A's  tendency  to  move)  and  pushes  A  backward,  thus  diminishing  its  for- 
ward tendency.* 

If,  for  instance,  a  stone  be  laid  upon  the  ground,  it  tends  to  more  downward,  but 
does  not  do  so,  because  a  repulsive  force  pushes  it  and  the  earth  apart  just  as  hard  as 
the  force  of  gravity  tends  to  draw  them  together. 

Similarly,  when  we  attempt  to  lift  a  moderate  weight  with  our  hand,  we  do  so  by 
giving  the  hand  a  tendency  to  move  upward.  If  the  hand  slips  from  the  weight, 
this  tendency  moves  the  hand  rapidly  upward  before  our  will  force  can  check  it. 
But  otherwise,  the  repulsive  force  generated  by  contact  between  the  hand  (tending 
upward)  and  the  weight,  moves  the  latter  upward  in  spite  of  the  force  of  gravity, 
and  pushes  the  hand  downward,  depriving  it  of  much  of  the  upward  velocity  which 
it  would  otherwise  have.  It  is  perhaps  chiefly  from  the  effort)  of  which  we  are 
conscious  in  such  cases,  that  we  derive  our  notions  of  "force." 

When  a  moving  billiard  ball.  A,  strikes  another  one,  B,  at  rest,  the  tendency 
of  A  to  continue  moving  forward  is  resisted  by  a  repulsive  force  acting  between  it 
and  B.  This  force  pushes  B  forward,  and  A  backward,  retarding  its  former  velocity. 
As  explained  in  Art.  23  (a),  the  repulsive  force  does  not  exist  in  either  body 

until  the  two  meet. 

(d)  The  repulsive  force  thus  generated  by  contact  between  two  bodies,  continues  to 
act  only  so  long  as  they  remain  in  contact,  and  only  so  long  as  tbey  tend  (from 
aome  extraneous  cause)  to  come  closer  together.  But  it  is  generally  or  always 
accompanied  by  an  additional  repulsive  force,  due  to  the  compression  of  the  particles 
of  the  bodies  and  their  tendency  to  return  to  their  original  positions.  This  elastic 
repulsive  force  may  continue  to  act  after  the  tendency  to  compression  has  ceased. 

(e)  Force  acts  eltlier  as  a  pull  or  as  a  pusli.  Thus,  when  a  weight 
is  suspended  by  a  hook  at  the  end  of  a  rope,  gravity  pulls  the  weight  downward,  the 
weight  pushes  the  hook,  and  the  hook  pulls  the  rope,  each  of  these  actions  being 
accompanied,  of  course,  by  its  corresponding  and  opposite  "reaction."  When  two 
bodies  collide,  each  pushes  the  other,  generally  for  a  very  short  time, 

(f)  !Eqiiality-  of  action  and  reaction.  A  force  alway-s  exerts  itself  equally 
upon  the  two  bodies  between  which  it  acts.  Thus,  the  force  (or  attraction)  of 
gravitation,  acting  between  the  earth  and  a  stone,  draws  the  earth  upward  just  as 
hard  as  it  draws  the  stone  downward ;  and  the  repulsive  force,  acting  between  a 
table  and  a  stone  resting  upon  it,  pushes  the  table  and  the  earth  downward  just  as 
hard  as  it  pushes  the  stone  upward.  This  is  the  fact  expressed  by  Newton's 
third  IsLW  of  motion,  that  "to  every  action  there  is  always  an  equal  and 
contrary  reaction."     For  measures  of  force,  see  Arts.  11,  12, 13. 

If  a  cannon  ball  in  its  flight  cuts  a  leaf  from  a  tree,  we  say  that  the  leaf  has  reacted 
against  the  ball  with  precisely  the  same  force  with  which  the  ball  acted  against  the 
leaf  That  degree  of  force  was  sufficient  to  cut  off  a  leaf,  but  not  to  arrest  the  ball. 
A  ship  of  war,  in  running  against  a  canoe,  or  the  fist  of  a  pugilist  striking  his 
opponent  in  the  face,  receives  as  violent  a  blow  as  it  gives ;  but  the  same  blow  that 
will  upset  or  sink  a  canoe^  will  not  appreciably  affect  the  motion  of  a  ship,  and  the 
blow  which  may  seriously  damage  a  nose,  mouth,  or  eyes,  may  have  no  such  effect 
4ipon  hard  knuckles. 

The  resistance  which  an  abutment  opposes  to  the  pressure  of  an  arch;  or  a  retain- 
ing-wall  to  the  pressure  of  the  earth  behind  it,  is  no  greater  than  those  pressures 
themselves ;  but  the  abutment  and  the  wall  are,  for  the  sake  of  safety,  made  capable 
of  sustaining  much  greater  pressures,  in  case  accidental  circumstances  should  pro- 
duce such. 

(gr)  In  most  practical  cases  we  liave  to  consider  only  one  of  the  two  bodies 
between  which  a  force  acts.  Hence,  for  convenience,  we  commonly  speak  as  if  the 
force  were  divided  into  two  equal  and  opposite  forces,  one  for  each  of  the  two  bodies, 
and  confine  our  attention  to  one  of  the  bodies  and  the  forc«  acting  upon  it,  neglect- 
ing the  other.  Thus  we  may  speak  of  the  force  of  steam  in  an  engine  as  acting 
upon  the  piston,  and  neglect  its  equal  and  opposite  pressure  against  the  head  of 
the  cylinder. 

(li)  That  point  of  a  body  to  which,  theoretically,  a  force  is  applied,  is  called  the 
point  of  application.  In  practice  we  cannot  apply  force  to  a  point  according 
to  the  scientific  meaning  of  that  word ;  but  have  to  apply  it  distributed  over  an.  ap- 
preciable area  (sometimes  very  large)  of  the  surface  of  the  body. 

*  We  ordinarily  express  all  this  by  saying  simply  that  A  pushes  B  forward,  and  this 
is  sufficiently  exact  for  practical  purpo.ses ;  but  it  is  well  to  recognize  that  it  is  merely 
a  convenient  expression  and  does  not  fully  state  the  facts,  and  that  every  force  neces- 
sarily consists  of  two  equal  and  opposite  pulls  or  pushes  exerted  between  two  bodies. 


334  FORCE   IN   RIGID   BODIES. 

For  the  present  we  shall  assume  that  the  line  of  action  of  the  force  passes 
through  the  center  of  gravity  of  the  body  and  forms  a  right  angle  with  the  sur- 
face at  the  point  of  application. 

Art*  7  (a).  Acceleration.  When  an  unresisted  force,  acting  upon  a  body, 
sets  it  in  motion  (i.  e.,  gives  it  velocity)  in  the  direction  of  the  force,  this  velocity 
increases  as  the  force  continues  to  act;  each  equal  interval  of  time  (if  the  force 
remains  constant)  bringing  its  own  equal  increase  of  velocity. 

Thus,  if  a  stone  be  let  fall,  the  force  of  gravity  gives  to  it,  in  the  first  in- 
conceivably short  interval  of  time,  a  small  velocity  downward.  In  the  next  equal 
interval  of  time,  it  adds  a  second  equal  velocity,  and  so  on,  so  that  at  the  end  of 
the  second  interval  the  velocity  of  the  stone  istwice  as  great,  at  the  end  of  the 
third  interval  three  times  as  great,  as  at  the  end  of  the  first  one,  and  so  on.  We 
may  divide  the  time  into  as  small  equal  intervals  as  we  please.  In  each  such 
interval  the  constant*  force  of  gravity  gives  to  the  sione  an  equal  increase  of 
velocity. 

Such  increase  of  velocity  is  called  acceleration.!  When  a  body  is  thrown  vertically 
upward,  the  downward  acceleration  of  gravity  appears  as  a  retardation  of  the  upward 
motion.  When  a  force  thus  acts  against  the  motion  under  consideration,  its  accelera> 
tion  is  called  negative. 

Art.  8  (a).  The  rate  of  acceleration  f  is  the  acceleration  which  takes 
place  in  a  given  time,  as  one  second. 

(to)  The  unit  rate  of  acceleration  is  that  which  adds  unit  of  velocity  in  a 
unit  of  time ;  or,  where  English  measures  are  used,  one  foot  per  second,  per  second. 

(c)  For  a  given  rate  of  acceleration,  the  total  accelerations  are  of  course  propor- 
tional to  the  times  during  which  the  velocity  increases  at  that  rate. 

Art.  9  (a).  Lavrs  of  acceleration.  Suppose  two  blocks  of  iron,  one  (which 
we  will  call  A)  twice  as  large  as  the  other  (a),  placed  each  upon  a  perfectly  frictionless 
and  horizontal  plane,  so  that  in  moving  them  horizontally  we  are  opposed  by  no  force 
tending  to  hold  them  still.  Now  apply  to  each  block, 

through  a  spring  balance,  a  pull  such  as  will  keep  the  pointer  of  each  balance  always 
at  the  same  mark,  as,  for  instance,  constantly  at  2  in  both  balances.  We  thus  have 
equal  forces  acting  upon  unequal  masses.^  Here  the  rate  of  acceleration  of  a  is 
double  that  of  A ;  for  vrlien  tlie  forces  are  equal  tlie  rates  of  accelera- 
ration  are  inversely  as  the  masses. 

In  other  words,  in  one  second  (or  in  any  other  given  time)  the  small  block  of  iron, 
a,  will  acquire  twice  the  increase  of  velocity  that  A  (twice  as  large)  will  acquire;  so 
that  if  both  blocks  start  at  the  same  time  from  a  state  of  rest,  the  smaller  one,  a,  will 
have,  at  the  end  of  any  given  time,  twice  the  velocity  of  A,  which  has  twice  its  mass. 

(to)  Again,  let  the  two  masses,  A  and  a,  be  equal,  but  let  the  force  exerted  upon  a 
be  twice  that  exerted  upon  A.  Then  the  rate  of  acceleration  of  a  will  (as  before)  be 
twice  that  of  A ;  for,  "wlien  tlie  masses  are  equal,  the  rates  of  accelera- 
ration  are  directly  as  tlie  forces. 

(c)  We  thus  arrive  at  the  principle  that,  in  any  case,  tlie  rate  of  acceleration 
is  directly  proportional  to  tlie  force  and  inversely  proportional 
to  ttoe  mass. 

*  We  here  speak  of  the  force  of  gravity,  exerted  in  a  given  place,  as  constant, 
because  it  is  so  for  all  practical  purposes.  Strictly  speaking,  it  increases  a  very  little 
as  the  stone  approaches  the  earth. 

t  Since  the  rate  of  acceleration  is  generally  of  greater  consequence,  in  Mechanics, 
than  the  total  acceleration,  or  the  "acceleration"  proper,  scientific  writers  (for  the 
sake  of  brevity)  use  the  term  "acceleration"  to  denote  that  rate,  and  the  term 
^^ total  acceleration"  to  denote  the  total  increase  or  decrease  of  velocity  occurring 
during  any  given  time.  Thus,  the  rate  of  acceleration  of  gravity  (about' 32.2  ft.  per 
second  per  second)  is  called,  simply,  the  "  acreleration  of  gravity."  As  we  shall  not 
have  to  use  either  expression  very  frequently,  we  shall,  generally,  to  avoid  misappre- 
hension, give  to  each  idea  its  full  name;  thus,  "total  acceleration"  for  the  whole 
change  of  velocity  in  a  given  case,  and  "  rate  of  acceleration  "  for  the  rate  of  that 
change. 

t  The  mass  of  a  body  Is  the  quantity  of  matter  that  it  contains. 


FORCE   IN    RIGID   BODIES.  335 

(d)  Hence,  if  we  make  the  two  forces  proportional  to  the  two  masses,  tne  rates 
of  acceluratioa  will  be  equal ;  or,  for  a  given  rate  of  acceleration)  tlie 
forces  must  l)e  directly  as  tlie  masses. 

(e)  Hence,  also,  a  greater  force  is  required  to  impart  a  given  velocity  to  a  given 
body  in  a  short  time  than  to  impart  the  same  velocity  in  a  longer  time.  For  instance, 
the  forward  coupling  links  of  a  long  train  of  cars  would  snap  instantly  under  a  pull 
sufficient  to  give  to  the  train  in  two  seconds  a  velocity  of  twenty  miles  per  hour,  sup- 
posing a  sufficiently  powerful  locomotive  to  exist.  In  many  such  cases,  therefore,  we 
have  to  be  contented  with  a  slow,  instead  of  a  rapid  acceleration. 

A  string  may  safely  sustain  a  weight  of  one  pound  suspended  from  our  hand.  If 
we  wish  to  impart  a  great  upward  velocity  to  the  weight  in  a  very  short  time,  we  evi- 
dently can  do  so  only  by  exerting  upon  it  a  great  force;  in  other  words,  by  jerking 
the  string  violently  upward.  But  if  the  string  has  not  tensile  strength  sufficient  to 
transmit  this  force  from  our  hand  to  the  weight,  it  will  break.  We  might  safely 
give  to  the  weight  the  desired  velocity  by  applying  a  less  force  during  a  longer  time. 

(f )  When  a  stone  falls,  the  force  pulling  the  earth  upward  is  (as  remarked  above) 
equal  to  that  which  pulls  the  stone  downward,  but  the  mass  of  the  earth  is  so  vastly 
greater  than  that  of  the  stone  that  its  motion  is  totally  imperceptible  to  ns,  and 
would  still  be  so,  even  if  it  were  not  counteracted  by  motions  in  other  directiona 
in  other  parts  of  the  earth.  Hence  we  are  practically,  though  not  absolutely,  right 
when  we  say  that  the  earth  remains  at  rest  while  the  stone  falls. 

(g)  But  in  the  case  of  the  two  billiard  balls  (Art.  5  c  p.  333),  we  can  clearly  see 
the  result  of  the  action  of  the  force  upon  each  of  the  two  bodies;  for  the  second 
ball,  B,  which  was  at  rest,  now  moves  forward,  while  the  forward  velocity  of  the 
first  one,  A,  is  diminished  or  destroyed,  its  backward  motion  thus  appearing  as  a 
retardation  of  ita  forward  motion.     And,  (since  the  same  force  acts  upon  both  balls) 

mass       mass         rate  of  acceleration       rate  of  negative  acceleration 
of  A   •    of  B    ■  •  of  B  •  of  A 

or  (since  the  force  acts  for  the  same  time  upon  both  balls) 

mass       mass         forward  velocity   .   loss  of  forward  velocity 
of  A   •    of  B   •  •  of  B  *  of  A 

(b.)  Bemark.  A  man  cannot  lift  a.  weight  of  20  tons;  but  if  it  be  placed  upon 
proper  friction  rollers,  he  can  move  it  horizontally,  as  we  see  in  some  drawbridges, 
turntables,  &c. ;  and  if  friction  and  the  resistance  of  the  air  could  be  entirely  removed, 
he  could  move  it  by  a  single  breath;  and  it  would  continue  to  move  forever  after  the 
force  of  the  breath  had  ceased  to  act  upon  it.  It  would,  however,  move  very  slowly, 
because  the  force  of  the  single  breath  would  have  to  diffuse.itself  among  20  tons  of 
matter.  He  can  move  it,  if  it  be  placed  in  a  suitable  vessel  in  water,  or  if  suspended 
from  a  long  rope.  Apowerful  locomotive  that  may  move  2000  tons,  cannot  lift  10  tons 
vertically. 

If  we  imagine  two  bodies,  each  as  large  and  heavy  as  the  earth,  to  be  precisely 
balanced  in  a  pair  of  scales  without  friction,  a  single  grain  of  sand  added  to  either 
Bcale-pan,  would  give  motion  to  both  bodies. 

Art.  10  (a).  The  constant  force  of  gravity  is  a  uniformly  accelerating  force 
when  it  acts  upon  a  body  faUing  freely  ;  for  it  then  increases  the  velocity  at  the  uni- 
form rate  of  .322  of  a  foot  per  second  during  every  hundredth  part  of  a  second,  or  32.2 
feet  per  second  in  every  second.  Also  when  it  acts  upon  a  body  moving  down  an  in- 
clined plane;  although  in  this  CRse  the  increase  is  not  so  rapid,  because  it  is  caused 
by  only  a  part  of  the  gravity,  while  another  part  presses  the  body  to  the  plane,  and  a 
third  part  overcomes  the  friction.  It  is  a  uniformly  retarding  force,  upon  a  body 
thrown  vertically  upward;  for  no  matter  what  may  be  the  velocity  of  the  body 
when  projected  upward,  it  will  be  diminished  .322  of  a  f<  ot  per  second  in  each 
hundredth  part  of  a  second  during  Its  rise,  or  32.2  feet  per  second  during  each 
entire  second.  At  least,  such  would  be  the  case  were  it  not  for  the  varying  resistance 
of  the  air  at  different  velocities.  It  is  a  uniformly  straining  force  when  it  causes  a 
body  at  rest,  to  press  upon  another  bodv ;  or  to  pull  upon  a  striner  by  which  it  is 
suspended.  The  foregoing  expressions,  like  those  of  momentum,  strain,  push,  pull, 
lift,  work,  &c.,  do  not  indicate  different  Mnds  of  force  ;  but  merely  different  kinds  of 
effects  produced  by  the  one  grand  principle,  force. 

(1>)  The  above  32.2  feet  per  second  is  called  the  acceleration  of  gravity  ;  and 
by  scientific  writers  is  conventionally  denoted  by  a  small  g ;  or,  more  correctly  speak- 


836  FORCE  IN   RIGID  BODIES. 

fng,  Biuce  the  acceleration  is  not  precisely  the  same  at  all  parts  of  th«  earth,  f^ 
denotes  the  acceleration  per  second,  whatever  it  may  be,  at  nay  particular  place. 

Art.  11  (a).  Relation  betTreen  force  and  mass.  The  mass  of  a  body 
is  the  quantity  of  matter  which  it  contains.  One  cubic  foot  of  water  has  twic9 
us  great  a  mass  as  half  a  cubic  foot  of  water,  but  a  less  mass  than  one  cubic 
foot  of  iron.  Thus,  the  size  of  a  body  is  a  measure  of  mass  between  bodies 
of  the  same  material,  but  not  between  bodies  of  different  materials. 

(b)  When  bodies  are  allowed  to  fall  freely  in  a  vacuum  at  a  given  place, 
they  are  found  to  acquire  equal  velocities  in  any  given  time,  of  whatever 
different  materials  they  may  be  composed.  From  this  we  know  (Art.  9(d), 
p.  335),  that  the  forces  moving  them  downward,  viz. :  their  respective  weights 
at  that  place,  must  be  proportional  to  their  masses. 

Thus,  in  any  given  place,  the  weight  of  a  body  is  a  perfect  measure  of  its  mass. 
But  the  loeight  of  a  given  body  changes  when  the  body  is  moved  from  one  level 
above  the  sea  to  another,  or  from  one  latitude  to  another;  while  the  mass  of 
the  body  of  course  remains  the  same  in  all  places.  Thus,  a  piece  of  iron  which 
weighs  a  pound  at  the  level  of  the  sea,  will  weigh  less  than  a  pound  by  a  spring 
balance,  upon  the  top  of  a  mountain  close  by,  because  the  attraction  between 
the  earth  and  a  given  mass  diminishes  when  the  latter  recedes  from  the  earth's 
center.  Or  if  the  piece  of  iron  weighs  one  pound  near  the  North  or  South 
Pole,  it  will,  for  the  same  reason,  weigh  less  than  a  pound  by  a  spring  balance 
if  weighed  nearer  to  the  equator  and  at  the  same  level  above  the  sea. 

The  difference  in  the  weight  of  a  body  in  different  localities  is  so  slight  as 
to  be  of  no  account  in  questions  of  ordinary  practical  Mechanics;*  but 
scientific  exactness  requires  a  measure  of  mass  which  will  give  the  same 
expression  for  the  quantity  of  matter  in  a  given  body,  wherever  it  may 
be;  and,  since  weighing  is  a  very  convenient  way  of  arriving  at  the  quantity 
of  matter  in  a  body,  it  is  desirable  that  we  should  still  be  able  to  express  the 
mass  in  terms  of  tlie  weight.  Now,  when  a  given  body  is  carried  to  a  higher 
level,  or  to  a  lower  latitude,  its  loss  of  weight  is  simply  a  decrease  in  the  force 
with  which  gravity  draws  it  downward,  and  this  same  decrease  also  causes 
a  decrease  of  the  velocity  which  the  body  acquires  in  falling  during  any 
given  time.  The  change  in  velocity,  by  Art.  9  (6),  p.  334,  is  necessarily  propor- 
tional to  the  change  in  weight. 

Therefore,  if  the  weight  of  a  body  at  any  place  be  divided  by  the  velocity 
which  gravity  imparts  in  one  second  at  the  same  place  ^and  called  s,  or  the 
acceleration  of  gravity  for  that  place),  the  quotientwUl  be  tne  same  at  aU  places, 
and  therefore  serves  as  an  invariable  measure  of  the  mass. 

(c)  By  common  consent,  the  nnlt  of  mass,  in  scientific  Mechanics,  is  said 
to  be  that  quantity  of  matter  to  which  a  unit  of  force  can  give  unit  rate  of 
acceleration.  This  unit  rate,  in  countries  where  English  measures  are  used, 
is  one  foot  per  second,  per  second.  It  remains  then  to  adjust  the  units  of  forc4 
and  of  mass.  Two  methods  (an  old  and  a  new  one)  are  in  use  for  doing  this. 
We  shall  refer'  to  them  here  as  methods  A  and  B  respectively. 

(d)  In  method  A,  still  generally  used  in  questions  of  statics,  the  unit 

of  force  is  fixed  as  that -force  which  is  equal  to  the  weight  of  one  pound  in  a 
certain  place;  t.  c,  the  force  with  which  the  earth  at  that  place  attracts  a 
certain  standard  piece  of  platinum  called  a  pound;  and  the  unit  of  mass  is 
not  this  standard  piece  of  metal,  but,  as  stated  in  (c),  that  mass  to  which  this 
unit  force  of  one  pound  gives,  in  one  second,  a  velocity  of  one  foot  per  second. 
Now  the  one  pound  attraction  of  the  earth  upon  a  mass  of  one  pound  will 
(Art.  1,  p.  330)  in  one  second  give  to  that  mass  a  velocity  =  gr  or  about  32  feet 

Ser  second ;  and  (Art.  9  (a),  p.334),  for  a  given  force  the  masses  are  inversely  as 
le  velocities  imparted  in  a  given  time.  Therefore,  to  give  in  one  second  a 
Telocity  of  only  one  foot  per  second  (instead  of  g  or  about  32)  the  one  pound 
nnit  of  force  would  have  to  act  upon  a  mass  g  times  (or  about  32  times)  that 
which  weighs  one  pound. 

This  could  be  accomplished,  with  an  Attwood's  machine.  Art.  16  (c),  p.  339, 
by  making  the  two  equal  weights  each  —  15  3^  lbs.  and  the  third  weight  =»  1  lb. 

*  The  greatest  discrepancy  that  can  occur  at  various  heights  and  latitudes, 
by  adopting  weight  as  the  measure  of  quantity,  would  not  be  likely  to  exctea 
1  in  300 ;  or,  under  ordinary  circumstances,  1  in  1000. 


FORCE   IN   RIGID   BODIES. 


337 


By  method  A,  therefore,  the  unit  of  mass  is  g  times  (or  about  32  times)  the 
mass  of  the  standard  piece  of  metal  called  a  pound;  i.  e.,  a  body  containing 
one  such  unit  of  mass  weighs  g  lbs.  or  about  32  lbs. ;  or,  l>y  metliod  A, 


Or, 


the  weight  of  any  given  body  _ 
in  lbs. 


=  a  V  *^®  inass  of  the  body, 
"y  -^       in  units  of  mass. 


the  mass  of  a  body,  in  units  of 
For  instance: 

mass  =3  — 

9 

in  a  body  weighing 
J^  pound 

the  mass  is  about 
^1    unit  of  mass 

1  " 

2  « 
32-      •♦ 

A    "    " 
1     "     " 

64        «    . 

2          «          M 

It  has  been  suggested  to  call  this  unit  of  mass  a  "  Matt." 
(e)  In  metliod  B,  the  mass  of  the  standard  pound  piece  of  platinum  is  taken 
as  the  unit  of  mass  and  is  called  a  pound;  and  the  force  which  will  give 
to  it  in  one  second  a  velocity  of  one  foot  per  second  is  taken  as  the  unit  offeree. 
This  small  unit  of  force  is  called  a  poundal.  In  order  that  it  may  in  one 
second  give  to  the  mass  of  one  pound  a  velocity  of  only  one  foot  per  second,  it 

must  (by  Art.  9  6),  be (ov  about    i  j  of  the  weight  of  said  pound  mass. 

Hence,  l>y  nietliod  B, 

the  mass  of  any  given  body,  in  pounds  =  the  weight  of  the  body  in  pouvdals 

and 

the  weight  of  a  body,  in  poundals  ==  gr  X  the  mass  of  the  body  in  pounds. 


the  mass  of  the  body  Is  about 


"or  instai'ce : 

in  a  body  weighing 
14  pouDdal  -=  ^^  pound 

1  " 

2  •' 
32 

-  1        « 

64        " 

-.2        « 

JL  pound 

1 


(f )  For  conventence,  we  sometimes  disregard  the  scientific  require- 
ment that  the  unit  of  force  must  be  that  which  will  give  unit  rate  of  accele- 
ration to  unit  mass,  and  take  a  pound  of  matter  as  our  unit  of  mass^  and  a 
pound  weight  as  our  unit  of  force.  Our  unit  of  force  will  then  in  one  second 
give  a  velocity  of  g  (or  about  32.2  feet  per  second)  to  our  unit  of  mass.  In 
Statics^  we  are  not  concerned  with  the  masses  of  bodies,  but  only  with  the 
forces  acting  upon  them,  including  their  weights. 

Art.  13  (a).  Impulse.  By  taking,  as  the  unit  of  force,  that  force  which,  in 
one  second,  will  give  to  unit  mass  a  velocity  of  one  foot  per  second,  we. have 
(by  Art.  9,  p.  334),  in  any  case  of  unbalanced /orce  acting  upon  a  mass  during  a 
fiven  time: 

force  X  time 

mass 

velocity  X  mass 

time 

force  X-  time 


Velocity    =• 


Force        = 


Mass  =- 


also 


22 


velocity 

mass  X  velocity 

force 

Force  X  time  -=  mass  X  velocity. 


Time         =- 


(1) 


(2) 
(3) 


(4) 


(5) 


338  FORCE   IN   RIGID   BODIES. 

To  ihe  product,  force  X  time,  in  equation  (5),  writers  now  give  the  name 
Impi&lse,  which  was  formerly  given  to  collision  (now  called  impaot).  See 
Art.  24  (ft).  The  term  impulse^  as  now  used,  conveys  merely  the  idea 

of  force  acting  through  a  certain  length  of  time.  Equation  (5)  tells  us  that  aa 
impulse  (the  product  of  a  force  by  the  time  of  its  action)  is  numerically  equal 
to  the  momentum*  which  it  produces.  Equation  (2)  tells  us  that  any  force  is 
numerically  equal  to  the  momentum  which  it  can  produce  in  one  second.  In 
other  words,  ttie  momentum  of  a  body  moving  with  a  given  velocity  is 
numerically  equal  to  the  force  which  in  one  second  can  produce  or  destroy 
that  velocity  in  that  body;  or,  a  force  is  numerically  equal  to  the  rate  per 
eeoond  at  which  it  can  produce  momentum.  Thus,  forces  are  proportional  to 
the  momentums  which  they  can  produce  in  a  given  time;  or,  in  a  given  time, 
equal  forces  produce  equal  momentums.  Therefore  a  force  must  always  give 
equal  and  opposite  momentums  to  the  two  bodies  between  which  it  acts. 

Art.  13  (a).  Tl&e  usual  way  of  measuring  a  force  is  by  ascertaining 
the  amount  of  some  other  force  which  it  can  counteract.  Thus  we  may  meas- 
ure the  weight  of  a  body  by  hanging  it  to  a  spring  balance.  The  scale  of  the 
balance  then  indicates  the  amount  of  tension  in  the  spring;  and  we  know  that 
the  weight  of  the  body  is  equal  to  the  tension,  because  the  weight  just  pre 
vents  the  tension  from  drawing  the  hook  upward. 

Thus,  forces  are  conveniently  expressed  in  weighiMf  as  in  pounds, 
tons,  &c.,  and  they  are  generally  so  measured  in  Statics,  and  in  our  following 
articles. 

(b)  A  force  may  toe  constant  or  variatole.  When  a  stone  rests  upon 
the  ground,  the  pull  of  gravity  upon  it  (i.  e.,  its  weight)  remains  constant, 
neither  increasing  nor  decreasing.  But  when  a  stone  is  thrown  upward  its 
weight  decreases  very  slightiy  as  it  recedes  from  the  earth,  and  again  increases 
as  it  approaches  it  during  its  fall.  In  this  case,  the  force  of  gravity,  acting 
upon  the  stone,  decreases  or  increases  steadily.  But  a  force  may  change 
suddenly f  or  irregularly,  or  may  be  intermittent;  as  when  a  series  of  unequiU 
blows  are  struck  by  a  hammer.  In  what  follows  we  shall  have  to  do  only  with 
forces  supposed  to  be  constant. 

Art.  14:  (a).  Density.  The  densities  of  materials  are  proportional  to  the 
masses  contained  in  a  given  volume,  as  a  cubic  inch ;  or  inversely  as  the  volume 
required  to  contain  a  given  mass.  Or,  since  the  weights  at  a  given  place  are 
proportional  to  the  masses,  the  densities  are  proportional  to  the  weights  per 
unit  of  volume  (or  "  specific  gravities  ")  of  the  materials.  Thus,  a  body  weigh- 
ing 100  lbs.  per  cubic  foot  is  twice  as  dense  as  one  weighing  only  60  lbs.  per 
cubic  foot  at  the  same  place. 

Art.  15  (a).  Inertia.  The  inability  of  matter  to  set  itself  in  motion,  or  to 
change  the  rate  or  direction  of  its  motion,  is  called  its  inertia,  or  inertness. 
When  we  say  that  a  certain  body  has  twice  the  inertia  (inertness)  of  a  smaller 
one,  we  mean  that  twice  tYiQ  force  is  required  to  give  it  an  equal  rate  of  accele- 
ration ;  and  that,  since  all  force  (Art.  5/),  acts  equally  in  both  direo 
tions,  we  experience  twice  as  great  a  reaction  (or  so-called  "  resistance")  from 
the  larger  body  a?  from  the  smaller  one.  The  "  inertia"  of  a  body  is  therefore 
ft  measure  of  the  force  required  to  produce  in  it  a  given  rate  of  jacceleration;  or, 
which  is  the  same  thing,  it  is  a  measure  of  the  mass  of  the  body.  We  may 
therefore  consider  "inertia"  and  "mass"  as  identical. 

(to)  What  is  called  the  "resistance  of  inertia"  of  a  body,  is  simply  the 
reaction,  (*.  e.,  one  of  the  two  equal  and  opposite  actions)  of  whatever 
force  we  apply  to  the  body.  Hence,  its  amount  depends  not  only  upon  the 
mass  of  the  Dody,  but  also  upon  the  rate  of  acceleration  which  we  choose  to 

*The  momentum  of  a  body  (sometimes  called  its  "quantity  of  motion'*) 
is  equal  to  the  product  obtained  by  multiplying  its  mass  by  its  velocity.  If  we 
adopt  the  pound  as  the  unit  of  mass,  as  in  "method  B,"  Art.  11  (e),  the 

•proawci,  weight  in  pounds  X  velocity,  is  numerically  either  exactly  or  nearly 
the  same  as  the  product,  mass  in  pounds  X  velocity,  depending  upon  whether 
or  not  the  body  is  in  that  latitude  and  at  that  level  where  a  mass  of  one  pound 
is  said  to  weigh  one  pound.  But  the  product,  weight  in  poundals  X  velocity,  is 
exactly  a  times  (about  32.2  times)  the  product,  mass  in  pounds  X  velocity ;  alsc^ 
by  "meftiod  A,"  weight  in  pounds  X  velocity  =  gX  ««ws  in  "  matts*'  X  velocity. 


T.2.4 


FORCE   IN   RIGID   BODIES.  339 

gire  to  it.    Therefore  we  cannot  tell,  from  the  mass  or  weight  of  a  body  alone, 
what  its  "  resistance  of  inertia"  in  any  given  case  will  be. 

Art.  16  (a).  Forces  in  opposite  directions.  When  two  equal  and 
opposite  forces  act  upon  a  body  at  the  same  time,  and  in  the  same  straight  line, 
we  say  that  they  destroy  each  other's  tendencies  to  move  the  body,  and  it  remains 
at  rest.  If  two  unequal  forces  thus  act  in  opposition,  the  smaller  force  and  an 
equal  portion  of  the  greater  one  are  said  to  counteract  each  other  in  the  same 
way,  but  the  remainder  of  the  greater  force,  acting  as  an  unbalanced  or  unresisted 
force,  moves  the  body  in  its  own  direction,  as  it  would  do  if  it  were  the  only 
force  acting  upon  it. 

Thus,  when  we  move  bodies,  in  practice,  we  encounter  not  only  the  "resist- 
ance of  inertia"  {i.  e.,  we  not  only  have  to  exert  force  in  order  to  move  inert 
matter),  but  we  are  also  opposed  by  other /orces,  acting  against  us,  as  friction, 
the  resistance  of  the  air,  and,  often,  all  or  a  part  of  the  iveight  of  the  body.  By 
"resistances,"  in  the  following,  we  mean  such  resisting /orce^,  and  do  not  include 
in  the  term  the  "  resistance  of  ineriia." 

(b)  If  separated,  the  two  bodies,  A  and  B,  of  3  Rs  and  2  lbs  respectively,  would 

fall  with  equal  accelerations  =  g ;  each  unit,  — ,  of  mass  being  acted  upon  by  its 

own  weight,  W.    But,  connected  as  they  are,  A  will 

move  downward,  and  B  upward,  with  an  acceler-  T='2.4 

ation  =  only  f :  for  now  an  unbalanced  force  of 

5 
only  3  —  2  =  1  ft)  must  give  acceleration  to  a  mass     m  ^  a 
3  +  25  2  *  -i5.4 

of =  ~.    But,  to  give  to  a  mass,  B,  of  -,  an 

g  g  2  2 

accel  of  f ,  requires  a  force  of  -  .  f  =  -  ft)  =  0.4  ft).  , — ™  

5       ^  g    5       6  III   ill  B 

This,  plus  2  ft)s  (required  to  balance  the  weight  of 
B)  is  the  tension,  2.4  ft)s  existing  throughout  the 
cord.  Exerted  at  A,  this  tension  balances  2.4  of 
the  3  ft)s  weight  of  A.    The  remainder  (3  —  2.4  =  0.6  ft))  of  the  weight,  actinj? 

downward  upon  the  mass,  -,  of  A,  gives  to  it  the  required  acceleration  of  ^; 

f„-,  here '^^  =  0.6  ^  *=%  =  0.2  g  =  ?. 
mass  g  3  5 

Or  we  may  regard  the  total  tension,  2.4  ft)s,  in  the  cord  at  A,  as  acting  upon  A 

g 
and  giving  to  it  a  negative  or  upward  acceleration  of  2.4  -r-  -  =  0.8  g,  which, 

deducted  from  g  (the  acceleration  which  A  would  otherwise  have)  leaves 

Acceleration  =  g  —  0.8  g  =  0.2  g  =  f . 
5 
Let  W  =  weight  of  A 
w  =  weight  of  B 
F  =  net  force  available  for  acceleration  =  W  —  w 

M  =  combined  mass  of  both  bodies  = 

g 

ra  =  mass  of  B  =  - 
g 
a    =  acceleration 
T  =  tension  in  cord. 

Then:    a=™=(W  — w)- 

M 
_  w 

T  =  w+ma  =  wH a  = 

g 

(c)  An  "  Atwood's  Macliine"  consists  essentially  of  a  pulley,  a  flexible 
cord  passing  over  the  pulley,  two  equal  weights  (one  suspended  at  each  end  of 
the  cord),  and  a  third  weight,  generally  much  lighter  than  either  of  the  other 
two.  The  two  equal  weights  balance  each  other  by  means  of  the  pulley  and 
cord.  The  third  weight  is  laid  upon  one  of  the  other  two  weights.  The  force 
of  gravity,  acting  upon  the  third  weight,  then  sets  the  masses  of  the  three 
weights  in  motion  at  a  small  but  constantly  increasing  velocity.  In  order  to  do 
this  it  must  also  overcome  the  friction  of  the  pulley  and  cord,  and  the  rigidity 


340  FORCE    IN    RIGID   BODIES. 

of  the  latter;  but,  as  these  are  made  as  slight  as  possible,  they  are,  for  con- 
veuience,  neglected.  The  machine  is  used  for  illustrating  the  acceleration  given 
to  inert  matter  by  unbalanced  force,  and  forms  an  excellent  example  of  the  two 
distinct  duties  which  a  moving  force  generally  has  to  perform,  viz:  (1st)  the 
balancing  of  resistance,  and  (2nd)  acceleration, 

(d)  In  the  case  of  a  locomotive,  drawing-  a  train  on  a  level,  fric- 
tion and  the  resistance  of  the  air  are  the  only  resistances  to  be  balanced  ;  for  the 
weight  of  the  train  here  opposes  no  resistance.  Unless  the  force  of  the  steam  is 
moie  than  sufficient  to  balance  the  resistances,  it  cannot  move  the  train.  If  it 
exceeds  the  resistances,  the  excess,  however  slight,  gives  motion  to  the  inert 
matter  of  the  train.  If,  at  any  moment  while  the  train  is  moving,  the  force  of 
the  steam  becomes  just  equal  to  the  resistances  (whether  by  an  increase  of  the 
latter  or  by  diminishing  the  force)  the  train  will  move  on  at  a  unifoi-m  velocity 
equal  to  that  which  it  had  at  the  moment  when  the  force  and  resistance  were 
equalized;  and,  if  these  could  always  be  kept  equal,  it  would  so  move  on  forever. 

But  so  long  as  the  excess  of  steam  pressure  over  the  resistances  continues  to  act, 
the  velocity  is  increased  at  each  instant ;  for  during  each  such  instant  the  excess 
of  force  gives  a  small  velocity  in  addition  to  that  already  existing. 

On  a  level  railroad,  let 
P   =  the  total  tractive  force  of  the  locomotive  =  say  13  tons 
W  =  weight  of  locomotive  =  50  tons- 
"W  =  weight  of  train  =  336  tons 

R  =  resistance  of  locomotive  (including  internal  friction,  etc.)  =  3  tons 
r    =  resistance  of  train  =  1  ton 
F   =  net  force  available  for  acceleration  =  P  —  R  —  r  =  9'toM8 

-        .  ,,     .         W  +  \7       50+336       ,. 

M  ==  mass  of  engine  and  train  = ■  «  —  =  12 

*  g  32.2 

1.  X     •         ^       336        .„  ,. 
m  =  mass  of  train  =  —  —  ——  =  10.44 

g        32.2 
a    =  acceleration 
T   =  tension  on  draw-bar. 

F        9 
Then :  Acceleration  =  a  =»  ;rj:  —  r^  =  0.75  ft  per  second  per  second. 

The  tension  T  on  the  draw-bar  =  resistance  of  train  +  force  causing  accel- 
eration a,  or  T  =  r  +  m  a  =-  1  +  10.44  X  0.75  =  1  -f  7.83  =  8.83  tons. 
This  tension,  T,  pulling  backward  against  the  locomotive,  causes  there  a 

T  8  83  ff 

retardation,  or  negative  acceleration,  of  ^-j -^-  =    '  ^  ^  =  5.69  ft 

'  ^  mass  of  locomotive  50 

per  sec  per  sec,  and  thus  reduces,  by  that  amount,  the  acceleration  which  the 

(P  —  r)  2       10  X.  32  2 
locomotive  would  otherwise  have,  and  which  would  be  —  i — ■  =  — "j^ 

50  50 

=  6.44.    This,  less  5.69,  =  0.75  ft  per  sec  per  sec  =  acceleration  of  train. 

(e)  If  the  tractive  force  of  a  locomotive  exceeds  the  resistances,  due  to  friction, 
grades,  and  air,  the  velocity  will  be  accelerated ;  but  it  then  becomes  more  diffi- 
cult to  maintain  the  excess  of  force,  for  the  pistons  must  travel  faster  through 
the  cylinders,  and  the  boiler  can  no  longer  supply  steam  fast  enough  to  maintain 
the  original  cylinder  pressure.  Besides,  some  of  the  resistances  increase  with 
increase  of  velocity.  We  thus  reach  a  speed  at  which  the  engine,  although 
exerting  i,ts  utmost  force,  can  do  no  more  than  balance  the  resistances.  The 
train  then  moves  with  a  uniform  velocity  equal  to  that  which  it  had  when  this 
condition  was  reached. 

When  it  becomes  necessary  to  stop  at  a  station  some  distance  ahead,  steam  is 
shut  off,  so  that  the  steanj  force  of  the  engine  shall  no  longer  counterbalance  or 
destroy  the  resisting  forces;  and  the  number  of  the  resistances  themselves  is  in- 
creased by  adding  to  them  the  friction  of  the  brakes.  The  resistances,  thus 
increased,  are  now  the  only  forces  acting  upon  the  train,  and  their  acceleration 
is  negative,  or  a  retardation.  Hence,  the  train  moves  more  and  more  slowly,  and 
must  eventually  stop. 

(f)  Caution.  When  two  opposite  forces  are  in  equilibrium,  an  addition  to 
one  of  the  forces  does  not  always  form  an  unbalanced  force ;  for  in  many  cases 
the  other  force  increases  equally,  up  to  a  certain  point.  For  instance,  when  we 
attempt  to  lift  a  weight,  W,  its  downward  resistance,  R,  remains  constantly  just 
equal  to  our  upward  pull,  P,  however  P  may  vary,  until  P  exceeds  W.  Thus,  R 
can  never  exceed.  W,  but  may  be  much  less  than  it.  Indeed,  when  we  stop  pull- 
ing, R  ceases,  although  W  (the  attraction  between  the  earth  and  J,he  weighj)  of 


FORCE   IN    RIGID   BODIES.  341 

course  remains  unchanged  throughout.  Such  variation  of  resisting  force,  to  meet 
varying  demands,  occurs  in  all  those  innumerable  cases  where  structures  sustain 
varying  loads  within  their  ultimate  strength. 

Art.  17  (a).  Work.  Force,  when  it  moves  a  body,*  is  said  to  do  *'  work  " 
upon  it.  The  whole  work  done  by  the  force  in  moving  the  body  through  any  dis- 
tance is  measured  by  multiplying  the/orce  by  the  distance;  or:  Work  =  Force 
X  distance.  If  the/orre  is  taken  in  pounds,  and  the  distance  infect,  the  product 
(or  the  work  done)  will  be  in  foot-pounds ;  if  the  force  is  in  tons  and  the  distance 
in  inches,  the  product  will  be  in  inch-tons  ;  and  so  on.f 

Thus,  if  a  force  of  moves  a  body  through  we  have  work  = 

1  pound  10,000  feet  10,000  foot-pounds 

100  pounds  100    "  10,000         " 

10,000      "  1  foot  10,000        " 

or,  in  any  case,  if  the  force  be  F  pounds,  the  "whole  work  done  by  it  in  moving  a 
body  through  s  feet,  is  F  s  foot-pounds. 

(b)  The  foot-pound,  the  foot-ton,  the  inch-pound,  the  inch-ton,  etc.,  etc.,  are 
called  nnits  of  ivorli.t 

For  practical  purposes,  in  this  country,  forces  are  most  frequently  stated  in 
pounds,  and  the  distances  (through  which  they  act)  in  feet.  Hence  the  ordi- 
nary unit  of  worlt  is  the  foot-pound.  Tlie  metric  unit  of  worK 
is  the  kilog'rani-nieter,  i.  e.  1  kilogram  raised  1  meter  =  2.2046  pounds 
raised  3.2809  feet,  =  7.2331  foot-pounds.    1  foot-pound  ==  0.13825  kilogram-meter. 

(c)  In  most  cases,  a  portion  at  least  of  the  \¥ork  done  by  a  force  is  ex- 
pended in  overcoming;  resistances.  Thus,  when  a  locomotive  begins 
to  move  a  train,  a  portion  of  its  force  works  against,  and  balances,  the  resist- 
ances of  friction  or  of  an  up-grade,  while  the  remainder,  acting  as  unbalanced 
force  upon  the  inert  mass  of  the  train,  increases  its  velocity. 

An  upward  pull  of  exactly  one  pound  will  not  raise  a  one  pound  weight,  but 
will  merely  balance  the  downward  force  of  gravity.  If  we  increase  the  upward 
pull  from  one  pound  (=  16  ounces)  to  17  ounces,  the  ounce  so  added,  being 
unbalanced  force,  will  give  motion  to  the  mass,  and  Mill  accelerate  its  upward 
velocity  as  long  as  it  continues  to  act.  If  we  now  reduce  the  upward  pull  to  1 
pound,  thus  making  it  just  equal  to  the  downward  pull  of  gravity,  the  body  will 
move  on  upward  with  a  uniform  velocity  ;  btit  if  we  reduce  the  upward  force  to 

15  ounces  (=  ^|  pound),  then  there  will  be  an  unbalanced  downward  force  of  1 
ounce  acting  upon  the  body,  and  this  downward  force  will  generate  in  the  body 
a  downward  or  negative  acceleration  or  retardation,  and  will  destroy  the  upward 
velocity  in  the  same  time  as  the  upward  excess  of  1  ounce  required  to  produce  it. 

During  any  time,  while  the  17  ounces  upward  "  force"  were  acting  against  the 

16  ounces  downward  •'  resistance,"  the  product  of  total  upward  force  X  distance 
must  be  greater  than  that  of  resistance  X  distance.  The  excess  is  the  work  done 
in  accelerating  the  velocity,  by  virtue  of  which  the  body  has  acquired  kinetic 
energy  or  capacity  for  doing  work  in  coming  to  rest. 

On  the  other  hand,  while  the  upward  velocity  was  being  retarded,  the  product 
of  total  upward  force  X  dist  was  less  than  that  of  resistance  X  dist,  the  difference 
being  the  work  done  by  the  kinetic  energy  against  the  resistance  of  gravity. 

In  practice,  the  term  "  work"  is  usually  restricted  to  that po7-tion  of  the  work 
which  a  force  performs  in  balancing  the  resistances  which  act  against  it ;  in  other 
words,  to  the  work  done  by  so  much  of  the  force  as  is  equal  to  the  resistance. 

With  this  restriction,  we  have  work  =  force  X  dist,  =  resistance  X  dist. 

Thus,  if  the  resistance  be  a  friction  of  4  ft)s.,  overcome  at  every  point  along  a 
distance  of  3  feet;  or  if  it  be  a  weight  of  4  lbs.,  lifted  3  feet  high,  then  the  work 
done  amounts  to  4  X  3  =  12  foot-fi)s,  provided  the  initial  and  the  final  velocities 
are  equal. 

(d)  In  cases  nrhere  the  velocity  is  uniform,  as  in  a  steadily  running 
machine,  the  force  is  necessarily  equal  to  the  resistance ;  and  where  the  velocities 
at  the  beginning  and  end  of  any  work  are  equal  (as  where  the  machine  starts 
from  rest  and  comes  to  rest  again)  the  mean  force  is  equal  to  the  mean  resistance. 
In  such  cases,  therefore,  the  two  products,  mean  force  X  distance,  and  mean 
resistance  X  distance,  are  equal;  and  we  have,  as  before. 

Work  =  force  X  dist  =  resistance  X  dist. 

*  A  man  who  is  standing  still  is  not  considered  to  be  working,  any  more  than 
is  a  post  or  a  rope  when  sustaining  a  heavy  load;  although  he  may  be  support- 
ing an  oppressive  burden,  or  holding  a  car-brake  with  all  his  strength  ;  for  his 
force  moves  nothing  in  either  case. 

fThese  products  must  not  be  confounded  with  moments,  =  force  X  leverage. 


342  FORCE   IN    RIGID    BODIES. 

(f )  In  calculating  the  work  done  by  macliiner^^  etc.,  allowance  must  be  made  for 
this  expenditure  of  a  portion  of  the  work  in  overcoming  resistances.  Thus,  in  pump- 
ing water,  part  of  the  applied  force  is  required  to  balance  the  friction  of  the  different 
parts  of  the  pump;  so  that  a  steam  or  water  "power,"  exerting  a  force  of  lUO  lbs., 
and  moving  6  feet  per  second,  cannot  raise  100  lbs.  of  water  to  a  height  of  6  feet 
per  second.  Therefore  machines,  so  far  from  gaining  power,  according  to  the  popular 
idea,  actually  lose  it  in  one  sense  of  the  word.  In  starting  a  piece  of  machinery,  the 
forces  employed  have  (Ist)  to  balance,  react  against,  or  destroy  the  resisting  force 
of  friction  and  the  cohesive  forces  of  the  material  which  is  to  be  operated  on ;  and 
(2d)  to  give  motion  to  the  unresisting  matter  of  the  machine  and  of  the  material 
operated  on,  after  the  resisting  forces  which  had  acted  upon  them  have  thus  been 
rendered  ineffective.  But  after  the  desired  velocity  has  been  established,  the  forces 
have  merely  to  balance  the  resistances  in  order  that  the  velocity  may  continue  uniform. 

(g)  That  portion  of  the  work  of  a  machine,  etc.,  which  is  expended  against  fric- 
tion is  ^metimes  called  **  lost  work"  or  *'  prejudicial  work,*'  while  only 
that  portion  is  called  "  useful  -work  "  which  renders  visible  and  tangible  service 
in  the  ^hape  of  output,  etc.  Thus,  in  pumping  water,  the  work  done  in  overcoming 
the  fri9^on  of  the  i)ump  and  of  the  water  is  said  to  be  lost  or  prejudicial,  while  the 
useful  ^ork  would  be  represented  by  the  product,  weight  of  water  delivered  X  height 
to  whi(5i  it  is  lifted. 

The  distinction,  although  artificial,  and  somewhat  arbitrary,  is  often  a  very  con- 
venient one;  but  the  work  is  of  course  not  actually  "  lost,"  and  still  less  is  it  "pre- 
judicial ;"  for  the  water  could  not  be  delivered  without  first  overcoming  the  resist- 
ances. A  merchant  might  as  well  call  that  portion  of  his  money  lost  which  he 
expends  for  clerk-hire,  etc. 

(li)  For  a  given  force  and  distance^  ttie^work  done  is  independent  of  the 
time ;  for  the  product,  force  X  distance,  then  remains  the  same,  whatever  the  time 
may  be.  But  the  distance  through  which  a  given  force  will  work  at  a  given  velocity 
is  of  course  proportional  to  the  time  during  which  it  is  allowed  to  work.  Thus,  in 
order  to  lift  50  pounds  100  feet,  a  man  must  do  the  same  work,  (=  5000  foot-pounds) 
whether  he  do  it  in  one  hour  or  in  ten  ;  but,  if  he  exerts  constantly  the  same  force, 
he  will  lift  50  lbs.  ten  times  as  high  in  ten  hours  as  in  one,  and  thus  will  do  ten  times 
the  work.    Thus,  for  a  given  force,  tl»e  work  is  proportional  to  tlie  time* 

Art.  18  (a),  Power.  The  quantity  of  any  work  may  evidently  be  considered 
without  regard  to  the  time  required  to  perform  it ;  but  we  often  require  to  know  the 
rate  at  which  work  can  be  done ;  that  is,  how  much  can  be  done  within  a  certain 
time. 

The  rate  at  which  a  machine,  etc.  can  work  is  called  its  power.  Thus,  in  selecting 
a  steam-engine,  it  is  important  to  know  how  much  it  can  do  pei-  minute,  hour,  or  day. 
We  therefore  stipulate  that  it  shall  be  of  so  many  horse-powers;  which  means  nothing 
more  than  that  it  shall  be  capable  of  overcoming  resisting  forces  at  the  rate  of  so 
many  times  33,000  foot-pounds  per  minute  when  running  at  a  uniform  velocity,  i.  «., 
when  force  X  distance  =  resistance  X  distance. 

(to)  The  liorse-po-wer,  33,000  foot-pounds  per  minute,  or  550  foot-pounds  per 
second,  is  the  unit  of  power,  or  of  rate  of  Avork,  commonly  used  in  connec- 
tion with  engines.  The  metric  liorse-po-wer,  called  "force  di 
cheval,"  " cheval-vapeur,"  or  (German)  "  Pferdekraft,"  is  75  kilogram-meters  pel 
second  =  542.48  ft.-lbs.  per  sec.  =32,549  ft.-Jbs.  per  minute  =  0.9863  horse-power.  1 
horse-power  =  1.0138  "  force  de  cheval."  In  theoretical  Mechanics  the  foot-pound 
per  second  is  used  in  English  measure;  and  the  kilogram-meter  per  sec- 
ond in  metric  measure, 

1  foot-pound  per  second  =  0.13825  kilogram-meter  per  second. 
1  kilogram-meter  per  second  =  7.2331  foot-pounds  per  second. 

(c)  Up  to  the  time  when  the  velocity  becomes  uniform,  the  po^wer,  or  rate  ot  • 

-work,  of  the  train,  in  Art.  16  (d),  is  variable,  being  gradually  accelerated. 

For  in  each  second  it  overcomes  its  resistances  (and  moves  its  point  of  application) 
through  a  greater  distance  than  during  the  preceding  second.  Also,  after  the  steam  is 
shut  oft",  the  rate  of  work  is  variable,  being  gradually  retarded.  When  the  foi'ce  of 
the  steam  just  balances  the  resistances,  the  rate  of  work  is  uniform. 

(d)  Power  =  force  X  velocity.  Since  the  rate  of  work  is  equal  to  the  work 
done  in  a  given  time,  as  so  many  foot-pounds  per  second,  we  may  find  it  by  dividing  the 
work  in  foot-pounds  done  during  any  given  time  by  the  number  of  seconds  in  that 
lame.     Thus 

_  ,         ,        force  in  pounds  X  distance  in  feet 

Power  =  rate  of  work  = :- — ; — • 

time  in  seconds 


FORCE  IN   RIGID   BODIES.  343 

But  this  is  equivalent  to 

.    Power  =  rate  of  work  -  force  in  pound.  X  t^tTn^Ltla 

=  Corce  in  lbs.  X  velocity  in  feet  per  second. 

Or  if  we  treat  only  of  the  work  of  that  force  which  overcomes  resistances :  or  m 
*  where  the  velocity  is  either  uniform  throughout  or  the  same  at  the 


beginning  and  end  of  the  work; 

Power  rate  of  work  resistance,  v,      velocity, 

in  ft-lbs.  per  sec.  ""  in  fi^lbs.  per  sec.  ™      in  lbs.       ^  in  ft  per  sec. 

Thus  if  the  resistance  is  3300  lbs.  and  is  overcome  through  a  distance  of  10' 
feet  in  every  minute;  or  if  the  resistance  is  33  lbs.  and  is  overcome  through 
a  distance  of  1000  feet  per  minute,  the  rate  of  the  work  is  in  each  case 
the  same,  namely,  33,000  foot-pounds  per  minute,  or  one  horse-power;   for 

lbs.       vel.     lbs.      vel.  . 

3300  X   10  =  33  X  1000  -=  33,000  foot-pounds  per  mmute. 

(e)  The  same  "power"  which  will  overcome  a  given  resistance  through  a 
given  distance,  in  a  given  time,  will  also  overcome  any  other  resistance  through 
any  other  distance,  in  that  same  time,  provided  the  resistance  and  distance 
when  multiplied  together  give  the  same  amount  as  in  the  first  case.  Thus, 
the  power  that  will  lift  50  pounds  through  10  feet  in  a  second,  will  m  a  second 
lift  500  pounds,  1  foot;  or  25  pounds,  20  feet;  or  5000  pounds  ^  of  a  foot. 
In  practice,  the  adjustment  of  the  speed  to  suit  different  resistances,  is  usually 
effected  by  the  medium  of  cog-wlieels,  toelts,  or  levers.  By  means  of 
these  the  engine,  water-wheel,  horse,  or  other  motive  power,  exerting  a  given 
force  and  running  at  a  given  velocity,  may  be  made  to  overcome  small  resist- 
ances rapidly,  or  great  ones  slowly,  as  desired. 

Art.  19  (a).  Tlie  -work,  vrlitcli  a  bodjr  can  do  "hy  virtue  of  its 
motion;  or  (which  is  the  same  thing)  tlie  work  required  to  bring 
tlie  body  to  rest.    Kinetic  energy,  vis  viva,  or  •'living  force." 

As  already  remarked,  a  force  equal  to  the  weight  of  any  body,  at  any  place, 
will,  in  one  second,  give  to  the  mass  or  matter  of  the  body  a  velocity  =  g,  or 
(on  the  earth's  surface)  about  32.2  feet  per  second.  Or  if  a  body  be  thrown 
Mpward  with  a  velocity  =  gr,  its  weight  will  stop  it  in  one  second. 

Since,  in  the  latter  case,  the  velocity  at  the  beginning  and  at  the  end  of  the 
second  are,  respectively,  =  g  feet  per  second,  and  ==  0,  the  mean  velocity  of  th8 

fcody  is  -1—  feet  per  second.   Therefore,  during  the  second  it  will  rise  ^—  feet, 

or  about  16  feet.  In  other  words,  the  work  which  any  body  can  do,  by  virtue 
of  being  thrown  vertically  upward  with  an  initial  velocity  (velocity  at  the 
Start)  of  g  feet  per  second,  is  equal  to  the  product  of  its  weight  multiplied  bf 

-|-feet.    Or, 

work  in  foot-pounds    =.    weight    X      ^ 

Notice  that  in  this  case  (since  the  initial  velocity  v  is  equal  to  gf),    ^      ».  1. 

g 

Suppose  now  that  the  same  body  be  thrown  upward  with  double  the  former 
velocity;  i.  6.,  with  an  initial  velocity  equal  to  2  g  (or  about  CA  feet  per  second). 
Since  gravity  requires  (Art.  8  c),  two  seconds  to  impart  or  destroy  this 

velocity,  the  body  will  now  move  upward  during  two  seconds,  or  twice  as  long 
a  time  as  before.  But  its  mean  velocity  now  is  (7,  or  twice  as  great  as  before. 
Therefore,  moving  for  double  the  time  and  with  double  the  velocity,  it  will 
travel /oMr  times  as  far,  overcoming  the  same  resistance  as  before  (viz.:  iia 
own  weight)  through /owr  times  the  distance. 

Thus,  by  making  its  initial  velocity  t?  =  2  ^,  i.  e.,  by  doubling  its  JL.,  making 

it  =  2,  we  have  enabled  the  body  to  do  four  times  the  work  which  it  could 

do  when  its  -^  was  1;  so  that  the  work  in  the  second  case  is  equal  to  the 


344  FORCE  IN   RIGID   BODIES. 


tofWbs.   -    -«-'>'    ^    ^ 


product  of  that  in  the  first  case  multiplied  by  the  square  of      ^,;  or, 
»    weight    X    -SL    X     (-^)' 

2  ^  g  ^ 

-    weight    X    -f-     X    -?^ 
2  ff  ^ 

=    weight    X    -^ 
2g 

And  it  is  plain  that  this  would  be  ithe  case  for  any  oi^er  velocity.  Now  the 
total  amount  of  the  work  which  the  body  can  do,  is  independent  of  the 
amount  of  the  resistance  against  which  it  is  done;  for  if  we  increase  the 
resistance  we  diminish  the  distance  in  the  same  proportion,  so  that  their 
product,  or  the  amount  of  worlc,  remains  the  same.  The  above  formula, 
therefore,  applies  to  ail  cases;  i.  e.,  the  total  amount  of  work,  in  foot 
pounds,  which  any  body  will  do,  against  any  resistance,  by  virtue  of  its  motion 
alone,  in  coming  to  rest,  is 
Work  =  weight  of  moving  body,  in  lbs.  X  square  of  its  velocity  in  ft  per  sec^d 

=  weight  of  moving  body,  in  lbs.  X  fall  in  ft  required  to  give  the  velocity 
^  weightof  moving  body,  in  lbs,  s^  square  of  its  velocity  in  ft  per  second 
9  2 

In  these  equations,  the  weight  is  that  which  the  body  has  in  any  given  place, 
and  g  is  the  acceleration  of  gravity  at  that  same  place. 

(to)  Since  the  weight  of  a  body  .^  .^^  ^^^^  (Art.  11,  p.  336) ,  the  last  formula 
9 
becomes,  by  "method  A,"  Art.  11  (d), 

Mrork        ^  mass  of  moving  body  w  square  of  its  velocity  in  ft  per  second 
in  foot-pownds  in  "waiii"  '^  2 

and  by  "method  B,"  Art.  11  («), 

vvorls:         _  mass  of  moving  body  y,  square  of  its  velocity  in  ft  per  second 
in  foot-poundals  ~  in  pounds  ^  2 

(c)  In  the  above  equations  the  ie/i  hand  side  represents  the  work  (or  resis- 
tance overcome  through  a  distance)  in  any  given  case,  while  the  right  hand 
side  represents  the  Iclnetic  energy  of  the  body,  by  which  it  is  enabled  to  do 
that  work.  Some  writers  call  this  energy  "vis  viva,"  or  "  living  force"  a 
name  formerly  given  (for  convenience)  to  a  quantity  just  double  the  energy, 
or  =  mass  X  velocity^. 

(d)  As  an  illustration  of  the  foregoing,  take  a  train  weighing  1,120,000 
pounds,  and  moving  at  the  rate  of  22  feet  per  second.  The  kinetic  energy 
•f  such  a  train  is 

energy    »    weight    X    — r — ~  ;         or, 

1,120,000  lbs.  X  -^  =  8,400,000  ft.-lbs. 
64.4 

That  is,  if  steam  be  shut  off,  the  train  will  perform  a  work  of  8,400,000  ft.-lbe. 
in  coming  to  rest.  Thus,  if  the  sum  of  all  the  resistances  (of  friction,  air, 
grades,  curves,  etc.)  remained  constantly  ==  5000  lbs.,*  the  train  would  travel 

8,400,000  ft.-lbs.    ^  jggQ  f^^ 
5000  lbs. 

(e)  We  thus  see  that  the  total  quantity  of  work  which  a  body  can  do  by  virtue 
of  its  motion  alone,  and  without  assistance  from  extraneous  forces,  is  in  pro- 
portion to  the  weight  of  the  body  and  to  the  square  of  its  velocity  when  it 
begins  to  do  the  work.  For  example,  suppose  that  a  tram,  at  the  moment 
when  steam  is  shut  off,  has  a  velocity  of  10  miles  an  hour  and  that  the  kinetio 
energy,  which  that  velocity  gives  it,  will  by  itself  carry  the  train  agamst  th« 

•In  practice,  this  would  not  be  the  case. 


FORCE  IN  RIGID   BODIES.  345 

resistances  of  the  road,  etc.,  for  a  distance  of  one  quarter  of  a  mile  before  it 
stops.  Then,  if  steam  be  shut  oflf  while  the  train  is  moving  at  5,  20,  30  or  40 
miles  per  hour  (i.  e.  with  J^,  2, 3  or  4. times  10  miles  per  hour)  the  train  will 
travel  JL.»  1»  2  J^  or  4  miles  (or  J^,  4,  9  or  IG  times  ^  mile)  before  coming  to 

rest* 

But  the  rate  of  work  done  is  proportional  simply  to  the  resistance  and  the 
v^ocity  (Art.  18d,  p.  342).  Therefore,  the  locomotive  whose  steam  is  shut  ofl 
at  20,  30  or  40  miles  pelr  hour,  will  require,  for  running  its  4, 9  or  16  quarters 
«f  amile,  but  2, 3  or  4  times  as  many  seconds  as  itrequired  at  10  miles  per  hour. 

The  same  principle  applies  to  all  cases  of  acceleration  or  of  retardation.f 

For  instance,  in  the  case  of  a  falling  body,  the  distance  through  which  it 
must  fall  in  order  to  acquire  any  given  velocity  is  as  the  square  of  that 
velocity,  but  the  time  required  is  simply  as  the  velocity.  Also,  if  a  body  is 
thrown  vertically  upward  with  any  given  velocity,  the  height  to  which  it  will 
rise  by  the  time  gravity  destroys  that  velocity,  will  be  as  the  square  of  the 
velocity,  l5ut  the  time  will  be  simply  as  the  velocity. 

Art.  20  (a).  The  momentuni  of  a  moving  body  (or  the  product  of  its 
mass  by  its  velocity)  is  the  rate,  in  foot-pounds  per  second,  at  which  it  works 
against  a  resisting  force  equal  to  its  own  weight,  as  in  the  case  of  a  body  thrown 
vertically  upward.  At  the  instant  when  it  comes  to  rest,  its  momentum,  or  rate 
of  work,  is  of  course  =  nothing.  Therefore  its  mean  rate  of  work,  or  mean 
momentum,  is  one-half  of  that  which  it  has  at  the  moment  of  starting. 

Thus,  suppose  such  a  body  to  weigh  5  lbs.  Then,  whatever  its  velocity  niay 
be,  5  pounds  is  the  resisting  force,  against  which  it  must  work  while  coming 
lo  rest.    Let  the  initial  velocity  be  96  feet  per  second.    Then  its 

momentum  =  mass  X  velocity  —  5  X  96  —  480  foot-pounds  per  second* 
and,  while  coming  to  rest,  its 

mean  momentum  =»  mass  X  T^  Qc^ty  ^  240  foot-pounds  per  second. 

Now,  in  falling,  the  weight  of  the  body  (5  lbs.),  would  give  it  a  velocity  of  9d 
feet  per  second  in  about  three  seconds.  Consequently,  in  rising,  it  will  destroy  its 
velocity  in  the  same  time.  In  other  words,  the  time  —  — \e  oci  y  _  ^  ve  ocity 

acceleration  g 

BB  £6  «=  3.  Three  seconds,  therefore,  is  the  time  during  which  it  can  work. 
Now,  if  the  mean  rate  of  work  in  foot-pounds  per  second  (at  which  a  body 
can  work  against  a  resistance)  be  multiplied  by  the  time  during  which  it  can 
continue  so  to  work,  the  product  must  be  the  total  work  done.    Or,  in  this  case, 

work         mean  rate  of  work  ^  time,         _  940  v  ^  L  720  foat.nnnnrl. 

in  ft-lbs.  ■"   in  ft-lbs.  per  sec.    ><  or  No.  of  sees.  -  240  X  ^  =«  7^0  loot-pounds. 

-  weight  X^^^^^^X^^^^ 

2  g 

—  weight  X  ^^^^^^^^  ,  as  in  Art.  19  (a),  =  5  X  -^  -=  720  ft-pound«. 

(b)  We  may  notice  also  that  since,  in  the  case  of  a  falling  body,  or.  of  one 

thrown  upward,  7^  ^^^  ^  is  the  time  during  which  it  must  fall  in  order  to 

g 
acquire  a  given  velocity,  or  during  which  it  must  rise  in  order  to  lose  it, 
therefore, 

velocity  ^  yelocity  ^  ^^^^  velocity  X  time  =  distance  traversed; 
2  g 

so  that 

weight  X  I^l^in^'  =  weight  X  I2l!!2i*L  x  l^^^S}^  » 

2^  2.  g 

weight  X  distance  traversed      «=»  the  work. 

*  This  supposes,  for  convenience,  that  the  resistances  remain  uniform  through- 
out, and  are  the  same  in  all  the  cases,  which,  however,  would  not  hold  good  in 
practice. 

t  Retardation  is  merely  acceleration  in  la  direction  opposite  to  tlaat  of  the 
motion  which  we  happen  to  be  considering. 


346  FORCE  IN   RIGID   BODIES. 

Art,  ai  (a).  Energy  is  In.lestmctible.  Energy,  expended  in  work,  is 
not  destroyed.  It  is  either  transterred  to  other  bodies,  or  else  stored  up  in  the 
body  itself;  or  part  may  be  thus  transferred,  and  the  rest  thus  stored.  But, 
altliough  energy  cannot  be  destroyed,  it  may  oe  rendered  useless  to  us.  Thus, 
a  moving  train,  in  coming  to  rest  on  a  level  track,  transfers  its  kinetic  energy 
into  other  kinetic  energy;  namely,  the  useless  heat  due  to  friction  at  the  rails, 
brakes  and  journals ;  and  this  heat,  although  none  of  it  ia  destroy ed^  is  dissipated 
in  the  earth  and  air  so  as  to  be  practically  beyond  our  recovery. 

Art.  Ji'4  (a).  Potential  energy,  or  possible  energy,  may  be  defined  as 
stored-up  energy.  We  lift  a  one-pound  body  one-foot  oy  expending  upon  it 
one  foot-pound  of  energy.  But  this  foot-pound  is  stored  up  in  the  "  system  " 
(composed  of  the  earth  and  the  body)  as  an  addition  to  its  stock  of  potential 
energy.  For,  while  the  stone  falls  through  one  foot,  the  system  will  acquire 
a  kinetic  energy  of  one  foot-pound,  and  will  part  with  one  foot-pound  of  its 
potential  energy. 

(b)  The  potential  energy  of  a  "system"  of  bodies  (such  as  the  earth  and  a 
weight  raised  above  it,  or  the  atoms  of  a  mass  of  powder,  or  those  of 
a  bent  spring)  depends  upon  the  relative  positions  or  those  bodies,  and 
upon  their  tendencies  to  change  those  positions.  The  kinetic  energy  of  a 
system  (such  as  the  earth  and  a  moving  train  of  cars)  depends  upon  the  masses 
of  its  bodies  and  upon  their  motion  relatively  to  each  other. 

Familiar  instances  of  potential  energy  are— the  weight  or  spring  of  a  clock 
when  fully  or  partly  wound  up,  and  whether  moving  or  not;  the  pent-up  water 
in  a  reservoir;  the  steam  pressure  in  a  boiler;  and  the  explosive  energy  of 
powder.  We  have  mechanical  energy  in  the  case  of  the  weight  or  springs  or 
water;  heat  energy  in  the  case  of  the  steam,  and  chemical  energy  in  that 
of  the  powder. 

(c)  In  many  cases  we  may  conveniently  estimate  the  total  potential  energy 
of  a  system.  Thus  (neglecting  the  resistance  of  the  air)  the  explosive  energy 
of  a  pound  of  powder  is  =-  the  weight  of  any  given  cannon  ball  X  the  height 
to  which  the  force  of  that  powder  could  throw  it,  =»  tlie  weight  of  the  ball  X 
(the  square  of  the  initial  velocity  given  to  it  by  the  explosion)  -i-  2o.  But  in 
other  cases  we  care  to  find  only  a  certain  definite  portion  of  the  total  potential 
energy.  Thus, the  total  potential  energy  of  a  clock-weight*  would  not  be 
exhausted  until  the  weight  reached  the  center  of  the  earth;  but  we  generally 
deal  only  with  that  portion  which  was  stored  in  it  by  winding-up,  and  which 
it  will  give  out  again  as  kinetic  energy  in  running  down.  This  portion  is  =  the 
weight  X  the  height  which  it  has  to  run  down  >=  the  weight  X  (the  square  of 
the  velocity  which  it  would  acquire  in  falling/reeZ^/ through  that  height)  -f-  2^g. 

(d)  There  are  many  cases  of  energy  in  which  we  may  hesitate  as  to  whether 
the  term  "kinetic"  or  "potential"  is  the  more  appropriate.  Thus,  the  pres- 
sure of  steam  in  a  boiler  is  believed  to  be  due  to  the  violent  motion  of  the 
particles  of  steam,  which  bombard  the  inner  surface  of  the  boiler-shell;  so 
that,  from  this  point  of  view,  we  should  call  the  energy  of  steam  kinetic.  But, 
on  the  other  hand,  the  shell  itself  remains  stationary;  and,  until  the  steam  is 
permitted  to  escape  from  the  boiler,  there  is  no  outward  evidence  of  energy 
in  the  shape  of  work.  The  energy  remains  stored  up  in  the  boiler  ready  for 
use.   From  this  point  of  view,  we  may  call  the  energy  of  steam  potential  energy. 

(e)  It  seen\s  reasonable  to  suppose  that  further  knowledge  as  to  the  nature 
of  other  forms  of  energy,  apparently  potential  (as  is  that  of  steam),  mighl 
reveal  the  fact  that  all  energy  is  ultimately  kinetic. 

Art.  23  (a).  There  is  much  confusion  of  ideas  in  regard  to  those 
actions  to  which,  in  Mechanics,  we  give  the  names,  **  force,"  *•  energy," 
"  power,"  etc.  This  arises  from  ihe  fact  that  in  every-day  language  these 
terms  are  used  indiscriminately  to  express  the  same  ideas. 

Thus,  we  commonly  speak  of  the  "  force  "  of  a  cannon-ball  flying  through  the 
air,  meaning,  however,  the  repulsive  force  which  ivoidd  be  exerted  between  the 
ball  and  a  building,  etc.  with  which  it  might  come  into  contact.  This  force 
would  tend  to  move  a  part  of  the  building  along  in  the  direction  of  the  flight 
of  the  ball,  and  would  move  the  ball  backward  ;  (i.e.,  would  retard  its  forward 
motion).  But  this  great  repulsive  "/o^ce"  does  not  exist  until  the  ball  strikes 
the  building.  Indeed,  we  cannot  even  tell,  from  the  velocity  and  weight  of  the 
ball,  what  the.amount  of  the  force  will  be,  for  this  depends  upon  the  strength, 
etc.,  of  the  building.  If  the  building  is  of  glass,  the  force  may  be  so  slight  as 
scarcely  to  retard  the  motion  of  the*  ball  perceptibly,  while,  if  the  building  is  an 

*  For  convenience  we  may  thus  speak  of  the  energy  of  a  system  of  bodies  (the 
earth  and  tlie  clock-weight)  as  residing  in  only  one  of  the  bodies. 


FORCE   IN    RIGID   BODIES.  347 

earth  embankment,  the  force  will  be  much  greater,  and  may  retard  the  motion 
of  the  ball  so  rapidly  as  to  entirely  stop  it  before  it  has  gone  a  foot  farther. 

The  moving  ball  has  great  (kinetic)  energy;  but  the  only  force  that  it  exerts 
during  its  flight  is  the  comparatively  very  slight  one  required  to  push  aside  the 
particles  of  air. 

The  energy  of  the  ball,  and  therefore  the  total  work  which  it  can  do,  are  inde- 
pendent  of  the  nature  of  the  obstruction  which  it  meets  ;  but  since  the  work  is 
the  product  of  the  resistance  ottered  and  the  distance  through  which  it  can  be 
overcome,  the  distance  must  be  inversely  as  the  resistance  ottered  ;  or  (which  is 
the  same  thing)  inversely  as  the  force  required  of,  and  exerted  by,  the  ball  in 
balancing  that  resistance. 

Since  work,  in  ft.-tbs.  =  force,  in  fibs.,  X  distance  traversed,  in  feet,  we  have 

force  in  lbs  = work,  in  ft.-ft>s.  ^       rate  of  work, 

'  '      distance  traversed,  in  feet       in  ft.-lbs.  per  foot. 

Art.  24  (a).  An  impact,  blow,  stroke  or  collision  takes  place  when  a 
•loving  body  encounters  another  body.  The  peculiarity  of  such  cases  is  that 
the  time  of  action  of  the  repulsive  force  due  to  the  collision  is  so  skvH  that  gen- 
erally it  is  impossible  to  measure  it,  and  we  therefore  cannot  calculate  the  force 
from  the  momentum  produced  by  it  in  either  of  the  two  bodies :  but  since  both 
bodies  undergo  a  great  change  of  velocity  (i,  e.,  a  great  acceleration)  during  this 
short  time,  we  know  that  the  repulsive  force  acting  between  them  must  be  very 
great. 

We  shall  consider  only  cases  of  direct  impact,  or  impact  where  the  centers 
of  gravity  of  the  two  bodies  approach  each  other  in  one  straight  line,  and  where 
the  nature  of  the  surfaces  of  contact  is  such  that  the  repulsive 

force  caused  by  the  impact  also  acts  through  those  centers  and  in  their  line  of 
approach. 

(b)  This  force,  acting  equally  upon  the  two  bodies  (Art.  Sjf),  for  the 
same  length  of  time  (namely,  the  time  during  which  they  are  in  contact),  neces- 
sarily produces  equal  and  opposite  changes  in  their  momentums  (Art.  12,  p.  338). 
Hence,  the  total  momentum  (or  product,  mass  X  velocity)  of  the  two  bodies  is 
always  the  same  after  impact  as  it  was  before. 

(c)  But  the  relative  behavior  of  the  two  bodies,  after  collision,  depends  upon 
their  elasticity.  If  they  could  be  perfectly  inelastic,  their  velocities,  after  im- 
pact, would  be  equal.  In  other  words,  they  would  move  on  together.  If  they 
could  be  perfectly  elastic,  they  would  separate  from  each  other,  after  collision, 
with  the  same  velocity  with  which  they  approached  each  other  before  collision. 

(cl)  Between  these  two  extremes,  neither  of  which  is  ever  perfectly  realized  in 
practice,  there  are  all  possible  degrees  of  elasticity,  with  corresponding  differences 
in  the  behavior  of  the  bodies.  The  subject,  especially  that  of  indirect  impact,  is 
a  very  complex  one,  but  seldom  comes  up  in  practical  civil  engineering. 

(e)  "In  some  careful  experiments  made  at  Portsmouth  dock-yard,  England,  a 
man  of  medium  strength,  and  striking  with  a  maul  weighing  18  fibs.,  the  handle 
©f  which  was  44  inches  long,  barely  atarted  a  bolt  about  Yq  of  an  inch  at  each 
blow ;  and  it  required  a  quiet  pressure  of  107  tons  to  press  the  bolt  down  the 
same  quantity  ;  but  a  small  additional  weight  pressed  it  completely  home." 


348 


GRAVITY — FALLING   BODIES. 


GRAVITY,  FAI.I.IXO  BODIES. 

Bodies  falliiig;>  vertically.  A  body,  falling  freely  in  racuo 
from  a  state  of  rest,  acquires,  by  tlie  end  of  tlie  fir^t  second,  a  velocity  of  about 
32.2  feet  per  second;  and,  in  each  succeeding  second,  an  addition  of  velocity,  or 
acceleration,  of  about  32.2  feet  per  second.  In  other  words,  tlie  velocity  receives  in 
each  second  an  acceleration  of  about  a2.2  feet  per  second,  or  is  accelerated  at  th« 
rate  of  about  32.2  feet  per  second,  per  second.  This  rate  is  generally  called  (for 
brevity,  see  foot-note,f  p.  334) ,  simply  the  acceleration  of  gravity  (but  see  * 
belowj,  and  is  denoted  by  g.  It  increases  from  about  32.1  f>.et  per  second,  per 
second,  at  the  equator,  to  about  32.5  at  the  poles.  In  the  latitude  of  London  it  is 
32.19.  These  are  its  values  at  sea-level ;  but  at  a  height  of  5  miles  above  that  level 
k  is  diminished  by  only  abuut  i  part  in  400.  For  most  practical  purposes  it  may  be 
taken  at  32.2. 

Caution.  Owing  to  tbe  resistance  of  the  air  none  of  the  follow- 
ing rules  give  perfectly  accurate  results  in  practice,  especially  at  great  vels. 
Thp  greater  the  specific  gravity  of  the  body  the  better  will  be  the  result.  The  air 
resists  botfi  rlsiii;;  and  fiiHin^  bodies. 

If  a  body  be  tliroiFn  vertically  upwards  with  a  given  vel,  it  will 
rise  to  the  same  height  from  wliich  it  inust  have  fallen  in  order  to  acquire  said 
vel;  and  its  vel  will  be  retarded  in  each  second  32.2  ft  per  sec.  Its  average  ascend' 
ing  velocity  will  be  half  of  that  with  which  it  started ;  as  in  all  other  cases  of 
uniformly  retarded  vel..  In  falling  it  will  acquire  the  same  vel  that  it  started 
up  with,  and  in  the  same  time.    See  above  Caution. 


Acceleration  acquired* 
in  a  given  time  =  9  X  time 

in  a  given  fall  from  rest     =  y'  2  g  xTall. 
in  a  given  fall  frotn  rest  )  _  twice  the  fall 
and  given  time  j  ~~         time 

Time  required 

acceleration 


for  a  given  acceleration     = 


for  a  given  fall  from  rest  = 


/fall 


fall 


fall 


^  final  velocity 
fall 


for  a  given  fall  from  rest  ■>  _ 

or  otherwise  J  ~~  mean   vel      ^^  (initial  vel  +  final  vel) 

Fall 

in  a  given  time  (starting  from  rest)  =  time  X  H  ^'^^^  ^^^  ==  ^^°^^  ^  X  j^Sf 

in  a  given  time  (starting)       ,.       .  ,  i       ^.       ^ .  initial  vel  -f  final  vel 

^    ^        ^         ^,         .     °  [=  time  X  mean  vel  =  time  X tt 

from  rest  or  otherwise)  )  2 

reqd  for  a  given  acceleration  ")  _  acceleration^ 

(starting  from  rest)  j  "~  2  g 

during  any  one  given  second  (counting  from  rest) 

=  g  X  (number  of  the  second  (1st,  2d,  &c)  —  ^  j 
during  any  equal  consecutive  times  (starting  from  rest)  oc  1,  3,  5,  7,  9,  &c. 

At  tlie  end  of  tlie 

4th.    5th.    6th.     7th.      8th.     9th.      10th. 
seconds 


''^"'wfLTf-'}     »^'-  2.1. 


Velocity  ;  ft  per  sec. 
Dist  fallen  since  end 
of  preceding  sec  ;  ft. 

Total  dist  fallen ;  ft. 


32.2 

64.4 

96.6 

128.8 

161.0 

193.2 

225.4 

257.6 

289.8 

16.1 

48.3 

80.5 

112.7 

144.9 

177.1 

209.3 

241.5 

273.7 

16.1 

64.4 

144.9 

257.6 

402.5 

579.6 

788.9 

1030.4 

1304.1 

322.0 
305.9 
1610.0 


*  By  "  acceleration,"  in  this  article,  we  mean  the  totat  acceleration ;  t.  e.,  the  whol« 
change  of  velocity  occurring  in  tbe  givea  time  or  Mi.  J^'or  tbe  rate  of  acceleration 
we  use  simply  tbe  letter  g. 


DESCENT    ON    INCLINED    PLANES.  349 

Descent  on  inclined  planes.  When  a  body,  U,  is  placed 
upon  an  inclined  plane,  AC,  its  whole  weight  W  is  not  employed  in  giving  it 
velocity  (as  in  ihe  case  of  bodies  falling  vertically) 
but  a  portion,  P,  of  it  (=  W  X  cosine  of  o  —  W  X 
cosine  of  a*)  is  expended  in  perpendicular  pressure 
against  the  plane;  while  only  S,  (=  W  X  sine  of  o 
=  W  X  sine  of  a,*)  acts  upon  U  in  a  direction  parallel 
to  the  surface  A  C  of  the  plane,  and  tends  to  slide  it 
down  that  surf. 

The  acceleration,  generated  in  a  given  body  in  a 
given  time,  is  proportional  to  the  force  acting  upon 
the  body  in  the  direction  of  the  acceleration 

Hence  if  we  make  W  to  represent  by  scale 
the  acceleration  g  (say  32.2  ft  per  sec)  which  grav 
would  give  to  U  in  a  sec  if  falling  freely,  then  S  -will 
give,  by  the  same  scale,  the  acceleration  in  ft  pei- 
sec  which  the  actual  sliding  force  S  would  give  to  U  in  one  sec  if  there  were 
no  friction  between  U  and  the  plane.  •  We  have  therefore 

theoretical  acceleration  down  the  plane  =  g  x  sine  of  a. 

Therefore  we  have  only  to  substitute  "(7.  sin  a"  in  place  of  "5';"  and  the 
sloping  distance  or  "slide"  A  C  in  place  of  the  corresponding  vertical  distance 
or  "  fall "  A  E  in  the  equations,  in  order  to  obtain  the  accelerations  etc  as 

follows: 

on  an  inclined  plane  wifliont  friction. 

In  the  following,  the  slides  A  €  are  in  feet,  the  times  in 

seconds,  and  the  velocities  and  accelerations  in  feet  per 

second. f 

Accelerationfof  sliding  velocity 

i«  Q  o-itroTi  tir^no   -  ^^^^  ^^^^cl  acQulred  in  falling)  ^  ^.    „ 
in  a  given  time  -      ^^^^  ^^^.^"^  ^^^^  ^^^^^  ^.^|  j  X  sin  a- 

=  g.  sin  a  X  time 

in  a  given  slide,  as  A  C,  ">  ^    slide 

from  rest  i       3^  time 

("vert  accel  acquired  in  falling"|  _ 

=<  freely  thro  the  corresponding  >=  j/  2  ^r  AB 
t    vert  ht  A  E  J 

«a  y  2  g.  sin  a  X  slide 

Time  required 

,.,.  ,       ..  sliding  acceleration 

for  a  given  sliding  acceleration  = ^ 

^  ^  g.sma 

for  a  given  slide,  as  A  C,  from  slide  _     /      slide 

rest  ' 


^  final  sliding  velocity        ^  /^9-  sin  a 

time  reqd  to  fall  freely  thro  the  correspond- 

^  ing  vert  ht  A  E 

sin  a 
for  a  given  slide,  from>  slide  _  slide 


rest  or  otherwise        J       mean  sliding  vel       3^  (initial  +  final  sliding  vels) 

horizontal  stretch,  as  EC,  ^_ 

.  _    base  E  C    _       of  any  length,  as  A  C  \/  AC2  — A  E2 

"~  length  A  C  ~  that  length  ~^ 

Sine  a       =  ^^^^S^IA?  —  fall,  A  E,  in  any  given  length,  AC 
length  A  C  ~  that  length 

*  Because  0  and  a  are  equal. 
fBy  acceleration,  in  this  article,  we  m.ati  the  lotai  acceleration,  t.  e.,  the  wholo 
chana^e  in  velocity  occurring  in  the  given  time  or  slide.   For  the  rate  of  accelwation 
ipe  use  simply  the  letter  g. 


350 


GRAVITY — PENDULUMS. 


Slide,  as  A  C 
in  a  given  time,  starting  from  rest  =  time  X  K  final  sliding  vel 

=  time  2  X  K  ^.  sin  a. 
in  a  given  time,  s<;arting  from  rest      ^. 
or  otherwise  =  "me  X  mean  sliding  vel 

=  time  X  li  (initial  -f  final,  sliding  vels) 

required  for  a  given  sliding  aceel-  _  sliding  accelerationg 
eration  (starting  from  rest)  2  g.  sin  a 

-  g  X  [sin  a  -  (cos  a.  coefF  fric)]  "  in  place  of  "  g.  sin  a  "  in  the  above  equations. 
Because 

Friction  =  Perpendicular  pressure  P  X  coefficient  of  friction 

=  weight  W  X  cosine  a  X  coefficient  of  friction 

retardation  of  friction  =  gX  cosine  a  X  coefficient  of  friction. 

Resultant  slidinjs^  acceleration 

=  theoretical  sliding;  accel  (due  to  the  sliding  force,  S)  —  retardation  of  fric 
=  {g.  sin  a)  —  {g.  cosine  a.  coeff  fric) 

=  ^  X  ^sin  a  —  (cosine  a.  coeff  fric)~| 

If  The  retardation  of  friction  (=  g.  cos  a  X  coeff  fric)  is  not  less  than  the  total 
or  theoretical  accel  ("p.  sin  a")  the  body  cannot  slide  down  the  plane. 


PENDULUMS. 


The  numbers  of  vibrations  which  diff  pendulums  will  make  in  any  given  place*  in 
It  given  time,  are  inversely  as  the  square  roots  of  their  lengths;  thus,  if  one  of  them 
is  4,  9,  or  16  times  as  long  as  the  other,  its  sq  rt  will  be  2,  3,  or  4  times  as  great ;  but 
its  number  of  vibrations  will  be  but  ]/^,  %,  or  %  as  great.  The  times  in  which  diff 
pendulums  will  mak«  a  vibration,  are  directly  as  the  sq  rts  of  their  lengths.  Thus, 
if  one  be  4,  9,  or  16  times  as  long  as  the  other,  its  sq  rt  will  be  2,  3,  or  4  times  as 
great;  and  so  also  will  be  the  time  occupied  in  one  of  its  vibrations. 

The  length  of  a  pendulum  vibrating  seconds  at  the  level  of  the  sea,  in  a  vacuum, 
in  the  lat  of  London  (513^°  North)  is  39.1393  ins ;  and  in  the  lat  of  N.  York  (40%° 
North)  39.1013  ins.  ^I'j  the  ecpiator  about  -^-q  inch  shorter ;  and  at  the  poles,  about  -^-^ 
Inch  longer.  Approximately  enough  for  experiments  which  occupy  but  a  few  sec, 
we  may  at  any  place  call  the  length  of  a  seconds  pendulum  in  the  open  air,  39  ins ; 
half  sec,  9^  ins  ;  and  may  assume  that  long  and  short  vibrations  of  the  same  pen- 
dulum are  made  in  the  same  time ;  which  they  actually  are,  very  nearly.  For  meas- 
uring depths,  or  dists  by  sound,  a  sulficiently  good  sec  pendulum  may  be  made  of  a 
pebble  (a  small  piece  of  metal  is  better)  and  a  piece  of  thread,  suspended  from  8 
common  pin.   The  length  of  39  ins  should  be  measured  from  the  cpntre  of  the  pebble. 


PKJ^TDULTJMS,    ETC,  351 

In  starting  the  vibrations,  the  pebble,  or  bob,  mxist  not  be  thrown  into  motion,  but 
merely  let  drop,  after  extending  the  string  at  the  proper  height. 

To  find  the  leiig>t;ti  of  a  pendulum  reqd  to  make  a  given  number  of 
vibrations  in  a  min,  divide  375  by  said  reqd  number.  The  square  of  the  quot  will  be 
the  length  in  ins,  near  enough  for  such  temporary  purposes  as  the  foregoing.  Thus, 
for  a  pendulum  to  make  100  vibrations  per  min,  we  have  |^^4  =  3.75 ;  and  the  square 
of  3.75  =  14.06  ins,  the  reqd  length. 

To  find  tbe  number  of  vibrations  per  min  for  a  pendulum  of 
given  length,  in  ins,  take  the  sq  rt  of  said  length,  and  div  375  by  said  sq  rt.    Thus, 

for  a  pendulum  14.06  ins  long,  the  sq  rt  is  3.75 ;  and  ^   =  100,  the  reqd  number. 

Rem.  1.  By  practising  before  the  sec  pendulum  of  a  clock,  or  one  prepared  as  just 
stated,  a  person  will  soon  learn  to  count  5  in  a  sec,  for  a  few  sec  in  succession ;  and  will 
thus  be  able  to  divide  a  sec  into  5  equal  parts ;  and  this  may  at  times  be  useful  for 
very  rough  estimating  when  he  has  no  pendulum. 

Centre  of  Oscillation  and  Percussion. 

Rem.  2.  When  a  pendulum,  or  any  other  suspended  body,  is  vibrating  or  oscillating 
backward  and  forward,  it  is  plain  that  those  particles  of  it  which  are  far  from  the 
point  of  suspension  move  faster  than  those  which  are  near  it.  But  there  is  always 
a  certain  point  in  the  body,  such  that  if  nil  the  particles  were  concentrated  at  it,  so 
that  all  should  move  with  the  same  actual  vel,  neither  the  number  of  oscillations, 
nor  their  angular  vel,  would  be  changed.  This  point  is  called  the  center  of  osciUa- 
tion.  It  is  not  the  same  as  the  cen  of  grav,  and  is  always  farther  than  it  from  the 
point  of  suspension.  It  is  also  the  centre  of  percussion  of  the  suspended  vibrating 
body.  The  dist  of  this  point  from  the  point  of  susp  is  found  thus  :  Suppose  the  body 
to  be  divided  into  many  (the  more  the  better)  small  parts;  the  smaller  the  better. 
Find  the  weight  of  each  part.  Also  find  the  cen  of  grav  of  each  part ;  also  the  dist 
from  each  such  con  of  grav  to  the  poiat  of  susp.  Square  each  of  these  dists,  and 
mult  each  square  by  the  wt  of  the  corresponding  small  part  of  the  body.  Add  the 
products  together,  and  call  their  sump.  Next  mult  the  weight  of  the  entire  body 
by  the  dist  of  its  cen  of  grav  from  the  point  of  susp.  Call  the  prod  g.  Divide  p  hyg» 
This  jo  is  the  moment  of  inertia  of  the  body,  and  if  divided  by  the  wt  of  the 
k)ody  the  sq  rt  of  the  quotient  will  be  the  Radius  of  Oy  ration. 

Ang-ular  Velocity. 

When  a  body  revolves  around  any  axis,  the  parts  which  are  farther  from  that 
axis  move  faster  than  those  nearer  to  it.  Tlierefore  we  cannot  assign  a  stated 
linear  velocity  in  feet  per  second,  or  miles  per  hour  etc,  that  shall  apply  to  every 
part  of  it.  But  every  part  of  the  body  revolves  around  an  entire  circle,  or 
throutrh  an  angle  of  360°,  in  the  same  time.  Hence,  all  the  parts  have  the  same 
velocity  in  degrees-  per  second,  or  in  revolutions  per  second.  This  is  called  the 
angular  velocity.  Scientific  writers  measure  it  by  the  length  of  the  arc  de- 
scribed by  any  point  in  the  body  in  a  given  time,  as  a  second,  the  length  of  the 
arc  being  measured  by  the  number  of  times  the  length  of  its  oion  radius  is  con- 
tained in  it.    When  so  measured, 

Angular  velocity     _  li:aear  velocity  (in  feet  etc)  per  sec 
in  radii  per  second  -        length  of  radius  (in  feet  etc) 

Here,  as  before,  the  angular  velocity  is  the  same  for  all  the  points  in  the  body, 
because  the  velocities  of  the  several  points  are  directly  as  their  radii  or  dis- 
tances from  the  axis  of  revolution. 

Ih  each  revolution,  each  point  describes  the  circumference  of  the  circle  in 
which  it  revolves  =  2  7rr  (tt  =  3.1416  etc;  r  =  radius  of  said  circle).  Conse- 
quently, if  tlie  body  makes  n  revolutions  per  second,  the  length  of  the  arc  de- 
scribed by  each  point  in  one  second  is  2  7rr?i;  and  the  angular  velocity  of  the 
body,  or  linear  velocity  of  any  point  measured  in  its  own  radii,  is 

a  =  =  2  TT  w  =  say  6.2832  X  revs  per  second  =  say  .1047  X  revs  pp'  i^inute. 

Moment  of  Inertia. 

Suppose  a  body  revolving  around  an  axis,  as  a  grindstone;  or  oscillating,  like 
a  pendulum.  Suppose  that  the  distance  from  the  axis  of  revolution  (which,  in 
the  pendulum,  is  the  point  of  suspension)  to  each  individual  particle  of  the 
body,  has  been  measured;  and  that  the  square  of  each  such  distance  has  been 
multiplied  by  the  weight  of  that  particle  to  which  said  distance  was  measured. 


352 


MOMENT    OF    INERTIA. 


The  sum  of  all  these  products  is  the  moment  of  inertia  of  the  body.    Or 


Moment 
of  inertia 


-{ 


■.%d^w. 


thpsnm       •    )  (weight  square  of  dist 

forallthen^rticlesr     ^^      1       ''^'       X    «t  Partielefrom 
lor  all  the  particles  j  (particle     axis  of  revolution 


Scientific  writers  frequently  use  the  mass  of  each  particle ; 

its  weii^ht  .  ,     „ .  .      .        ,     . 

t  e,  -, --: 7-7 — -^ ~. r — r^^FT^  instead  of  its  weight,  m  calculatiug 

acceleration  {g)  of  gravity,  or  about  32.2  ° 

the  moment  of  inertia. 

In  practice  we  may  suppose  the  body  to  be  divided  into  portions  measuring 
a  cubic  inch  (or  some  other  small  size)  each  ;  and  use  these  insteaO  of  the  theo- 
retical infinitely  small  particles.  The  smaller  these  portions  are  taken,  the 
more  nearly  correct  will  be  the  result. 

When  tl)e  moment  of  inertia  of  a  mere  surface  is  wanted  (instead  of  that  of  a 
body),  we  suppose  the  surface  to  be  divided  into  a  number  of  small  areas^  and 
use  ihem  instead  of  the  weights  of  the  small  portions  of  the  body. 


Moment  of  inertia  = 


weight  of  body. 


square  of 


area  of  surface       '^^^"«  «^'  gyration 


Table  of  Radii  of  Oyration. 


Body 


Revolving; 
around 


Any  body  or 
figure 


Solid  cylin- 
der 


ditto 

ditto,  infinitely 

short  (circular 

surface; 

Hollow  cyl- 
inder   > 

ditto,  infinitely 
thin 

ditto,  of  any 
thickness 

ditto,  infinitely 
thin 

ditto,  infinitely 
thin  and  infinitely 
short   (circumfer- 
ence of  a  circle) 

Solid   spliere 


any  given  axis 


its  longitudinal 
axis 


a  diam,  midway 
between  its  ends 


a  diameter 

its  longitudinal 
axis 

ditto 

a  diam  midway 
between  its  ends 

ditto 

a  diameter 
a  diameter 


Radius  of  Gyration 


Vmoment  of  inertia  around  the  given  axis 
weight  of  body,  or  area  of  surface 

radius  of  cylinder  X  -\/-|- 
—  radius  of  cylinder  X  about  .7071 


4 


'length  2 


4 


radius^  of  cylinder 
.2         '  4 

radius  of  cylinder 
2 

[inner  rad2  +  outer  rad^ 


4 


radius  of  cylinder 

inner  rad^  +  outer  rad^      length* 
4  +  ~l2~" 

Vradius^  of  cylinder     lengths 
2  +      12 


radius  of  cylinder  X  \ 

'=  radius  of  cylinder  X  about  .7071 


/radius^  of  sphere 

-  radius  of  sphere  X  1/^4" 

=  radius  of  sphere  X  about  .63246 


RADII    OF    GYRATION. 


353 


Table  of  Radii  of  Oyration.— Continued. 


Body 


Revolving^ 
around 


Radius  of  Oyration 


Hollow 
sphere  of  any 

thickness 

ditto,  thin 

ditto,  infinitely 

thin  (spherical 

surface) 

Straight  lin^, 

ab 


Solid  cone 

Circular 
plate,  of  rect- 
angular cross  sec- 
tion 

Circular 
ring,  of  rectan- 
jular  cross  section 

Square,  rect- 
ang^le  and 
other  sur- 
faces 


a  diameter 
ditto 

ditto 

any  point, «,  in  its 
length 

either  end,  a  or  6 

its  center,  e 
its  axis 


3  Solid  cylin- 
der 


e  Hollow  cylin- 
der 


4 


2  (outer  rad^  —  inner  rad^) 


5  (outer  rad^  —  inner  rad^) 
approx  (outer  rad  +  inner  rad)  X  4085 

radius  of  sphere  X  'v-v" 
«=  radius  of  sphere  X  about  .8165 


4 


/axs  ^  xb» 


3  ab 


length  abX  -yj-Y- 
—  length  abX  abont  .5775 


ac  X 

»=  length  abX  about  .2887 

radius  of  base  of  cone  X  l/^3~ 
-=  radius  of  base  of  cone  X  -5477 


For   the    thickness  of   plate  or    ring, 
.  measured  perpendicularly  to  the  plane 
of  the  circumference,  take  the  length  of 
the  cylinder. 


For  least  radius  of  gyration,  or  that  around  the  longett  axii, 
see  p  496  and  497. 


23 


354  CENTRIFUGAL   FORCE. 


CENTRIFUGAI.  FORCE. 

When  a  body  a,  Fig.  1,  moves  in  a  circular  path  abd,  it  tends,  at  each  point;  as 
a  or  b,  to  move  in  a  tangent  at  or  bi'  to  the  circle  at  that  point.  But  at  each 
point,  as  a,  etc.,  in  the  path,  it  is  deflected  from  the  tangent  by  a  force  acting 
toward  Llie  center,  c,  of  the  circle.  This  force  may  be  the  tension  of  a  string,  ca, 
or  the  attraction  between  a  planet  at  c  and  its  moon  a,  or  the  inward  pressure 
of  the  rails,  a  6,  on  a  curve,  etc.,  etc.  Like  all  force,  it  is  an  action  between  two 
bodies,  tending  either  to  separate  them  or  to  draw  them  closer  together,  and  act- 
ing equally  upon  both.  (See  Art.  5  (fc),  p.  332).  In  the  case  of  the  string,  it  pulls 
the  body  a,  toward  the  center,  c,  and  the  nail  or  hand,  etc.,  at  c,  toward  the  body 
at  a  or  6,  etc. ;  i.  e.,from  the  center.  In  the  case  of  a  car  on  a  curve  it  pushes  the 
car  toward  the  center,  and  the  rails  frojn  the  center.  The  pull  or  push  on  the 
revolving  body  toward  the  center  is  called  the  centripetal  force;  while  the 
pull  or  push  tending  to  move  the  deflecting  body  from  the  center  is  called  the 
ceiitrifug-al  force.  These  two  "  forces,"  being  merely  the  two  "sides  "  (as 
it  were)  of  the  same  stress,  are  necessarily  equal  and  opposite,  and  can  only  exist 
together.  The  moment  the  stress  or  tension  exceeds  the  strength  (or  inherent 
cohesive  force)  of  the  string,  etc.,  the  latter  breaks.  The  centripetal  and  centrif- 
ugal forces  therefore  instantly  cease ;  and  the  body,  no  longer  disturbed  by  a 
deflecting  force,  moves  on,  at  a  uniform  velocity,*  in  a  tangent,  at  or  bt',  etc.,  to 
its  circular  path  ;  i.  e.,  at  right  angles  to  the  direction  which  the  centrifugal  force 
had  at  the  moment  it  ceased. 

{a).  A  sing>1e  revolving  body,  a,  Fig.  1.    Let 
/  =  the  centrifugal  or  centripetal  force,  in  pounds. 
W  =  the  weight  of  the  body  a,  in  pounds, 

K  =  the  radius  ca  of  the  path  of  the  center  of  gravity  of  the  body  a,  in  feet, 
V  =  the  uniform  velocity  of  the  body  a  in  its  circular  path  abd,  in  feet  per 

second, 
n  =  the  number  of  revolutions  per  minute, 
g  =  the  acceleration  of  gravity  =  say  32.2  feet  per  second  per  second, 

900  g  =  about  28980. 
n  =  circumference  -f-  diameter  =  say  3.1416.  ir^  =  about  9.8696. 

Then,  for  the  centrifug^al  force,  /: 

If  we  have  the  velocity  v  in  feet  per  second :  f  =  W  ;^—  t  •  •  •    (1) 

R  g 

If  we  have  the  number  ?i  of  revolutionsper  minute  :/=  W-^ J   ...    (2) 

/  =  about  .0003406  WR7i«  §   .  .   .    (3). 

*  Neglecting  friction,  gravity,  the  resistance  of  the  air,  etc. 

t  For  let  at,  Fig.  1,  represent  the  amount  and  direction  of  the  velocity  v  of  the  body 
at  a  in  feet  per  second.  Then  at  the  end  of  one  second  the  body  will  have  reached 
the  point  b  (the  arc  ab  being  made  =  at),  and  the  amount  and  direction  of  its 
velocity  at  b  will  then  be  represented  by  the  line  bt'  =  at  in  length,  but  dilTeriMg  in 
direction.  Drawing  cu  and  cu'  at  the  center,  equal  and  parallel  respectively  to  ai 
and  bt',  we  find  that  the  change  in  the  direction  of  the  motion  {i.e.,  the  acceleration 
toward  the  center)  during  the  second  is  represented  by  the  arc  uu';  and,  since  angle 
acb  =  angle  ucu',  we  have  the  proportion,  radius  R  or  ac  :  ab  or  at ::  cu^  or  ai :  arc 
uu\     In  other  words,  the  acceleration  mm'  in  one  second,  or  rate  of  acceleration,  is  = 

at^         v^ 

—  =  —  ;  ana,  for  the  force  causing  that  acceleration,  we  have 

A  ii 

/  =  mass  of  body  X  rate  of  acceleration  =  mass  of  body  X    tT 
X  By  formula  (1),/=.  W  ^^  ^.    But  .  =  —^  ;  and  v'^  =  — ^go^— 

^  Txr  'T^  I^""  W''  XTr'T'^Bn* 

Hence, /=^W^---^^  =  W^^. 

§  Formula  (3)  is  obtained  from  (2)  by  substituting  the  va^^e8  9.8696  and  28980  for 
w*  and  900  g  respectively. 


CENTRIFUGAL   FORCE. 


355 


(6>  "Wheels  and  discs.    Suppose  the  rim  of  a  wheel  to  be  cut  into  very  ishort 
slices,  as  shown  (much  exaggerated)  at  a,  Fig.  2.    Then  for  each  slice,  as  a,  by 

formula  (1):  /=  weight  W  of  slice  X  :^~  ;*  and  if  each  slice  were  connected 


n    m  '  c      n 
Fig.  7 


with  the  center  by  a  separate  string,  the  sum  of  the  stresses  in  all  the  strings 
(neglecting  friction  between  adjacent  slices)  would  be : 


F  =  sum  of  centrifugal  forces  of  all  the  slices  t  =  weight  of  rim  X 


^g' 


.   (4). 


But  the  stress  with  which  we  are  usually  concerned  in  such  cases  (viz. :  flie 
tension  in  tlie  rim  itself  in  the  direction  of  a  tangent  to  its  own  cir- 
cumference) is  much  less  than  the  theoretical  quantity  F  ol»tained  from  formula 

(4),  being  in  fact  only  ■  of  it.    For  suppose  first  that  the  same  thin  rim  is 

cut  only  at  two  opposite  points  m  and  n,  Fig.  3,  and  that  its  two  halves  are  held 
together  only  by  tne  string  S. 


*  If  the  rim  is  very  thin  in  proportion  to  its  diameter  mn,  we  may  take  the  center 

ef  gravity  of  each  slice  as  being  in  a  circle  mn  midway  between  the  inner  and  outer 

,         r  i.1,      •           J.^    ^  -^       inner  radius  -f  outer  radius     ^  .        »  .  , , 

edges  of  the  nm,  so  that  R  = .     In  a  rjm  of  appreciable 

thickness,  this  is  not  the  case,  because  each  slice  is  a  little  thicker  at  its  outer  than  at 
Its  inner  end.  See  Fig.  6.  Hence  its  center  of  gravity  is  a  little  outside  of  the  curved 
line  win,  Fig.  2. 

t  In  a  perfectly  balanced  rim  (i.  e.,  a  rim  whose  center  of  gravity  coincides  with  its 
center  of  rotation,  as  in  Fig.  3)  the  centrifugal  forces  of  the  particles  on  one  side  of  c 
counterbalance  those  on  the  opposite  side.  Here,  too,  R  =  0  Hence,  as  a  whole, 
such  a  rim  has  no  centrifugal  force  ;  i.  e.,  no  tendency  to  leave  the  center  in  any  one 
direction  by  virtue  of  its  rotation.  But  if  the  two  centers  do  not  coincide  (Fig.  4), 
then  the  rim  is  a  single   revolving  body,  and  its  centrifugal  force  is :  /  =  weight 

1)2 

of  entire  rim  X  :;;" ;  where  R  is  the  distance  between  the  two  centers,  and  v  the 

Rj7' 
velocity  of  the  center  of  gravity  a.    The  force  /  acts  in  the  line  joining  the  two 
tenters. 


356  CENTRIFUGAL   FORCE. 

Then :  * 

F 
semi-circumference  mzn  :  diameter  mn  -  '•  '^  '  pull  on  the  string  S ; 

so  that 
pull  on   _  half  weight       J^        2  _  weight          v^        F_     F 
strings  S  "      of  rim           R^  ^   tt  ~  of  rim  ^  Ugn"  it~  3.1416  •  •  •  ^  )' 
and  if  the  rim  is  now  made  complete  by  joining  the  ends  at  m  and  n,  and  if  the 
string  S  is  removed,  then  the  pull  on  the  string  by  formula  (5)  will  be  equally- 
divided  between  m  and  n.    Hence  each  cross-section,  as  m  or  w,  of  the  rim,  will 
sustain  a  tensile  stress  equal  to  half  the  pull  on  the  string ;  or 
^         .        .        .            F            F            weight  of  rim  X  v^ 
tension  m  rim  =  -  =  --  =  6.2832  Rg •   <«'• 

The  centripetal  force,/,  Fig.  2,  holding  any  part  o  of  the  rim  to  its  circular  path, 
is  the  resultant  of  the  two  equal  tensions  at  the  ends  of  that  part. 
For  the  stress  per  square  inch  of  cross-section  of  rim,  we  have  : 
.^    ^  tension  in  rim 

unit  stress  =  r ^ 1 . jr-^ ; ; j— - 

area  A  of  cross-section  of  rim,  in  square  inches 
_        F       _  weight  of  rim  X  v^  ,-. 

~  6.2832  A  ~  6.2832  KB.g         ^  '' 

We  shall  arrive  at  the  same  result  if  we  reflect  that  the  pull  in  the  string  S 
or  the  sum  of  the  two  tensions  at  m  and  w,  is  equal  to  the  centrifugal  force/ of 
either  half  of  the  rim,  revolving,  as  a  single  body,  about  the  center  c.  Find  the 
center  of  gravity  G  of  the  half  rim,  and  then,  in  formula  (1),  use  the  velocity  of 
that  point,  and  the  radius  cG  instead  of  velocity  at  z  and  radius  cz  respectively ; 
thus: 

p«U  in  string  =  /=  -'„f,f^r,/r^=  weight  of  half-rim  X  -^'-f^^i 

and  half  of  this  is  the  tension  in  each  cross-section  of  the  rim.f 

If  the  rim  were  infinitely  thin,  cG,  Fig.  3,  would  be  0.6366  cz. 
If  its  thickness  must  be  taken  into  consideration,  and  if  it  is  of  rectangular 
cross-section,  find  the  centers  of  gravity  g  and  g\  Fig.  6,  of  the  whole  semicircular 
segment  cz  and  of  the  small  segment  ch  respectively  {eg  =  0.4244  cz,  and  eg'  = 
0.4244  cb.  Then 

, ,,  .^  area  of  entire  segment  cz 

g'^  =  gg>y^    -^—-J^^^^~ . 

For  rims  of  other  than  rectangular  cross-section,  use  formulse  (4),  (5)  and  (6). 

In  a  disc,  sucli  as  a  ;s:rindstonc,  the  tension  in  each  full  cross-section 
mn.  Fig.  7,  is  equal  to  the  centrifugal  force  /  of  half  the  disc.  Let  W  =  weight 
of  half  disc.  The  distance  cG  from  the  center  c  to  the  center  of  gravity  G  of 
the  half  disc,  is  cG  =  0.4244  cz ;  and  the 

*  In  Fig.  2,  let  the  centrifugal  force  of  any  slice,  o,  be  represented  by  the  diagonal, 
/,  of  a  rectangle,  whose  sides,  H  and  V,  are  respectively  parallel  and  perpendicular 
to  the  given  diameter  mn.  Then  H  and  V  represent  the  components  of  /  in  those 
two  directions.  The  equal  and  opposite  horizontal  components  H,  of  o  and  of  the 
corresponding  slice  o\  being  parallel  to  mn,  have  no  tendency  to  pull  the  rim  apart  at 
m  or  n.  Hence,  the  pull  on  a  string  S,  Fig.  3,  perpendicular  to  mn,  is  the  sum  of  the 
components  V  of  all  the  slices.  For  each  very  thin  slice,  Fig.  5  (greatly  exaggerated) 
we  have  (since  angle  A  =  angle  A') : 

Length  I  .  its  horizontal  . .  centrifugal  force  .    its  vertical 
of  slice    •  projection,  J)    **        /,  of  slice        *  component  V. 
Hence,  for  the  entire  half-rim  mn.  Fig.  3  (made  up  of  such  slices),  we  have : 

♦v.«  o,iT«  r.f  *>,o  ^oT,fr4f     t^®  s"°^  of  the 
Length  mzn      its  horizontal  ,  f^ill^f  oUfT.!  .  vertical  compo- 

of  half-rim    •  projections,™    =  ^^t^rthelf/iit '  t^tsTicS?" 
which  is  identical  with  the  proportion  at  top  of  page. 

t  The  rims  of  revolving  wheels  are  usually  made  strong  enough  to  resist  the  tension 
dtie  to  the  centrifugal  force,  without  aid  from  the  spokes,  which  thus  have  merely  to 
support  the  weight  of  the  wheel.  But  if  the  rim  breaks,  the  centrifugal  forces  of  its 
fragments  come  entirely  upon  the  spokes ;  and,  since  the  breakage  is  always  irregu- 
lar, some  of  the  spokes  will  always  receive  more  than  their  share. 


CENTRIFUGAL    FORCE. 


357 


tension  In  mn  =  f  =  W     ,'  ^    '   = 

rad.  cGXg 


■  W 


0.4244  (vel.  at  g)a 

czXg 
0.4244  ir^  n^  cz 

900  g 


0.42448  (vel.  at  g)" 
0.4244  czXg 

.    .    .  (8). 


=  W 


(9). 


The  stress  per  square  inch  in  any  full  section  mn  is 
tension  in  mn 


unit  stress  = 


area  of  cross-section  in  square  inches 


=  W 


0.4244  (velocity  at  z)^ 


diam.  mn,  ins.  X  thickness,  ins.  X  cz  X  g  ' 

^ 0.4244  7r»  n^  cz 

diam.  m»i,  ins.  X  thickness,  ins.  X  900  g   ' 


.  (10). 
•  (11). 


Fig.  5 


^n    nv      c      n 
Fig.  7 


/=  the  centripetal  force,  in  pounds,  acting  upon  a  single  revolving  body,  a, 
Figs.  1,  2,  4  and  5,  or  upon  the  half-rim  or  half-disc,  Figs.  3,  6  and  7 
=  the  centrifugal  force  exerted  by  such  body. 

F  =  the  sum  of  the  centrifugal  forces  /,  of  all  the  particles  of  a  rim,  Fig.  3. 
W  =  the  weight  of  the  body,  in  pounds. 

R  =  the  radius  ca,  Figs.  1,  4  and  5,  of  the  path  of  the  center  of  gravity  of  the 
body. 

V  =  the  uniform  velocity  of  the  body  in  its  circular  path,  in  feet  per  second. 

M  =  tlie  number  of  revolutions  per  minute. 

g  =  the  acceleration  of  gravity  =  say  32.2  feet  per  second.    900  g  =  about 


■■  say  3.1416.    n^  =  about  9.8696. 


circumference 


diameter 

Ilia  rolling'  "wlieel,  each  point  in  the  rim,  during  the  moment  when  it 
touches  the  ground,  is  stationary  witk  respect  to  the  earth;  but  each  particle  has 
the  same  velocity  abovi  the  center  as  if  the  latter  were  stationary,  and  hence  the 
centrifugal  force  has  no  effect  upon  the  weight. 


358  STATICS. 


STATICS. 

FORCES. 

1.  Statics  Defined.  The  science  of  statics,  or  of  equilibrium  of  forces; 
takes  account  of  those  very  numerous  cases  where  the  forces  under  con* 
sideration  are  in  equilibrium,  or  balanced.  It  embraces,  therefore,  all  cases 
of  bodies  which  are  said  to  be  "at  rest."* 

2.  In  the  problems  usually  presented  in  civil  engineering,  a  certain 
given  force,  or  certain  given  forces,  applied  to  a  stationary  *  body  (as  a  bridge 
or  building)  tend  to  produce  motion,  either  in  the  structure  as  a  whole  or  in 
one  or  more  of  its  members ;  and  it  is  required  to  find  and  to  apply  another 
force  or  other  forces  which  will  balance  the  tendency  to  motion,  and  thus 
permit  the  structure  and  its  members  to  remain  at  rest.     See  1[  33,  below. 

3.  Equilibrium.  Suppose  a  body  to  be  acted  upon  by  certain  forces. 
Then  those  forces  are  said  to  be  in  equilibrium,  when,  as  a  whole,  they  pro- 
duce no  change  in  the  body's  state  of  rest  or  of  motion,  either  as  regards  its 
motion  as  a  whole  along  any  particular  line  (motion  of  translation),  or  as 
regards  its  rotation  about  any  point,  either  within  or  without  the  body. 
In  such  cases  the  body  also  is  said  to  be  in  equilibrium.     See  1  84,  below. 

4.  A  body  may  be  in  equilibrium  as  regards  the  forces  under  consideration, 
even  though  not  in  equilibrium  as  regards  other  forces.  Thus,  a  stone,  held 
between  the  thumb  and  finger,  is  in  equilibrium  as  regards  their  two  equal 
pressures,  even  though  it  may  be  lifted  upward  by  the  excess  of  the  muscular 
force  of  the  arm  over  the  attraction  between  the  earth  and  the  stone.  Simi- 
larly, on  a  level  railroad,  a  car  is  in  equilibrium  as  regards  gravity  and  the 
upward  resistance  of  the  rails,  although  the  horizontal  pull  of  the  locomotive 
may  exceed  the  resistance  to  traction. 

5.  Molecular  Action.  Any  force,  applied  to  a  body,  is  in  fact  made 
up  of  a  system  of  forces,  often  parallel  or  nearly  so,  applied  to  the  several 
particles  of  the  body.  Thus,  the  attraction  exerted  by  the  earth  upon  a 
grain  of  sand  or  upon  the  moon  is,  strictly  speaking,  a  cluster  of  nearly  par- 
allel forces  exerted  upon  the  several  particles  of  those  bodies ;  but,  for  con- 
venience, and  so  far  only  as  concerns  their  tendency  to  move  the  body  as  a 
whole,  we  conceive  of  such  forces  as  replaced  by  a  single  force,  equal  to 
their  sum  and  acting  in  one  line.  In  thus  considering  the  forces,  we  as- 
sume that  the  bodies  are  absolutely  rigid,  so  that  each  of  them  acts  as  a 
single  "  material  particle"  or  **  material  point.'* 

6.  Transmission  of  Force.  Theupwardpressureof  the  ground,  upon 
a  stone  resting  upon  it,  acts  directly  only  upon  those  particles  which  are 
nearest  to  the  ground.  These,  in  turn,  exert  a  (practically)  equal  upward 
force  upon  those  immediately  above  them,  and  so  on ;  and  the  force  is  thus 
transmitted  throughout  the  stone. 

7.  Rigid  Bodies.  In  treating  of  bodies  as-  rigid,  we  assume  that  the 
intermolecular  forces  hold  the  several  particles  absolutely  in  their  original 
relative  positions. 

It  is  not  the  material  that  resists  being  broken,  but  the  forces  which  hold  its 
particles  in  their  places.  Thus,  a  cake  of  ice  may  sustain  a  great  pressure ; 
but  its  particles  yield  readily  when  its  cohesive  forces  are  destroyed  by  a 
melting  temperature. 

8.  Force  Units,  The  force  units  generally  used  in  statics  are  those  of 
weight,  as  the  pound  and  the  kilogram.     See  Conversion  Tables,  p.  235. 

In  statics  we  have  no  occasion  to  consider  the  masses  of  bodies  (except 

*  Strictly  speaking,  absolute  rest  is  scarcely  conceivable,  since  all  bodies 
are  actually  in  motion  (see  Art.  3,  p.  331),  so  that  unbalanced  forces  produce 
merely  changes  in  the  states  of  motion  oi  bodies.  Yet,  for  a  body  to  be  at 
rest,  relative  to  other  bodies,  is  a  very  common  condition,  and,  in  practical 
statics,  we  usually  regard  the  body  under  consideration  as  being  at  rest 
relatively  to  the  earth  or  to  some  other  large  body,  so  that  the  change  of 
state  of  motion,  due  to  the  action  of  unbalanced  force  upon  it,  consists  in  a 
change  from  relative  rest  to  relative  motion.     See  ^  33,  below. 


FORCES.  359 

in  so  far  as  these  determine  their  weights,  or  the  force  of  gravity  exerted  upon 
them),  bodies  being  regarded  merely  as  the  media  upon  and  through  which 
the  forces  under  consideration  are  exerted.  Hence  we  require,  in  statics, 
no  units  of  mass;  and,  as  the  bodies  are  regarded  as  being  "at  rest,"  no  units 
of  time,  velocity,  acceleration,  momentum,  or  energy. 

9.  Forces,  how  Determined.  A  force  is  fully  determined  when  we 
know  (1)  its  amount  (as  in  pounds,  or  in  some  other  weight  unit),  (2)  its 
direction,  (3)  its  sense  (see  ^  10),  and  (4)  its  position  or  its  point  of  applica- 
tion. 

10.  When  a  force  is  represented  by  a  line,  the  length  of  the  line 
may  be  made  to  represent  by  scale  the  amount  of  the  force,  and  its  direction 
and  position  may  often  be  made  to  indicate  those  of  the  force,  while  the  sense 
of  the  force  may  be  shown  by  arrows  or  letters  affixed  to  the  lines,  or  by  the 
signs,  +  and  — . 

Thus,  the  directions  of  the  forces  represented  by  lines  a  and  h,  Fig.  1,  are 
vertical,  and  those  of  c  and  d  are  horizontal.  The  sense  of  a  is  upward,  of  h 
downward,  of  c  right-handed,  of  d  left-handed.  Thus,  a  and  h  are  of  like 
direction,  but  of  opposite  sense ;  and  so  with  c  and  d.  In  treating  of  vertical 
or  horizontal  forces,  we  usually  call  upward  or  right-handed  forces  posi- 
tive, and  downward  or  left-handed  forces  negative,  as  indicated  by  the 
signs,  +  and  — ,  in  Fig.  1,  When  a  force  is  designated  by  two  letters,  at- 
tached to  the  line  representing  it,  one  at  each  end  of  the  line,  the  sense  of  the 
force  may  be  indicated  by  the  order  in  which  the  letters  are  taken.  Thus,  in 
Fig.  1,  having  regard  to  the  directions  of  the  arrows,  we  have  forces,  e  f,  hg, 
k  I,  and  n  m. 

11.  Line  of  Action,  etc.  The  point  (see  ^  5)  at  which  a  force  P,  Fig.  2, 
is  supposed  to  be  applied,  as  a,  is  called  its  point  of  application.  But 
the  force  is  transmitted,  by  the  particles,  throughout  the  body  (see  H  6),  and 


'A  :i '-' 


fa 


the  effect  of  the  force,  as  regards  the  body  as  a  whole,  is  not  changed  if  it 
be  regarded  as  acting  at  any  other  point,  as  b,  in  its  line  of  action.  We 
may  therefore  regard  any  point  in  that  line  as  a  point  of  application  of  the 
force.  For  instance,  the  tendency  to  move  the  stone,  Fig.  2,  as  a  whole,  will 
not  be  changed  if,  instead  of  pushing  it,  at  a,  we  apply  a  pull  (in  the  same 
direction  and  in  the  same  sense)  at  b;  and  if  a  weight,  P,  be  laid  upon  the 
top  of  the  hook,  at  b,  Fig.  3,  it  will  have  the  same  tendency,  to  move  the 
hook  as  a  whole,  as  it  has  when  suspended  from  the  hook  as  in  the  Fig. 

A  force  cannot  actually  be  applied  to  a  body  at  a  point  outside  of  the  sub- 
stance of  the  body,  as  between  the  upper  and  lower  portions  of  the  hook  in 
Fig.  3,  yet  this  portion  also  of  the  line  a  6  is  a  part  of  the  line  of  action  of  the 
force.  The  vertical  force,  exerted  by  the  weight,  P,  is  transmitted  to  b  by 
means  of  bending  moments  in  the  bent  portion  of  the  hook. 

12.  Stress.  (See  Art.  1,  Strength  of  Materials,  p.  454.)  Opposing 
forces,  applied  to  a  body  by  contact  (see  Art.  5  c,  p.  332),  cause  stress,  or  the 
exertion  of  intermolecular  force,  within  it,  or  between  its  particles,  tending 
to  pull  them  apart  (tension)  or  to  press  them  closer  together  (compression). 
The  stress,  due  to  two  equal  opposing  forces,  is  equal  to  one  of  them. 

Tension  and  Compression.  Ties,  Struts,  etc.  If  the  action  of 
the  forces  tends  to  pull  farther  apart  the  particles  of  the  body  upon  which 
they  act,  the  stress  is  called  a  tension  or  pull,  or  a  tensile  stress.  If  it 
tends  to  press  them  closer  together,  the  stress  is  called  a  pressure,  com- 
pression or  push,  or  a  compressive  stress.  A  long  slender  piece  sustaining 
tension  is  called  a  tie.  One  sustaining  compression  is  called  a  strut  or 
post.  One  capable  of  sustaining  either  tension  or  compression  is  called  a 
tie-strut  or  strut-tie. 


360 


STATICS. 


MOMENTS. 

13.  Moments.  If,  from  any  point,  o,  or  o',  Fig.  4,  a  line,  o  c  or  o'  s,  be 
drawn  normally  to  the  line  of  action,  n  m,  of  a  force,  Pi,  whether  the  point,  o 
or  o',  be  within  or  outside  of  the  body  upon  which  the  force.  Pi,  is  acting,  said 
line,  o  c  or  o'  s,  is  called  the  arm  or  leverage  of  the  force  about  such  point; 
and  if  the  amount  of  the  force,  in  lbs.,  etc.,  be  multiplied  by  the  length  of  the 
arm,  in  ft.,  etc.,  then  the  product,  in  ft.-lbs.,  etc.,  is  called  the  moment  of 
the  force  about  that  point.*  The  moment  represents  the  total  tendency  of 
the  force  to  produce  rotation  about  the  given  point.  A  force  has  evidently 
no  moment  about  any  point  in  its  line  of  action. 

14.  Sense  of  Moments.  Since  the  moment  of  Pi  about  o.  Fig.  4, 
tends  to  cause  rotation  (about  that  point)  in  the  direction  of  the  motion  of 
the  hands  of  a  clock,  as  we  look  at  the  clock  and  at  the  figure,  or  from  left  to 
right,  as  indicated  by  the  arrow  on  the  circle  around  o,  it  is  called  a  clock- 
wise or  right-hand  moment;  but  the  moment  of  the  same  force  about  o' 
tends  to  produce  rotation  from  right  to  left.  Hence  it  is  called  a  counter- 
clockwise or  left-hand  moment,  as  is  also  that  of  P2  about  o.  Right- 
hand  or  clockwise  moments  are  conventionally  considered  as  positive, 
or  +,  and  left-hand  or  counter-clockwise  moments  as  negative,  or  — . 

15.^  The  plane  of  a  moment  is  that  plane  in  which  lie  both  the  line 
of  action  and  the  arm  of  the  force. 

16.  The  resultant  or  combined  tendency  of  two  or  more  moments  in 
the  same  plane  is  equal  to  the  algebraic  sum  of  the  several  moments.  Thus, 
Fig.  4,  if  the  forces.  Pi,  P2,  and  Pg,  are  respectively  6,  5,  and  3  lbs.,  and  if 
the  arms,  o  c,  oy,  and  o  e,  of  their  moments  about  o  are  respectively  7,  6,  and 
8  ft.,  we  have 

Pi  .  o  c  —  P2  .  o  2/  -f  P3  .  o  e 
=  6X7— 5X  6  +  3X3 
=      42        —       30         +         9         =21  ft.-lbs. 


k — -m ■» 

I        K— w— ^ 


«*    w 


? 


Fig.  4. 


Tig.  6. 


17.  If  the  algebraic  sum  of  the  moments  is  zero,  they  are  in  equilibrium 
and  tend  to  cause  no  rotation  of  the  body  about  the  given  point. 

Thus,  in  Fig.  5,  where  W  is  the  weight,  and  G  the  center  of  gravity  of  the 
body,  and  R  the  upward  reaction  of  the  left  support,  a,  taking  moments 
about  the  right  support,  b,  we  have  R  Z  —  W  x  =  zero;  or  R  Z  =  W  x.  Hence, 

W  X 
having  W,  x  and  I,  to  find  R,  we  have  R  =  — ,    . 

Similarly,  in  Fig.  6,  where  W  =  weight  of  beam  alone,  and  g,  the  center  of 
gravity  of  W,  is  at  the  center  of  the  span  I,  so  that  the  leverage  b  g  of  the 

weight  of  the  beam  about  6,  is  =  -^f  we  take  moments  about  b,  thus: 


R  I  +  O  o 


R  = 


W^—  M  m  — 

M  m  +  N  n  +  W 


•  Oo 


I 


♦Note  that  a  very  small  force  may  have  a  great  moment  about  a  point, 
while  a  much  greater  force,  passing  nearer  to  the  same  point,  may  have  a 
smaller  moment  about  it ;  or,  passing  through  the  point,  no  moment  at  all. 


MOMENTS. 


361 


In  Fig.  7,  where  W  is  the  weight  of  the  beam  itself,  and  w  its  leverage,  tak- 
ing moments  about  b,  we  have 

+  RZ  +  Oo  —  Nn  —  Ww;  +  Mm  =  0; 


Hence, 


Reaction  at  a   =   R   = 


Ww  +  Nn  — -Mm  —  Oo 

I 


^  In  any  case,  if  W  be  the  combined  weight  and  G  the  common  center  of 
gravity,  of  the  beam  and  its  several  loads,  and  x  the  horizontal  distance  of 
that  center  from  the  right  support,  b ;  and  if  I  be  the  span,  R  the  reaction  of 
the  left  support,  a,  and  R'  that  of  the  right  support,  b,  we  have 


R   = 


Ifx  = 


Ras  =  - 


Wrc, 
I     ' 
R'. 


and  R'    =   W 


R. 


Fig.  7. 

Note  that  the  moments  of  two  or  more  forces,  about  a  given  point, 
may  be  in  equilibrium,  while  the  forces  themselves  are  not  in  equilibrium. 
See  ^  84,  below. 

18.  Center  of  Moments.  So  far  as  concerns  equilibrium  of  moments, 
it  is  immaterial  what  point  is  selected  as  a  center  of  moments ;  but  it  is  gen- 
erally convenient  to  take  the  center  of  moments  in  the  line  of  action  of 
one  (or  more,  if  there  be  concurrent  forces,  see  ^  19)  of  the  unknown  forces, 
for  we  thus  eliminate  that  force  or  those  forces  from  the  equation. 

CLASSIFICATION  OF  FORCES. 

19.  Classification  of  Forces.  Concurrent,  Colinear,  Coplanar, 
and   Parallel   Forces.     Forces  are  called  concurrent  when  their  lines  of 


Fig.  8. 


Tig.  9. 


action  meet  at  one  point,  as  a,  b,  e,  d,  e  and  /,  or  /  and  g,  Fig.  8 ;  non-concur- 
rent when  they  do  not  so  meet,  as  c  and  g ;  colinear  when  their  lines  of  action 
coincide,  as  a  and  b,  or  c  and  d;  non-colinear  when  they  do  not  coincide,  as 
b  and  /;  coplanar  when  their  lines  of  action  lie  in  one  plane,*  as  a,  b,  c,  d  and 
c,  or  6,  /  and  g,  etc. ;  non-coplanar,  as  c  and  g,  or  6,  /  and  d,  when  they  do  not 
lie  in  one  plane;  parallel  when  their  lines  of  action  are  parallel,  as  b  and  g\ 
non-parallel  when  those  lines  are  not  parallel,  as  b  and  /. 


♦Acting  upon  a  plane,  as  in  Fig.  9,  must  not  be  confounded  with  acting  in 
that  plane,  as  in  Figs.  70,  etc. 


362 


STATICS. 


Any  two  parallel  forces  must  be  coplanar.  Three  or  more  parallel  forces 
may  or  may  not  be  coplanar.  Any  two  concurrent  forces  must  be  coplanar. 
Three  or  more  concurrent  forces  may  or  may  not  be  coplanar.  Any  two 
coplanar  forces  must  be  either  parallel  or  concurrent. 

COMPOSITION  AND  RESOLUTION  OF  FORCES. 

^  30.  Resultant.  A  single  force,  which  can  produce,  upon  a  body  con- 
sidered as  a  whole,  the  same  effect  as  two  or  more  given  forces  combined,  is 
called  the  resultant  of  those  forces.  Thus,  in  Fig.  10  (6),  a  downward  pres- 
sure, G,  =  w  +  W,  is  the  resultant  of  the  downward  pressures  w  and  W; 
and,  in  Fig.  11  (6),  a  downward  pressure,  =  W  —  w,  is  the  resultant  of  the 
downward  pressure  W  and  the  upward  pull  w  of  the  left-hand  string.* 

31.  Component.  Any  two  or  more  forces  which,  together,  produce, 
upon  a  body  considered  as  a  whole,  the  same  effect  as  one  given  force,  are 
called  the  components  of  that  force,  which  thus  becomes  their  resultant. 
Thus,  in  Fig.  10  (b),  w  and  W  are  the  components  of  the  total  force,  G,  = 
w  +  W.  In  Fig.  11  (6),  +  W  (=  5)  and  ty  (==  —  3)  are  the  components  of  G.* 
33.  If  we  take  into  account  the  resultant  of  any  given  forces,  those  forces 
(components)  themselves  must  of  course  be  left  out  of  account,  as  regards 
their  action  upon  the  body  as  a  whole;  although  we  may  still  have  to  con- 
sider their  effect  upon  its  particles.  Vice  versa,  if  the  forces  (components) 
are  considered,  their  resultant  must  be  neglected. 


(a) 


Fig.  10. 


=3 


>J=3 


(C) 


Fig.  11. 


33.  Anti-resultant.  The  anti-resultant  of  one  or  more  forces  is  a  single 
force  which,  acting  upon  any  body  or  system  of  bodies  considered  as  a  whole, 
produces  an  effect  equal,  but  opposite,  to  that  of  their  resultant.  In  other 
words,  the  anti-resultant  is  the  force  required  to  hold  the  given  force  or 
forces  in  equilibrium.  Thus,  in  Fig.  10  (5),  the  upward  reaction,  G,  of  the 
ground,  is  the  anti-resultant  of  the  two  downward  forces,  w  and  W ;  and  the 
downward  resultant,  W  -j-  w,  oi  W  and  w,  is  the  anti-resultant  of  G.  In 
Fig.  11  (6),  G  (upward)  is  the  anti-resultant  of  W  (downward)  and  w  (acting 
upward  through  the  left-hand  string).  Similarly,  this  upward  pull  of  w  is 
the  anti-resultant  of  W  and  G. 

34.  In  any  system  of  balanced  forces  (forces  in  equilibrium),  any  one  of 
the  forces  is  the  anti-resultant  of  all  the  rest ;  and  any  two  or  more  of  them 
have,  for  their  resultant,  the  anti-resultant  of  all  the  rest.  In  such  a  system, 
the  resultant  (and  the  anti-resultant)  of  all  the  (balanced)  forces  is  zero. 

35.  Anti-component.  The  anti-components  of  a  given  force,  or  of  a 
given  system  of  forces,  are  any  two  or  more  forces  whose  resultant  is  the  anti- 
resultant  of  the  given  force  or  of  the  given  system  of  forces. 

36.  Composition  and  Resolution  of  Forces.  The  operation  of 
finding  the  resultant  of  any  given  system  of  forces  is  called  the  composition  of 
forces ;  while  that  of  finding  any  desired  components  of  a  given  force  is  called 
the  resolution  of  the  force. 


*  For  convenience,  we  here  reverse  the  convention  of  H  10. 


COLINEAR  FORCES. 


863 


Colinear  Forces. 

27.  Let  the  vertical  line,  w.  Fig,  10  (6),  represent,  by  any  convenient 
scale,  the  weight  of  the  upper  stone  in  Fig.  10  (a),  and  W  that  of  the  lower 
stone.  Then,  w  +  W,  =  G,  =  the  combined  length  of  the  two  lines,  gives, 
by  the  same  scale,  the  combined  weight  of  the  two  stones,  and  a  vertical  line 
G,  coincident  with  them,  equal  to  their  sum,  and  pointing  upward,  would 
represent  their  anti-resultant,  or  the  reaction  of  the  ground. 


(a) 


(b) 


(€) 


c 

\ 

\ 

V 

/ 

/ 

a 

\ 

\ 

A 

/ 

/ 

&  J 

m 

m 

Z=l€ 


Zive  10  10  10 
Dead_2^  _2  2 
Total  12    12   12 


10    10   10 

_2      2     _ 
12    12   12 


2    n= 


X+2> 

>--12 


n==s6< 


12 


^24 


Tig.  12. 

28.  Similarly,  if,  at  each  panel  point  of  the  lower  chord  in  the  bridge 
truss  in  Fig.  12  (o),  we  have  2  tons  dead  load  (weight  of  bridge  and  floor, 
etc,*)  and  10  tons  live  load  (train,  vehicles,  cattle,  passengers,  etc.),  the  com- 
bined length  of  the  two  lines  in  Fig.  12  (6),  L  =  10,  and  D  =  2,  gives  the  total 
panel  load  of  12  tons. 

29.  In  Fig.  11  the  pressure,  5  lbs.,  of  W  upon  the  ground,  is  diminished  by 
the  3  lbs.  upward  pull  of  the  cord,  transmitted  from  the  smaller  weight  to, 
leaving  2  lbs.  upward  pressure  to  be  exerted  by  the  ground  in  order  to  main- 
tain equilibrium.  The  upward  reaction,  R,  of  the  pulley  is  =  ly  +  W  —  G 
=  3+5  —  2  =«6.     This  is  represented  graphically  in  Fig.  11  (c). 

30.  In  the  truss  shown  in  Fig.  12  (o),  the  total  dead  and  live  load  is  =•  6 
X  12  =  72  tons,  and  half  this  total  load,  or  36  tons,  rests  upon  each  abut- 
ment. Hence,  to  preserve  equilibrium,  each  abutment  must  exert  an  up- 
ward reaction  of  36  tons;  but,  in  order  to  ascertain  how  much  of  these  36 
tons  is  transmitted  through  the  end-post,  a  c,  we  must  deduct  from  it  the  12 
tons  which  we  assume  to  be  originally  concentrated,  as  dead  and  live  load, 
at  the  panel  point  a;  for  this  portion  is  evidently  not  transmitted  through 
a  c.  Accordingly,  in  Fig.  12  (c),  we  draw  R  upward,  and  equal  by  scale  to 
36  tons;  and,  from  its  upper  end,  draw  p  downward  and  =  12  tons.  The  - 
remainder  of  R,  «=  R  —  p  ==  36  —  12  =  24  tons,  is  then  the  pressure  trans- 
mitted through  a  c. 

31.  Colinear  forces  are  called  similar  when  they  are  of  like  sense,  and 
opposite  when  of  opposite  sense.  The  same  distinction  applies  to  result- 
ants. 


-^-h 


Fig.  13. 


d        0 


32.  For  equilibrium,  under  the  action  of  colinear  forces,  it  is, 

of  course,  necessary  that  the  sum  of  the  forces  acting  in  one  sense  be  equal  to 
the  sum  of  those  acting  in  the  opposite  sense,  or,  in  other  words,  that  the 
algebraic  sum  of  all  the  forces  be  zero.  Thus,  in  Fig.  13,  if  the  forces  are  in 
equilibrium,  the  sum,  6  a  +  a  o,  of  the  two  right-handed  forces  must  be 
equal  to  the  sum,  ed-^dc  +  co,  oi  the  three  left-handed  forces.  Or,  con- 
sidering the  right-handed  forces,  b  a  and  a  o,  as  positive,  and  the  left-handed 
forces,  e  d,  d  c  and  c  o,  as  negative,  as  in  ^  10,  we  have,  as  the  condition  of 
equilibrium  of  colinear  forces: 

ha-\-ao  —  oc  —  cd  —  de  —  0. 


*  The  dead  load  is,  of  course,  never  actually  concentrated  upon  one  chord, 
as  here  indicated;  but  it  is  often  assumed,  for  convenience,  that  it  is  so 
concentrated. 


364  STATICS. 

In  other  words,  the  algebraic  sum  of  all  the  forces  must  be  zero ;  or,  more 
briefly, 

2  forces  =   0, 

where  the  Greek  letter  S  (sigma),  or  sign  of  summation,  is  to  be  read  "The 
sum  of — ." 

33.  Two  equal  and  opposite  forces,  acting  upon  a  body,  are  com- 
monly said  to  keep  it  at  rest ;  but,  strictly  speaking,  they  merely  prevent  each 
other  from  moving  the  body,  and  thus  permit  it  to  remain  at  rest,  so  far  as 
they  are  concerned ;  for  they  cannot  keep  it  at  rest  against  the  action  of  any 
third  force,  however  slight  and  in  whatever  direction  it  may  act;  and  the 
body  itself  has  no  tendency  to  move. 

34.  Unequal  Opposite  Forces.  If  two  opposite  forces,  acting  upon 
a  body,  are  unequal,  the  smaller  one,  and  an  equal  portion  of  the  greater 
one,  act  against  each  other,  producing  no  effect  ujjon  the  body  as  a  whole ; 
while  the  remainder,  the  resultant,  moves  the  body  in  its  own  direction. 


Concurrent  Coplanar  Forces.     The  Force  Parallelogram. 

35.  Composition.  Let  the  two  lines,  a  o,  h  o,  in  any  of  the  diagrams  of 
Fig.  14,  represent,  in  magnitude,  direction  and  sense,  concurrent  forces 
whose  lines  of  action  meet  at  the  point  o.  Then,  in  the  parallelogram,  acbo, 
formed  upon  the  lines  a  o,  b  o,  the  resultant  of  those  two  forces  is  repre- 
sented, in  magnitude  and  in  direction,  by  that  diagonal,  R,  which  passes 
through  the  point,  o,  ex  concurrence.  The  parallelogram,  a  c  b  o,  is  called  a 
force    parallelogram. 


\«' 

""W 

\^ 

^ 

\\ 

c< 

\ 

V 

V 

\b 

/ 

'.&' 

Fig.  14. 

36.  Resolution.  Conversely,  to  find  the  components  of  a  given  force, 
o  c,  Fig.  14,  when  it  is  resolved  in  any  two  given  directions,  o  a,  ob,  draw  the 
lines,  o  a\  o  b',  in  those  directions  and  of  indefinite  length,  and  upon  these 
lines,  with  the  diagonal  R  =  o  c,  construct  the  force  parallelogram  a  c  b  o. 
The  sides,  o  a,  o  b,  of  the  parallelogram  then  represent  the  required  compo- 
nents in  amount  and  in  direction.   ■ 

37.  Caution.  The  two  forces,  a  o  and  b  o,  Fig.  14,  may  act  either  toward 
or  from  the  point  o;  or,  in  other  words,  they  may  act  either  as  pulls  or  as 
pushes ;  but  the  lines  representing  them  in  the  parallelogram,  and  meeting  at 
the  point,  o,  must  be  drawn,  either  both  as  pushes  or  both  as  pulls ;  and  the 
resultant,  R,  as  represented  by  the  diagonal  of  the  parallelogram,  will  be  a 
pull  or  a  push,  according  as  the  two  forces  are  represented  as  pulls  or  as 
pushes. 

38.  Thus,  in  Fig.  15  (a),  the  inclined  end-post  of  the  truss  pushes  obliquely 
downward  toward  o,  with  a  force  represented  by  a'  o,  while  the  lower  chord 
pulls  away  from  o,  toward  the  right,  with  a  force  represented  by  o  b'.  If, 
now,  we  were  to  construct,  in  Fig.  15  (a),  the  parallelogram  o  a'  c'  b\  we 
should  obtain  the  diagonal  o  c'  or  c'  o,  which  does  not  represent  the  true  re- 
sultant. In  fact,  as  one  of  the  two  forces  acts  toward,  and  the  other  from, 
the  point,  o,  we  could  not  tell  (even  if  R'  were  the  direction  of  the  resultant) 
in  which  sense  its  arrow  should  point. 

We  must  first  either  suppose  the  push,  a'  o,  in  the  end-post,  toward  o,  to  be 
carried  on  beyond  o,  so  as  to  act  as  a  pull,  o  a,  Fig.  15  (6)  (of  course,  in  the 
same  direction  and  sense  a&  before),  thus  treating  both  forces  as  pulls;  or 


FORCE  PARALLELOGRAM. 


365 


else  we  must  similarly  suppose  the  pull,  o  h',  in  the  chord,  to  be  transformed 
into  the  push,  h  o,  of  Fig.  15  (c),  thus  treating  both  forces  as  pushes.  In 
either  case  we  obtain  the  true  resultant,  R  (=  a'  b',  Fig.  15  a),  which,  in  this 
case,  represents  the  vertical  downward  pressure  of  the  end  of  the  truss  upon 
the  abutment. 


Figr.  15. 

Caution.  The  tensile  force,  exerted  at  the  end  of  a  flexible  tie,  neces- 
sarily acts  in  the  line  of  the  tie;  but,  in  general,  the  pressure,  exerted  at 
the  end  of  a  strut,  acts  in  the  line  of  the  axis  of  the  strut  only  when  all 
the  forces  producing  it  are  applied  at  the  other  end  of  the  strut.  Thus, 
in  Fig.  15  (d),  the  components,  R  and  H,  of  the  weight,  W,  do  not  coin- 
cide with  the  axis  of  the  beam  which  supports  the  load;  but  in  Fig.  15 
(e),  where  the  weight  acts  at  the  intersection  of  the  two  struts,  its  com- 
ponents, R  and  H,  do  coincide  with  the  axes  of  the  struts.  See  also  Figs. 
143  and  145  (6). 

39.  Demonstration.  The  rational  demonstration  of  the  principle  of 
the  force  parallelogram  is  given  in  treatises  on  Mechanics.  (See  Bibliog- 
raphy.) It  may  be  established  experimentally  as  indicated  in  Fig.  16, 
where  c  o  represents  by  scale  the  pull  shown  by  the  spring  balance  C,  while 
»  a  and  o  b  represent  those  shown  by  A  and  B  respectively. 


Tig,  16. 

40.  Equations  for  Components  and  Resultant.  Given  the 
amounts  of  the  forces,  a  and  c,  or  of  the  resultant,  R,  and  the  angles  formed 
between  them,  Fig.  17  (a),  we  have*: 


*  See  dotted  lines,  Fig.  17  (a),  noting  that  c' 
X,  and  a.  sin  (x  +  y)  =  R.  sin  y. 


c;  c.  sin  (x  -\-  y)  =  R.  sin 


366 


STATICS. 


.^            sin  (x 
K  =  c   ^ 

c  =  R 


y) 


sin  X 
sin  X 


sin  (X  -\-  y)' 


a  =  R 


,  sin  (3;  +  y) 
sin  y 
sin  y 
sin  (x  +3/)* 


If  the  angle  between  the  two  forces  is  90°,  Fig.  17  (6),  these  formulas  be- 
come: 

R  =      ^       =      "^ 

cos  y       cos  X 
c  =  R  cos  y,     a  =  R  cos  x. 


Fig.  17. 

41.  Position    and    Sense    of    Resultant.     Figs.   18.     If  the  lines 

representing  the  components  be  drawn  in  accordance  with  1[1[  37  and  38, 
and  if  a  straight  line,  mnor  m'  n\  be  drawn  through  the  point,  o,  of  concur- 
rence, in  such  a  way  that  both  forces  are  on  one  side  of  that  line,  then  the  line 
representing  the  resultant  will  be  found  upon  the  same  side  of  that  line  with 
the  components,  and  between  them;  and  it  will  act  toward  the  line,  mnor 
m'  n',  if  the  components  act  toward  it,  and  vice  versa.  The  resultant  is 
necessarily  in  the  same  plane  with  its  two  components. 


Fig.  18. 


Figr.  19. 


42.  If  one  of  the  components  is  colinear  with  the  force,  it  is  the  force  itself, 
and  the  other  component  is  zero.  In  other  words,  a  force  cannot  be  resolved 
into  two  non-colinear  components,  one  of  which  is  in  the  line  of  action  of  the 
force.  Thus  the  rope,  o  c,  Fig.  19,  may  receive  assistance  from  two  ad- 
ditional ropes,  pulling  in  the  directions  a  c,  and  c  6;  for  the  resultant  of  their 
pulls  may  coincide  with  o  c;  but,  so  long  as  o  c  remains  vertical,  no  single 
force,  as  c  a  or  c  6,  can  relieve  it,  unless  acting  in  its  own  direction  c  o. 

43.  In  Fig.  20,  the  load,  P,  placed  at  C,  is  suspended  entirely  by  the  verti- 
cal member  B  C,  and  exerts  directly  no  pull  along  the  horizontal  member, 
C  E.  Neither  does  a  pull  in  the  latter  exert  any  effect  upon  the  force  acting 
in  B  C,  so  long  as  B  C  remains  vertical.  But  the  tension  in  B  C,  acting 
at  B,  does  exert  a  thrust  o  a  along  B  D,  although  that  member  is  at  right 
angles  to  B  C ;  for  B  C  meets  there  also  the  inclined  member  A  B ;  and 
the  tension  o  d  is  thus  resolved  into  o  a  and  o  b,  along  B  D  and  B  A 
respectively.  The  horizontal  thrust,  o  a,  in  B  D,  is  really  the  anti-resultant 
of  the  horizontal  component,  d  b,  of  the  oblique  thrust  in  the  end-post  B  A, 
at  its  head,  B,  which  thrust  is==the  pull  in  A  E,  due  to  P. 


FORCE  TRIANGLE. 


367 


44.  In  Fig.  21,  the  tension,  o  c,  in  the  inclined  tie,  D  G,  is  resolved,  at  D, 
into  o  a  and  o  b,  acting  at  right  angles  to  each  other  along  D  F  and  D  E  re- 
spectively. 

45.  A  resultant  may  be  either  greater  or  less  than  either  one  of  its  two 
oblique  components,  but  it  is  always  less  than  their  sum.  If  the  components 
are  equal,  and  if  the  angle  between  them  =  120°,  the  resultant  is  equal  to  one 
of  them.  Therefore  the  same  weight  which  would  break  a  single  vertical 
rope  or  post,  would  break  two  such  ropes  or  posts,  each  inclined  60°  to  the 
vertical. 


Fig.  20. 


The  Force    Triangle. 

46.  The  Force  Triangle.  Inasmuch  as  the  two  triangles,  into  which  a 
parallelogram  is  divided  by  its  diagonal,  are  similar  and  equal,  it  is  suffi- 
cient to  draw  either  one  of  these  triangles,  a  o  c  or  b  o  c,  Figs.  14,  16,  18,  in- 
stead of  the  entire  parallelogram. 

47.  If  three  concurrent  coplanar  forces  are  in  equilibrium,  the  lines  rep- 
resenting them  form  a  triangle;  and  the  arrows,  indicating  their  senses, 
follow  each  other  around  the  triangle.  Thus,  in  Fig.  22  (a),  we  have,  acting 
at  o  and  balancing  each  other  there,  three  forces:  viz.,  (1)  the  vertical  down- 
ward force  o  c  of  the  weight,  acting  as  a  pull  through  the  rope  o  c,  (2)  the 
horizontal  thrust  a  o  through  the  beam  a  o,  and  (3)  the  upward  inclined 
thrust  6  o  of  the  strut  o  b,  all  acting  in  the  senses  (o  c,  a  o,b  o)  in  which  the 
letters  are  taken,  and  as  indicated  by  the  arrows. 

48.  ICach  of  the  forces  in  Fig.  22  (Jb)  and  (c)  is  the  anti-resultant  of  the 
other  two  in  the  same  triangle;  and,  if  its  sense  be  reversed,  it  becomes  their 
resultant.  Thus,  o  c,  Fig.  22  (6),  is  the  anti-resultant,  and  c  o  the  resultant, 
of  c  a  and  a  o;  and  o  c.  Fig.  22  (c),  is  the  anti-resultant,  and  c  o  the  resultant 
of  c  6  and  b  o,  c  b  being  parallel  to  a  o,  Fig  (6),  and  representing  the  thrust 
exerted  by  the  horizontal  beam  against  the  joint  o,  Fig.  (a).* 


(b)    (c) 


Tig.  22. 


*Fig.  22  (d)  and  (e),  representing  the  same  two  forces,  a  o,  b  o,  of  Fig. 
22  (a),  show  the  erroneous  resultant  (a  b)  obtained  if  the  lines  are  drawn 
with  their  arrows  pointing  both  toward  or  both  from  the  meeting-point  of  the 
lines.  See  1^  37,  38.  A  comparison  of  any  force  parallelogram,  as  that 
in  Fig.  18,  with  either  of  the  two  force  triangles  composing  it,  will  show 
that  this,  while  apparently  contradicting  Ifl  37  and  38,  is  merely  another 
statement  of  the  same  fact.  The  apparent  contradiction  is  due  to  the 
fact  that,  in  the  force  triangle,  the  lines  representing  the  forces  do  not 
meet  at  the  point,  o,  of  concurrence  of  the  forces. 


368 


STATICS. 


49.  Conversely,  if  the  thtee  sides  of  a  triangle  be  taken  as  representing, 
in  direction  and  in  amount,  three  concurrent  forces  whose  senses  are  such 
that  arrows,  representing  them  and  affixed  to  their  respective  sides  in  the 
triangle,  follow  each  other  around  it,  then  those  forces  are  in  equilibrium. 

50.  The  three  forces,  Fig.  23,  are  proportional,  respectively,  to  the 
sines  of  their  opposite  angles.     Thus : 


Force  a  :  force  h  :  force  c 
=  Sin  A  :  sin  B  :  sin  C.       Fig.  23. 


51.  Example.  In  Fig.  24,  the  half  arch  and  its  spandrel,  acting  as  a 
single  rigid  body,  are  assumed  to  be  held  in  equilibrium  by  their  combined 
weight,  W,  the  horizontal  pressure  h  at  the  crown,  and  the  reaction  R  of  the 
skewback,  which  is  assumed  to  act  through  the  center  of  the  skewback.  In 
the  force  triangle  c  s  t,  c  s,  acting  through  the  center  of  gravity  of  the  half 
arch  and  spandrel,  represents  the  known  weight  W,  and  s  t  is  drawn  hori- 
zontal, or  parallel  to  h  .  From  c,  where  h,  produced,  meets  the  line  of  ac- 
tion of  W,  draw  c  t  through  the  center  of  the  skewback.  Then  a  t  and  c  t 
give  "lis  the  amounts  of  h  and  R  respectively. 


Fig.  24. 


Fig.  25. 


52.  Example.  Let  Fig.  25  represent  a  roof  truss,  resting  upon  its  abut- 
ments and  carrying  three  loads,  as  shown  by  the  arrows.  Draw  a  R  ver- 
tically, to  represent  the  proportion  of  the  loads  carried  by  the  left  abut- 
ment, a,  or,  which  is  the  same  thing,  the  vertical  upward  reaction  of  that 
abutment.  Then,  drawing  R  c,  parallel  to  the  chord  member,  a  d,  to  inter- 
sect a  e  in  c,  we  have,  for  the  stresses  in  a  e  and  a  d,  due  to  the  three  loads: 
Stress  in  a  e  =  a  c 
*'       **  o  d  =  R  c 


Fig.  26. 

53.  While  any  two  or  more  given  forces,  as  o  6  and  h  c,  Fig.  26  (a)  (arrows 
reversed),  or  o  h'  and  h'  c,  or  o  a  and  a  c,  or  o  a'  and  a'  c,  can  have  but  one  re- 
sultant o  c ;  a  single  force,  as  o  c,  may  be  resolved  into  two  or  more  concur- 
rent components  in  any  desired  directions.  In  other  words,  there  is  an 
infinite  number  of  possible  systems  of  concurrent  forces  which  have  o  c  for 
their  resultant. 


EECTANGULAE  COMPONENTS. 


369 


Rectangular  Componentf. 

54.  Resolutes,  or  Rectangular  Components.  A  very  common  case 
of  resolution  of  forces  is  that  where  a  force,  as  the  pressure,  c  n,  of  the  post, 
Fig.  27,  is  to  be  resolved  into  components  at  right  angles  to  each  other,  as  are 
the  vertical  and  horizontal  components  c  t  and  t  n  in  Fig.  27  (a).  Two  such 
components,  taken  together,  are  called  the  resolutes  or  rectangular  compo- 
nents of  the  force.  The  joint,  o  d,  in  Fig.  27  (a),  is  properly  placed  at  right 
angles  to  c  n;  but  the  joint  ci  b,  Fig.  27  (6),  provides  also  against  accidental 
changes  in  the  direction  of  c  n.  In  Fig.  27  (6),  the  surfaces,  c  i  and  i  b,  are 
preferably  proportioned  as  the  components,  c  t  and  t  n.  Fig.  27  (a),  respec- 
tively, by  similarity  of  triangles,  cib,  ctn. 


Fig.  27. 


Fig.  28. 


55.  Example.  In  bridge  and  roof  trusses  it  is  often  required  to  find  the 
vertical  and  horizontal  resolutes  of  the  stress  in  an  inclined  member,  or  to 
find  the  stress  brought  upon  an  inclined  member  by  a  given  vertical  or  hori- 
zontal stress  applied  at  one  of  its  ends,  in  conjunction  with  another  stress 
(whose  amount  may  or  may  not  be  given)  at  right  angles  to  it. 

Thus,  in  Fig.  28,  the  tension  C  p  in  the  diagonal  C  rf  is  resolved  into  a  com- 
pression e  p  along  the  upper  chord  member  C  D*  and  a  compression  C  e  in  the 
post  C  c*  Adding  to  C  e  the  load  at  c,  and  representing  their  sum  by  /  c,  we 
have  tension  f  g  in  chord  member  c  d,  and  tension  c  g  in  the  diagonal  B  c. 
Making  B  h  =  c  ^,we  have  j  h,  compression  in  B  C,  and  B  ;,  compression  in 
the  end-post  or  batter  post  B  A.  But  the  load  at  b  also  sends  to  B,  through 
the  hip  vertical  B  6,  a  load  (tension)  equal  to  itself.  Representing  this  by 
B  k,  we  have  I  k  as  its  component  along  the  chord  member  B  C,  and  B  i  as  its 
component  along  the  end-post  B  A.  Now,  making  A  m  =  the  sum  of  B; 
and  B  I,  we  find  the  vertical  resolute  A  n  =  so  much  of  the  vertical  reaction 
of  the  abutment  as  is  due  to  the  three  loads  only,  and  the  horizontal  resolute 
m  n  =  the  corresponding  stress  in  the  chord  member,  A  c. 


Fig.  29. 


Fig.  30. 


56.  Example.  Inclined  Plane.  Again,  in  Fig.  29,  let  it  be  required 
to  find  the  two  resolutes  of  P  (the  weight  of  the  ball)  respectively  parallel  and 
perpendicular  to  the  inclined  plane.  The  former  is  the  tendency  of  the  ball 
to  move  down  the  plane,  and  is  called  the  tangential   component.     The 


*The  stress  thus  found  is  not  necessarily  the  total  stress  in  the  member. 
The  compression  in  C  c  (neglecting  its  own  weight  and  that  of  the  top  chord) 
is  due  entirely  to  the  tension  C  p  in  C  d,  acting  at  its  top,  and  hence  C  e  rep- 
resents the  total  compression  in  C  c;  but  e  p  is  only  a  portion  of  the  com- 
pression sustained  by  C  D ;  for  B  C  also  contributes  its  share  toward  this. 
24 


370 


STATICS. 


latter  is  the  pressure  of  the  ball  against  the  plane,  and  is  called  the  normal 
component. 

Here  we  have  only  to  draw  the  triangle  of  forces  o  a  c,*  drawing  o  c  =  P  to 
represent  the  weight  of  the  ball,  and  o  a  and  a  c  in  the  required  directions. 
Then  o  a  and  a  c  give  respectively  the  normal  and  the  tangential  components 
of  the  force,  P.f 

57.  If  we  suppose  the  inclined  plane  g  m,  Fig.  29,  to  be  frictionless,  and  if 
the  body  o  is  to  be  prevented  from  sliding  down  the  plane,  by  means  of  a  force 
applied  in  a  direction  parallel  to  the  plane,  that  force  must  be  =  c  a. 

Thus,  in  Fig.  30,  supposing  the  plane  o  m  to  be  frictionless,  we  have  a  c 
=   pressure  against  the  stop,  s. 

58.  Table  of  normal  and  tangential  components  for  different 
angles  of  inclination: 


Pres  on 

Tendency 

Inclination  or  Slope  of  the  Plane. 

vertical  height 
The  sloping  length  is  _      ^.^^^  ^^^  ^ 

Plane,  in 

parts  of  the 

wt.     Or,  nat. 

00a.  of  angle 

of  Plane. 

Pres.  on 

Plane,  in  fits 

per  ton. 

down  the 

Plane,  in 

parts  of  the 

wt.    Or.  nat. 

sine  of  angle 

of  Plane. 

Tendency 

down  the 

Plane,  in  lbs 

per  ton. 

Tert.     Hor. 

Ft.  per  mile. 

Deg.  Min. 

1    in       3. 

1760.00 

18      26 

.9487 

2125 

.3162 

708. 

1    in       4. 

1320.00 

14        2 

.9702 

2173 

.2425 

543. 

1    in       5. 

1056.00 

11       19 

.9806 

2196 

.1962 

439. 

1    in       6. 

880.00 

9      28 

.9864 

2210 

.1645 

368. 

1    in       8. 

660.00 

7        8 

.9923 

2223 

.1242 

278. 

1    in       9. 

588.66 

6      20 

.9939 

2226 

.1103 

247. 

1    in      10. 

528.00 

5      43 

.9950 

2229 

.0996 

223. 

1    in      11.4 

461.94 

5      00 

.9962 

2231 

.0872 

195. 

1    in      12. 

440.00 

4      46 

.9965 

2232 

.0831 

186. 

1    in      14.8 

369.23 

4      00 

.9976 

2232 

.0698 

156. 

1    in      15. 

352.00 

3      49 

.9978 

2233 

.0666 

149. 

1    in      19.1 

276.73 

3      00 

.9986 

2237 

.0523 

117. 

1    in      20. 

264.00 

2      52 

.9987 

.0500 

112. 

1    in      23.1 

229.04 

2      30 

.9990 

»' 

.0436 

97.7 

1    in      25. 

211.20 

2      17 

.9992 

2238 

.0398 

89.2 

1    in      28.6 

184.36 

2      00 

.9994 

.0349 

78.2 

1    in      30. 

176.00 

1      55 

" 

.0334 

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*  Or  o  6  c.  If  both  triangles  are  drawn,  we  have  the  force  parallelogram, 
o  a  c  b. 

fThe  line  a  c  (or  c  a)  is  called  the  projection  of  o  c  upon  the  inclined 
plane;  and  o  a  (or  a  o)  is  the  projection  of  o  c  upon  a  normal  to  the  inclined 
plane. 


STRESS   COMPONENTS. 


371 


59.  Equations.     In  Fig.  29, 

o  a  =    P  .  cos  c  o  a 
a  c    =   P  .  sin  c  o  a 
and,  since  the  angle  c  o  a  between  the  vertical  o  c  and  the  normal  component 
o  a  is  equal  to  the  angle  A  of  inclination  between  the  plane  g  m  and  the  hori- 
zontal g  n,  we  have : 

Normal  component,        o  a  =   P  .  cos  A. 
Tangential  compo^ient,  a  c   =   P  .  sin  A. 

60.  When  a  force  is  resolved  into  rectangular  components,  as  in  Figs.  29 
and  30,  each  of  these  components  represents  the  total  effort  or  tendency  which 
that  force  alone  can  exert  in  that  direction. 


Tig.  31. 

Thus,  in  Fig.  31,  the  utmost  force  which  the  weight  a  c  alone  can  exert 
perpendicularly  against  the  plane  is  that  represented  by  the  component  o  a. 
True,  if,  in  order  to  prevent  the  body  from  sliding  down  the  plane,  we  apply 
a  force  in  some  other  direction,  such  as  the  horizontal  one,  h  o,  instead  of  the 
tangential  one  5  o,  and  find  the  components  of  o  c  in  the  directions  h  o  and  o  a, 
we  shall  find  the  normal  component  o  d  greater  than  before ;  but  the  increase 
a  d  is  due  entirely  to  the  normal  component,  h  b,  of  the  horizontal  force  h  o. 
Thus,  the  only  effect  upon  the  body  o,  and  upon  the  plane,  of  substituting 
h  o  for  b  o,  is  to  add  the  normal  component,  h  b,  of  the  former,  to  that  (o  a) 
of  o  c. 

Stress  Components. 

61.  Stress  Components.  In  Fig.  32,  let  o  o  and  6  o  be  any  two  forces, 
and  c  o  their  resultant.  From  a  and  b  draw  a  a'  and  b  b'  at  right  angles  to 
the  diagonal  o  c  of  the  force  parallelogram  a  o  b  c,  and  construct  the  sub- 
parallelograms  (rectangles),  o  a'  a  a''  and  ob'  b  b" .  Each  of  the  original  com- 
ponents, o  a,  o  b,  is  thus  resolved  into  two  sub-components,  perpendicular  to 
each  other,  one  of  which  is  perpendicular  also  to  the  resultant,  o  c,  while  the 
other  coincides  with  o  c  in  position  and  in  sense.  Now,  perpendiculars,  let 
fall  from  the  opposite  angles  of  a  parallelogram  upon  its  diagonal,  are  equal. 


fc  --^ 


^ a 

1  /    > 

h'/ 

/ 

Figr.  32. 


Hence  the  two  colinear  forces,  o  a"  and  o  b",  acting  upon  the  body  at  o,  are 
equal  and  opposite  (although  the  lines,  a'  a  and  b'  b,  representing  them,  are 
not  opposite).  Hence  also  they  are  in  equilibrium,  and  their  only  effect 
upon  the  body  is  a  stress  of  compression  in  Fig.  32  (a),  and  of  tension 
in  Fig.  32  (6).  They  may  therefore  be  called  the  stress  coniponents.  The 
other  two  sub-components  (o  a'  of  o  a,  and  o  6'  of  o  b)  combine  to  form  the 
resultant  o  c,  which  is  equal  to  their  sum,  and  which  tends  to  move  the  body 
o  in  its  own  direction. 


372 


STATICS. 


62.  The  two  great  forces,  o  a,  oh,  in  Fig.  33  {h)  have  the  same  resultant, 
oc,  =  oc',  as  the  two  small  forces,  o  a'  oh',  in  Fig.  33  (a),  although  their 
stress  components,  a"  a,  =  h"  h,  are  much  greater. 

63.  It  often  happens  that  one  of  the  components  is  itself  normal  to  the 
resultant.  Thus,  in  Fig.  22,  where  o  c  is  vertical,  its  component,  o  a,  is  hori- 
zontal, and  the  perpendicular,  let  fall  from  a  upon  o  c,  represents  its  hori- 
zontal anti-component,  a  o.  Here  the  horizontal  and  the  inclined  l5eam 
sustain  equal  horizontal  pressures;  but  the  vertical  pressure,  o  c,  =  the 
weight,  W,  is  borne  entirely  by  the  inclined  beam. 


^^~--^_£ 

■ — -— ^._^ 

b 

Tig.  33. 


Fig.  34. 


64.  When,  as  in  Fig.  34,  the  resultant,  o  c,  forms,  with  one  of  the  original 
components,  o  a  and  o  h,  an  angle,  a  o  c,  greater  than  90°,  the  perpendiculars, 
a  a',  h  b',  from  a  and  b,  must  be  let  fall  upon  the  line  of  the  resultant  produced. 
Here,  however,  as  before,  the  two  equal  and  opposite  sub-components,  o  a" 
and  o  h",  are  in  equilibrium  at  o,  while  the  other  two  sub-components,  o  h'  and 
o  a\  go  to  make  up  the  resultant  o  c;  which,  however  (since  a  h'  and  o  a'  here 
act  in  opposite  senses)  is  equal  to  their  difference,  and  not  to  their  sum,  as  in 
Fig.  32. 

Fig.  34  shows  that  a  downward  force,  o  c,  may  be  so  resolved  that  one  of  its 
components  is  an  upward  f<^e,  o  a,  greater  than  the  original  downward  force, 
and  that  the  pressure,  o  b,  nas  a  component,  o  b'  or  V  h,  parallel  to  o  c,  and 
greater  than  o  c  itself;  for  b"  h  =  a  b'  =  o  c  -\-  c  h'. 

Applied  and  Imparted  Forces. 

65.  Applied  and  Imparted  Forces.  In  Fig.  29,  the  ball  is  free  to 
roll  down  the  inclined  plane.  Hence,  although  the  entire  weight  P  of  the 
ball  is  applied  to  the  body  g  m  n,  only  the  normal  component  o  a  is  imparted 
to  it  or  exerts  any  pressure  upon  it,  and  this  pressure  is  in  the  direction  o  a. 

But  in  Fig.  30,  the  body  g  m  n  receives  and  resists  not  only  the  normal 
component  o  a,  but  also  (by  means  of  the  stop  s)  the  tangential  component 
o  h;  and  the  entire  force  P,  or  o  c,  is  thus  imparted  to  the  body  g  m  n,  press- 
ing it  in  the  direction  o  c. 

Composition    and   Resolution  of    Concurrent    Forces  by  Means 
of  Co-ordinates. 

66.  In  Fig.  35  (a)  let  the  three  coplanar  forces  E,  F  and  G  act  through 
the  point  x.  Draw  two  lines,  H  H,  and  V  V,  Fig.  35  (6),  crossing  each 
other  at  right  angles,  as  at  o*  These  lines  are  called  rectangular  co-ordin- 
ates. From  o,  draw  lines  E  o,  F  o,  G  o,  parallel  to  E  x,  F  x,  Gx,  Fig.  35  (a), 
and  equal  respectively  to  the  forces  E,  F,  and  G  by  any  convenient  scale.  Re- 
solve each  of  these  forces,  Fig.  35  (6),  into  two  components,  parallel  to  H  H 
and  V  V  respectively.  Thus,  E  o  is  resolved  into  t  o  and  n  o,  F  o  into  u  o 
and  e  o,  G  o  into  i  o  and  m  o.     Then,  summing  up  the  resolutes,  we  have: 

Sum  of  horizontal  resolutes  =  u  o  —  io  —  to  =  —  so,  and 
Sum  of  vertical       resolutes  =  n  o  -\-  e  o  —  m  o  =  a  o; 


*It  is  only  for  convenience  that  the  co-ordinates  are  usually  drawn  (as  in 
Fig.  35)  at  right  angles.  They  may  be  drawn  at  any  other  angle  (see  Fig. 
36) ;  but,  in  any  case,  the  forces  must  of  course  be  resolved  into  components 
'parallel  to  the  co-ordinates,  whatev^  the  directions  of  those  co-ordinates  may 


COMPOSITION   AND   RESOLUTION. 


373 


and  —  8  0  and  a  o  are  the  resolutes  of  the  resultant,  R,  of  the  three  forces,  E, 
F  and  G. 

67.  When  a  system  of  (concurrent)  forces  is  in  equilibrium,  the  algebraic* 
sum  of  the  components  of  all  the  forces,  along  either  of  the  two  co-ordinates, 
is  zero.  Thus,  in  Fig.  35  (6)  or  36,  if  the  sense  of  R  be  such  that  it  shall  act  as 
the  anti-reeultant  of  the  other  three  forces  E,  F  and  G,  its  component,  o  s  or 
o  a,  along  either  co-ordinate,  will  be  found  to  balance  those  of  the  other 
forces  along  the  same  co-ordinate. 


/« 

F 

•^  w 

u 

o 

/          i      8    t 

H^ 

1 

n 

—  ^^^ 

(a) 


Fig.  35. 


(&) 


Hence  we  have  the  very  important  proposition  that:  When  a  system  of 
concurrent  coplanar  forces  is  in  equilibrium,  the  algebraic  sums  of  their  com- 
ponents, in  any  two  directions,  are  each  equal  to  zero. 


Figr.  36. 

68.  Conversely,  in  a  system  of  concurrent  forces,  if  the  algebraic  sums  of 
the  components  in  any  two  directions  are  each  equal  to  zero,  the  forces  are 
in  equilibrium. 

If  the  sum  of  the  components  in  one  of  any  two  directions  is  not  equal  to 
zero,  the  forces  cannot  be  in  equilibrium.  Thus,  in  Fig.  35  (6)  or  36  (6),  the 
sum  of  the  components,  along  either  one  (as  VV)  of  the  two  co-ordinates, 
may  be  zero;  and  yet,  if  the  sum  of  those  along  the  other  co-ordinate  is 
not  zero,  their  resultant,  or  algebraic  sum,  will  move  the  body,  on  which 
they  act,  in  the  direction  of  that  resultant. 


*The  components  being  taken  as  +  or  — ,  according  to  the  sense  of  each. 


374 


STATICS. 


69.  With  vertical   and   horizontal   co-ordinates,   the  condition  of 

equilibrium*  becomes: 

The  sum  of  the  horizontal  resolutes  must  be  equal  to  zero; 
The  sum  of  the  vertical       resolutes  must  be  equal  to  zero; 
or,  more  briefly: 

2  horizontal  resolutes  =  0 
2  vertical       resolutes  =  0 
Conversely,  if  these  conditions  are  fulfilled,  the  forces  are  in  equilibrium. 


Figr.  37. 


Fig.  38. 


Tig,  39. 


70.  Resultant   of  More  than   Two   Coplanar  Forces.     Where  it 

is  required  to  find  the  resultant  of  more  than  two  concurrent  and  coplanar 
forces,  as  in  Fig.  37,  we  may  first  find  the  resultant  Ri  of  any  two  of  them, 
as  of  P]  and  P2 ;  then  the  resultant,  Rq,  of  Ri  and  a  third  force,  as  P3 ;  and  so 
on,  until  we  finally  obtain  the  resultant  R  of  all  the  forces.  This  resultant  is 
evidently  concurrent  and  coplanar  with  the  given  forces. 

71.  It  is  quite  imniaterial  in  what  order  the  forces  are  taken. 
Thus,  we  may,  as  in  Fig.  38,  first  combine  P]  and  P3;  then  their  resultant  Ri 
with  P2,  obtaining  R2 ;  and,  finally,  R2  with  P4,  obtaining  R ;  or,  as  in  Fig.  39, 
we  may  first  combine  any  two  of  the  forces,  as  Pi  and  P2,  obtaining  their 
resultant  Ri ;  then  proceed  to  any  other  two  forces,  as  P3  and  P4,  and  obtain 
their  resultant  R2;  and  finally  combine  the  two  resultants,  Ri  and  R2,  ob- 
taining the  resultant  R. 

The  Force  Polygon. 

72.  The  Force  Polygon.  Comparing  Figs.  37  and  38  with  Figs.  40 
and  41,  respectively,  we  see  that  we  may  arrive  at  the  same  resultant  R  by 
simply  drawing,  as  in  Fig.  41,  lines  representing  the  several  forces  in  any 
order,  but  following  each  other  according  to  their  senses.  It  will  be  noticed 
that  this  is  merely  an  abbreviation  of  the  process  of  drawing  the  several  force 
parallelograms. 

73.  Resultant  and  Anti-resultant.  The  line, — R,  required  to  com- 
plete the  polygon,  represents  the  an^i-resultant  of  the  other  forces  if  its  sense 
is  such  that  it  follows  them  around  the  polygon,  as  in  Fig.  40.  If  its  sense  is 
opposed  to  theirs,  as  in  Fig.  41,  it  is  their  resultant,  R. 

74.  In  other  words,  if  any  number  of  concurrent  forces,  as  Pj,  P2,  P3,  P4 
and  R,  Figs.  37  and  38,  f  are  in  equilibrium,  the  lines  representing  them,  if 
drawn  in  any  order,  but  so  that  their  senses  follow  each  other,  will  form  a 
closed  polygon,  as  in  Fig.  40  (or  in  Fig.  41  if  the  sense  of  R  be  reversed). 

75.  Conversely,  if  the  lines  representing  any  system  of  concurrent 
coplanar  forces,  when  drawn  with  their  senses  following  each  other,  form  a 
closed  polygon,  as  in  Fig.  40,  those  forces  are  in  equilibrium. 


*With  non-concurrent  forces,  another  condition  must  be  satisfied.  See  ^  83. 
fR  is  here  regarded  as  tending  upward,  so  as  to  form  the  an^t-resultant  of 
the  other  forces. 


FORCE   POLYGON. 


375 


It  will  be  noticed  that  the  force  triangle,  and  the  straight  line  representing 
a  system  of  colinear  forces,  Figs.  10  and  11,  If^H  20,  etc.,  or  a  system  of 
parallel  forces,  Figs.  55,  etc.,  1[^  111,  etc.,  are  merely  special  cases  of  the 
force  polygon. 

76.  In  a  force  polygon.  Fig.  42,  any  one  of  the  forces  is  the  anti-resultant 
of  all  the  rest.  Any  two  or  more  of  the  forces  balance  all  the  rest;  or,  their 
resultant  is  the  anti»resultant  of  all  the  rest. 

If  a  line  acorh  d.  Fig.  42,  be  drawn,  connecting  any  two  corners  of  a  force 


polygon,  that  line  represents  the  resultant,  or  the  anti-resultant  (according  as 
its  arrow  is  drawn)  of  all  the  forces  on  either  side  of  it.     Thus: 

a  c  is  the  resultant  of  Pi  P2        and  the  anti-resultant  of  P3  P4  P5 

ca  -              "  "  P3  P4  P5        "                  "                 "  Pi  P2  ^ 

6  d  "              "  ••  P2  P3              "                 "                "  P4  P5  Pi 

d  6  "              ••  •'  P4  P5  Pi        "                 "                "  P2  P3 

77.  Knowing  the  directions  of  all  the  forces  of  a  system,  as  Pi P5, 

Fig.  42,  and  the  amounts  of  all  hut  two  of  them,  as  Po  and  P3,  we  may  find  the 
amounts  of  those  two  by  first  drawing  the  others,  P4,  P5  and  Pi,  as  in  the 
figure.  Then  two  lines  h  c  and  c  d,  drawn  in  the  directions  of  the  other  two 
and  closing  the  polygon,  will  necessarily  give  their  amounts. 


Fig.  43. 


Fig.  44. 


78.  If  any  two  points,  as  o  and  c,  Fig.  43,  be  taken,  then  the  force  or  forces 
represented  by  any  line  or  system  of  lines  joining  those  two  points  will  be 
equivalent  to  o  c.  Thus :  oc  =  oabc  =  odc  =  onpc  =  o  h  k  m  c  = 
o  h  m  c  =  o  f  c  =  o  g  c,  etc.,  etc. 

Similarly,  in  Fig.  42,  the  force  polygon  ah  c  d  e  ais  equivalent  to  the  force 
polygon  ah  f  d  e  a,  and  to  the  force  triangle,  ah  c  a,  each  being  =  zero. 

Non-concurrent  Coplanar  Forces. 

79.  Non-concurrent  Coplanar  Forces.  Fig.  44.  The  process  of 
finding  the  resultant  of  three  or  more  coplanar  but  non-concurrent  forces  is 
the  same  as  if  they  were  concurrent.  Thus,  let  Pi,  P2  and  P3  represent  three 
such  forces.*     We  may  first  find  the  resultant  Ri  of  any  two  of  them,  as  P2 


*Any  two  coplanar  non-parallel  forces,  as  Pi  and  P2,  or  P2  and  P3  are 
necessarily  concurrent  (see  *[f  19) ;  but  there  is  no  single  point  in  which 
the  three  forces  meet. 


376 


STATICS. 


and  P3;  and  then,  by  combining  Ri  with  the  remaining  force  Pi,  we  find  the 
resultant  R  of  the  three  forces.  Here  the  line  R  represents  the  resultant,  not 
only  in  amount  and  in  direction,  but  also  in  position.  That  is,  the  line  of 
action  of  the  resultant  coincides  with  R. 

80.  The  resultant  R  is  the  same,  in  amount  and  in  direction,  as  if  the 
forces  were  concurrent,  and  its  position  is  the  same  as  it  would  have  been  if 
their  point  of  concurrence  were  in  the  line  of  R.  If  there  are  more  than  three 
forces,  we  proceed  in  th^  same  way. 

81.  Conversely,  the  resultant  R,  or  any  other  force,  may  be  resolved 
into  a  system  of  any  number  of  concurrent  or  nonconcurrent  coplanar  forces, 
in  any  directions,  at  pleasure.  Thus,  we  may  first  resolve  R  into  Pi  and  Ri; 
then  either  of  these  into  two  other  forces,  as  Ri  into  Pa  and  P3,  and  so  on. 

83.  If  a  system  of  non-concurrent  coplanar  forces  is  in  equilibrium,  the 
forces  will  still  be  in  equilibrium  if  they  are  so  placed  as  to  be  concurrent; 
provided,  of  course,  that  their  directions,  senses  and  amounts  remain  un- 
.  changed ;  but  it  does  not  follow  that  a  system  of  forces,  which  is  in  equilib- 
rium when  concurrent,  will  remain  in  equilibrium  when  so  placed  as  to  be 
non-concurrent. 

Thus,  the  five  forces,  Pi P-„  Fig.  45  (a),  may  be  so  placed,  as  in  Fig. 

45  (6),  that  the  resultant  a  c,  of  Pi  and  P2,  does  not  coincide  with  the  re- 
sultant c  a  of  P3,  P4  and  P5,  but  is  parallel  to  it.  These  two  resultants  then 
form  a  couple.     (See  1[Tf  155,  etc.) 


Fig.  45. 


83.  Third  Condition  of  Equilibrium.  Hence,  the  conditions  of 
equilibrium  for  concurrent  forces,  stated  in  H  69, 

2  vertical        components   =   0 
2  horizontal  components   =   0 

do  not  suffice  for  non-iconcurrent  forces,  and  a  third  condition  must  be  added, 
viz.: — 

2  moments   =   0; 

i.  e.,  the  moments  of  the  forces,  taken  about  any  point,  must  be  in  equilib- 
rium. 

A  system  of  forces  in  equilibrium  has  no  resultant ;  hence  it  has  no  moment 
about  any  point.  In  other  words,  the  moments  of  the  forces,  as  well  as  the 
forces  themselves,  are  in  equilibrium. 

84.  The  resultant  of  a  system  of  unbalanced  non-concurrent 
forces,   acting  upon  a  body,  may  be  either 

(1)  a  single  force,  acting  through  the  center  of  gravity  of  the  body;  or 

(2)  a  couple;  i.  e.,  two  equal  and  parallel  forces  of  opposite  sense  (see 
It  155,  etc.) ;  or 

(3)  either  (a)  a  single  force,  acting  through  the  center  of  gravity  of  the 
body,  and  a  couple ;  or  (b)  a  single  force,  acting  elsewhere  than  through  the 
center  of  gravity  of  the  body. 

In  Case  (3),  the  two  alternative  resultants  are  interchangeable;  i.  e.,  a 
single  force,  acting  elsewhere  than  through  the  center  of  gravity  of  the  body, 
may  always  be  replaced  by  an  equivalent  combination  consisting  of  an  equal 


CORD    POLYGON. 


377 


parallel  force,  acting  through  the  center  of  gravity  of  the  body,  and  a  couple, 
and  vice  versa.    See  11[  161,  etc. 

The  resultant  gives  to  the  body,  in  Case  (1),  motion  of  translation  in  a 
straight  line,  without  rotation;  in  Case  (2),  rotation  without  translation; 
and  in  Case  (3),  both  translation  and  rotation.     See  foot-note  (*),  ^  1. 

85.  The  force  polygon,  H  72,  Figs.  40,  etc.,  and  the  method  by  co- 
ordinates, Tf  66,  Fig.  35,  therefore,  give  us  only  the  amount,  direction  and 
sense  of  the  resultant  of  non-concurrent  forces,  and  not  its  position.  To  find 
the  position  of  the  resultant  of  non-concurrent  forces,  we  may  have  recourse 
to  a  figure,  like  Fig.  44,  where  the  forces  are  represented  in  their  actual  posi- 
tions, or  to  the  cord  polygon,  1[^  86,  etc.,  Fig.  46. 

The  Cord  Polygon. 

86.  In  the  force  triangle  any  two  of  the  three  lines  may  be  regarded  as 
representing,  by  their  directions,  the  positions  of  two  members  (two  struts 
or  two  ties,  or  one  strut  and  one  tie)  of  indefinite  length,  resisting  the  third 
force ;  while  their  lengths  give  the  amounts  of  the  forces  which  those  mem- 
bers must  exert  in  order  to  maintain  equilibrium. 


%4  XA4  V\ 


Fi^.  86  (repeated). 


87.  Thus,  in  Fig.  26  (6),  are  shown  four  different  systems,  of  two  mem- 
bers each,  inclined  respectively  like  the  forces  c  b  and  6  o  in  Fig.  26  (a)  and 
balancing  the  third  force  o  c.  The  stresses  in  these  two  members  are  given 
by  the  lengths  of  the  lines  c  b  and  6  o  in  Fig.  26  (o) . 

The  members  acting  as  struts  are  represented,  in  Fig.  26  (6),  as  abutting 
against  flat  surfaces,  while,  those  acting  as  ties  are  represented  as  attached 
to  hooks,  against  which  they  pull. 

In  Fig.  26  (c)  and  (d)  are  indicated  systems  of  members,  inclined  like  the 
forces  c  a'  and  a'  a,  ca  and  a  o,  respectively,  of  Fig.  26  (a),  by  which  the  third 
force  o  c  might  be  supported. 

88.  In  the  force  polygon  ahcd  ea,  Fig.  46  (6),  representing  the  four 
forces,  Pi,  Ps,  P3,  P4,  of  Fig.  46  (a),  if  w-e  select,  at  pleasure,  any  point  o 
(called  the  pole)  and  draw  from  it  a  series  of  straight  lines  o  a,  o  b,  etc.  (called 
rays),  radiating  to  the  ends,  a,  b,  c,  etc.,  of  the  lines  P],  P2,  etc.,  representing 
the  forces,  we  shall  form  a  series  of  force  triangles,  a  o  b,  b  o  c,  etc. 

Thus,  in  the  triangle  ab  o  we  have  the  force  Pi,  or  a  b,  balanced  by  the  two 
forces  o  a  and  b  o;  in  the  triangle  b  c  o,  the  force  Pg,  or  b  c,  balanced  by  the 
two  forces  o  b  and  c  o;  and  so  on. 

89.  The  Cord  Polygon.  If,  now,  in  Fig.  46  (a),  we  draw  the  lines  a 
and  b,  parallel  respectively  to  the  rays  o  a  and  o  6  of  Fig.  46  (6)  and  meeting 
in  the  line  representing  the  force  Pi,  they  will  represent  the  positions  of  two 
tension  members  of  indefinite  length,  which  will  balance  the  force  Pi  by  ex-, 
erting  forces  represented,  in  amount  as  well  as  in  direction,  by  the  rays  o  a 
and  b  o.  Fig.  46  (6).  Again,  taking  pole  o^  Fig.  46  (b),  instead  of  o,  we  have 
a'  and  b\  Fig.  46  (a'),  parallel  respectively  to  the  rays,  o'  a  and  o'  b,  and  rep- 
resenting a  pair  of  struts  performing  the  same  duty. 

90.  Similarly,  the  lines  b  and  c.  Fig.  46  (a),  parallel  respectively  to  rays  o  b 
and  0  c,  represent  two  tension  members,  which,  with  stresses  equal  respec- 
tively to  o  6  and  c  o,  Fig.  46  (6),  balance  the  force  Pg. 


378 


STATICS. 


91.  We  thus  obtain,  finally,  a  system  of  five  tension  members,  a  b  c  d  «, 
Fig.  46  (a),  which,  if  properly  fastened  at  the  ends  a  and  e  respectively,  will, 
by  exerting  forces  represented  respectively  by  the  rays,  o  a,  o  b,  o  c,  etc..  Fig. 
46  (6),  balance  the  four  given  forces  Pi,  P2,  P3  and  P4. 

93.  The  figure  a  b  c  d  e,  Fig,  46  (a),  is  called  a  cord  polygon,  funicular 
polygon,  or  equilibrium  polygon. 

93.  Resultant,    Anti-resultant.    Amount  and  Direction.     In  the 

force  polygon,  Fig.  46  (6)  or  (d),  the  line  e  a,  joining  the  end  of  the  last  force- 
line  de  with  the  beginning  of  the  first  one  a  b,  represents  the  anti-resultant  of 
the  given  system  of  four  forces,  and  a  e  their  resultant.  Evidently,  there- 
fore, the  rays  a  o  and  o  e,  which  represent  two  components  of  a  e,  represent 
also,  in  direction  and  in  amount,  two  forces  which  would  balance  e  a,  or  which 
would  be  equivalent  to  the  given  system  of  (four)  forces. 


Fig^s.  46  (a),  {a')  and  (6). 


94.  Position  of  Resultant.  Hence,  in  the  cord  polygon.  Fig.  46  (a), 
the  intersection,  i,  of  the  cords  a  and  e,  parallel  respectively  to  the  rays  o  a 
and  c  o,  is  a  point  in  the  line  of  action  of  the  resultant  R ;  and,  if  we  imagine 
a  %  and  e  i  to  be  rigid  rods,  and  apply,  at  i,  a  force,  —  R,  equal  and  parallel  to 
o  e,  but  of  opposite  sense,  that  force  will  be  the  anti-resultant  of  the  (four) 
given  forces,  and  we  shall  have  a  frame-work  b  c  dioi  cords  and  rods,  kept  in 
equilibrium  by  the  action  of  the  five  forces.  Pi,  P2,  P3,  P4  and  —  R. 

95.  By  choosing  other  positions  of  the  pole,  as  o',  Fig.  46  (6),  or  by  differ- 
ently arranging  the  given  forces,  as  in  Fig.  46  (c),  we  merely  change  the 
shape  of  the  cord  polygon,  and  (in  some  cases)  reverse  the  sense  of  the 
stresses  in  the  members.  Thus,  in  Fig,  46  (a),  all  the  stresses  are  tensions,  or 
pulls ;  while  in  Fig.  46  (c)  a,  b,  d  and  e  are  tensions  or  pulls,  and  c  is  a  com- 
pression or  push. 

96.  In  constructing  the  cord  polygon.  Fig.  46  (a),  (a'),  (c),  and  (e),  care 
must  be  taken  to  draw  the  cords  in  their  proper  places ;  and  for  this  it  is  nec- 
essary to  remember,  simply,  that  the  two  rays  pertaining  to  any  particular 
force  line  in  the  force  polygon.  Fig.  46  (&),  represent  those  members  which, 
in  the  cord  polygon,  Fig.  46  (a),  take  the  components  of  that  force. 


COED  POLYGON. 


379 


Thus,  o  a  and  h  o,  Fig.  46  (Jb),  pertain  to  the  force  Pi;  o  6  and  c  o  to  the 
force  Po.  Hence,  in  Fig.  46  (a)  or  (c)  we  draw  a  and  h  (parallel  respectively 
to  o  a  and  h  o)  meeting  in  the  line  of  action  of  Pi ;  b  and  c  (parallel  respect- 
ively to  o  6  and  c  o)  meeting  in  the  line  of  action  of  Po,  etc.,  etc. 

97.  Each  ray  in  the  force  polygon,  Fig.  46  (6),  including  the  outside  ones, 
is  thus  seen  to  pertain  to  two  force?,  and  each  force  has  two  rays.  The  two 
cords,  parallel  respectively  to  the  two  rays  of  any  force,  must  be  drawn  to 
meet  in  the  line  of  action  of  that  force;  and  each  cord  must  join  the  lines  of 
action  of  the  two  forces  to  which  its  parallel  ray  pertains.  The  lines,  a,  b,  c. 
etc.,  in  the  cord  polygon.  Fig.  46  (a)  and  (c),  give  merely  the  inclinations  of 
members  which,  as  there  arranged,  would  sustain  the  given  forces.  The 
lengths  of  these  lines  have  nothing  to  do  with  the  amounts  of  the  stresses. 
These  are  given  by  the  lengths  of  the  corresponding  rays  in  the  force  polygon. 
Fig.  46  (6). 


Figs.  46  (c),  (d)  and  (c). 


98.  Tf  the  anti-resultant  force,  —  R,  is  not  applied,  the  cords  a  and  e  may 
be  supposed  fastened  to  firm  supports,  against  which  they  exert  stresses  rep- 
resented, in  amount  and  in  direction,  by  the  rays  a  o  and  o  e  respectively. 
But  the  resistances  of  those  two  supports  are  plainly  equal  and  opposite  to 
those  stresses,  or  equal  to  o  a  and  e  o  respectively.  Hence,  their  resultant  is 
the  anti-resultant,  —  R,  of  the  four  original  forces. 

99.  If,  Fig.  46  (e),  the  two  end  members  a  and  e  were  attached  merely  to 
two  ties,  V  and  V,  parallel  to  the  anti-resultant,  — R,  they  would  evidently 
draw  the  ends  of  those  ties  inward  toward  each  other.  To  prevent  this,  let 
the  strut  k  be  inserted,  making  it  of  such  length  that  the  ties  V  and  V  may 
remain  parallel  to  —  R,  and  draw  o  k.  Fig.  46  (6),  parallel  to  k.  Then  a  k 
and  k  e  give  the  stresses  in  V  and  V  respectively. 

100.  If  the  anti-resultant,  —  R,  found  by  means  of  the  force  polygon,  be 
applied  in  a  line  passing  through  the  intersection  of  the  outer  (initial  and 
final)  members  in  the  cord  polygon,  all  the  forces,  includinc:  of  course  the 
anti-resultant,  will  be  in  equilibrium.  In  other  words,  coplanar  forces  are 
in  equilibrium  if  they  may  be  so  drawn  as  to  form  a  closed  force  polygon,  and 
if  a  closed  cord  polygon  may  be  drawn  between  them.  But  if  the  anti-re- 
sultant be  applied  elsewhere,  we  shall  have  a  couple,  composed  of  the  anti- 
resultant,  —  R,  and  the  resultant  R  of  the  forces. 


380 


STATICS. 


Concurrent  Non-coplanar  Forces. 

101.  Any  two  of  the  concurrent  forces,  as  o  a  and  o  c.  Fig.  47  (a)  or  (6),  are 
necessarily  coplanar.  Find  their  resultant,  o  r,  which  must  be  coplanar  with 
them  and  with  a  third  force  o  b.  Then  the  resultant,  R,  of  o  r  and  o  6  is  the 
resultant  of  the  three  forces.  If  there  are  other  forces,  proceed  in  the  same 
way. 

103.  No  three  non-coplanar  forces,  whether  concurrent  or  not,  can  be  in 
equilibrium. 

103.  Force  Parallelopiped.  The  resultant  of  any  three  concurrent 
non-coplanar  forces,  o  a,  o  b,  o  c.  Figs.  47,  will  be  represented  by  the  diagonal 
o  R,  of  a  parallelopiped,  of  which  three  converging  edges  represent  the  three 
forces. 

104.  Methods  by  Models.  (a)  For  three  forces.  Construct  a 
box.  Fig.  47  (a)  or  (b),  with  three  convergent  edges  representing  the  three 
forces  in  position  and  amount.  Then  a  string  o  R,  joining  the  proper  corners, 
will  represent  the  resultant. 


Fig.  47. 

Or,  let  ao,  bo,  c  o,  Fig.  48  (a),  be  three  forces,  meeting  at  o.  Draw  on 
pasteboard  the  three  forces  a  o,  b  o,  c  o,  as  in  Fig.  48  (6),  with  their  actual 
angles  a  o  b,  b  o  c,  c  o  a,  and  find  the  resultant  w  o  of  the  middle  pair,  6  o  and 
c  o.  Cut  out  neatly  the  whole  figure,  a  o  a  c  lo  b  a.  Make  deep  knife- 
scratches  along  o  b,  o  c,  so  that  the  two  outer  triangles  may  be  more  readily 
turned  at  angles  to  the  middle  one.  Turn  them  until  the  two  edges  o  a,  o  a 
meet,  and  then  paste  a  piece  of  thin  paper  along  the  meeting  joint  to  keep 


(«) 


(ft) 

Fig.  48. 


(c) 


them  in  place.  Stand  the  model  upon  its  side  o  6  ty  c  as  a  base,  and  we  shall 
have  the  slipper  shape  a  o  b  w,  P'ig.  48  {c);  o  w  being  the  sole,  and  a  ob  the 
hollow  foot.  In  the  model,  the  force  a  o  and  the  resultant  w  o  oi  the  other 
two  forces,  are  now  in  their  actual  relative  positions.  To  find  their  resultant, 
cut  out  a  separate  piece  of  pasteboard,  ^  a  o  w,  with  R  a  and  R  w  parallel 
respectively  to  w  o  and  a  o.  Draw  upon  each  side  of  it  the  diagonal  R  o. 
Paste  this  piece  inside  the  model,  with  its  lower  edge  w  o  on  the  line  w  o,  Fig. 
48  (6),  and  its  edge  a  o  in  the  corner  a  o.  This  done,  R  o  represents  the  re- 
sultant oi  a  o,b  o,  c  o,  Fig.  48  (a),  in  its  actual  position  relative  to  them. 

105.  (b)  For  four  forces,  as  a  o,  6  o,  c  o,  rfo,  in  Fig.  49.  Draw  them  as  in 
Fig.  49  (a),  with  their  angles  a  o  b,  b  o  c,  etc.  Draw  also  the  resultants  v  o,  of 
a  o  and  b  o;  and  w  o,  of  c  o  and  d  o.  Then  cut  out  the  entire  figure,  as  before, 
and  paste  together  the  two  edges  a  o,  a  o.  Hold  the  model  in  such  a  way 
that  two  of  its  planes  (as  a  o  6  and  b  o  c)  form  the  same  angle  with  each  other 


NON-COPLANAE  FORCES. 


881 


as  do  the  two  corresponding  planes  between  the  forces.  Then  we  have  the 
two  resultants  v  o,  w  o.  Fig.  49  (6),  in  their  actual  relative  positions.  Cut  out  a 
separate  piece  of  pasteboard  K  v  o  w,  Fig.  49  (6),  draw  the  diagonal  R  o  on 
each  side  of  it,  and  paste  it  inside  the  model,  with  o  v  and  o  w  on  the  corre- 
sponding lines  of  the  model.  Then  R  a  will  represent  the  resultant  of  the 
four  forces,  a  o,  bo,  c  o,  do,  in  its  actual  position  relative  to  them. 

The  model  may  be  made  of  wood,  the  triangles  a  o  b,  b  o  c,  etc.,  being  cut 
out  separately,  the  joining  edges  bevelled,  and  then  glued  together. 


Non-concurrent  Non-coplanar  Forces. 

106.  Non-concurrent  Non-coplanar  Forces.  Fig.  50  (a).  (For  par- 
allel non-coplanar  forces,  see  ^1  110,  etc.)  Resolve  each  force  into  two  rec- 
tangular components,  one  normal  to  an  assumed  plane,  the  other  coin- 
ciding with  the  plane.*  Find  the  resultant  of  the  (coplanar)  components 
coinciding  with  the  plane,  by  methods  already  given,  and  that  of  the  normal 
(parallel)  components,  by  ^1f  110,  etc.  If  these  two  resultants  are  coplanar, 
they  are  also  concurrent,  and  their  resultant  (which  is  the  resultant  of  the 
system)  is  readily  found. 

107.  If  not,  let  V,  Fig.  50  (6),  be  the  resultant  normal  to  the  plane,  and  H 
the  resultant  lying  in  the  plane.  By  ^  162,  substitute,  for  H,  the  equal  and 
parallel  force  H',  meeting  V  at  O,  and  the  couple  H  .  O  a,  and  find  the  result- 
ant, R',  of  V  and  H'.  The  system  of  forces  is  thus  reduced  to  the  single  force 
R'  and  the  couple  H  .  O  a.     For  Couples,  see  1  155. 


108.  Moments  of  Non-coplanar  Forces.  The  action  of  the  weight 
W  of  the  wall.  Fig.  51  (a),  and  of  the  non-coplanar  forces  Pi  and  Pg,  may  be 
represented  as  in  Fig.  51  (6),  where  the  axle  o'  c'  represents  the  edge  a  c 
about  which  the  wall  tends  to  turn,  while  the  bars  or  levers  represent  the 
leverages  of  the  forces.  So  far  as  regards  the  overturning  stability  of  the 
wall,  regarded  as  a  rigid  body  and  as  capable  of  turning  only  about  the  edge 
a  c,  it  is  immaterial  whether  an  extraneous  force,  as  P],  is  applied  at  p  or  at 
q;  but  it  is  plainly  not  immaterial  as  regards  a  tendency  to  swing  the  wall 
around  horizontally,  or  to  fracture  it;  or  as  regards  pressures  (and  conse- 
quent friction)  between  the  axle  a'  c'  and  its  bearings.  For  equilibrium.  Pi  m 

=  Pg  ^  -i-  W.  -r-.     Here  a  torsional  or  twisting  stress  is  exerted  in  the  axle, 
♦Wires,  stuck  in  a  board  representing  the  plane,  will  facilitate  this. 


382 


STATICS. 


and  the  pressures  of  its  ends  in  the  bearings  are  more  or  less  modified ;  but, 
so  far  as  merely  the  equilibrium  of  the  moments  is  concerned,  we  may  sup- 
pose all  of  the  forces  and  their  moments  to  be  shifted  into  one  and  the  same 
plane,  as  in  Fig.  51  (c). 

109.  In  cases  like  that  represented  in  Fig.  51,  it  is  usual,  for  convenience, 
to  restrict  ourselves  to  a  supposed  vertical  slice,  s,  1  foot  thick,  and  to  the 
forces  acting  upon  such  slice ;  supposing  the  weight  of  the  slice  to  be  concen- 
trated at  its  center  of  gravity,  and  the  extraneous  forces  to  be  applied  in  the 
same  vertical  plane  with  gravity.  In  effect,  we  are  then  dealing  with  a 
slice  indefinitely  thin,  but  having  the  weight  of  the  1-ft.  slice. 


m\ 

'^''' 
3.^ 

f' 

srj^^^X^ 

<A 

h 

Fl^.  51. 


PARALLEL   FORCES. 

110.  The  resultant  of  any  number  of  parallel  forces,  whether 
they  are  in  the  same  plane  or  not,  and  whether  in  the  same  direction  or  not, 
is  parallel  to  them  and  =  their  algebraic  sum. 

Coplanar  Parallel  Forces. 

111.  The  resultant  of  any  number  of  coplanar  parallel  forces 

is  in  the  same  plane  with  them,  whether  the  forces  are  of  the  same  or 
of  opposite  sense;  and  the  leverages,  or  arms,  of  such  forces,  and  of  their 
resultant,  about  any  given  point  in  the  same  plane,  are  in  one  straight  line. 
Thus,  in  Fig.  56  (a),  where  the  five  forces,  a,  h,  c,  d  and  e  are  in  one  plane,  their 
resultant,  R,  is  in  that  same  plane;  and  the  leverages  of  the  forces,  and 
of  R,  about  any  point,  as  h  or  v,  in  the  same  plane,  are  in  the  straight  line  R  v. 


Fig.  53. 


112.  The  resultant,  R,  or  anti-resultant,  Q,  Fig.  52,  of  two  parallel 
forces,  a  and  h,  intersects  any  straight  line,  u  v,  joining  the  directions  of 
the  two  forces.  Hence,  if  three  parallel  forces  are  in  equilibrium,  they  are 
in  the  same  plane.  In  Fig.  52  (o),  the  two  forces,  a  and  b,  are  of  like 
sense.  R  is  then  between  a  and  b,  and  R  =  6  +  a.  In  Fig.  52  (b),  a  and 
6  are  of  opposite  sense.     R  is  then  not  between  a  and  b,  and  R  =  6  —  a. 


PARALLEL   FORCES. 


383 


113.  To  find  the  position  of  the  resultant,  draw  and  measure  any  straight 
line,  u  V,  joining  the  lines  of  action  of  the  forces.  It  is  immaterial  whether 
w  V  is  perpendicular  to  said  directions,  or  not.  The  line  representing  the 
resultant  cuts  u  v,  and  its  position  is  found  thus : 

b  ,       ,  a 

uz  =  u  V  X   .^;      and  v  t  =  u  v  X  -^-. 


Figr.  53. 


^=1 


114.  This  may  be  conveniently  done  by  making  u  v  equal,  by  any  conve- 
nient scale,  to  the  sum  of  the  forces,  as  in  Fig.  53,  where  u  v  =  42.  Then 
make  u  i  equal,  by  the  same  scale,  to  the  force  at  v,  or  v  i  equal  to  the  force  at 
u.  Then  a  line,  R,  Fig.  52  (a),  drawn  through  i  parallel  to  a  and  h,  gives  the 
position  and  direction  of  their  resultant;  and  its  amount  is  equal  to  the  sum 
of  a  and  b;  or  R  =  a  +  6.  In  other  words,  if  a  force,  Q,  parallel  to  a  and  6, 
and  equal  to  their  sum,  but  of  opposite  sense,  be  applied  to  the  body  any- 
where in  a  line  passing  through  i,  it  will  balance  a  and  b,  or  will  be  their  anti- 
resultant. 


*^i 


-"^K 


\ 


I 


w^m 


(a) 


Fig.  55. 


(&) 


115.  The  position  of  the  resultant,  so  found,  satisfies  the  condition  of 
equilibrium  of  moments:  thus,   b.vi  —  a.ui  =  zero. 

If  the  two  forces  are  equal,  their  resultant  R  is  evidently  midway  between 
them. 

116.  In  the  common  steelyard.  Fig.  54,  the  two  forces  a  and  6,  of 
Fig.  52  (a),  are  represented  by  the  two  weights,  a  =  3  pounds  at  u,  and  b  == 
1  pound  at  v,  with  leverages  ui  and  vi  respectively,  as  2  : 6,  or  as  1  : 3. 


384 


STATICS. 


It  will  be  noticed  that  in  Fig.  56  (a)  the  resultant,  R,  owing  to  the  posi- 
tions and  amounts  of  the  several  forces,  falls  outside  of  the  system  of  given 
forces. 

117.  Figs.  55  to  58  illustrate  the  application  of  the  cord  polygon  (^t  86 
to  100)  to  coplanar  parallel  forces.  Here  the  force  polygon  is  necessarily  a 
straight  line. 


, 

a 

h 

d 

\ 

ii 

1                       1 
!        •"         ! 

^^^^^^^^^ 

mm^^mwM 

j       > 

\ — - 

1^ 

--1 — r- 

' 

["  1  / 

i\ 

! 

(         1  <' 

m^ 

^ 

^n' 

^p 

-^ 

'        e 

(a) 


Figr-  56. 


118.  Resolution.  Let  Fig.  57  (a)  represent  a  beam  bearing  a  single 
concentrated  load,  a,  elsewhere  than  at  its  center;  and  let  it  be  required 
to  find  the  pressure  on  each  of  the  two  supports,  w  and  x. 


Fig.  57. 


Draw  X  a,  Fig.  57  (6),  to  represent  the  load  a  by  scale,  and  rays  X  O,  o  O, 
to  any  point  O  not  in  the  line  X  a.  In  Fig.  (a),  from  any  point,  ^,  in  the 
vertical  through  the  point,  a,  where  the  load  is  applied,  draw  i  s  and  i  r, 
parallel  respectively  to  O  X  and  O  a.  Join  r  s,  and  in  Fig.  (6)  draw  O  w  par- 
allel to  rs.  Then  the  two  segments,  w  a  and  X  w,  of  X  a,  give  by  scale  the 
pressures  upon  the  two  supports,  w  and  x  respectively.  The  greater  pres- 
sure will  of  course  be  upon  the  support  nearest  to  the  load ;  but  we  may 
be  guided  also  by  remembering  that  the  segment  X  w,  adjoining  the  radial 
line  O  X  in  Fig.  (b)  represents  the  pressure  on  that  support,  x,  Fig.  (a), 
which  pertains  to  the  line  i  s  parallel  to  O  X ;  and  vice  versa. 

119.  Fig.  58  represents  a  case  where  there  are  several  loads  on  the 
beam.  Here  tHe  intersection,  i,  of  the  lines  h  s  and  k  r,  Fig.  (a),  drawn 
parallel  respectively  to  O  X  and  c  O,  Fig.  (6)  shows  the  position  of  the 
resultant  of  the  three  loads.     Here,  as  in  Fig.  57,  we  join  r  a.  Fig.  (a), 


PARALLEL   FORCES. 


385 


and  draw  O  w,  Fig.   (6),  parallel  to  r  s.     Then  X  w,  Fig.   (6),  gives  the 
pressure  upon  x,  and  w  c  that  upon  w. 


(a) 


Fig.  58. 


Non-coplanar  Parallel  Forces. 

120.  Non-coplanar  Parallel  Forces.  Fig.  59  (a).  Between  the 
lines  of  action  of  any  two  of  the  forces,  as  a  and  b,  draw  any  straight  line,  u  v, 
and  make 

u  I  =  u  V   X  r  ;      or     V  I  =  u  V  X  T  ' 

a  -t  b  a  +  6 

Through  i  draw  R',  parallel  to  a  and  b,  and  equal  to  their  sum.  Then 
is  R'  the  resultant  of  a  and  b.  Then,  from  any  point,  i,  in  the  line  of  action 
of   R',   draw  i  z  to  any  point,  2,   in  the  line  of  action  of  c,   and  make 

c  R' 

ik  =  i  z  X  — -^^,  ;  or  zk  =  i  z  X  — -^^,  .     Through  k  draw  R  parallel  to 

a,  b  and  c,  and  equal  by  scale  to  their  sum.  Then  is  R  the  resultant  of  the 
three  forces,  a,  b  and  c.  If  there  are  other  forces,  proceed  in  the  same  way 
with  them. 


c=6 


Fig.  59. 


121.  In  Fig.  59  (a)  we  have  shown  the  forces,  a  and  c,  acting  upon  surfaces 
raised  above  the  general  plane,  merely  in  order  to  illustrate  the  fact  that  it  is 
not  at  all  necessary  that  the  forces  be  supposed  to  act  upon  or  against  a  plane 
surface. 

122.  Although  Fig.  59  (a)  illustrates  the  method  of  finding  the  resultant 
of  non-coplanar  parallel  forces,  yet  it  plainly  does  not  give  the  actual  relative 
positions  of  the  forces  and  their  resultant ;  because  it  is  necessarily  drawn  in  a 
kind  of  perspective,  and  therefore  all  the  parts  cannot  be  measured  by  a 
scale.  The  true  relative  positions  may  of  course  be  represented  in  plan,  as 
by  the  five  stars,  a,  b,  c,  i  and  k.  Fig.  59  (6) ,  corresponding  to  the  points  where 

25 


386 


STATICS. 


the  forces  and  resultants  intersect  some  one  chosen  plane.  But  it  is  now 
impossible  to  represent  the  forces  themselves  by  lines.  They  must  there- 
fore be  stated  in  figures,  as  is  here  done.  It  is  then  easy  to  find  the  positions 
of  the  resultants,  as  before. 

133.  If  there  are  also  forces  acting  in  the  opposite  direction,  as 
d  and  e.  Fig.  59  (a),  find  their  resultant  separately.  We  thus  obtain,  finally, 
two  resultants  of  opposite  sense.  These  resultants  may  be  equal  or  unequal, 
and  colinear  or  non-colinear.  If  they  are  non-colinear,  see  ^  84,  and  Couples, 
tt  155,  etc. 

134.  Method  by  projections.  Fig.  60.  First  find  the  projections, 
a',  b'  and  c'  of  the  forces,  a,  b  and  c,  upon  any  plane,  as  x  y,  parallel  to 
them ;  and  then  their  projections,  a" ,  b",  and  c",  upon  a  second  plane,  x  v, 
parallel  to  them  and  normal  to  the  first.  Find  the  position,  R',  of  the  re- 
sultant of  a',  b'  and  c',  in  plane  x  y,  and  that  R",  of  a",  b"  and  c",  in  plane 
X  V.  Now,  as  the  lines,  a',  b\  c',  and  a",  b",  d',  are  projections  of  the  forces, 
a,  b  and  c,  so  R',  R",  are  projections  of  the  resultant,  R,  of  the  forces.  The 
position  of  R  is  therefore  at  the  intersection  of  two  planes,  R  R'  and  R  R", 
perpendicular  to  the  planes,  x  y  and  x  v,  and  standing  upon  the  projections 
R'  and  R",  of  the  resultant,  R.     R  =  a  +  6  +  c. 


CENTER  OF  GRAVITY. 

125.  If  a  body.  Fig.  1,*  or  a  system  of  bodies,  Fig.  2,  be  held  successively 
in  different  positions,  (a),  b),  etc.,  the  resultant  of  the  parallel  forces  of  grav- 
ity, acting  upon  its  particles  and  indicated  by  the  arrows  in  the  figures,  will 
occupy  different  positions,  relatively  to  the  figure  of  the  body  or  system. 
That  point,  where  all  these  positions,  or  lines  of  gravity,  meet,  is  called  the 
center  of  gravity  of  the  body  or  system.  Thus,  if  a  homogeneous  cylinder 
be  stood  vertically  upon  either  end,  the  line  of  gravity  will  coincide  with 
the  axis  of  the  cylinder;  but  if  the  cylinder  be  then  laid  upon  its  side,  the 
line  of  gravity  will  intersect  the  axis  at  right  angles  and  will  bisect  it. 
Hence,  in  the  cylinder,  the  center  of  gravity  is  at  the  center  of  the  axis. 

126.  About  the  center  of  gravity  the  moments  of  all  the  forces  of  gravity 
are  in  esquilibrium,  in  whatever  position  the  body  or  system  may  be.  Hence, 
the  body,  or  system,  if  suspended  by  this  point,  and  acted  upon  by  gravity 
alone,  will  balance  itself;  i.  e.,  if  at  rest  it  will  remain  at  rest;  or,  if  set  ii* 
motion  revolving  about  its  center  of  gravity,  and  then  left  to  itself,  it  will 
continue  to  revolve  about  that  center  indefinitely  and  with  uniform  angulaf 
velocity.  Or,  if  suspended  freely  from  any  point,  it  will  oscillate  until  the 
center  of  gravity  comes  to  rest  vertically  under  such  point. 


*  Figs.  1  to  45,  relating  to  Center  of  Gravity,  are  numbered  independently 
of  the  rest  of  the  series  of  figures  relating  to  Statics. 


CENTER   OF   GRAVITY. 


387 


127.  In  some  bodies,  such  as  the  cube,  or  other  parallelopiped,  the  sphere, 
etc.,  the  center  of  gravity  is  also  the  center  of  the  weight  of  the  body;  but 
very  frequently  this  is  not  the  case.  Thus,  in  a  body  a  b,  Fig,  2,  with  its 
center  of  gravity  at  G,  there  is  more  weight  on  the  side  a  G,  than  on  the  side 
G6. 


Fig.  1. 

Stable,  Unstable,   and  Indifferent  Equilibrium. 

128.  A  body  is  said  to  be  in  stable  equilibrium  when,  as  in  the  pendulum, 
it  is  so  suspended  that,  if  swung  a  little  to  either  side,  it  tends  to  oscillate 
until  it  comes  to  rest  again,  with  its  center  of  gravity  vertically  under  the 
point  of  suspension. 

129.  It  is  said  to  be  in  unstable  equilibrium  when,  as  in  the  case  of  an 
egg,  stood  upon  its  point,  it  is  so  supported  that,  if  swung  a  little  to  either 
side,  and  left  to  itself,  it  swings  farther  out  from  the  vertical  and  eventually 
falls. 

130.  It  is  said  to  be  in  indifferent  equilibrium  when,  as  in  the  case  of  a 
grindstone,  supported  by  its  horizontal  axis,  or  of  a  sphere  resting  upon  a 
horizontal  table,  it  is  so  suspended  or  supported  that,  if  made  to  rotate  about 
its  center  of  gravity  and  then  left  to  itself,  it  will  continue  in  that  state  of  rest 
or  of  angular  motion  in  which  it  is  left. 


mmmm. 


(a)  (b) 

Fig.  2. 


General    Rules. 

131.  The  following  general  rules  (1)  to  (6),  form  the  basis  of  the  special 
rules,  (7)  to  (39). 

In  speaking  of  the  center  of  gravity  of  one  or  more  bodies,  we  shall  assume, 
for  simplicity,  that  they  are  homogeneous  (i.  e.,  of  uniform  density  through- 
out) and  of  the  same  density  with  each  other.  The  center  of  gravity  is  then 
the  same  as  the  center  of  volume,  and  we  may  use  the  volumes  of  the  bodies 
(as  in  cubic  feet,  etc.)  in  the  rules,  instead  of  their  weights  (as  in  pounds,  etc.). 

In  applying  these  general  rules  to  surfaces,  use  the  areas  of  the  surfaces, 
and  in  applying  them  to  lines,  use  the  lengths  of  the  lines,  in  place  of  the 
weights  or  volumes  of  the  bodies. 

In  all  of  the  rules  and  figures,  pp.  388  to  398,  G  represents  the  center  of 
gravity,  except  where  otherwise  stated. 


388 


FORCE   IN   RIGID   BODIES. 


(1).  Any  tw^o  bodies^  Fig.  3. 
if  each  body/ 
g  and  g' ;  and 


,  ,  .  ,      „         Having  found  the  center  of  gravity,  g,  gr', 

of  each  body,  by  means  of  the  rules  given  below :  then  G-  is  in  the  line  joining 


weight  of  g' 


sum  of  weights  of  g  and  gf 


g'G-gg'  X 


weight  of  g 


sum  of  weights  of  g  and  / 


Fig.  3 


(3).  Any  nnmlier  of  bodies,  as  a,  b  and  c,  Fig.  4,  whether  their  center* 
of  gravity  are  in  the  same  plane  or  not. 

First,  by  means  of  rule  (1>  find  the  center  of  gravity,  g,  of  any  two  of  the 
bodies,  as  a  and  b.  Then  the  center  of  gravity,  G,  of  the  three  bodies,  a,  b 
and  c,  is  in  the  line  grgr' joining  g  with  thts  center  of  gravity,  g'  of  c;  and 


gG  =  gg'  X 


weight  of  c 


g'G^gg'  X 


sum  of  weights  of  a,  b  and  c  ' 
sum  of  weights  of  a  and  b     . 
sum  of  weights  of  a,  b  and  c 
and  so  on,  if  there  are  other  bodies. 

(3).  In  many  cases,  a  single  complex  body  may  be  supposed  to  be  divided 
into  parts  whose  several  centers  of  gravity  can  be  readily  found.  Then  the 
center  of  gravity  of  the  whole  may  be  found  by  the  foregoing  and  following 
rules.    Thus,  in  Fig.  5,  we  may  find  separately  the  centers  of  gravity  of  the 


two  parallelopipeds  and  of  the  cylinder  between  them  (each  in  the  center  of 
its  respective  portion  of  the  whole  solid) ;  and  in  Fig.  6  the  centers  of  gravity 
of  the  square  prism  and  the  square  pyramid  (the  latter  by  rule  (36), 
and  then,  knowing  in  either  case  the  weights  of  the  several  parts,  find  their 
common  center  of  gravity  as  directed  in  rules  (1)  and  (2). 


CENTER   OF   GRAVITY.  389 

(4).  Any  liolloiv  body,  or  body  containing  one  or  more  openings.  Fig.  7, 
Find  the  common  center  of  gravity,  g\  of  the  openings  by  rule  (1)  or  (2),  anc 


jnig.  r 


the  center  of  gravity,  gr,  of  the  entire  figure,  as  though  it  had  no  opentoga 
Then  G  is  in  the  line  fir  y',  extended,  and 

qG  ^  oQf  sy sum  of  volumes  of  openings 

volume  of  entire  body  —  volumes  of  openings 
•  Q^     ^v^  volume  of  entire  body 


volume  of  entire  body  —  volumes  of  openings 


Bemark.  For  convenience,  we  have  shown  the  several  centers  of  gravity. 
g,  g,'  G,  upon  the  surface  of  the  figure.  In  the  real  solid  (suwosed  to  be  ol 
uniform  thickness)  they  would  of  course  be  in  the  middle  of  its  thickness- 
and  immediately  under  the  positions  shown  in  the  figure. 

(5).  In  any  line,  figure  or  body,  or  in  any  system  of  lines,  figures  or  bodies,  any 
plane  passing  through  the  center  of  gravity  is  called  a**  plane  of  gravity" 
for  said  line,  etc.,  or  system  of  lines,  etc.  The  intersection  of  two  such  planes 
of  gravity  is  called  a  "  line  of  gfravlty.'*  The  center  of  gravity  is  (1st)  the 
intersection  of  two  lines  of  gravity;  (2nd)  the  intersection  of  three  planes 
of  gravity,  or  (3rd)  the  intersection  of  a  plane  of  gravity  with  a  line  of  gravity 
not  lying  in  said  plane. 

If  a  figure  or  body  has  an  axis  or  plane  of  symmetry  (i.  e.,  8  'ine  or  plane 
dividing  it  into  two  equal  and  similar  portions)  said  axis  or  plane  is  a  line  or 
plane  of  gravity.  If  a  figure  or  body  has  a  central  point,  said  point  is  the 
center  of  gravity. 

In  Fig.  1,  the  string  represents  a  line  of  gravity ;  and  any  plane  with 

which  the  string  coincides  is  a  plane  of  gravity.  Thus  O  may  often  be  con- 
veniently found,  especially  in  the  case  of  a  flat  body,  by  allowing  it  to  hang 
freely  from  a  string  attached  alternately  at  different  comers  of  it,  or  by  bal- 
ancing it  in  two  or  more  positions  over  a  knife-edge,  etc.,  and  finding  G  ia 
either  case  by  the  intersection  of  the  lines  or  planes  of  gravity  thus  found. 

(6).  Tbe  grapbic  method  of  finding  the  resultant  of  parallel  forces 
may  often  be  advantageously  used  for  finding  the  center  of  gravity  of  a  com- 
pound body  or  figure,  or  of  a  system  of  bodies  or  figures,  when  the  centers  of 
gravity  of  the  several  parts  are  known. 

Thus,  in  Fig.  8,  let  o,  b  and  c  represent  three  figures  or  bodies  whose  centers 
of  gravity  are  in  one  plane.  Draw  vertical  lines  through  said  centers,  and 
construct  the  polygon  of  forces,  xabc,  Fig.  9,  making  the  lin<-s  a? a,  a 6,  etc., 

Sroportional  to  the  weights  of  a,  b  and  c;  and  from  any  convenient  point  O 
raw  radial  lines  Ox,  Oa,  etc.  In  Fig.  8,  draw  inh,inn,  npy  and  p k,  parallel 
Tespectively  to  O  a;,  O  a,  O  6,  O  c.  Then  a  vertical  line,  i  G,  drawn  through  the 
"intersection,  %oi  mh  and  p  Ar,  is  a  line  of  gravity  of  the  system  or  figure.  If 
the  body  or  figure  is  symmetrical^  as  in  the  cross  section  of- a  T  rail,  I  oeam  or 
deck  beam,  etc.,  the  axis  of  symmetry,  dividing  the  figure,  etc.  into  two  simi- 
lar and  equal  parts,  is  also  a  line  of  gravity,  and  its  intersection  with  the  line 
tG-  already  found  is  the  required  center  of  gravity  G.  In  such  cases  it  is 
generally  most  convenient  to  draw  the  lines  through  the  several  centers  of 
gravity  perpendicular  to  the  axis  of  symmetry,  so  that  the  line  of  gravity 
found  will  also  be  perpendicular  to  it. 

But  if,  as  in  Fig.  8,  the  body  or  figure,  etc.,  is  not  symmetrical,  we  must  find 
a  second  line  of  gravity,  the  intersection  of  which  with  the  first  will  give  the 
center  of  gravity,  G.  To  do  this,  repeat  the  process,  drawing  another  set  of 
parallel  lines  through  the  several  centers  of  gravity,  Fig.  8.  It  will  be  most 
convenient  to  draw  them  horizontally,  or  at  right  angles  to  those  already  drawn, 
and  in  the  following  instructions  we  suppose  this  to  be  dona. 


390 


FORCE   Iiq-  RIGID   BODIES. 


Then  draw  a  second  funicular  polygon,  m^n'p^i',  Fie.  8,  making  the  lines, 
m'n/  etc.,  perpendicular  (instead  of  parallel)  to  the  radial  lines  O  x,  etc.,  Fig.  9; 
and  draw  the  second  line  of  gravity,  i'  G,  through  i',  perpendicular  to  the  first* 
Then  G  is  at  the  intersection  of  the  two  lines  of  gravity. 


y-Qa 

//  \\m, 


\  4F-rf -\1~1    / 


N(i 


^ ^^^^ 


I^XS.  8 


fi^iSi.9 


The  drawing  of  the  second  funicular  polygon  is  often  less  simple  than  tha* 
of  the  first,  because  in  the  second  the  parallel  lines  through  the  several  centers 
of  gravity  do  not  necessarily  follow  each  other  in  the  same  order  as  in  the  first. 
Bear  in  mind  that  the  two  lines  (as  n' p' ^  n'  m!)  meeting  in  the  parallel  line 
(as  6nO  pertaining  to  any  given  pait,  6,  of  the  tigure,  must  be  perpendicular 
respectively  to  those  radial  lines  (O  a,  O  6)  which  meet  the  ends  of  the  line, 
a  6,  that  represents  that  same  part. 

Figs.  10  and  11  show  the  application  of  the  same  process  to  an  irregular  fig- 
ure composea  of  three  rectangles,  o,  6  and  c  The  lettering  is  the  same  as  in 
Figs.  8  and  9;  but  in  Fig.  10  it  happens  that  t'  and  p'  of  the  second  funicular 
polygon  fall  upon  the  game  point. 


B^igr.  lO 


Kig.  11 


If  the  centers  of  gravity  of  the  several  bodies,  or  of  the  several  parts  of  the 
body,  etc.,  are  in  more  than  one  plane,  we  must  find  their  projections  upon 
certain  planes,  and  apply  the  process  to  those  projections. 


CENTER  OP   GRAVITY. 


391 


Special  Rules. 
132.    Special  Rules,  derived  from  the  general  rules,  (1)  to  (6), 

Liines. 
(7).  Straiglit  line.    G  is  in  the  line,  and  at  the  middle  of  its  length. 
(8).  Circular  arc,*  aob,  Figs.  12  and  13  (center  of  circle  at  c).    G  is  in  th« 
line  CO  joining  the  center  of  the  circle  with  the  middle  of  the  arc,  and 


c G  =■  radius  ac  X 


chord  a  b 
length  of  arc  aob  ' 


Fig.  IS 
<9a).  If  the  arc  is  a  semi-olrcle,* 


cG 


radius  ac  X 


^ias.  13 


radius  ac  X  0.6366. 


=  .15 

« 

t<  . 

'«    =.6ft3  so 

==.20 

« 

"  > 

«    =.660  so 

=  .25 

«t 

u  . 

«    =.657  «o 

(8  6).  Approximate  rules  for  distance  8  G,  Fig.  12,  from  chord  to  center  of 
gravity. 

If  rise  so  =.  .01  chord  a  b;  sG=  .6fi6  so    If  riae  so  =-  .30  chord  ab;  sQ=-  .653  s  o 

•■  .10      "         "  ;    «    =  .665  so      "     "      "  =  .35      "         "  ;    "    =-  .649  8  O 

«<  «  =.40  "  ♦*;  *•  =-.645  8  0 
«  «  =,45  «  «;  «•  =..64ls  a 
«      «  =.50     •'         '*;  «    =- .637*0 

(9).  Triangle,  a  6  c,  Fig.  14.  The  center  of 
gravity,  G,  of  its  three  sides*  is  the  center  of  the 
circle  inscribed  by  a  triangle,  d  e/,  wl;>ose  corners 
are  in  the  centers  of  the  sides  of  the  given  triangle. 

(lOj.  Paralleloi^ram  (square,  rectangle, 
rhombus  of  rhomboid).  The  center  of  gravity 
of  the  four  sides*  is  at  the  intersection  of  the 
liiagonals. 

(11).  Circle,  ellipse,  or  regular  polygon. 

The  center  of  gravity  of  the  outline  or  circumfer- 
ence* IS  the  center  of  the  figure. 

(13).  Reisular  prism,  right  or  oblique,  and  ri{^lit  regular  pyramid, 

or  frustum.    The  center  of  gravity  of  the  edges*  is  thf^  center  of  the  axis. 
in  the  »rtsw,  the  position  of  G  is  not  affected  by  either  including  or  excludinc 
the  sides  of  bolh  of  the  polygons  forming  the  ends. 
(12  a).  Cycloid.*    tiee  p.  194. 


Surfaces. 
A.  Plane  surfitoes. 

We  now  treat  of  the  centers  of  gravity  of  plane  surfaces,  which  may  b© 
regarded  as  infinitely  thin  flat  bodies.  The  rules  for  surfaces  may  be  used 
also  for  actual  flat  bodies,  in  which,  however,  the  center  of  gravity  (s  in  the 
middle  of  the  thickness,  immediately  under  the  points  found  by  the  rules. 

(13)  Parallelogram  (square,  rectangle,  rhombus  or  rhomboid),  circle, 
ellipse  or  regular  polygon.  G  is  the  center  of  the  figure;  or  the  inter- 
section of  any  two  diameters,  or  the  middle  of  any  diameter.  In  a  Parallelo- 
gram, G  is  the  intersection  of  the  two  diagonals. 

(14).  Triangle,  Fii^.  15.  G  is  at  the  intersection  of  lines  (as  a  c  and  cd) 
drawn  from  any  two  angles,  a  and  c,  to  the  centers,  e  and  d,  of  the  sides,  6« 

*  We  are  now  treating  of  lines  only;  not  of  the  surfaces  bounded  by  them, 
For  surfaces,  see  rules  (13),  etc.,  eta. 


392 


FORCE  IN   RIGID   BODIES. 


and  a  6,  respectively  opposite  to  said  angles.  Such  lines  are  called  "  medial  linea." 
•C*  "•  /^  o«;     rf<*  =»  J^  cd;    /a  '^libf  (J  being  the  middle  of  ae), 
b 

Figi  VIS 


a'      h'G'  C 

(14o),  Fig.  15.  Or,  on  either  one  of  the  sides  (as  a  ft),  meeting  at  any  angle,  a, 
make  ao  ^  }4  ab.  Draw  op  parallel  to  the  other  side,  a c.  Then  o  G  «=  i^ 
op,  and  G  is  at  the  intersection  of  op  with  any  medial  line,  as  ae,  etc. 

(14:6).  Fig.  16.    If  aa\  bb\  cc^  and  GG'  are  the  distances  of  the  three  cor- 
ners and  of  G  from  any  straight  line  or  plane  a'  c' ;  then 
CtG'         —  K  {aa'  +  66'  +  cc'). 

This  gives  us  the  position  of  the  line  of  gravity  G  G''.  In  the  same  wa3r  we 
find  the  distance  G  G''  of  G  from  any  second  line  or  plane,  6"  c".  This  gives 
us  the  position  of  a  second  line  of  gravity  G  G'.  G  is  at  the  intersecton  of 
G  G'  and  G  G''. 

(14:  c).  Fig.  17.  The  distance  Gn  of  Gin  any  direction  from  any  side,  as  a  « 
(extended  if  necessary)  is  =-  ^  the  distance  w'  6  measured  in  a  parallel  direo* 
tion  from  the  same  side  to  the  opposite  angle,  6. 


It  follows  firom  this  that  the  shortest  distance,  G  o,  of  G  from  any  side  (as 
oc)  is  —  3^  the  shortest  distance,  o'  6,  from  the  same  side  to  its  opposite  angle  b. 

It  foHows  also  that  p  G  =  3^  p  6,  as  in  Rule  (14). 

(15).  Trapeziiun  or  trapexold,  Fig.  18.  For  trapezoids,  see  also  Rule 
(16).  Draw  the  two  diagonals,  o  c  and  bd.  Divide  either  of  them,  as  a  c,  into 
two  equal  parts,  am  a.nd  cm.  From  6,  on  b  d,  lay  off  6  »  —  d  s  (or  from  d  lay  ofl 
dn  —  0  8).    Join  mn.    G  is  in  m n,  and 

mG         =^%mn, 
(G  is  the  center  of  gravity  of  the  triangle  a  e  n). 


^ig- 18 


Fig.  19 


(15  a).  Or,  Fig.  19,  find  first  the  centers  of  gravity,  m  and  n,  of  the  two  tri- 
angles, cb  d  and  a  6  d,  into  which  the  trapezium  is  divjded  by  one  of  its  diago- 
nals, b  d.  Join  m  n.  Then  find  the  centers  of  gravity,  o  and  p,  of  the  two 
triangles,  d  a  c  and  b  a  c,  into  which  the  trapezium  is  divided  by  its  other 
diagonal,  a  c.    Join  op.    Then  G  is  the  intersection  of  m  w  and  op. 


CENTER   OF   GRAVITY. 


393 


(16),  Trapezoid  only 9  Fig.  20.  O  is  in  the  line  ef  joining  the  centers, 
«and/,  of  the  two  parallel  sides,  a  6  and  cd.  To  find  its  position  in  said  line, 
prolong  either  parallel  si(Je,  as  a  6,  in  either  direction,  say  toward  i;  and  make 
hi  equal  to  the  opposite  side,  cd.  Then  prolong  said  opposite  side,  cd,  in  the 
oi)posite  direction,  making  dh'='  ah.  Join  h i.  Then  Gr  is  the  intersection  of 
hi&ndef.    Or 


fO 


ef         2ab  +  cd 
3     ^     06  -f  cd 


^  en   ^^  2ab  +  ed 

or    oG       __x___. 


h   e  a 


n.'=.^^h 


(ID.  Regnlar  polygon.    G  is  the  center  of  the  figure. 

(11' a).  Irregular  polygon.  If  the  polygon  be  divided  into  any  two 
portions,  as  by  any  diagonal,  G  must  be  in  the  line  (of  gravity)  joining  the 
centers  of  gravity  of  those  two  portions.  If  we  again  divide  the  whole  polygon 
into  two  other  parts  by  another  diagonal,  and  join  the  centers  of  gravity  of 
those  two  parts,  Cr  is  the  intersection  of  the  two  lines  of  gravity. 

(176).  Or  we  may  divide  the  polygon  into  triangles,  find  the  center  of 
gravity  of  each  triangle,  by  Rules  (14),  etc.,  and  then  find  O  by  general  Rul« 
5).  (2)  or  (6). 


inig..si 


(IS).  Cirenlar  sector,  aobCy  Fig.  21.    (Center  of  circle  at  c). 

^^  2  chord  a  5  radius^  x  chord 

c  w  —         ■—  radius  ac  X  r-         —  »  ^^ — 

3  arc  a  o  6  3  X  area 


For  length  of  arc,  see  p.  141. 
(18a).  If  the  sector  is  a  sextant^ 

eO        —  radius  X  —        -»  radius  X  0.6386, 
ft 

(18&).  If  the  sector  is  a  qnadrant.  Fig.  22, 

cO  —         -^  radius  X  ^    -~         —  radius  X  0.6002. 


4  •vX  2 

—  radius  X  

3  IT 


c«  —        a:  G  —        -i  radius  X  — 
3  IT 


(18  e).  If  the  sector  is  a  semi-circley 

^  4       ,.  1 

c  »         ""  -;r  radius  X  

3  T 


—  radius  X  0.4244 


^  (approximately)  radius  X  -^  • 


394 


FORCE   IN   RIGID   BODIES. 


(19).  ClrculMr  se^m«iit,  a  o  6  s,  Fig.  23.    (Center  of  bircle  at  c). 
cube  of  chord  a  b 


cG  — 


12  X  area  of  segment 


(19  a).  If  the  segment  is  a  semi-circle, 

4  I 

«C>        —  -r  radius  X =-  radius  X  0.4244 

o  jr 

14 


•  (approximately)  radius  X  -—• 


Fig.  S3 


1»0).  Cycloid,  Fig.  24.    (Vertex  at  v). 


(ai).  Parabola,  a  6  c,  Fig.  25.  «c      IPig.  S5 
is  the  base;    ax  and  ca;,  ordinates; 
and  the  height  or  axis,  6  a:,  an  ab- 
scissa.  Center  of  gravity  at  G,  in  the 
axis  X  6,  and 

2 
X  G        —  -r  a?  5. 
5 

(/SI  a).  Semi-parabola,  a  6  a;  or 

ebx.    Center  of  gravity  at  G^,  and 


xQ 


xb; 


GCy         -^-^aa 


(aa).  £:ilipse,  mnop.  Fig.  26.     The  center  of  gravity,  c,  of  the  whote 
•Uipse  is  at  the  center  of  the  figure. 


JFig.  S6 


G  is  the  center  of  gravity  of  the  quarter  ellipse,  one. 
G'    «  **         "  "       «   half  "       nop, 

Q//    «  «  a  «         M       u 


p. 

mTio, 


4  1  14 

C  G'  a.  —  o  c  X  "=  0.4244  oe  '-^  (approximately)  -—-  o  c. 

3  IT  <*5 

C&'-^G/G-^^en  X  -^^ 0.4244  cw  —  (approximately)  -^  0«k 

3  IT  o3 


CENTER   OF   GRAVITY, 


395 


(83).  Any  plane  figure.  Draw  the  figure  to  scale  on  stout  card-board. 
Cut  it  out  and  balance  it  in  two  or  more  positions  over  the  edge  of  a  table  or 
on  a  knife-edge;  and  mark  on  it  the  several  positions  of  the  supporting  edge. 
Where  these  intersect  is  the  center  of  gravity.  Considerable  care  is  of  course 
necessary  to  obtain  very  close  results  by  this  method.  Before  balancing  the 
card,  its  upper  edges  should  be  marked  off  into  small  equal  spaces.  Otherwise 
it  will  be  difficult  to  locate  the  positions  of  the  supporting  edge.  The  paper 
on  which  the  figure  is  prepared  must  of  course  be  so  stiff  that  the  figure  will 
not  bend  when  balanced  on  the  knife-edge.    See  Rule  (5). 

B.    SSarfaees  of  ISolids.* 

(34).  Curved  surface  *  of  spliere  or  splieroid  (ellipsoid).  G  is  the  center 
of  the  figure. 

(S5).  Curved  surface*  of  any  splierical  zone,  as  a  splierical  segmemtf 
lienilspliere .  etc.,  Figs.  27.    G  is  the  center  of  the  axis  or  height,  ao.f 

In  the  liemispliere,       oO       =  }i  radius.f 


Fig.  sr 


(86).  Right  or  oblique  prism,  whose  ends  are  either  regular  figures 
or  parallelograms  (this  includes  the  cnbe  and  other  parallelopipeds) ;  and 

right  or  oblique  cylinder  (circular  or  elliptic).  Surface*  (either  including 
both  or  excluding  both  of  the  two  parallel  ends).  G  is  the  center  of  the  axis, 
or  line  joining  the  centers  of  the  two  parallel  ends. 

(87).  Curved  surface*!  of  right  cone,  Fig.  28  (circular  or  elliptic),  or  slanting 
surfaces*!  of  right  regular  pyramid.  Fig.  29.   G  is  in  the  axis  oa  (the  line 
joining  the  apex  and  the  center  of  the  base);  and 
oOt       -=  %  oa. 

In  an  oblique  cone  or  pyramid,  the  perpendicular  distance  of  G*  from  the 
base  is  one-third  of  the  perpendicular  height,  as  in  the  right  cone  and  pyramid; 
but  does  not  lie  in  the  axis. 


(88).     Frustums  with  top  and  base  parallel,  Figs.  30  and  31        Curved  sur 
face*t  of  frustum  of  right  cone  (circular  or  elliptic);   or  slanting  surfaces *t  of- 
frustum   of  right   regular  pyramid.     G  is  in  the  axis'o  a  (the  line  joining  the 
centers  of  the  two  parallel  ends) ;  and  .  j  b 

o  G  =  -L  0  a  X  ^^^^""^^Q^Q^^Q  of  o  -f  2  circumference  of  a. 
3  circumference  of  o  -f  circumference  of  a. 

*  We  treat  now  of  the  surfaces  of  solids,  not  of  their  contents  or  volumes  or 
weights.    For  these,  see  Rules  (29),  etc. 
t  If  the  top  or  1>ase  is  to  be  included,  see  Rules  (1)  and  (2), 


396 


FORCE   IN   RIGID   BODIES. 


In  the  conic  frustum,  Fig.  30,  we  may  use  the  radii  of  the  two  ends;  and 
in  the  frustum  of  a  regular  pyramid.  Fig.  31,  any  side  of  each  end  (as 
b  c  and  d  e)  instead  of  the  circumferences. 


T^is.30  ITig.  31 

Solids. 

In  the  following  rules  for  center  of  gravity  of  solids,  the  solid  is  supposed 
to  be  homogeneous;  i.  e.,  of  uniform  density  throughout;  so  that  the  center  of 
gravity  is  the  center  of  magnitude  or  of  volume. 

(39).  Spl&ere  and  spheroid  (ellipsoid).    G  is  the  center  of  the  body. 

(30).  Hemisphere,  Fig.  32.  (Center  of  sphere  at  c).  Height  c  T  =  radius 
cb.    G  is  in  the  axis,  c  T,  and 


cG       =■ 


cT 


■■  — -  radius  eb. 


(31>  Splierical  sector,  Fig.  33.    (Center  of  sphere  at  c). 

c G     =  —  (radius  ch ^). 


(33).  Spherical  segment,  a  m  6  T,  Fig.  34.    Center  of  sphere  at  c. 
of  base  aim.    Rise  or  height  of  segment  =  wT  —  ^.    G  is  in  the  axis  wi 
3        (2  radius  c  b  of  sphere  —  height  h)^ 
^      '^^  4  3  radius  c  5  of  sphere  —  height  h 

_  height,  h        2  (radius  m  6  of  base)^  +  (height,  h)^ 


Center 
T;  and 


height,  h 
■"  4 


3  (radius  m  6  of  base)^  +  (height,  h)^ 

4  X  radius  c  6  of  sphere  —  height,  h 
3  X  radius  c  6  of  sphere  ~  height,  k  ' 


(33).  Spherical  zoue.  Fig.  35. 

ot         2  (radius  o 6  of  base)^  -f  4  (radius  t c  of  top)2  +  (height  o  <)» 
2^3  (radius  o  b  of  base)2  +  3  (radius  t  c  of  top)2  -|-  (height  o  t)^  ' 


oG  = 


(34).  Prism,  regular  or  irregular,  right  or  oblique  (including  the  cube 
and  other  parallelopipeds),  and  cylinder,  circular  or  elliptic,  etc.,  regular 
or  irregular,  right  or  oblique.  G  is  the  center  of  the  axis  joining  the  centers 
of  gravity  of  the  two  ends. 


CENTER   OF    GRAVITY. 


397 


(34  a).  A  flat  body,  such  as  an  iron  plate,  etc,  may  be  treated  as  a  very- 
short  cylinder  or  prism.    See  (34) 

(35).  Ungula  of  a  cylinder,  circular,  or  elliptic  (provided  one  of  the  axes 
©f  the  ellipse  coincides  with  the  oblique  cutting  plane);  right  or  oblique. 
Figs.  36  and  37. 

C 


I«et  O  T  be  the  axis  (joining  the  centers  of  gravity  of  the  ends),  and  X  N  • 
line  drawn  parallel  to  the  axis,  in  the  plane,  A  B  C  D,  passing  through  the 
•xis  and  through  the  uppermost  and  lowermost  points  C  and  D  of  the  oblique  " 
cutting  pUne.    Then  the  position  of  G  in  the  plane  A  B  C  D,  is  found  thua: 


OX 


OB 


XG 


2 


•^  2h  +  a  * 

=  l(2h  +  a+i_J^). 
4  ^  2h  +»/ 


(35  a).  Figs  38  and  39.    If  the  oblique  plane  G  D  meets  the  base,  A  6,  at  A,  so 
that  *»»0,  while  C  D  remains  a  complete  ellipse  or  circle,  this  becomes 


OX       - 


OB 


XG 


XN 
2 


-•A* 


(36).  Cone,  Figs.  40  and  41,  circular, 
elliptic,  etc.,  right  or  oblique;  or  pyra- 
mid, regular  or  irregular,  right  or 
oblique.  The  center  of  gravity  G  is  in 
the  axis  O  T,  drawn  from  the  apex,  or 
top,  T,  to  the  center  of  gravity  O  of  the 
base;  and 

OT 
OG       =^. 


T^ig.  40 


(37).  Frustum  of  a 'cone.  Figs.  42  and  43,  circular  or  elliptic,  right  or 
oblique ;  or  of  a  pyramid,  regular  or  irregular,  right  or  oblique ;  provided  the 
two  ends  A  B  and  C  D  are  parallel. 

Call  the  area  of  the  large  end  A,  and  that  of  the  small  end  a  ;  and  let  h  be 
the  height  O  Z  of  th'3  frustum,  measured  along  its  axis.    Then 


398 


FORCE  IN   RIGID   BODIES, 


G  is  in  the  axis  O  Z,  which  joins  the  centers  of  gravity  O  and  Z  of  the  t 
•nds;  and  its  distance  from  the  base,  A  B,  measured  along  the  axis,  is 


OG     -      iL 
4 


A    +_2   V^Ai 


+    3  a 


A    + 


/a« 


(37  a).  In  a  frustum  of  a  circular  cone,  right  or  oblique,  with  parallel 
ends,  this  becomes 

2L     X    R^    +    2  R  r    +    3  r8 


OG     « 


R2    +       Rr    + 


r2  ■ 


where  R  and  r  are  the  radii  of  the  large  and  small  ends  of  the  frustum 
respectively. 

(38).  Figs.  44  and  45.  Frustum,  A  B  C  D,  of  a  cone,  circular,  elliptic, 
etc.,  right  or  oblique;  or  of  a  pyramid,  regular  or  irregular,  right  or  oblique; 
whether  the  ends  are  parallel  or  not.  By  rule  (36)  find  the  center  of  gravity 
N  of  the  entire  pyramid  (or  cone,  as  the  case  may  be)  A  B  T,  of  which  the 
frustum  forms  the  lower  part;  and  the  center  of  gravity  S  of  the  smaller 
pyramid  or  cone  D  C  T  (==  entire  pyramid  or  cone,  minus  the  frustum).  Also 
find  the  volume  of  each:  thus, 


Volume  of  pyramid  or  cone   =.   area  of  base  X  perpendicular  height  ^ 

•ad 

Volume  of  volume  of  volume  ^f 

the  frustum  =    entire  pyramid  —      smaller 
A  B  C  D               or  cone,  A  B  T  one,  DOT 

Then  the  center  of  gravity  G  of  the  frustum  A  B  C  D  is  in  the  extension  oC 
the  line  S  N ;  and  ...  .,  ^«« 

volume  of  smaller  pyramid  or  cone,  DOT 
NG    =.    SN    X  volume  of  frustum,  A  B  C  D  ' 

(39).   Paraboloid.    G  is  in  the  axis,  and  at  one-third  of  its  length  £ro«i 
the  base. 


CENTER   OF   PRESSURE.  399 

LINE  OF  PRESSURE. 
CENTER  OF  FORCE  OR  OF  PRESSURE. 

Position  of  Resultant. 

133.  In  Tf  ^  133  to  154  we  discuss  the  position  of  the  resultant,  or  line  of 
pressure,  of  a  system  of  parallel  forces  acting  against  a  surface.  For  the 
changes  in  that  position  within  a  structure,  due  to  the  action  of  non-parallel 
forces,  see  Arches,  Dams,  etc.,  1[t  251,  etc. 

134.  In  a  system  of  parallel  forces,  acting  against  a  surface,  the  line  of 
pressure,  or  pressure  line,  is  the  position  of  the  resultant  of  the  forces;  and 
the  center  of  force  or  center  of  pressure  is  the  point  where  the  pressure  line 
meets  that  surface  against  which  the  forces  act. 

135.  If  the  lengths  of  the  lines  which  represent  the  forces  be  taken  as  rep- 
resenting weights,  to  scale,  theii  the  position  of  the  pressure  line  is  the  line  of 
gravity  (see  (5),  ^  131)  corresponding  to  those  weights. 

136.  Thus,  in  Fig.  55  (a),  1  117,  if  the  three  forces,  a,  b  and  c,  be  taken 
as  weights,  represented  to  scale  by  the  arrows,  a,  b  and  c,  respectively,  then 
the  resultant  R  of  the  three  forces  occupies  the  position  of  the  line  of  gravity 
of  the  three  weights. 

137.  Again,  in  a  mass  of  sand,  Fig.  61,*  with  an  irregular  surface,  we  may 


Fig.  61. 

suppose  the  mass  to  consist  of  innumerable  vertical  columns  of  sand,  of 
different  heights,  and  exerting  pressures  proportional  to  those  heights.  Here, 
also,  the  pressure  line  is  the  vertical  line  of  gravity  of  the  mass,  and  the  cen- 
ter of  pressure  against  the  base  of  the  containing  box  is  the  point  where  said 
pressure  line  meets  that  base. 

138.  Although  we  are  usually  concerned  with  forces  acting  against  sur- 
faces, so  that  the  lines  representing  the  forces  form  a  solid  and  not  merely  a 
surface,  yet,  ii.  a  majority  of  the  cases  which  occur  in  civil  engineering,  we 
may,  for  con/cnience,  regard  the  forces  as  concentrated  in  a  single  plane, 
and  therefore  as  acting  against  a  mere  line. 

139.  Thus,  in  the  case  of  an  arch,  pressing  against  its  skewback,  the  pres- 
sure is  ordinarily  distributed  over  all  or  a  considerable  part  of  the  bearing 
surface  of  the  skewback;  but  we  may,  for  convenience,  regard  it  as  concen- 
trated in  a  single  plane,  midway  between,  and  parallel  to,  the  two  faces  of 
the  arch. 

140.  Similarly,  in  the  case  of  the  water  pressure  against  the  back  of  a  dam 
(or  against  a  small  strip  of  the  back,  extending  from  the  water  surface  to  the 
bottom,  or  to  any  other  depth),  the  water,  of  course,  presses  upon  the  entire 
surface  of  such  strip;  but  we  may,  for  convenience,  regard  the  pressure  as 
concentrated  in  a  vertical  plane  normal  to  the  back  of  the  dam  and  meeting 
it  in  the  vertical  axis  of  the  assumed  strip. 

141.  We  have  just  seen  (tH  138  to  140)  that,  when  a  system  of  parallel 
pressures  acts  against  a  surface,  they  may  often  be  assumed  to  act,  in  one 
plane,  against  a  single  line — viz.,  the  intersection  of  that  plane  with  the  sur- 
face. It  also  frequently  happens  that  such  forces  are  so  distributed  along 
that  line  that  the  lines  representing  the  forces  are  either  ojf  equal  length  or  of 
lengths  increasing  uniformly  from  one  end  of  the  line  to  the  other. 

♦Following  Fig.  60,  of  Parallel  Forces,  ^  124.  Figs.  1  to  45,  illustrating 
Center  of  Gravity,  are  numbered  independently  of  the  rest  of  the  series  ol 
figures  relating  to  Statics. 


400 


STATICS. 


143.  Thus,  in  the  case  of  water  resting  upon  a  horizontal  surface,  Fig.  62, 
the  pressure  is  uniformly  distributed,  and  the  diagram,  Fig.  (6),  representing 
the  pressures,  is  a  rectangle  bounded  by  a  horizontal  line,  and  its  center  of 
gravity,  G,  is  at  the  center  of  the  figure.  Hence,  the  center  of  pressure,  c,  is 
at  the  center  of  the  line  a  b,  or  I. 

Here  the  unit  pressure,  p,  is  uniform,  and  R  =  p  Z. 


JB 


(a) 


mi; 


]p(b) 


I 

Figr.  62. 


Fi^.  63. 


143.  But  when  the  water  presses  horizontally  against  a  vertical  or  in- 
clined surface,  a  b,  Fig.  63,  the  unit  pressure  increases  uniformly  from  zero, 
at  the  water  surface,  b,  to  a  max  at  the  bottom,  a  ;  and  the  hor  pressures 
are  represented,  in  Fig.  (6),  by  the  ordinates  of  the  triangle  b'  a'  d.  Since 
the  resultant  passes  through  the  center  of  gravity,  G,  of  the  triangle,  the 
center  of  pressure,  c,  is  at  such  a  depth  that  c  a  =  i  a  fe,  and  c'  a'  =  ^  h. 
See  Rule  (14  c)  under  Center  of  Gravity . 

Here  the  mean  horizontal  unit  pressure,  p,  is  half  the  maximum  horizontal 
pressure  at  a,  and  the  total  horizontal  pressure  ia  =  p  h. 


(a) 


Figr.  64. 


144.  Again,  if  we  consider  only  the  water  pressures  against  a  certain  part, 
a  b,  Fig.  64,  of  the  depth  of  the  back  of  a  dam,  the  diagram.  Fig.  (b),  repre- 
senting the  horizontal  unit  pressures,  becomes  a  trapezoid,  composed  of  a 
rectangle  b'  d,  and  a  triangle  e  d  f,  with  their  centers  of  gravity  at  g  and  g' 
respectively ;  and  the  center  of  pressure,  c,  on  a  b,  is  opposite  their  common 
center  of  gravity  (center  of  gravity  of  trapezoid),  G.  If  h  be  the  vertical 
depth  of  the  portion  considered,  then 


-  =  4 


See  Rule  (16)  under  Center  of  Gravity, 
under  Hydrostatics. 


2¥  e  -\-  a'  f 
b'  e  -\-  a'  f 

See  also  Center  of  Pressure, 


Distribution  of  Pre'ssmre. 
145.  Conversely,  if  two  surfaces,  as  those  of  a  masonry  joint,  are  in  such 
contact  that  the  pressure  is,  or  may  be  regarded  as,  regularly  distributed, 
and  if  the  position  of  the  resultant  is  known,  the  rectilinear  figure,  represent- 
ing the  distribution  of  pressure,  may  be  drawn  by  means  of  the  principles 
just  stated. 


DISTRIBUTION    OF   PRESSURE. 


401 


I 


146.  In  Figs.  65  to  68  inclusive,  let 

o    =  the  center  of  the  joint  a  b  between  the  two  surfaces; 
R  =  the  total  pressure  =  resultant  of  all  the  pressures ; 
c    =  point  of  application  of  resultant,  R ; 
ab  =  the  length  of  the  joint ; 
■  o  c  =  the  distance  of  the  center  of  pressure  from  the  center  of  the 
joint; 

!  —  —  X  =  a  c  =  distance  of  center  of  pressure  from  nearest  end  of 

joint; 

R 


the  mean  unit  pressure  = 


r 


pa  =  the  maximum  unit  pressure ; 
pb  =  the  minimum  unit  pressure. 
1^  147  to  154  apply  equally  whether  the  surface  is  horizontal,  vertical  or 
inclined,  and  whether  the  forces  are  normal  or  inclined  to  it. 

147.  If  X  is  not  greater  than  —»  or,  in  any  case,  if  the  joint  is  capable  of 
o  ^  ■ 

Buetaining  tension,  as  well  as  compression,  we  have : 

6  X 
Maximum  unit  pressure  =  Pa  =  P  (1  +  ~z~^' 

6  X, 


Minimum  unit  pressure  =  Pb  =  P  (1  • 
I 


). 


If  a;  exceeds  — ,  and  if  the  joint  is  incapable  of  resisting  tension,  see  tH 
151,  152,  154. 


Fi^.  65. 


Tig.  66. 


148.  Demonstration.  In  Fig.  66,  where  the  parallelogram  a'  d  repre- 
sents the  total  pressure  R  as  it  would  be  if  uniformly  distributed  along  I,  we 
see  that  the  moment  of  R,  about  o,  which  changes  the  parallelogram  a'  d  into 
the  trapezoid  a'  ¥  n  m,  is  equivalent  to  a  couple  (see  Couples,  ^  155,  etc.) 
composed  of  two  forces — viz.,  a  pressure,  /  (not  shown)  distributed  over  o  a 
and  represented  by  the  shaded  triangle  on  the  left,  and  a  tension,  — /,  or 
diminution  of  pressure,  distributed  over  o  b  and  represented  by  the  triangle 
on  the  right.  The  forces,  /  and  — /,  act  through  the  centers  of  gravity  of 
these  two  triangles  respectively;  and  the  distance  of  each  of  these  centers 
of  gravity  from  the-center,  o,  of  the  joint,  measured  parallel  to  the  joint, 

2       I 
is  =  :^  .  ^.    Hence  the  distance  between  the  two  centers  of  gravity,  meas- 

26 


402 


STATICS. 


2  I 
ured  parallel  to  the  joint,  is  =  — .     Let  x  be  the  eccentricity,  c  o,  of  R, 

measured  along  the  joint,  and  let  Ar  and  Ac  (not  shown)  be  the  lever  arms 
of  R  and  of  the  couple,  respectively,  about  the  center,  o,  of  the  joint. 
Then,  since  R  is  parallel  to  /  and  — /,  Ae  to  Ac,  and  x  to  I,  we  have: 

Ar   :  Ac  =  X  :  -5-. 

2  I 
If  R  is  normal  to  the  joint,  we  have:  Ab  =  x;  and  Ac  =  — . 


Figr*  65  (repeated). 

moment  of  R 


Now 
Hence, 


/  = 


arm  of  couple 
R^ 

3 


Fig.  66  (repeated). 

R.Ar 

Ac    * 


R    3a;  Zx 

=  7--2-  =  PT-- 


The  mean  additional  pressure  on  o  a  (or  mean  tension  on  o  6)  is  =  ^-r 

7  • 
and  the  corresponding  maximum  additional  pressure  is 

^    f  4   ,       4       3x  6a: 

■KT  t    J  ,       6  X  ,,    ,    6  a:. 

Now  Pa   =P  +  /m  =  p4-p-y-=P(l+  ^) 


and  Pb 


-/m 


V  ■ 


6  X 


M  6  X, 

p(i-7-). 


149.  If,  as  in  Fig.  65,  the  center  of  pressure,  c,  is  at  the  center,  o,  of  the 
surface,  we  have  x  =  o  c  =  zero,  and  the  pressure,  R,  is  uniformly  distrib- 
uted over  the  surface, 

150.  "  The  Middle  Third."     If,  as  in  Fig.  67,  x  =  i;—i.  e.,  if  the  re- 

o 
sultant,  R,  of  all  the  forces,  meets  the  surface  at  the  edge  of  the  middle  third 
of  that  surface,  then  pa  =  2  p;  and  Pb  =  0.     See  ^^  143  and  148. 

151.  When,  as  in  Fig.  68  (a),  x  exceeds  — , — i.  e.,  when  the  center  of  pres- 

0 
sure,  c,  falls  beyond  the  middle  third  of  the  surface  of  pressure,  a  portion, 
s  b,  of  the  surface,   is  in  tension,   the  maximum  tension,   Pb,  Fig.  68  (6), 
6x,  ^  .  ^^    ,    6  X. 


being  =  p  (1 —)  as  above;  maximum  pressure  =  p  (1  - 


- ) ,  and  total 


pressure  on  a  s  =     — " —    =  R  plus  the  tension  in  s  b  ;  but  if,  as  usually 


DISTRIBUTION  OP  PRESSURE. 


403 


happens  in  masonry,  the  surfaces  are  incapable  of  resisting  tension,  the  total 
pressure,  R,  is  simply  concentrated  upon  a  portion  a  v.  Fig.  68  (c),  of  the  sur- 
face, o  V  being  =  3  y. 

Then,  mean  unit  pressure  on  a  v  ==  ^ —  =  ^-~  \ 

p»  =  23^  =  2  r^- 


Fig.  B7. 


Figr.  68. 


153.  Hence,  in  a  joint  incapable  of  sustaining  tension,  Fig.  68  (c),  if  pa  =" 

2  — -  is  the  maximum  permissible  unit  stress,  the  distance  a  c,  from  the  cen* 

ter  of  pressure,  c,  to  the  nearest  end  of  the  joint,  must  not  be  less  than  y  "^ 
2R 

3  Pa' 

153.  If  the  joint  can  j-esist  tension.  Fig.  68  (6),  we  substitute,  in  the  equa* 

6  X  I 

tion,  Pa  =  P  (1  +  -y),  the  value  of  a?  =  -^  —  V,  and,  solving  for  y,  we  have 

2  ,       Pa  Z 
^==■3^-- 6^- 

154.  The  influence  diagrams,  Fig.  69  (see  %^  339,  etc.,  and  Trusses, 
^1^  79,  etc.),  show  the  changes  in  the  maximum  and  minimum  unit  pres- 
sures, Pa  and  Pb,  as  the  center  of  pressure,  c,  recedes  from  the  center,  o,  of 
the  joint.  The  diaecrams  are  constructed  for  a  mean  unit  pressurp,  p,  of  1. 
If  the  surfaces  of  the  joint  are  capable  of  sustaining  tension,  every  part  of 
the  joint  always  sustains  either  pressure  or  tension;  and  (see  dotted  lines. 


404 


STATICS. 


6  X 
Fig.  69)  the  maximum  unit  pressure,  Pa  =  P  (1  +  -7  ).  see  I  146,  increases 


4R 


proportionally  with  x;  becoming  =  4  p  =  — ,—  when  c  reaches  the  end,  a, 


of  the  joint,  and  y  =  — . 


The  maximum  tension,  2>b.  is  then  =  2  p  = 


2R 

I 


But  if  the  surfaces  are  incapable  of  sustaining  tension  (see  solid  lines,  Fig. 

69),  the  increase  of  Pa  is  proportional  to  x  only  so  long  as  a;  <  — ; — i.  e.,  so 

long  as  the  resultant  of  all  the  pressures  falls  within  the  middle  third  of  the 
base  a  b.  When  that  limit  is  exceeded,  the  maximum  unit  pressure,  Paf 
begins  to  increase  more  rapidly  than  does  the  distance,  x,  of  c,  from  the 
center,  o,  of  the  joint,  the  diagram  becoming  a  rectangular  hyperbola;  so 
that,  if  the  resultant  could  be  actually  applied  at  the  very  edge  of  the 
joint,  the  unit  pressure  there  would  become  infinite. 


Values  ofije  =  oc  =distance  from 
center  of  joint-to  center  of  pressure 

Fig.  69. 

COUPLES. 

155.  Couples.  Two  equal  parallel  forces,  p  and  q,  or  p'  and  q\  Fig.  70,* 
of  opposite  sense,  are  called  a  couple.  A  couple  has  no  tendency  to  move 
the  body  t  as  a  whole  in  any  straight  line.  In  other  words,  the  two  forces, 
forming  a  couple,  can  have  no  resultant.  Their  only  tendency  is  to  make  the 
body  revolve  about  its  center  of  gravity,  G,  and  in  the  plane  of  the  couple 
— i.  €.,  the  plane  in  which  the  two  forces  lie.  A  body  with  a  fixed  axis  can 
revolve  only  in  a  plane  normal  to  that  axis.  The  actual  plane  of  rotation  of 
a  free  body  depends  upon  the  distribution  of  mass  in  the  body,  and  is  not 
necessarily  the  plane  of  the  couple. 


*  Figs.  70  to  75  are  supposed  to  be  seen  in  perspective,  and  the  forces  are 
supposed  to  act  in  the  planes  shown. 
t  See  foot-note  (*),  K  1. 


COUPLES. 


405 


156.  The  moment  of  a  couple  is  equal  to  the  product  of  one  of  the 
two  forces,  v  or  q,  into  the  perpendicular  distance,  d,  between  the  two  forces. 
Or,  in  our  figures, 

moment  of  couple  =  p  .  d  =  q  .  d. 

157.  Graphic  Representation  of  Couples.  A  couple,  M  or  N,  Fig. 
70,  is  indicated,  in  amount,  in  direction  and  in  sense,  by  a  line,  L  or  L'. 
normal  to  the  plane  of  the  couple,  so  placed  that,  looking  along  it  toward 
that  plane,  the  couple  appears  positive  or  right-handed,  and  of  such  length 
as  to  represent,  by  scale,  the  moment  of  the  couple.  In  Fig.  70,  the  two 
couples  M  and  N  are  of  opposite  sense.  Hence  the  lines  L  and  W,  repre- 
senting them,  project  in  opposite  directions  from  their  respective  planes. 


Fig.  70. 


Fig.  71. 


158.  Composition  of  Couples.  If  the  lines,  L  and  L',  Fig.  71,  repre- 
sent two  couples,  in  accordance  with  ^  157,  then  the  line  R,  completing  the 
triangle,  will,  in  the  same  way,  represent  their  resultant  or  anti-resultant. 
As  drawn,  with  its  arrow  following  those  of  the  other  two  sides,  it  represents 
their  anft-resultant.  _  For  their  resultant,  the  arrow  on  R,  and  that  indicating 
the  direction  of  rotation,  must  be  reversed. 

159.  Equality  of  Couples.  Two  couples,  M  and  N,  in  the  same  plane. 
Fig.  72  or  Fig.  73,  or  in  parallel  planes.  Fig.  70,  are  equal  if  their  moments 
are  equal,  whether  or  not  the  forces  of  one  of  the  couples  be  equal  or  parallel 
to  those  of  the  other.  In  Fig.  73,  the  two  couples,  M  and  N,  are  of  like  sense; 
in  Figs.  70  and  72,  of  opposite  sense. 


Fig.  72. 


Fig.  73. 


160.  Since  a  couple  has  no  resultant  (^  155),  it  can  have  no  anti-resultant; 
i.  c,  no  single  force  can  balance  a  couple  and  thus  preserve  equilibrium. 
(But  see  ^  168.)  To  do  this  requires  an  equal  and  opposite  couple.  Thus, 
in  Fig^  72  the  couple  M  is  balanced  by  the  equal  and  opposite  couple  N.  If, 
as  in  Fig.  72,  the  two  couples  are  in  the  same  plane,  and  if  we  find  first  the 
resultant  of  either  pair  of  non-parallel  forces,  as  p  and  p',  and  then  those  of 
the  other  pair,  q  and  g',  we  shall  find  these  resultants  equal  and  opposite, 
maintaining  equilibrium. 

161.  Any  couple,  as  M,  Fig.  73,  may  be  replaced  by  any  other 
equal  couple,  N,  in  the  same  plane  or  in  a  parallel  plane,  and  of  like  sense. 

163.  If,  to  a  force,  P,  Fig.  74  (a),  we  add  a  couple,  M,  Fig.  74  (6),  in 
the  same  plane  with  the  force,  we  may  replace  the  couple,  M,  by  an  equal 
and  like  couple,  N,  Fig.  (c),  composed  of  the  forces,  — P  and  P',  each  =  P, 
placing  — P  opposite  P,  as  shown.  Then  P  and  — P  counteract  each  other,  and 
we  have  left  only  P',  equal  and  parallel  to  P ;  and,  since  Pd  =  M,  we  have 


406  STATICS. 

d  =  — .     In  other  words,  the  effect  of  the  addition  of  the  couple,  M,  Fig.  (b), 

to  the  forcfe,  P,  is  simply  to  shift  the  line  of  action  of  P,  parallel  with  itself, 
through  the  distance,  d.  If  the  couple  M  is  left-handed,  as  in  the  figure,  P 
will  be  shifted  to  the  right  (looking  in  its  own  direction),  and  vice  versa. 

163.  Conversely,  the  force,  P',  Fig.  (c),  is  equivalent  to  the  combination 
of  force  P  and  couple  M,  Fig.  (6) . 


164.  Again,  having  only  the  force  P',  Fig.  (c),  if  we  apply,  at  a  distance, 
d,  from  P',  the  two  opposite  forces,  P  and  — P,  each  equal  and  parallel  to  P', 
we  shall  thus  substitute,  for  P',  the  equal  and  parallel  force,  P,  and  a  couple 
=  Pd  =  M. 

165.  Hence,  also,  the  combination  of  the  force  P  and  the  couple  M,  Fig. 
(6),  is  equivalent  to  the  combination  of  the  force  P  and  the  couple  N,  Fig. 
(c). 

166.  If  the  moment  of  the  couple,  M,  Fig.  (6),  or  N,  Fig.  (c),  be  equal  and 
opposite  to  the  moment  of  the  force  P  about  the  center  of  gravity,  G,  of  a 

body,  we  have  d  =  p  =  distance  from  P  to  G.     In  other  words,  the  effect  of 

such  a  couple  is  to  shift  the  force,  P,  parallel  with  itself,  to  a  line  passing 
through  the  center  of  gravity,  G. 

167.  Hence,  the  effect  of  a  force,  P,  Fig.  (a),  applied  to  a  body  at  a  dis- 
tance, d,  from  its  center  of  gravity,  G,  is  equivalent  to  the  combined  effect  of 
an  equal  and  parallel  force,  P',  Fig.  (c),  applied  at  the  center  of  gravity,  and 
a  couple  (as  M,  Fig.  b)  =  Fd,  and  of  like  sense,  applied  to  any  part  of  the 
body  in  a  plane  parallel  to  P  and  P'. 

168.  It  will  be  seen  that,  although  (^  160)  no  single  force  can  balance  a 
couple  and  establish  equilibrium,  yet,  if  a  force,  P,  be  so  applied  that  its 
moment,  Fd,  about  the  center  of  gravity,  G,.  of  the  body,  is  equal  and  oppo- 
site to  the  moment  of  the  couple,  it  will  counteract  the  tendency  to  rotatioQf 
due  to  the  couple,  and  substitute  for  it  a  motion  of  translation  only. 


Fig.  75. 

169.  Thus,  in  Fig.  75,  where  the  force,  p,  acts  through  the  center  of 
gravity,  G,  of  the  body,  let  a  force,  — q,  equal  and  opposite  to  q,  be  applied 
in  the  same  line  with  it.  Then  rotation  will  be  prevented,  and  the  body  will 
move  *  under  the  action  of  p  (=  the  resultant  of  the  three  forces),  which  acts 
through  the  center  of  gravity,  G,  of  the  body.  The  rotation  will  similarly 
be  prevented  if  a  force  less  than  q  be  applied  farther  from  G  than  g  is ;  or  if  a 
force  greater  than  q  be  applied  nearer  G  than  q  is ;  provided  always  that  the 
moment  of  said  third  force,  about  G,  be  equal  and  opposite  to  that  of  the 
couple  p  q.  But  in  the  first  case  the  resultant  of  the  three  forces  (being  always 
equal  to  the  third  force)  will  be  less,  and  in  the  second  case  greater,  than  p. 

170.  If,  to  a  couple,  be  added  a  third  force,  colinear  with  one  of  the  forces 
of  the  couple,  we  have  the  case  of  two  unequal  parallel  forces  of  opposite 
sense.     See  H  112,  under  Parallel  Forces. 

*  See  foot-note  (*),  If  1. 


FKICTION.  ^  407 


FRICTION.* 

171.  When  one  rough  body  rests  upon  another,  the  projections  and  de- 
pressions, forming  the  roughnesses  of  their  surfaces  of  contact,  interlock 
to  a  greater  or  less  extent ;  and,  in  order  to  slide  one  over  the  other,  we  must 
expend  a  portion  of  the  sliding  force,  either  in  separating  the  bodies  (as  by  lift- 
ing the  upper  one)  suflSciently  to  clear  the  projections,  or  in  breaking  off  some 
of  the  projections  and  clearing  the  others. 

173.  Even  the  most  highly  polished  flat  surface,  as  xy.  Fig.  76,  is  not  (as  it 
appears  to  the  eye)  a  plane,  but  is,  in  fact,  a  more  or  less  jagged  surface,  as 
would  appear  under  a  sufficiently  powerful  microscope;  so  that  the  force,  a  b, 
instead  of  forming  the  apparent  angle,  ah  x,  with  one  smooth  surface,  x  y,  of 
application,  really  becomes  a  series  of  parallel  forces,  as  c,  d  and  e,  which  form 
other  angles  with  a  number  of  surfaces,  m  m,  nn,  etc.,  of  application,  inclined 
(often  in  different  directions)  to  the  general  surface,  x  y,  as  shown.  Among 
these  surfaces  may  be  some,  as  m  m,  at  right  angles  to  the  applied  force;  and, 
the  force  c  will  be  imparted  to  them  in  its  original  direction,  although  applied 
obliquely  to  the  apparent  surface,  x  y.  In  the  case  of  the  two  forces,  d  and  e, 
applied  to  the  surfaces,  n  n  and  s  s,  if  the  sliding  tendencies  along  the  two 
surfaces  are  equal  and  act  in  opposition  to  each  other,  the  combined  resistance 
of  the  two  surfaces,  n  n  and  s  s,  is  directly  opposite  to  the  forces,  as  would  be 
that  of  a  single  surface  at  right  angles  to  those  forces. 


173.  It  is  of  course  entirely  out  of  the  question  to  ascertain  the  exact 
resistance  of  each  such  microscopic  projection  in  any  given  case.  Instead  of 
this,  we  find  by  experiment  the  combined  resistance  which  all  of  the  projec- 
tions, in  a  given  case,  offer  to  the  sliding  force,  and  give  to  this  resistance  the 
name  of  friction. 

174.  Friction  always  tends  to  prevent  relative  motion  of  the  two  bodies 
between  which  it  acts;  i.  e.,  motion  of  one  of  the  bodies  relatively  to  the  other. 
In  doing  so,  however,  it  tends  equally  to  cause  relative  motion  f  between 
each  of  those  two  and  a  third,  or  outside  body.  Thus,  the  fric  between  a  belt 
and  the  pulley  driven  by  it  tends  to  prevent  slipping  between  them;  but  t-hus 
tends  to  make  the  belt  slip  on  the  driving  pulley,  and  sets  the  driven  pulley 
and  its  shaft  in  motion  relatively  to  the  bearing  in  which  the  shaft  revolves. 
This  motion  is  resisted  by  the  fric  between  journal  and  bearing;  and  this  fric, 
in  turn,  tends  equally  to  make  the  bearing  revolve  with  the  journal,  and  to 
make  the  belt  slip  on  the  driven  pulley. 

175.  The  fric  between  two  bodies  at  rest  relatively  to  each  other  is  called 
static  friction,  or  fric  of  rest.  That  between  two  bodies  in  relative  motion 
is  called  liinetic  friction  or  fric  of  motion. 

176.  The  ultimate  or  maximum  static  fric  between  two  bodies, 
as  U  and  L,  Fig.  77  (or  the  greatest  fric  resistance  which  they  are  capable 
of  opposing  to  any  sliding  force  when  at  rest),  is  equal  to  a  force  (as  that  of 

*  **  Friction"  (meaning  rubbing)  is  a  misnomer  in  so  far  as  it  implies  that 
rubbing  must  take  place  in  order  to  produce  the  resistance.  For  we  meet 
this  resistance,  not  only  during  rubbing,  but  also  before  motion  (or  rubbing) 
takes  place.     * '  Resistance  of  roughness  "  would  better  express  its  nature. 

t  See  foot-note  (*),  t  1. 


408  ' 


STATICS. 


the  wt  F)  which  is  just  upon  the  point  of  making  TJ  begin  to  slide  upon  L.* 
Thus  fric,  like  other  forces,  may  be  expressed  in  weights,  as  in  lbs. 

177.  A  resistce  cannot  exceed  the  force  which  it  resists. t  Therefore  if  F 
is  less  than  the  ult  static  fric  between  U  and  L,  the  frictional  resistce  actually 
exerted  by  them  is  also  less.  When  F  is  =  the  ult  fric  (and  U  is  therefore  on 
the  point  of  sliding)  the  actual  resistce  is  =  the  ult  stat  fric.  If  F  exceeds 
the  ult  stat  fric,  the  excess  gives  motion  to  U. 

178.  If,  when  a  body  is  in  motion,  all  extraneous  forces  and  resistces  are 
removed  or  kept  in  equilib,  it  moves  at  a  uniform  vel.  Hence,  if  the  force,  F, 
Fig,  77,  is  just  =  the  ultimate  kinetic  fric  between  U  and  L,  their  vel  is  uni- 
form. If  F  exceeds  this,  the  excess  accelerates  the  vel.  If  the  ult  kinetic 
fric  exceeds  F,  the  excess  retards  the  vel.  Thus  the  actual  frictional 
resistce  exerted  by  two  bodies  in  relative  motion  is  =  their  ult  kinetic 
fric  =  that  force  (as  F)  which  can  just  maintain  their  relative  vel  uniform. 

179.  Hence,  if  the  hor  surf  S  upon  which  L  rests,  could  be  made  perfectly 
frictionless,  the  pres  of  L  against  the  lug  m  (which  would  then  always  be  = 
the  actual  fric  resistce  between  U  and  L)  would  also  be  =  their  ult  fric  so  long 
as  U  continued  in  motion  over  L,  and  might  therefore  be  greater  or  less  than 
or  =  F;  but  when  U  was  at  rest  the  pres  against  m  would  be  =  F,  and  less 
than  (or  at  most  just  = )  the  ult  fric. 


Coefficient  of  Friction. 

180.  Since  no  surface  can  be  made  absolutely  smooth,  some  separation  of 
the  two  bodies  must  in  all  cases  take  place  in  order  to  clear  such  projections 
as  exist.  Hence  the  fric  is  always  more  or  less  affected  by  the  amount  of  the 
perp  pres  which  tends  to  keep  them  together. 

181.  The  ratio  of  the  ult  fric,  in  a  given  case,  to  the  perp  pres,  is  called 
the  coefficient  of  friction  for  that  case.     Or, 


Coefl&cient  of  friction  = 


ultimate  friction 


and 


Ultimate  friction  = 


perpendicular  pressure 
perp  pres  X  coeffoffric. 


Thus,  if  a  force  F,  Fig.  77,  of  10  lbs,  just  balances  the  ult  fric  between  U 
and  L,  and  if  the  wt  of  U  (the  perp  pres  in  this  case  since  the  surf  between  XJ 

and  L  is  hor)  is  50  lbs,  then  the  coeff  of  fric  between  U  and  L  is  =»        ^ 

=  0.2. 


^\ 


Fig.  77. 


Fig.  78. 


182.  The  coeff  Is  usually  expressed  decimally,  or  by  a  common 
fraction;  but  sometimes,  as  in  the  case  of  railroad  cars  and  engines,  in  lbs 
(of  fric)  per  ton  (of  perp  pres).     Or  by  the  "angle  of  fric"  in  degs  and  mins. 


*  We  here  neglect  the  fric  of  the  string  and  pulley,  and  assume  that  all  the 
force  of  the  wt  F  is  transmitted  by  the  string  to  U. 

t  If  a  resisting  force  exceeds  the  force  resisted,  the  excess  is  not  resistce, 
but  motive  force. 


ANGLE   OF   FRICTION. 


409 


183.  Angle  of  Friction.  In  Fig.  78,  let  W  =  the  weight  of  the  body, 
P  =  its  pressure  normal  to  the  plane,  and  S  =  the  component  tending  to 
slide  the  body  down  the  plane. 

When  the  angle  a  is  such  that  the  body  is  just  on  the  point  of  sliding  down 
the  plane,  it  is  called  the  angle  of  friction,  or  angle  of  repose.  The  friction 
F  and  the  sliding  force  S  are  then  equal. 

But  p  ==  p"  ==  r»  ^  coefficient  of  friction  =  tan  a.     Hence  F  =  P  tan  a 

=  W  cosin  a  .  tan  a. 

184.  Frictional  Stability.  Let  R,  Fig.  79,  be  the  resultant  of  all  the 
forces  pressing  a  body  against  a  plane,  and  N  a  normal  to  the  plane.  If  the 
angle  i  between  R  and  N  exceeds  the  angle  of  friction  (a.  Fig.  78)  between  the 
two  surfaces  in  contact,  the  body  will  slide  on  the  plane,  but  not  otherwise. 
If  i  does  not  exceed  the  angle  of  friction,  the  entire  resultant  R  will  be  im- 
parted to  the  plane  and  in  its  own  direction,  and  not  merely  its  normal  com- 
ponent V,  as  would  be  the  case  if  the  surfaces  were  frictionless. 


Fig.  79. 


Fig.  80. 


185.  To  find  the  coeff  of  kinetic  fric,  allow  one  of  the  bodies,  U, 
Fig.  80,  to  slide  down  an  inclined  plane  A  C  formed  of  the  other  one  and  hav- 
ing any  convenient  known  steepness  ACE  greater  than  the  angle  of  fric  (1[ 
183).  Note  the  vert  dist  A  E  through  which  U  descends  in  sliding  any  dist 
asAC(AE  =  ACX  sine  of  A  C  E) ;  also  its  actual  sliding  \e\  in  ft  per  sec 
on  reaching  C.  Calculate  the  vert  dist  A  D  through  which  it  would  nave  to 
descend  along  the  plane  (from  A  to  B)  to  acquire  that  vel  if  there  were  no  fric. 


AD  == 


velocity^  in  ft  per  sec 
twice  the  accel  g  of  grav 


-*)■ 


Find  DE  (=  A  E 

(E  C  =  A  C  X  cosine  ACE 


A  D),  and  the  hor  dist  E  C  corresponding  to  A  C 

i/ac2- 


A  E2;.     Then 
Coeflf  of  the  average  fric  in  sliding  from  A  to  C 


DE 

'  EC 


because,  if  we  let  A  E  represent  the  total  sliding  force  expended  (in  accelera- 
tion and  in  overcoming  the  fric),  then  A  D  represents  the  portion  of  A  E  ex- 
pended on  vel,  and  D  E  that  expended  on  fric,  and,  since  C  E  represents  the 
perpendicular  pressure  (^  183), 


DE 

EC  " 


friction 


=  coefif. 


prep  pres 

186.  Or,  find  sine  and  tangent  of  A  C  E ;  and  the  dist  A  C  (  =  time^  in 
sees  X  i  gf  *  X  sine  of  A  C  E)  through  which  U  would  slide  in  a  given  time 
if  there  were  no  fric.  Measure  the  dist  A  B  through  which  it  actually  slides  in 
that  time ;  and  find  BC  =  AC  —  AB.     Then 

coeff  of  the  average  }       +„„-nn-i^       +o„Amr\/J^ 

...       ,.  ,.        -  K  4.     -D  C  =  ^^^  D  C  E  ==  tan  A  O  E  X  -j-?^ 

fric  in  sliding  from  A  to  B  J  AC 

because 


*  g  =  about  32.2  ft  per  second  per  second. 


410  FRICTION. 

(1st)    AC:AB:BO    ::    AS:AD:DS 

'-''Z  iT^irll^Tlroe  ''  t^«  -tual  velocity  :  the    frictional    retardation 

sliding  force  employed     the  friction,  or  the  sliding 
: :  the  total  sliding  force  :    in  giving  the  actual     ;    force  required  to  balance 
velocity  the  friction. 

And,  if  A  E  is  =  the  total  sliding  force,  then  K  C  is  =  the  perpendicular  pressure, 
And  T)  "p 

=  the  coefficient  of  friction  =  tangent  of  D  0  E. 
EC 
(2nd)  Owing  to  the  similarity  of  the  two  triangles,  A  B  D  and  A  C  E,  we  have 

A  C  :  B  C  : :  A  E  :  D  E  : :  —  :  5^  : :  tangent  ACE:  tangent  D  C  E. 
EC  EC 
187.  In  1831  to  1834,  Oen'l  Artbnr  Morin*  experimented  with 
pressures  not  exceeding  about  30  lbs  per  sq  in ;  and  arrived  at  the  following 
couciusions  in  regard  to  sliding  fric  where  the  perp  pres  is  considerably  less 
than  would  be  necessary  to  abrade  the  surfs  appreciably.  These  were  for  a  long 
time  generally  regarded  as  constituting  the  tliree  fundamental  laws  of 
fric. 

Ist.  The  ult  fric  between  two  bodies  is  proportional  to  the  total  perp  force 
which  presses  them  together;  i  e, the  coeff  is  independent  of  t^e  perp 
pres  and  of  its  intensity  {pres  per  unit  of  surf ).    Hence 

2d.  For  any  given  total  perp  pres,  tlie  coeff  is  independent  of  the 
area  of  surf  in  contact. 

If  upon  a  hor  support  we  lay  a  brick,  measuring  8X4X2  ins,  first  upon  its 
long  edge  (8X2  ins)  and  then  upon  its  side  (8X4  ins),  we  double  the  area  of 
contact,  while  the  total  pres  (the  wt  of  the  brick)  remains  the  same,  and  thus  re- 
duce the  pres  per  sq  in  by  one-half.  Consequently  (the  coeff  remaining  practically 
the  same)  we  have  only  half  the  fric  per  sq  in.  But  we  have  twice  as  many  sq  ins 
of  contact,  and  therefore  the  same  total  fric. 

But  if  we  can  increase  or  diminish  the  area  of  contact  without  affecting  the  pres 
per  sq  in,  the  total  pres  will  of  course  vary  as  the  area,  and  the  total  fric  will  vary 
m  the  same  proportion,  for  the  coeff  remains  the  same.  Thus,  if  we  place  two 
similar  sheets  of  paper  between  the  leaves  of  a  book  (taking  care  not  lo  place 
both  sheets  between  the  same  two  leaves)  and  then  squeeze  the  book  in  a  letter- 
copying  press,  it  will  require  about  twice  as  much  force  to  pull  out  both  sheets 
as  to  pull  out  only  one  of  them. 

3d.  Although  the  coeff  of  static  fric  between  two  bodies  is  often  much  greater 
than  their  coeff  of  kinetic  fric ;  yet  the  coeff  of  kinetic  fric  is  inde- 
pendent of  the  vel. 

This  applies  also  (approx)  to  the/He,  and  hence  to  the  worifc  (in /compounds  etc) 
of  overcoming  fric  through  a  given  dist;  for  then  the  work  (  ==  resistce  X  dist)  is 
independent  of  the  vel.  But  in  a  given  time,  the  dist  (and  consequently  the 
work  also)  of  course  varies  as  the  vel. 

188.  (a)  Some  kinds  of  surfaces  appear  to  interlock  their  projections 
much  more  perfectly  when  at  rest  relatively  to  each  other,  than  when  in  even 
very  slow  motion  ;  and  in  somie  cases  the  degree  of  interlocking  seems  to  in- 
crease with  time  of  contact.  Hence  there  is  often  a  great  diff  in  amount  between 
fric  of  rest  and  fric  of  motion.  Thus,  Gen'l  Morin  found  that  with  oak  upon 
oak,  fibres  of  the  two  pieces  at  right  angles,  the  resistce  to  sliding  while  still  at 
rest,  and  after  being  for  "  some  time  in  contact,"  was  about  one  eighth  greater 
than  when  the  pieces  had  a  relative  vel  of  from  1  to  5  ft  per  sec. 

(b)  But  experience  shows  that  even  very  slight  jarring  suffices  to  remove  this 
diff;  and  since  all  structures,  even  the  heaviest,  are  subject  to  occasional  jarring, 
(as  a  bridge,  or  a  neighboring  building,  or  even  a  hill,  during  the  passage  of  a 
train  ;  or  a  large  factory  by  the  motion  of  its  machinery ;  or  in  numberless  cases, 
by  the  action  of  the  wind)  it  is  expedient,  in  construction,  not  to  rely  on  fric  for 
stahilUy  any  further  than  the  coeff  for  moving  fric  will  justify.  When  it  is  to  be 
regarded  as  a  resistce,  which  we  must  provide  force  for  overcoming,  it  should  be 
taken  at  considerably  more  than  our  tabular  statement. 

*  See  his  "Fundamental  Ideas  of  Mechanics",  translated  hj  Jos.  Bennett;  D.  Appl«ton  k  Ckk, 
New  York.  1860. 


FRICTION. 


Table  of  inoTingr  friction,  of  perfectly  smootb,  clean,  an4 
dry,  plane  surfaces,  chiefly  from  Morin. 


Materials  Experimented  -with. 


Coeffof 
Fric ;  or 
Propor- 
tion of 
Fric  to  the 
Pros. 


Oak  ou  oak  ;  all  the  fibers  parallel  to  the  motion 

"  "       moving  fibres  at  right  angles  to  the  others ;  and  to  the  motion. . . 

'  •  "       all  the  fibres  at  right  angles  to  the  motion 

"  "       moving  fibres  on  end ;  resting  fibres  parallel  to  the  motion 

"         cast  iron,  fibres  at  right  angles  to  motion 

Elm  on  oak,  fibres  all  parallel  to  motion 

Oak  on  elm,       "  "  "        

Elm  on  oak,  moving  fibres  at  right  angles  to  the  others,  and  to  motion 

Ash  on  oak,  fibres  all  parallel  to  motion 

Fironoak,        "    "  "  "        

Beech  on  oak    "     *'  "  "        

Wrought  iron  on  oak,  fibres  parallel  to  motion 

Wrought  iron  on  elm,      '*  "        "      "       

Wrought  iron  on  cast  iron,  fibres  parallel  to  motion 

"  "on  wrought  iron,  fibres  all  parallel  to  motion 

Wrought  iron  (Tn  brass 

W"rought  iron  on  soft  limestone,  well  dressed 

"  "      "   hard        "  *'        "        

"  "      *'      *'  "  '*        "         wet 

"  "     or  steel  on  hard  marble,  sawed.    By  the  writer about. . 

*'          "      •'     "      "  smoothly  planed,  and  rubbed  mahogany,  fibres  par- 
allel to  motion '. 

"  "      "     "      "   smoothly  planed  wh  pine 

Cast  iron  on  oak,  fibres  parallel  to  motion 

"      *»     "  elm,      "  "        "        "     

"      "     "  cast  iron 

<<      i<     li  brass 

Steel  on  cast  iron 

Steel  on  steel.     By  the  writer 

Steel  on  brass 

Steel  on  polished  glass.    By  the  writer about. . 

"    quite  smooth,  but  not  polished;  on  perfectly  dry  planed  wh  pine,  fibres 

parallel  to  motion , about. . 

"    quite  smooth, but  not  polished;  on  perfectly  dry  planed  and  smoothed 

mahogany,  fibres  parallel  to  motion about. . 

Yellow  copper  on  cast  iron 

"  "       on  oak 

Brass  on  cast  iron 

"      on  wrought  iron,  fibres  parallel  to  motion 

"      on  brass 

"     on  perfectly  dry  planed  wh  pine,  fibres  parallel  to  motion about. . 

"      **          '*         "         "      and  smoothed  mahogany,  fibres  parallel  to  mo- 
tion   about. . 

i*olished  marble  on  polished  marble.    By  the  writer average 

"  "       on  common  brick "       

Common  brick  on  common  brick •'       

Soft  limestone  well  dressed,  ou  the  sam^ 

Common  brick,  on  well-dressed  soft  limestone 

"  "       "      "        "        hard        "        

Oak  across  the  grain,  on  soft  limestone,  well  dressed 

"         "         "       "        "  hard        "  "  "       

Hard  limestone  on  hard  limestone,  both  "  "       

"  "  "  soft  "  "      "  "       

Soft         "  "  hard  "  •'      "  *'      

Wood  on  metal,  generally,  .2  to  .62 mean.. 

Wood,  very  smoofA,  on  the  same,  generally,  .25  to  .5  "     .. 

Wood,     "         "         on  metal,  "  .2    to  .62 "     .. 

Metal  on  metal,  very  smooth,  dry        "  .15  to  .22 "     .. 

Masonry  and  brickwork,  dry  "  ,6    to  .7  "     .. 

"         "  "  with  wet  mortar about.. 

"         "  "  "     slightly  damp  mortar "     .. 

"      on  dry  clay "     .. 

"       "   moist"    «•     .. 

Marble,  sawed  ;  on  the  same ;  both  dry.    By  the  writer •• averaga    "     .. 

"  "         "    "        "        both  damp "     ..* "  "     .. 

"  •'        on  perfectly  dry  planed  wh  pine.     "     ..• "  "     .. 

"  "        on  damp  planed  wh  pine "     ..* "     .. 

"      polished,  on  perfectly  dry  planed  wh  pine    ••     «'     .. 

White  pine,  perfectly  dry ;  planed ;  on  the  same ;    all  the  fibres  parallel  to 

motion about.. 

"        "      damp,  planed ;  on  the  same "     .. 


*  But  after  a  few  trials  the  surfaces  become  so  much  smoother  as  to  reduce  the  angles  as  much  a 
from  2°  to  5° ;  the  sliding  blocks  weighing  about  30  tts  w  'h. 


412 


FRICTION. 


189.  Recent  experiments,  with  much  greater  variations  of  pres  and  of 
vel,  and  with  more  delicate  apparatus  for  detecting  slight  changes  in  the  coeff, 
although  giving  conflicting  results,*  show  that  the  three  laws  in  ^  187 
are  far  from  correct  for  surfs  moving  at  liigli  vels,  and  under  great  pres;  and 
that  they  are  only  approximately  correct  for  ordinary  vels  and 
pressures ;  for  the  coeff  is  found  to  vary  both  with  the  intensity  of  the  pres  and 
with  the  vel,  as  also  with  the  temperature.*  But  in  the  cases  with  which  the 
civil  engineer  has  mostly  to  deal,  slight  diflfs  in  the  character  of  the  surfs,  or 
even  in  the  dampness  of  the  air,  will  often  cause  much  greater  changes  of  coeff 
than  tjhose  due  to  any  probable  changes  of  pres,  vel  and  temp:  so  that,  within 
the  limits  of  abrasion,  we  may  generally  take  Morin's  rules  as  sufficiently  cor- 
rect for  such  cases. 


.28 

.a7  I 


100       90        80        70       60         50       40        30 
Telocity  of  Sliding,  in  inches  per  second. 

Line  A  shows  coeffs  at  diff  vels  under  a  pres  of  1.58  l>s  per  sq  in. 
"     B  "  "  "  1.59      "  " 

.<     Q  «  ..  <.  1^60      "  « 

«     D  *'  "  "  1.61      "  " 

..     E  u  .«  u  417      u  u 

It  will  be  seen  that  at  low  vels  the  coeff  decreased  when  the  pres  per  sq  in  was 
almost  imperceptibly  increased;  but  this  diff  disappeared  as  the  vel  increased. 
At  vels  from  4  to  120  ins  per  sec,  the  coeff  generally  decreased  as  the  vel  in- 
creased ;  rapidly  at  first,  but  more  slowly  as  the  vel  became  greater.  This  agrees 
with  other  recent  expts.  But  at  very  low  vels  (.08  to  5  ins  per  sec)  Prof.  Kimball 
found  the  coeff  (line  E)  increasing  very  rapidly  with  the  vel. 

We  have  made  the  scale  of  coeffs  large  in  order  to  show  their  variations,  which 
are  so  slight  that  they  would  otherwise  be  scarcely  perceptible.  Less  delicate 
expts  would  have  failed  to  show  them  at  all. 

191.    (a)  In  1878  Capt.  Douglas  Chiton  and  Mr.  C^eorse  West* 

tnglioase,  Jr.,  made  careful  experiments  in  England  to  ascertain  the  effect  ol 
fHction  in  connection  with  rail^vay  brakes.  %    The  friction  and  pressure  wen 


190.    Prof.  A.  S.  Kimball,  of 

the  Worcester    (Mass)    Inst    of   Industrial 
Science,  has  made  some  very  delicate  experi- 

A 

j 

r 

- 

ments  upon  the  fric  between  surfs  of  pine 
wood.f     The  results  are    given  in   Fig  3, 
merely  to  show  how  the  coeff  varied  with 
vel  and  pres.    Our  table  gives  a  coeff  of  .4 

1  ] 

B 

/ 

/ 

/ 

/ 

/ 

D 

for  pine  on  pine. 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

T^. 

IT 

is 

\ 

/. 

/ 

y 

/ 

\ 

/ 

<> 

.^ 

^ 

\ 

^ 

'J> 

^ 

^ 

y 

^___ 

1 

^ 

,.^ 

v^ 

^ 

^ 

^ 

C- 

p^ 

== 

— 

=J 

■^ 

^ 

»  This  is  not  surprising  in  view  of  the  extent  to  which  the  coeff  is  affected  by  the  nature  of  the 
surf.  If  the  shape  of  the  minute  projections  is  such  that  they  fit  into  each  other  as  perfectly  under 
small  pressures  as  under  great  ones,  and  if  they  are  too  strong  to  be  broken  by  the  pressures  applied^ 
the  coeff,  as  stated  in  the  1st  law,  should  be  independent  of  the  pres.  But  if  high  pres  wedges  the 
projections  of  one  body  more  closely  between  those  of  the  other,  the  coeff  should  increase  under  such 
pres.  On  the  other  hand,  if  the  higher  pres  breaks  down  the  projections  while  the  lower  ones  are 
unable  to  do  so,  the  coeff  should  decrease  under  the  higher  pres.  The  particles  thus  broken  off  may 
either  act  as  a  lubricant  and  thus  still  further  reduce,  the  fric  and  its  coeff,  or  (if  angular  and  hard) 
may  increase  it.  Change  of  area  of  contact,  under  a  given  total  pres,  may,  by  affecting  the  intensity 
Of  the  pres,  make  changes  in  the  coeff  similar  to  those  just  mentioned. 

At  high  vels  the  roughnesses  have  not  time  to  interlock  as  perfectly  as  at  low  vels.  Hence  we 
should  expect  a  less  coeff  at  high  vels.  But  high  vel  generally  increases  the  number  of  projection!* 
broken  away  ;  and  these  may  either  increase  or  diminish  the  coeff,  as  explained  above.  High  vel 
often  indirectly  affects  it  by  increasing  the  temperature. 

t  Silliman's  Journal  (American  Journal  of  Science)  March  1876  and  May  1877. 

+  See  Proc,  Instn  of  Mechl  Engrs,  London,  June  and  Oct  1878  and  April  1879;  and  "  Bngineer. 
lag,"  London,  18T8 ;  vol.  25,  pp  432,  469,  490 ;  vol.  26,  pp  153,  386,  395. 


FRICTION. 


413 


antomatically  recorded  by  means  of  hydraulic  gauges.  With  cast  iron  brake-blocks 
and  stf  el-tired  wooden  wheels,  43%  inches  in  diameter,  they  found  coeflBcients  about 
as  shown  in  Pig.  4. 

The  points  in  lines  A,  B  and  C  show  the  average  brake  coeflfs,  or  coeffs  of  slid- 
ing fric  between  the  tread  of  a  rolling  wheel  and  the  brake-block. 

Speed  of  Car,  in  miles  per  hour. 
40         35         30         25        20  15 


{§.95 


"2-20 

a 
|.io. 

^.05 


X ^ ^:::::zzzt 

---------       -_   ^       ^£^-«-          ^           -           ^   . 

-           -           ---.--=5=-'    :      zt .•'--'^    :     :   :  :::'"    : 

-    -    -      ;  3=  =■  ^                        -lJ- ' 

"  *'''      "  5:                v^-  T     -          -.--,' 

„"'.''.     "_:::     :          ,^       x     -              -     -     J 

___     ..^:__i.  +±.g_-_,,,.r— ttj- ^—— 

|!=:=:::i;;=;==:r?!==:=;=p;;i^^^ 

!__:5,,..  =  =-!:.S!: 

__^:^ 

:.05O 


55 


5U 


10 


45         40         35         3U        25         20 
Speed  of  Car,  in  miles  per  hour. 

Line  A  shows  brake  coeffs  obtained  immed'y  after  application  of  brake 
"     B  "  "  5  sees  "  " 

.i     c  "  "       ,        15    " 

*'    D  shows  rail  coeffs  or  coeffs  of  sliding  fric  between  the  tread  of  a  dic- 
ing or  '■'■  skidding'''  wheel  (held  fast  by  the  brake)  and  the  rail. 

(b)  From  lines  A  B  and  C  it  appears  that  the  brake  coeff  obtained  at  a 
given  length  of  time  after  the  application  of  the  brake  was  g:enerally  greater 
at  lour  than  at  big^li  vels.  But  where  the  vel  was  maintained  uniform 
the  brake  coeff  diminisbed  as  block  &ncl  urtaeel  remained 
long^er  in  contact.  Thus,  lines  A  and  B  show  that  at  37^  miles  per  hour 
the  brake  coeff  was  .154  when  the  brake  was  first  applied  (point  g)^  but  fell  to 
.096  in  5  sees  {%).  Line  A  (immed'y  after  application)  shows  a  higher  brake  coeff 
(.132  at/)  at  47 J  miles  than  line  B  (5  sees  after  application)  shows  at  37^  miles 
(.096  at  X). 

The  diminution  of  the  rail  coeff  with  length  of  time  of  application  of  brake, 
was  scarcely  noticeable. 

(c)  When  the  brake  fric  (owing  to  the  reduction  of  vel  and  consequent  in- 
crease of  coeff)  becomes  =:  the  "  adhesion  "  or  static  fric  between  the  rail  and 
the  tire  of  the  rolling  wheel,  the  vel  of  rotation  rapidly  falls  below  that  due  to 
the  vel  of  the  car ;  i  e,  tbe  i¥heel  beg-ins  to  "  skid  "  or  slide  along  the 
rail:  and  in  from  .75  to  3 sees  the  rotation  of  the  wheel  ceases  entirely. 

(d)  The  rail  coeff,  line  D,  is  g-enerally  much  less  than  the 
brake  coeff,  lines  A,  B  and  C.  The  pres  on  the  rail  (  =  the  wt  on  a  wheel) 
"Was  about  5000  ibs  per  sq  in,  or  greatly  in  excess  of  the  limit  of  abrasion.  That 
at  the  brake  was  about  200  fcs  per  sq  in.  A  few  expts  were  made  with  brake 
blocks  having  but  \  of  the  usual  area  of  contact,  and  therefore  3  times  the  pres 
oer  sq  in  under  a  given  total  pres.  They  failed  to  show  conclusively  that  this 
caused  any  marked  change  in  the  coeff. 

(e)  The  rail  coeff,  line  D,  like  the  brake  coeff,  increases  as  the  vet 
diminishes ;  slowly  at  first,  but  much  more  rapidly  as  the  speed  becomes 
less;  until,  at  the  moment  of  stopping,  it  is  generally  even  greater  than  the 
brake  coeff  just  before  skidding.  With  steel  tires  on  iron  rails  at  high  vels  it  was 
somewhat  greater  than  on  steel  rails,  but  this  diff  disappeared  as  the  vel  dimin- 
ished. 

(f )  liOComotives  overcome  resistces  =  from  ^  to  ^  or  more  of  the  wt  on 
all  the  drivers;  %  e,  they  have  a  coeff  of  .33  or  more,  although  the  experimental 
coeff  for  steel  on  steel  in  motion  at  low  pres,  is  only  about  .15.  But  the  cases  are 
so  diff  that  a  similarity  in  their  coeffs  could  hardly  be  expected.  The  great  wt, 
say  from  2  to  6  or  even  7  tons,  on  a  driver,  is  concentrated  on  a  surf  (where  the 
wheel  touches  the  rail)  about  2  ins  long  X  about  \  inch  wide,  or  =  say  1  sq  in. 
The  pres  per  sq  in  thus  greatly  exceeds  not  only  that  upon  which  the  tables  are 
based,  but  also  the  limit  of  abrasion.    Besides,  any  point  in  the  tread,  during 


414 


FRICTION. 


the  instant  when  it  is  acting  as  the  fulcrum  for  the  steam  pres  in  the  cyl,  i» 
stationary  upon  the  rail.    Its  fric  (miscalled  "  adhesion  ")  is  therefore  static. 

Capt.  Galton  found  that  the  coeff  of  *'  adhesion  "  was  independent  of 
the  vel,  and  depended  only  on  the  character  of  the  surfs  in  contact.  With  a 
four-wheeled  car  having  about  5000  Ihs  load  on  each  wheel,  it  was  generally  over 
.20  on  dry  rails ;  in  some  cases  .25  or  even  higher.  On  wet  or  greasy  rails,  with- 
out sand,  it  fell  as  low  as  .15  in  one  case,  but  averaged  about  .18.  With  sand 
on  wet  rails  it  was  over  .20.  Sand  applied  to  dry  rails  before  starting  gave  .35 
and  even  over  .40  at  the  start,  and  an  average  of  about  .28  during  motion ;  but 
sand  applied  to  dry  rails  while  the  car  was  in  motion  was  apt  to  be  blown  away 
by  the  movement  of  the  car  and  wheels. 

(g)  Owing  to  the  constancy  of  the  coeflf  of  "  adhesion  "  under  given  conditions 
of  tire  and  rail,  the  brake  fric  necessary  to  *'  skid  "  the  wheels  in  any  case  was 
also  practically  constant  for  all  vels.  But  at  high  vels,  owing  to  the  lower  brake 
coeff,  a  higher  brake  pres  was  reqd  to  produce  this  fixed  amount  of  brake  frie 
The  skidding  also  reqd  a  longer  time  than  at  low  speeds. 

193.  If  the  pres  is  sufficient  to  produce  abrasion  (indeed,  while  it  is 
much  less)  the  fric  often  varies  greatly,  but  no  precise  law  has  yet  been  discov- 
ered for  estimating  it.  Rennie  gives  the  following  table  of  coefTs  of  frie 
of  dry  surfaces,  under  pressures  g^radually  increased  up  to 
the  limits  of  abrasion.  It  will  be  noticed  that  in  this  table  the 
coeff  g^enerally  increases  with  the  intensity  of  the  pres : 

Coeffs  of  friction  of  dry  surfaces,  under  pressures  grrad-^ 
ually  increased  up  to  the  limits  of  abrasion.  (By  G.  Rennie,  C  E.> 


Pres.  in  Lba. 

Wrought  Iron 

Wrought  Iron 

Steel 

Brass 

per 

on 

on 

on 

on 

Square  Inch, 

Wrought  Iron. 

Cast  Iron. 

Cast  Iron. 

Cast  Iron. 

.32.5 

.140 

.174 

.166 

.157 

186 

.250 

.275 

.300 

.225 

224 

.271 

.292 

.333 

.219 

336 

.312 

.333 

.347 

.215 

448 

s 

M5 

.354 

.208 

560 

.367 

.358 

.233 

672 

.376 

.403 

.2.33 

709 

.434 

.2.34 

784 

.232 

821 

.273 

193.  (a)  Rolling;  friction,  or  that  between  the  circumf  of  a  roll- 
ing body  and  the  surf  upon  which  it  rolls,  is  somewhat  similar  to  that  of  a 
f union  rolling  upon  a  rack.  In  disengaging  the  interlocking  projections,  or  in 
ifting  the  wheel  over  an  obstacle  o,  Figs  5  and  6,  the  motive  force  F,  instead  of 
dragging  one  over  the  other,  as  in  Fig76,  p.  407,  acts  at  the  end  of  a  bent  lever 
F  E,  W  Figs  5  and  6,  the  other  end  W  of  which  acts  in  a  direction  perp  to  the 
contact  surf;  and  in  practical  cases  of  rolling  fric  proper  the  leverage  RW  of 
the  resisting  wt  of  the  wheel  and  its  load  is  very  much  less,  in  proportion  to 
that  (FR)  of  the  force  F,  than  in  our  exaggerated  figs.  Hence  the  force  F  reqd 
to  roll  a  wheel  etc  is  usually  very  much  less  than  would  be  necessary  to  slide  it. 
(b)  There  are  usually  two  irays  of  applying;  the  force  in  overcom- 
ing rolling  fric :  1st  (Fig  5)  at  the  axis  of  the  rolling  body ;  as  the  force  of  a 

horse  is  applied  at    the    axle  of    a 
-p,  wagon-wheel;  or  that  of  a  man  at  the 

axle  of  a  wheel-barrow :  2d  (Fig  6)  at 
the  circumf;  as  when  workmen  pusli 
along  a  heavy  timber  laid  on  top  of 
two  or  more  rollers;  or  as  the  ends  of 
an  iron  bridge-truss  play  backward 
and  forward  by  contraction  and  ex- 
pansion, on  top  of  metallic  rollers  or 
balls  (p725).  In  Fig  5  we  have,  in  ad« 
dition  to  the  rolling  fric  of  the  cir- 
cumf of  the  wheel  on  its  support,  the 
sliding  fric  of  the  axle  in  its  bearing. 
In  Fig  6  we  have  only  rolling  fric^ 


(\ 

R 

J 

^4 

^v^,^^ 

m 

^ 

^ 

Fig.S 


Fig.  6 


but  at  both  top  and  bottom  of  the  wheel. 

(c)   When  the  obstacles  o  are  very  small,  as  in  the  case  of  cart-wheels  on 
smooth  hard  roads,  or  of  car-wheels  on   iron   or  steel  rails,   the   leverage 


FRICTION. 


415 


(FR)  of  F  becomes,  practically,  in  Fig  5  the  radius,  and  in  Fig  6  the  diam,  of  the 
wheel;  while  that  (RW)  of  the  resistce  is  very  small.  Hence,  neglecting  axle 
fric  in  Fig  5,  the  force  F  reqd  to  overcome  rolling  fric  in  such  cases  is  directly 
as  the  wt  W  of  and  on  the  wheel,  and  inversely  as  the  diam  of  the  wheel. 

The  few  expts  that  have  been  made  upon  the  coeffs  of  rolling  fric,  apart  from 
axle  fric,  are  too  incomplete  to  serve  as  a  basis  for  practical  rules. 

(d)  The  fric  (or  "  acHiesion  ")  between  wheel  and  rail,  which  enables  a 
locomotive  to  move  itself  and  train,  or  which  tends  to  make  a  car-wheel  revolve 
notwithstanding  the  pres  of  the  brake,  is  a  resistce  to  the  sliding  of  the  wheel 
on  the  rail ;  and  is  therefore  not  rolling  but  sliding  fric  ;  static  when  the  wheels 
either  stand  still  or  roll  perfectly  on  the  rails ;  and  kinetic  when  they  slip  or 
*•  skid  ". 

194,  The  IViction  of  liquids  moving  in  contact  with  solid  bodies 
Is  independent  of  ttie  pressure,  because  the  "lifting"  of  the  particles 
of  the  fluid  over  the  projections  on  the  surf  of  the  solid  body,  is  aided  by  the 
pres  of  the  surrounding  particles  of  the  liquid,  which  tend  to  occupy  the  places 
of  those  lifted.  Hence  we  have,  for  liquids,  no  coeff  of  fric  corresponding  with 
that  (=  resistce  -^  pres)  of  solids.  The  resistce  is  believed  to  be  directly  as  the 
area  of  surf  of  contact.  Recent  researches  indicate  that  Resistce  =  a  coeff  X 
area  pf  surf  X  ^el»  in  which  both  n  and  the  coeff  depend  upon  the  vel  and 
upon  the  character  of  the  surf;  and  that  at  low  vels  n  =  1,  but  that  at  a  certain 
"critical"  vel  (which  varies  with  the  circumstances)  n  suddenly  becomes  =  2, 
owing  to  the  breaking  up  of  the  streatn  into  marked  counter  currents  or  eddies. 
The  resistance  of  fluid  fric  arises  principally  from  the  counter  currents  thus  set 
in  motion,  and  which  must  be  brought  into  compliance  with  the  direction  of 
the  force  which  is  urging  thp  stream  forward. 

195.  Table  of  coefficients  of  moving^  friction  of  smootli 
plane  surfaces,  when  kept  perfectly  lubricated.    (Morin.) 


Oak  on  oak,  fibres  parallel  to  motion 

"     "     "     fibres  perpendicular  to  motion 

"  on  elm,  fibres  parallel  to  motion 

"  on  cast  iron,  fibres  parallel  to  motion 

"  on  wrought  iron,  "        "        "       

Beech  on  oak,  fibres  *'        "        "      

Elm  on  oak,         "  "        "        '•       

"  on  elm,         "  ,     '♦        "        "       

•*  cast  iron,      "  "        "        "       

■Wrought  iron  on  oak,  fibres  parallel,  greased  and  wet,  .256. 

"         "      "    "     fibres  parallel  to  motion 

"         "     on  elm,      "  '*        "        "      

"         "     on  cast  iron,  "        ".       "     

"         "     on  wrought  iron,  "        "        "      

*'  •'     on  brass,  fibres     "        "        "      

Cast  iron  on  oak,  fibres  parallel  to  motion 

"      •'      "     "        "  "         "        *'    greased  and  wet,  .218 

"     "     on  elm,     "  "         "        "     

"      "     on  cast  iron,  with  water,  .314 

"      "     on  brass 

Copper  on  oak,  fibres  parallel  to  motion 

Yellow  copper  on  cast  iron 

Brass  on  cast  iron 

"      on  wrought  iron 

"      on  brass  

Steel  on  cast  iron 

"     on  wrought  iron 

"     on  brass  

Tanned  oxhide  on  cast  iron,  greased  and  very  wet,  .365 

"  "       on  brass 

"  "       on  oak,  with  water,  .29 


Dry 
Soap. 

Olive 
Oil. 

Tal- 
low. 

Lard. 

.164 

.075 

.067 

.083 

.072 

.136 

.••;; 

.073 
.080 
.098 
.0.55 

.066 

.137 

.070 

.060 

.139 

.066 

.214 

.085 

.055 

.078 

.076 

.066 

.103 

.076 

.070 

.082 

.081 

.078 

.103 

.075 

.189 

.075 

.078 

.075 

.061 

.077 

.197 

.064 

.100 

.070 

.078 

.103 
.069 

.075 

.066 

.072 

.068 

.077 

.086 

.072 

.081 

.058 

.079 

.105 

.081 

.093 

.076- 

.053 

.056 

.133 

.159 

.191 

.241 

.091 
.056 


The  launching  friction  of  the  wooden  frigate  Princeton  \ras  found  by  a 
committee  of  the  Franklin.  In8ti^u^B  in  1844,  to  average  about  .067  or  one-fifteenth 
of  the  pressure  during  the  first  .75  of  a  second  and  .022  or  one  forty-fifth  for  the 
next  4  seconds  of  her  motion.  The  slope  of  the  ways  was  1  in  13,  or  4  degrees  24 
minutes.  They  were  heavily  coated  with  tallow.  Pressure  on  them  =  15.84  lbs. 
per  square  inch,  or  2280  ft)8.  per  square  foot.  In  the  first  .75  of  a  second  the  vessel 
slid  2.5  inches;  in  the  next  4  seconds  16  feet  6.5  inches;  total  for  4.75  seconds  15.7* 
feet 


416 


FRICTION. 


196.  The  ftrictlon  of  lubricated  surfaces  varies  greatly  with 
the  character  of  the  surfs  and  with  that  of  the  lubricant  and  the  manner  of  Ita 
application.  If  the  lubricant  is  of  poor  quality,  and  scantily  and  unevenly  ap- 
plied under  great  pres,  it  may  wear  away  in  places  and  leave  portions  of  the  dry 
surfs  in  contact.  The  conditions  then  approximate  to  those  of  unlubricated 
surfaces.  But  if  the  best  lubricants  for  the  purpose  are  used,  and  supplied  reg- 
ularly and  in  proper  quantity,  so  as  to  keep  the  surfs  always  perfectly  separated, 
the  case  becomes  practically  one  of  liquid  friction,  and  the  resistce  is  very 
small.  Between  these  two  extremes  there  is  a  wide  range  of  variations  (see 
table,  ^  197  (d)),  the  coeflF  being  affected  by  the  smallest  change  in  the  condi- 
tions. Where  any  degree  of  accuracy  is  reqd,  we  would  refer  the  reader  to  the 
experimental  results  given  in  Prof.  Thurston's  very  exhaustive  work,*  devoted 
exclusively  to  this  intricate  subject. 

197.  (a)  Expts  by  Mr.  Arthur  M.  Wellington  upon  the  fric  of 
lubricated  journals  t  gave  a  gradual  and  continuous  increase  of  coeff  as 
the  vel  of  revolution  diminished  from  18  ft  per  sec  (  =  a  car  speed  of  12  miler 
per  hour)  to  a  stop.  This  increase  was  very  slight  at  high  vels,  but  much  more 
rapid  at  low  ones  ;  as  in  Figs  3  and  4.  At  vels  from  2  to  18  ft  per  sec  the  coeff 
was  much  less  under  high  pressures  than  under  low  ones;  but  at  starting  there 
was  little  diff  in  this  respect.  The  coeff  increased  rapidly  as  the  tempera- 
ture rose  from  100°  to  120°  and  150°  Fahr. 

(b)  Prof.  Thurston,  also  experimenting  with  lubricated  journals,  t 
found  that  at  starting,  the  coeff  increased  with  increase  of  pres,  as  it  did  also 
when  in  motion,  if  the  pres  greatly  exceeded  the  max  (say  500  to  600  lbs  per  sq 
in)  allowable  in  machinery.  He  also  found  that  at  high  vels  the  coeff  increased 
very  slowly  (instead  of  continuing  to  decrease)  as  the  vel  increased. 

(c)  Prof.  Thurston  gives  the  following  approx  formulae  for  journal 
friction  at  ordinary  temperatures,  pressures  and  speeds,  with  journal  and 
bearing  in  good  condition  and  well  lubricated: 


Coelf  for  starting:  =  (.015  to  .02)  X  if^pres  in  Bbs  per  sq  in. 


Coeff  when  the  shaft      ,  ^,  ,     -^,  ^^  l^    vel  in  ft  per  min 
is  revolving       - (-02  to  .03)  X  -%=z=— ~zzz=zr 
V^pres  in  ros  per  sq  in. 

At  pressures  of  about  200  Rs  per  sq  in : 

Temperature  of  minimum      ._  ^^   3. — ,  .     ., r- 

fric ;  in  Fahr  degs  — 15  X  l/vel  in  ft  per  mm 

Caution.  Tlie  leverage,  with  which  journal  fric  Resists  motion,  in- 
creases \¥itli  tbe  diam  of  the  journal. 

(d)  Tbe  following  figures,  selected  from  a  table  of  experimental  results  ^ven 
by  Profo  Thurston,  merely  show  the  extent  to  wbich  the  coen  of 
journal  fric  is  aflfected  by  pres,  vel  and  temperature;  and 

hence  the  risk  incurred  in  rigidly  applying  general  rules  to  such  cases.  In 
these  expts  the  character  of  journal  and  bearing,  the  lubricant  and  its  method 
of  application,  remained  the  same  throughout.  Where  these  vary,  still  further, 
and  much  greater,  variations  in  the  coeff  may  occur. 

Steel  journal  in  bronze  bearing,  lubricated  with  standard 
sperm  oil. 


fa 
U  a 


130° 
90° 


Speed  of  revolution 

)  feet  per  minute  1 100  feet  per  minute  1500  ft  per  mm|l200  ft  per  min 


200       100   I     4 
lbs  per  sq  in 


Coeff 
.0160 
.0056 


Coeff 
.0044 
.0031 


Coeff 
.125 


Pressures. 

200    I    100    I     4 
lbs  per  sq  in 


Coeff 
.0087 
.0040 


Coeff 
.0019 
.0019 


Coeff 
.0630 
.0630 


200       100 
lbs  per  sq  in 


200       100 
lbs  per  sq  in 


Coeff  Coeff 
.0053  .0037 
.0075     .0061 


Coeff 
.0065 
.0100 


Coeff 
.0075 
.0150 


*  Friction  and  Lost  Work  in  Machinery  and  Mill  Work.    John  Wiley  &  Sons,  New  York, 
t  Traua  Amer  See  of  Civil  Engrs,  New  York,  Dec.  1884. 
t  Journal  of  the  Franklin  Institute.  Hoi.  lil*» 


FRICTION. 


417 


(e)  Where  the  force  is  applied  first  on  one  side  of  the  jour- 
nal and  then  on  the  opposite  side,  as  in  crank  pins,  the  fric  is  less 
than  where  the  resultant  pres  is  always  upon  one  side,  as  in  fly-wheel  shafts; 
because  in  the  former  case  the  oil  has  time  to  spread  itself  alternately  upon  both 
sides  of  the  journal. 

(f )  Friction  rollers.  If  a  journal  J,  in- 
stead of  revolving  on  ordinary  bearings,  be  sup- 
ported on  friction  rollers  R,  R,  the  force  required 
to  make  J  revolve  will  be  reduced  in  nearly  the 
same  proportion  that  tbe  diam  of  the  axle  o  or 
0  of  the  rollers,  is  less  than  the  diam  of  the 
rollers  themselves. 

Mr.  Wellington  experimented  with  a  patent 
bearing  on  this  principle,  invented  by  Mr.  A. 
Higley.  Diam  of  rollers  RR,  8  ins;  of  their 
axles  0  0  If  ins ;  of  the  journal  c,  3^  ins.  Here, 
theoretically, 

.     .  .    ^    diam  of  axles  0  0        If  ins 

fric  of  patent  journal  =  fric  of  3^  in  journal  X  ^^^^^1^-^^^^-  Yi^s 

or  as  1  to  4.6.  Under  a  load  of  279  lbs  per  sq  in,  Mr.  Wellington  found  it  about 
as  1  to  4  when  starting  from  rest;  and  about  as  1  to  2  at  a  car  speed  of  10  miles 
per  hour. 

198.  (a)  Resistance  of  railroad  rolling  stock.  This  con- 
sists of  rolling  fric  between  the  treads  of  the  wheels  and  the  rails  (the  treads 
also  sometimes  slide  on  the  rails,  as  in  going  around  curves) ;  of  sliding  fric  be- 
tween the  journals  and  their  bearings,  and  between  the  wheel  flanges  an<3  the 
rail  heads;  of  the  resistce  of  the  air;  and  of  oscillations  and  concussions,  which 
consume  motive  power  by  their  lateral  and  vert  motions,  and  also  increase  the 
wheel  and  journal  fries. 

Its  amount  depends  greatly  upon  the  condition  of  the  road-bed  and  rails  (as 
to  ballast,  alignment,  surf,  spaces  at  the  joints,  dryness  etc);  upon  that  of  the 
rolling  stock  (as  to  wt  carried,  kind  of  springs  used,  kind  and  quantity  of  lubri- 
cant, condition  and  dimensions  of  wheels  and  axles  etc) ;  upon  grades  and  curv- 
ature; upon  the  direction  and  force  of  the  wind;  and  upon  many  minor  con- 
siderations.   Experiments  give  very  conflicting  results. 

(b)  During  the  summer  of  1878,  Mr.  TVelling-ton  experimented  with 
loaded  and  empty  box  and  flat  freight  cars,  passenger  and  sleeping  cars,  and  at 
speeds  varying  from  0  to  35  miles  per  hour.  The  cars  were  started  rolling  (by 
grav)  down  a  nearly  uniform  grade  of  .7  foot  per  100  feet,  or  36.5  feet  per  mile, 
and  6400  ft  long.  Their  resistces  were  calculated  as  in  ^  185.  "The  rails 

were  of  iron,  60  lbs  per  yd,  and  the  track  was  well  ballasted  and  in  good  line  and 
surf,  but  not  strictly  first  class."  The  following  approx  figures  are  deduced 
from  Mr.  Wellington's  expts  upon  cars  fitted  with  ordinary  journals:* 

Car  Resistance  in  pounds  per  ton  (2340  lbs)  of  iveight  of 
train,  on  straight  and  level  track  in  good  condition. 


Empty  cars 

Loaded  cars 

Speed  of 
train  in 

Oscilla- 

Oscilla- 

miles per 

Axle, 

tion 

Axle, 

tion 

hour 

tire  and 
flange 

and 
con- 
cuss'n 

Air 

Total 

tire  and 
flange 

and 

con- 

cuss'n 

Air 

Total 

0 

14 

0 

0- 

14 

18 

0 

0 

18 

10 

6 

.6 

.4 

7 

4 

.6 

.4 

5 

20 

6 

2.7 

1.3 

10 

4 

2. 

1. 

7 

30 

6 

5.3 

2.7 

14 

4       1      4.7 

2.3 

11 

(c)  With  the  Higley  patent  anti-fric  roller  journal,  the  resistce  to  ^/aWin^r  was 
but  about  4  lbs  per  ton. 

(d)  About  midway  in  the  track  experimented  upon,  was  a  curve  of  1°  de- 
flection angle  (5780  ft  rad)  3000  ft  long,  with  its  outer  rail  elevated  3  to  4  ins 


*  Transactioiis>  American  Society  of  Civil  En^'ineers.  Feb  1879. 


27 


418 


FRICTION. 


above  the  inner  one.  The  rise  of  the  outer  rail  was  begun  on  the  tangent,  about 
500  ft  before  reaching  the  curve.  In  the  first  500  ft  of  the  curve  the  resistce  was 
greater  than  that  encountered  just  before  reaching  the  curve,  by  from  .6  to  2.1 
(average  1.1)  lbs  per  ton.  In  the  last  500  ft  of  the  curve  this  excess  had  diminished 
to  from  .2  to  .9  (average  .6)  lbs  per  ton.  Owing  to  the  continuance  of  the  down 
grade  on  the  curve,  the  vel  increased  as  the  train  traversed  the  curve ;  but  it 
does  not  clearly  appear  whether  the  decrease  in  curve  resistce  was  due  to  the 
increase  in  vel,  or  to  the  fact  that  the  oscillations  caused  by  enteriag  the  curve 
gradually  ceased  as  the  train  went  on. 

(e)  Mr.  P.  H.  Dudley,  experimenting  with  his  "  dynagraph  "*  ob- 
tained results  from  which  the  following  are  deduced: 

Train  Resistance  in  ponnds  per  ton  (2240  lbs)  of  weig^lit  otf 
train,  including^  g^rades. 


description  of  train 


Loaded 
cars 


29 
37 
25 


Empty     jWeighttons 
cars  (2240  lbs) 


Trip 


Toledo  to  Cleve- 
land.   95  miles 

Cleveland  to  Erie 
95.5  miles 

Erie  to  Buffalo. 
88  miles 


Average 
speed. 

Miles  per 
hour 


20 
20 
20 


Average 
resist- 
ance. 


8.34 
7.67 
8.89 


**  With  the  long  and  heavy  trains  of  the  L.  S.  &  M.  S.  Ry,  of  600  to  650  tons,  it 
reqd  less  fuel  with  the  same  engine  to  run  trains  at  18  to  20  miles  per  hour  than 
it  did  at  10  to  12  miles  per  hour",  owing  to  the  fact  that  at  the  higher  speeds 
steam  was  used  expansively  to  a  greater  extent,  and  hence  more  economically. 
199.  The  work,  in  ft-lbs,  reqd  to  overcome  fric  through 
any  dist,  is  =  the  fric  in  lbs  X  the  dist  in  ft.  In  order  that  a  body,  started  slid- 
ing or  rolling  freely  on  a  hor  plane  and  then  left  to  itself,  may  do'this  work  ;  ie, 
may  slide  or  roll  through  the  given  dist,  its  kinetic  energy  ( ='  its  wt  in  lbs  X  its 
vel2  in  ft  per  sec  -f-  2gf)  must  =  the  fir.st-named  prod.  Conversely,  tlie  dist 
in  ft  through  which  such  a  body  will  slide  or  roll  on  a  hor  plane,  is 

its  kinetic  energy  in  ft-lbs,  at  start 
fric  in  lbs 
wt  of  body  in  lbs  X  Initial  vel^  in  ft  per  sec     initial  vel^  in  ft  per  sec 
*"      wt  of  body  in  lbs  X  coeff  of  fric  X  2gf~  ^     coeffoffric  X  2  ^  ^ 

Tbe  time  reqd,  in  sees,  is  ==  ' 


dist  in  ft 


mean  vel,  in  ft  per  sec     J  initial  vel  in  ft  per  sec 

Suppose  two  similar  locomotives,  A  and  B,  each  drawing  a  train  on  a  level 
straight  track ;  A  at  10  miles,  and  B  at  20  miles,  per  hour.  The  total  resistce  of 
each  eng  and  train  (which,  for  convenience,  we  suppose  to  be  independent  of 
vel)  is  1000  lbs.  Hence  the  force,  or  total  steam  pres  in  the  two  cyls  reqd  tQ 
balance  the  fric  and  thus  maintain  the  vel,  is  the  same  in  each  eng.'  In  travel- 
ing ten  miles  this  force  does  the  same  amount  of  work  (1000  lbs  X  10  miles 
=  10000  pound-miles)  in  each  eng,  and  with  the  same  expenditure  of  steam  in 
each ;  although  B  must  supply  steam  to  its  cyls  ttvice  as  fast  as  A,  in  order  to 
maintain  in  them  the  same  pres.  Id  one  hour  the  force  in  A  does  10000  lb-miles 
as  before,  but  that  in  B  does  (1000  lbs  X  20  miles  =  )  20000  lb-miles,  and  with 
twice  A's  expenditure  of  steam. 

But  in  fact  the  resistce  of  a  given  train  is  much  greater  at  higher  vels.  See 
table,  If  198  (b)  And  even  if  we  still  assumed  the  resistce  to  be  the  same  at 
both  vels,  B  must  exert  more  force  than  A  in  order  to  acquire  a  vel  of  20  miles 
per  hour  while  A  is  acquiring  10  miles  per  hour. 

*  An  inst  for  measuring  the  strain  on  the  draw-bar  of  a  locomotive,  or  tbe  foroc  which  the  lattetl 
exerts  upon  the  train, 
t  y=  acceleration  of  gravity  =  say  32.2  ;  2g=  say  64.4. 


LEVERS. 


419 


200.  Natural  Slope.  When  granular  materials,  as  sand,  earth,  grain, 
etc.,  are  deposited  loosely,  as  when  they  are  shoveled  from  a  cutting  or 
dumped  from  a  cart,  the  angle,  formed  between  a  level  plane  and  the  sloping 
surface  of  the  pile  of  material,  is  called  the  natural  slope.  This  angle  de- 
pends upon  the  friction  and  adhesion  between  the  separate  particles  of  the 
material,  and  often  varies,  in  one  and  the  same  material,  from  time  to  time, 
with  changes  in  weather  conditions,  etc.,  especially  with  dampness. 


Fig:.  85. 


201.  Any  force,  p.  Fig.  85,  acting  upon  a  body,  B,  will  suffice  to  move  the 
body  (see  foot-note  (*),  H  1),  provided  it  exceeds  the  sum,  S,  of  all  resist- 
ances, including  friction  between  B  and  the  surface  upon  which  B  rests,  or  if 
it  forms,  with  any  other  force  or  forces,  P,  a  resultant,  R,  greater  than  S. 

If,  before  the  application  of  p,  the  body  is  already  in  uniform  motion,  P  is 
=  S ;  and  any  force,  p,  however  small,  will  suffice  to  change  the  direction  of 
motion.  This  accounts  for  the  ease  with  which  a  revolving  shaft  may  be  slid 
longitudinally  in  its  bearings,  and  for  the  fact  that  a  cork  may  be  more 
easily  drawn  if  we  first  give  it  a  twisting  motion  in  the  neck  of  the  bottle. 

LEVERS. 

202.  Classes  of  tevers.  Figs.  86.  Levers  are  classe-  according  to 
the  relative  positions  of  "power, "  *  "weight"  *  and  fulcrum    iS  follows: 


^,      K 


)(&) 


l^ 


QW 
Fig:.  86. 


W 


(C) 


Fig.  (a),  Class  1.  Fulcrum  R  between  power  w  and  weight  W; 
"  (6),  "  2.  Weight  W  between  power  w  and  fulcrum  R; 
"     (c),        "      3.  Power  W  between  weight  w  and  fulcrum  R. 

In  class  2,  the  leverage  of  the  power  is  necessarily  greater  than  that  of  the 
weight.    In  class  3,  vice  versa. 

203.  In  Fig,  86,  taking  the  moments  of  the  forces  about  any  point  at 
pleasure,  as  o,  we  have,  for  equilibrium : 

Fig.  (a),  W  .  Zw  —  R  ?R  +  1^  .  Zw  =  0; 
Fig.  (6),  W.Zw  —  R^R-w?.^w  =  0; 
Fig.-  (c),   W  .lw  —  B,ln-}-<u}  .1^  =  0. 


*  When  levers  are  used  for  lifting  weights  or  for  overcoming  other  resifit- 
ances,  the  force  applied  is  called  the  "power,"  and  the  resistance  to  be  over- 
come is  called  the  "weight." 


420 


STATICS. 


304.  Compound  levers,  Fig.  87,  may  be  used  where  there  is  not  room 
for  the  arms  of  a  single  lever  of  sufficient  length.  In  a  compound  lever, 
neglecting  friction, 


weight  _  product  of  lengths  of  power  arms  _  8  X  10  X  2 
power         product  of  lengths  of  weight  arms         2X1X4 


160 


-  20. 


The  three  levers  of  Fig.  87,  taken  separately  and  beginning  at  the  power 
end,  give: 

weight  _  _8   _  4.  IP  _  iQ.  ^  _  J  ; 
power  2  '1  '4  2' 

and  4  X  10  X  -^  =  20.  as  before. 


W^20LJ 


P  =  l 


i 


2^ 


-Th 


Fig.  87. 


Fig.  88. 


205.  Toothed  or  Cog  Gearing.  Wheels  and  Pinions.  Fig.  88. 
These  are  a  series  of  continuous  compound  levers.  The  power  is  usually  ap- 
plied to  a  crank,  c,  and  the  weight  is  attached  to  a  drum,  d.  The  larger 
wheel,  w,  on  a  given  shaft,  is  called  the  wheel ;  the  smaller  one,  p,  the  pinion. 

Let  c  =  the  radius  of  the  crank,  d  =  that  of  the  drum,  m  =  the  product 
of  the  radii  of  the  pinions,  and  n  =  the  product  of  the  radii  of  the  wheels. 
Then,  neglecting  friction, 

weight  _     c  .  n 

power         Tti  .d 

Instead  of  the  several  radii,  we  may  of  course  use  the  corresponding  diam- 
eters or  circumferences;  and,  as  the  teeth  are  necessarily  of  equal  "pitch" 
(length,  measured  along  the  circumference),  the  number  of  teeth  on  a  wheel 
or  pinion  is  usually  taken  instead  of  the  radius. 

When  the  ratio, ,  is  great,  the  system  is  said  to  be  of  high  gear. 

power 
When  that  ratio  is  small,  we  have  low  gear. 

Compound  levers  and  gearing  are  used  for  converting  low  into  high  veloc- 
ity,, as  well  as  for  lifting  great  weights  by  means  of  small  powers.  When 
used  for  increasing  the  velocity,  the  positions  of  power  and  of  weight  are  the 
reverse  of  those  shown  in  Figs.  87  and  88. 

206.  Whenever  the  power  and  the  weight  balance  each  other,  either 
in  a  single  lever,  or  in  a  connected  system  of  levers  or  leverages,  of  any 
kind  whatever,  then  if  we  suppose  them  to  be  put  into  motion  about  the 
fulcrum,  their  respective  velocities  will  be  in  the  same  proportion  or  ratio 
as  their  leverages ;  that  is,  if  the  leverage  of  the  power  is  2,  5,  or  50  times  as 
great  as  that  of  the  weight,  the  power  will  move  2,  5,  or  50  times  as  fast 
as  the  weight.  Therefore,  by  observing  these  velocities,  we  may  determine 
the  ratio  of  the  leverages.  The  weight  and  the  power  are  to  each  other, 
therefore,  inversely  as  their  velocities,  as  well  as  inversely  as  their  leverages. 

207.  No  mechanical  advantage  is  gained  by  merely  increasing  the  length 
of  a  lever,  as  by  curving  it,  as  at  abo,  Fig.  4,  H  13,  or  by  giving  it  an  in- 
clination to  the  line  of  action  of  the  power,  P,  as  at  o  m,  o  i^  or  o  n. 


LEVERAGE. 


421 


208.  Thus,  in  Fig.  89,  representing  a  bent  lever,  a  f  h,  the  length  of  the 
lever,  or  of  any  of  its  members,  as  /  h,  must  not  be  confounded  with  the  arm 
or  leverage  of  the  force  acting  upon  the  lever.  These  may  or  may  not  be 
equal.  Thus,  the  member  /  6  is  much  longer  than  the  member  /  a;  yet,  if 
the  arms,  /  a  and  /  c,  of  the  forces  or  weights  are  equal,  the  weights  n  and  m 
must  also  be  equal  in  order  to  insure  equilibrium. 


Fi^.  89. 


209.  If  the  weight  m  be  removed,  a  force  c,  or  s,  or  y,  or  d,  with  leverage 
=  c',  8\  y',  d',  respectively,  may  be  applied  at  any  point,  as  h,  to  balance  the 
moment  of  n.     In  any  case  this  force  must  be  such  that 

force  X  its  leverage  =  n.a  f. 
Hence, 

n  .  a  f 


force  ■■ 


leverage  of  force' 


210.  Hence  also  the  force  required  is  least  when,  as  at  y,  it  is  perpendicular 
to  the  length  of  the  member  /  b;  for  the  leverage  (which  evidently  cannot  ex- 
ceed f  b)  is  then  greatest.  The  force  required  increases  as  it  deviates  in 
either  direction  from  the  line  b  y  (perpendicular  to  /  b)  and  approaches  more 
nearly  to  the  direction  of  /  6  itself;  for  its  leverage  then  constantly  decreases. 
No  force,  however  great,  could  balance  the  moment  of  n  about  /,  if  applied 
in  the  direction  fb,  orb  f;  for  such  a  force  would  have  no  moment  about  /. 

211.  Similarly,  in  Fig.  90,  the  moment,  about  a,  of  a  load  W,  placed  at  6, 
is  =  W .  a  c,  or  the  same  aa  if  it  were  placed  at  c,  and  not  =  W  .  a  b. 


Figr.  90. 


Fig.  91. 


212.  In  Figs.  91,  also,  the  moments  W.  o  e  and  W.  o'e',  of  the  equal 
weights,  W  and  W,  are  equal.  But  if  forces,  p  and  p',  be  applied  in  direc- 
tions perpendicular  to  the  longer  beam,  o  t,  the  leverage  o  t  of  p  becomes 
about  6  times  that  (o'  t')  of  p\  Hence  a  force,  p,  applied  at  t,  has  about  the 
same  bending  moment  as  a  parallel  force  =  6  p,  applied  at  e'. 


422 


STATICS. 


STABILITY. 

213.  Stability.  Figs.  92.  If  the  resultant,  R,  Fig.  (a),  of  the  force  P 
and  weight  W,  falls  beyond  the  base,  as  shown,  then  the  overturning  moment 
of  P,  Fig.  92  (&),  about  the  toe  n,  will  exceed  the  moment  of  stability  of  the 
weight  W  about  the  same  point,  and  the  body  will  overturn  about  n.  If  not, 
it  will  stand. 

314.  Assuming  stability  against  overturning,  the  body  will  slide  if  the 
horizontal  component,  h,  of  R,  Fig.  92  (a),  exceeds  the  frictional  and  other 
resistances. 

215.  In  practice,  the  toe,  n,  or  the  ground  beneath  it,  might  yield  if  the 
stone  revolved  upon  it,  or  if  R  fell  near  n  (see  ^U  145,  etc.) ;  but  this  is  a 
question  of  strength  of  materials.  Cement,  clamps,  etc.,  between  the  base 
and  the  ground,  would  add  a  third  force,  and  thus  change  the  problem. 


Fig.  92. 


Tig.  93. 


Fig^.  94. 


216.  Owing  to  the  greater  leverage,  l^,.  Fig.  93,  of  W  about  a,  the  moment 
of  stability  is  much  greater  about  a  than  about  b. 

217.  In  Fig.  94,  let  G  =  2  lbs.;  f?  =  1  lb.;  leverages  =  3,  4  and  6  ft.,  as 
shown.  Then  the  moment  of  stability  of  the  rectangular  body,  G,  against 
a  horizontal  force,  P,  is  =  3G  =  3X2  =  6  ft. -lbs. ;  and  the  moment  of  the 
lower  triangular  body,  ^,  is  =  4^  =  4X1=4  ft.-fbs. ;  so  that,  although  the 
larger  body  weighs  twice  as  much  as  the  smaller  one,  yet  its  moment  of 
stability  is  only  1.5  times  as  great. 


p 

■x-> 

/ 

/ 

/ 

<! 

/ 

^P 

>' 

^^^ 

;^ 

4//J^//////M 

^ — h — > 

^  / 


w 


/ 


(«) 


(6) 

Fig-.  95. 


218.  Work  of  Overturning.  In  Figs.  95  (a)  and  (6),  let  the  shaded 
portion  of  each  figure  be  of  lead,  and  the  remainder  of  wood,  and  let  the 
center  of  gravity  of  the  entire  body,  in  each  case,  be  at  G.  Then,  since  the 
weight,  W,  is  the  same  in  both  cases,  as  is  also  its  leverage  of  stability,  about 

o,  =  y,  the  moment  of  stability,  =  —  6.W,  is  the  same  in  both  cases,  as  is 

also  the  force,  P,  required  to  balance  that  moment  when  applied  at  a  given 
elevation,  e.  As  overturning  proceeds,  the  weight,  W,  remaining  un- 
changed, the  leverage  and  moment  of  stability,  and  the  overturning  moment 
required,   decrease,   becoming    =   0  when  the  bodies    reach   the  positions 


STABILITY. 


423 


shown  by  the  dotted  lines.  If  the  elevation,  e,  remains  constant,  the  force, 
P,  required  for  overturning,  decreases  in  the  same  proportion  as  the  lever- 
age, etc. 

319.  But  in  order  that  the  bodies  may  be  overturned  by  the  force  of  grav- 
ity alone,  they  must  be  brought  into  the  positions  shown  by  the  dotted  lines. 
This  requires  that  the  weights  of  the  bodies  be  lifted  through  a  height  =  the 
distance,  h,  through  which  their  centers  of  gravity,  G,  are  raised.     Hence 


work    of    overturning 


W.A. 


Since  h  is  greater  in  Fig.  (6),  the  work  of  overturning  is  greater  in  that  case. 

In  civil  engineering  we  are  generally  concerned  with  the  amount  of  the 
force  which  will  begin  overturning,  rather  than  with  the  amount  of  work 
required  to  complete  the  overthrow. 

220.  Stability  against  overturning  is  of  course  affected,  and  may  be  in- 
creased, by  forces  other  than  the  weight  of  the  body  itself.  Thus,  the 
stability  of  a  bridge  pier  is  ordinarily  increased  by  the  weight  of  the  bridge 
itself  if  this  be  brought  upon  the  pier  symmetrically.  Otherwise  the  weight 
of  the  bridge  may  either  increase  or  diminish  the  stability  of  the  pier,  accord- 
ing to  circumstances. 

221.  The  coefficient  of  stability,  in  any  given  case,  is  the  ratio  of  the 
moment  of  stability  to  the  overturning  moment.     Or, 


Coefficient  of  stability  = 


moment  of  stability 
overturning  moment' 


222.  Let  the  weight,  W,  of  the  stone  in  Fig.  96  be  10  lbs,,  G  its  center  of 
gravity,  and  o  g  =  2  feet.  Then  the  moment  of  the  weight  about  o,  or  the 
moment  of  stability  about  o,  is  10  X  2  =  20  ft.-lbs,;  and,  if  o  n  =  5  feet,  a 

20 
force  P   =   —    =  4  lbs.,  will  just  hold  in  equilibrium  the  moment  of  the 

5 
weight,  so  that,  except  at  the  corner,  o,  no  pressure  will  be  exerted  upon  the 
base  o  m,  although  the  stone  remains  in  contact  with  the  base.     If  the  force 
P  exceeds  4  lbs.,  the  stone  will  begin  to  turn  about  o.     If  P  is  less  than  4  lbs., 
the  stone  will  exert  a  pressure  upon  the  base  o  m. 

Let  the  stone  be  supported  at  o  and  at  m  only.  The  leverage  of  the  sup- 
porting force  R,  at  m,  is  =  the  length  o  m  of  the  base,  =  I.  Let  P  =  1  and 
base  o  m  =  4.5  ft.     Then,  for  equilibrium, 


W  .  o  g  —  P. on  —  R.om  =  0; 
20  ft.-lbs.  —  1  X  5   =  4.5  X  R; 
20-5 
^  ~  -475" 


-  =  3.33 lbs. 


In  other  words,  a  vertical  upward  force,  R,  of  3.33  ..  .  lbs.,  at  m,  will 
maintain  equilibrium. 


-^^ 

^ 

o/  ff 

W 

m. 

K 

r — > 

If 

Fig.  96. 


223.  In  Fig.  97  (6),  let  g  be  the  center  of  gravity  of  the  load  W  and  the 

W    eg 
table,  combined.  Fig.  97  (a).     Then,  upward  reaction  of  6  =  — r-^r^.    Those 


of  a  and  c  may  be  similarly  found. 


ih 


424 


STATICS. 


224.  In  Fig.  98,  let  h  be  the  horizontal  force  exerted  at  the  crown  by  the 
left-hand  half  of  the  arch,  against  the  half-arch  shown,  and  e  its  leverage 
about  o.  Let  W  be  the  weight  of  the  half-arch  with  its  spandrel,  acting  as  a 
single  rigid  body,  and  I  its  leverage  about  o.     Then,  for  equilibrium,  we  have 


h  .  e  =  W  .  I ;  OT  h  = 


W  .1 


Stability   on   Inclined   Planes. 

225.  Stability  on  Inclined  Planes.  Fig.  99.  Here,  as  in  If  213,  if  the  re- 
sultant, R,  of  the  force  P  and  weight  W,  falls  beyond  the  base, — i.  e.,  if  the 
overturning  moment  exceeds  the  moment  of  stability, — the  body  will  over- 
turn.    If  not,  it  will  stand. 

The  force,  P,  in  any  given  direction,  required  to  prevent  overturning,  is 
=  the  anti-resultant.  A,  of  weight  W  and  reaction  II ;  and  reaction  II  = 
anti-resultant  of  force  P  and  weight  W. 

226.  Neglecting  friction,  as  in  Fig.  99  (a),  R  will  be  normal  to  the  plane. 
Taking  friction  into  account.  Fig.  99  (b),  R  may  form,  with  a  normal,  N,  to 
the  plane,  an  angle,  a,  not  exceeding  the  angle  of  friction  between  the  body 
and  the  plane.    R  may  be  either  uphill  or  downhill  from  N. 

227.  In  Fig.  100,  the  body  B  has  less  stability  against  overturning  about 
its  toe,  a,  than  has  the  similar  body.  A,  when  the  force,  n,  tends  to  upset  it 
downhill ;  but  a  greater  stability  than  A  against  overturning  about  c  under 
the  action  of  a  force  tending  to  upset  it  uphill. 

228.  The  body  C,  which  would  upset  if  upon  a  horizontal  base,  would  be 
stable  against  overturning  if  placed  upon  an  inclined  plane,  as  at  D.  Assum- 
ing ao  =  tc,a given  upward  vertical  force  would  have  the  same  overturning 


Fig.   100. 

moment,  whether  applied  at  a  or  at  c.  But  a  given  horizontal  force,  applied 
at  any  given  height,  as  at  g,  has  a  greater  leverage,  g  o,  when  pushing  down- 
hill than  when  pushing  uphill.     In  the  latter  case  its  leverage  is  only  g  t. 

229.  Structures  built  upon  slopes  are  liable  to  slide.  This  may  be  ob- 
viated by  cutting  the  slope  into  horizontal  steps,  as  at  d  y.  Fig.  E;  but  the 
vertical  faces  of  such  steps  break  the  bond  of  the  masonry ;  and,  moreover, 
the  joints  being  more  numerous,  and  the  mortar  therefore  in  greater  quan- 
tity, on  the  deeper  side,  s  d,  than  cJn  the  shallower  uphill  side,  e  y,  the  struc- 
ture is  liable  to  unequal  settlement,  the  downhill  side  settling  most  and  tend- 
ing to  split  away  from  the  uphill  portion,  as  might  be  the  case  with  a  founda- 


THE    CORD. 


425 


tion  firm  in  some  parts  and  compressible  in  others.  Hence,  when  circum- 
stances permit,  it  is  preferable  to  level  off  the  foundation,  as  at  d  v;  or,  if  the 
structure  has  to  withstand  downhillward  pressures,  v  shguld  be  lower  than 
d,  and  the  courses  of  masonry  laid  with  a  corresponding  inclination. 

THE    CORD. 
230.  The   Cord.     Figs.  101  (a)  and  (6)  and  102  (a)  and  (6).      In  f  H  230 
to  239  we  deal  with  cords  supposed  to  be  perfectly  flexible,  inextensible, 
frietionless,  weightless  and  infinitely  thin. 


Figr.  101. 

331.  Let  P  be  the  external  force  applied  to  the  cord  at  the  knot  or  pin,  o, 
and  let  R  be  the  resultant  of  the  stresses,  8\  and  s^,  or  o  a  and  o  b,  in  the  two 
segments,  o  m  and  o  n,  of  the  cord.  Then,  for  equilibrium,  R  must  be  equal 
to  and  colinear  with  P. 

833.  Knowing  the  amount  of  P  (  =  R),  the  tensions  S\  and  82  may  be 
found  by  means  of  If  36;  and,  vice  versa,  given  si  and  S2,  we  may  find  R 
( =  P)  by  1  35.     Or  see  H  40. 

233.  If,  as  in  Figs.  101  (a)  and  (6),  the  force  P  be  applied  to  the  cord,  at  o, 
by  means  of  a  fixed  knot,  incapable  of  sliding  along  the  cord,  so  that  the  seg- 
ments, o  m  and  o  n,  of  the  cord,  are  of  fixed  lengths,  and  the  angle,  x  -\-  y, 
between  them,  of  fixed  magnitude,  then  the  force  may  be  applied  in  any 
direction,  as  P  or  P',  passing  between  the  two  segments  of  the  cord ;  and  the 
components,  8\  and  sa,  will  be  equal  only  when  R  (P  produced)  forms  equal 
angles,  x  and  y,  with  the  two  segments  of  the  cord.  If  the  direction  of  the 
force,  as  P'',  coincides  with  either  segment,  as  o  n,  of  the  cord,  that  segment 
transmits  the  entire  force,  P'',  and  the  other  segment  none. 


Flff.   102. 

234.  But  if,  as  in  Figs.  102  and  103,  the  force  P  be  applied  to  the  cord  by 
rneans  of  a  frietionless  ring,  slip-knot,  pin  or  pulley,  etc.,  then,  for  equilib- 
rium, the  two  stresses,  s\  and  so,  must  be  equal,  as  must  also  the  two  angles, 
X  and  y;  and,  if  we  suppose  the  direction  of  the  force  P  to  be  changed,  as  to 
P',  the  pin  and  the  cord  will  readjust  themselves,  as  indicated  by  the  dotted 
lines  in  Fig.  103,  until  the  pin  finally  comes  to  rest  at  that  point,  o',  where 
the  angles,  x'  and  y',  are  equal,  and  also  the  stresses,  si'  and  s^'. 


426 


STATICS. 


^U 


a'\ 


.+-.--' 


^-rK"^ 


Fig.   103. 

335.  Even  though  the  pin  or  pulley  be  rigidly  fixed  to  some  external  ob- 
ject, as  at  o.  Fig.  104,  yet,  if  there  is  no  friction  at  its  axle,  or  between  it 
and  the  cord,  the  components,  Si  and  so,  will  still  be  equal,  and  their  resultant, 
R,  will  bisect  the  angle,  x  -\-  y,  between  them.  In  other  words,  the  angles, 
X  and  y,  will  be  equal. 


Fig.  104. 


Fig.  105.    ^ 


336.  When  the  pin  is  movable.  Figs.  102  and  103,  to  find  the  position,  o. 
Fig.  105,  which  it  will  assume.  From  the  end,  n,  of  one  of  the  segments,  o  n, 
of  the  cord,  draw  n  v  parallel  to  P.  From  the  end,  m,  of  the  other  segment, 
with  radius  =  m  o  -j-  o  n,  =  length  of  cord,  describe  an  arc,  cutting  nvind. 
Bisect  ndine.  Draw  e  o  normal  to  n  v,  intersecting  mdino.  Then  o  is  the 
required  point. 

337.  Whether  o  be  a  fixed  knot  or  a  movable  pin  or  pulley,  it  is  always  in 
the  circumference  of  an  ellipse  whose  foci  are  at  the  ends,  m  and  n,  of  the 
cord. 


Fig.  106 


338.  From  the  foregoing  it  follows  that,  if  o,  Fig.  106,  be  a  fixed  knot,  and 
if  the  other  pins  or  pulleys,  etc.,  are  frictionless,  the  stress  a  o,  or  8\,  will  be 
transmitted  uniformly  throughout  the  left  segment  of  the  cord,  from  o  to  its 
end  at  m;  and  h  o,  or  sa,  throughout  the  right  segment,  from  o  to  n. 


FUNICULAR   MACHINE. 


427 


239.  Caution.  Note  that,  in  Fig,  107  (6),  the  stresses  in  all  the  cords  are 
twice  as  great  as  the  stresses  in  the  corresponding  cords  in  Fig.  107  (a), 
although  each  Fig.  shows  a  load  =  4  suspended  from  the  pulley.  Thus,  if 
the  weight  be  that  of  a  man,  hanging  by  the  rope,  and  if  the  rope,  in  Fig.  (a), 
be  just  sufficiently  strong  to  hold,  it  will  break  if  he  gives  one  end  of  the  rope 
to  another  man  to  hold,  or  makes  it  fast,  as  in  Fig.  (6). 


v////////////////^^^^^^ 


2 


4 


(a) 

Fig.   107 


(b) 


The   Funicular   Machine. 

240.  When  the  angles,  x  and  y,  Figs.  101,  etc.,  are  very  great,  a  very  small 
force,  P,  will  balance  a  very  great  stress,  siors^,  in  the  cord.  When  x  =  y 
=  90°,  we  have  cos  x  =  cos  y  =  0,  and  si  =  s^  =  infinity,  however  small  P 
may  be.  If  a  line,  m  n,  joining  the  ends  of  the  cord,  is  horizontal  or  inclined, 
the  weight  of  the  cord  itself  acts  as  a  force  P.     Hence 

"There  is  no  force,  however  great,  can  stretch  a  cord,  however  fine,  into 
a  horizontal  line  that  shall  be  absolutely  straight." 

241.  The  funicular  machine  takes  advantage  of  the  fact  that,  when  the 
total  angle,  x  -j-  y,  between  the  two  segments  of  the  cord,  approaches  180°, 
a  small  force,  P,  may  balance  great  stresses,  si  and  82.  Thus,  in  Fig.  108,  let 
W  represent  a  heavy  boat  (seen  in  plan)  which  is  to  be  hauled  ashore.  One 
end  of  a  rope  being  made  fast  to  the  bow  of  the  boat,  the  rope  is  passed 
around  one  smooth  post,  n,  to  another,  m,  around  which  it  is  given  one  or 
more  whole  turns ;  and  a  man  stands  at  the  end,  e,  to  take  in  the  slack ;  while 
others,  taking  hold  of  the  rope  between  m  and  n,  pull  it,  in  the  direction  of  P, 
into  a  position  m  o  n.  If  the  two  angles,  x  and  y,  are  equal,  the  component 
in  the  segment  o  n  exceeds  P,  so  long  as  the  angle  x  exceeds  60°,  and  a 
pull,  equal  to  this  component  (except  in  so  far  as  it  is  reduced  by  the  rigidity 
of  the  rope  and  by  its  friction  against  the  post  n),  is  exerted  upon  the  boat  at 
W,  drawing  it  a  short  distance  up  the  beach.  The  rope  is  then  straightened 
again,  from  m  to  n,  by  taking  in  the  slack  at  e,  and  the  operation  is  repeated 
as  often  as  may  be  necessary. 


yp 


Fig.  108. 

The    Toggle    Joint. 

242.  The  toggle  joint.  Fig.  109,  is  simply  an  inversion  of  the  funicular 
machine  with  a  fixed  knot,  the  force  P  and  the  components,  S\  and  So,  being 
pushes  or  compressions,  instead  of  pulls  or  tensions.  The  joint  being  unable 
to  move  along  the  arms,  the  force  P  may  be  applied  in  any  direction  at  pleas- 
ure, but  it  is  usually  exerted  in  a  direction  forming  approximately  equal 
angles  with  the  two  arms. 


428 


STATICS. 


The  Pulley. 

343.  Figs.  110  show  the  relations  of  stresses  and  weights  in  several  ar- 
rangements of  fixed  and  movable  pulleys.  Thus,  in  (a),  1  lb.  balances 
1  lb.,  in  (6)  2  lbs.,  in  (c)  and  in  {d)  4  lbs.  In  each  case,  if  the  bodies  or 
weights  be  set  in  motion,  their  velocities  are  inversely  as  their  masses.  See 
^  206. 


Fisr.  110. 


244.  The  simple  pulley.  Fig.  110  (a),  is  used  simply  for  convenience  of 
changing  direction  of  stress,  for  the  forces  at  the  two  ends  of  the  cord  are 
equal;  but  in  the  compound  pulley,  Figs.  110  (6),  (c),  (d),  a  small  force  (the 
"power"),  moving  rapidly,  at  one  part  of  the  rope,  balances  a  greater  force 
(the  "weight"),  moving  slowly,  at  another  part.  Hence,  the  compound 
pulley  is  used  for  the  purpose  of  overcoming  great  resistances  slowly,  by 
means  of  small  forces,  moving  rapidly. 

245.  To  set  such  a  system  in  motion  *  (i.  e.,  to  raise  the  "weight")  re- 
quires that  the  equilibrium  be  disturbed  by  making  the  "power"  exceed  the 
stress  in  the  cord  due  to  the  "weight."  But  the  motion,  once  generated, 
will  continue  indefinitely  if  the  "power"  is  made  sufficiently  greater  than  th« 
"weight"  to  balance  the  resistances  of  friction,  etc. 

The  Loaded  Cord  or  Chain. 

246.  In  Figs.  Ill  the  principle  of  the  cord  polygon,  f  ^  86,  etc.,  is  applied 
to  the  case  of  a  flexible  cord  or  chain,  sustaining  four  loads,  pi  .  .  .  p^, 
at  fixed  points,  and  exerting  a  horizontal  t  puU,  H,  at  its  lower  end,  and  an 
inclined  pull,  R,  at  its  upper  end.  The  loads,  pi  .  .  .  p^,  are  represented 
by  the  vertical  line,  0-4,  Fig.  Ill  (a) ;  the  horizontal  pull,  H,  by  0-c;  the 
amount  and  direction  of  the  inclined  pull,  R,  at  the  upper  end  of  the  cord,  by 
4-c,  and  the  tensions  in  the  segments,  1-2,  2-3  and  3-4,  by  the  rays,  1-c,  2-c 
and  3-c,  respectively. 

247.  The  horizontal  tension,  H  (  =  the  horizontal  component  of  the  ten- 
sion in  each  segment),  is  uniform  throughout  the  cord;  but  the  vertical 
component  of  the  tension  in  any  segment  is  equal  to  the  sum  of  the  loads 
between  that  segment  and  the  pulley,  m.     Thus,  the  vertical  component 


*  See  foot-note  (*).  H  1. 
t  See  foot-note  (*),  If  249. 


LOADED  CORD. 


429 


(0-2,  Fig.  a)  of  the  tension,  c-2,  in  segment  2-3,  is  =  pi  +  P2 ;  that  in  seg- 
ment 3-4  is  0-3  =  Pi  -t-  P2  +  P3.  etc. 

248.  If  all  the  loads  (including  W)  be  increased  in  the  same  proportion,  as 
indicated  by  the  dotted  lines  in  Fig.  Ill  (a),  or  diminished  in  the  same  pro- 
portion, the  new  triangles,  c'  4'  0,  etc.,  Fig.  (a),  will  be  similar  to  the  old, 
and  the  profile  of  the  cord.  Fig.  (6),  will  remain  unchanged,  although  the 
stresses  in  its  segments  will  of  course  be  increased  or  diminished  in  the  same 
proportion. 

349.  In  Fig.  Ill  we  make  the  weight,  W,  which  is  necessarily  equal  to 
the  horizontal*  pull,  H    (see  The  Cord,  1[*[f  230,  etc.),   equal  also  to  the 


Fiff.  112. 


sum  of  the  loads,  pi  .  .  .  p4.  When  this  is  the  case,  the  cord  segment,  a-4, 
next  to  the  support,  a,  and  the  corresponding  line,  c-4.  Fig.  (o),  will  be  in- 
clined 45°  to  the  vertical. 

250.  But  if,  while  the  loads,  pi  .  .  .  P4,  remain  unchanged,  we  raise  the 
pulley  m,  so  as  to  keep  H  horizontal,*  we  shall  obtain  a  flatter  curve,  as  in 
Fig.  112;  and,  for  equilibrium,  H  (=  W)  must  be  made  greater  than  the 
sum  of  pi  .  .  .  P4.  On  the  other  hand,  if  we  place  the  pulley,  m,  lower 
than  in  Fig.  Ill  (still  keeping  H  horizontal),  we  obtain  a  deeper  curve,  as  in 
Fig.  113;  and  H  (  =  W)  must  be  made  less  than  the  sum  of  pi  .   .  .  P4. 


*  In  Figs.  Ill,  112  and  113  we  suppose  the  weight,  W,  and  the  position  of 
the  pulley,  m,  to  be  so  adjusted,  relatively  to  the  support,  a,  that  the  pull,  H, 
shall  remain  horizontal. 


430 


STATICS. 


ABCHES,  DAMS,  ETC.     THRUST  AND  RESISTANCE   LINES. 

The  Arch. 

In  ^1^  251  to  257  are  given  the  elements  of  the  commonly  accepted  theory 
of  the  arch.  For  practical  considerations,  see  Kl  258  to  266,  and  Stone 
Bridges. 

251.  If  Figs.  Ill,  112  and  113  be  inverted,  the  cord  segments  will  repre- 
sent struts,  sustaining  compression,  as  do  the  stones  of  a  masonry  arch,  Fig. 
114. 

Thrust  Line.     Resistance  Line. 

353.  In  the  case  of  an  arch.  Fig.  114,  assuming  *  that  the  horizontal  thrust 
H,  at  the  crown,  m,  and  the  reaction,  R,  of  the  skewback,  a,  act  at  the  center 
(or  at  some  other  definite  point)  of  crown  and  of  skewback,  respectively, 
their  amounts,  and  the  direction  of  the  reaction,  R,  may  be  found  by  means 
of  the  Force  Triangle,  t  51,  or  by  Moments,  1[  224.  (See  If  257.)  We  then 
suppose  the  half -arch  and  its  spandrel  to  be  divided,  by  vertical  planes,  *  Fig. 


Fi^.  114. 

114  (6),  into  a  number  of  segments,  as  shown;  and,  finding  the  weight  and 
the  center  of  gravity  of  each  such  segment  (see  1f^  257  and  266),  we  treat 
these  segments  as  we  treated  the  loads,  pi  .  .  .  p4,  of  Figs.  Ill  to  113,  lay- 
ing them  off  from  0  to  6,  Fig.  114  (a),  and  laying  off  0-c  horizontal  and  ==  H. 
The  rays,  c-1,  c-2,  etc.,  then  give,  theoretically,*  the  directions  and  amounts 
of  the  pressures  exerted  by  the  segments,  1,  2,  etc.,  respectively. 

The  broken  line,  m  d  e  f  .  .  .  .a.  Fig.  114  (6),  thus  formed,  is  called  the 
thrust  line,  or  line  of  resultants.  It  corresponds  with  the  cord  polygons  of 
Figs.  Ill  (6),  etc.* 

353.  The  resistance  line  is  a  broken  line  joining  the  points  where  the 
several  resultants,  forming  the  thrust  line,  cut  the  respective  joints  between 
the  arch  stones. 

354.  When  the  planes,  by  which  the  arch  is  supposed  to  be  divided  into 
segments,  are  vertical,*  as  in  Fig.  114  (6),  and,  indeed,  in  most  actual 
arches,  the  thrust  and  resistance  lines.  Fig.  115,  practically  coincide;  but  if 
these  planes  are  far  from  vertical,  as  in  Fig.  115,  the  two  lines  separate,  the 
resistance  line  being  always  the  outer  one. 

Thus,  in  Fig.  115  (where  the  thrust  line  is  shown  solid,  and  the  resistance 
line  dotted),  noticing  where  resultant  a  cuts  joint  A,  where  resultant  h  cuts 
joint  B,  etc.,  it  will  be  seen  that  the  two  lines  practically  coincide  as  far  as  to 
joint  C,  where  they  begin  to  diverge. 


*  See  Practical  Considerations,  ^^  258,  etc. 


ARCHES. 


431 


255*  In  ^  252  we  assumed  that  the  arch  and  its  spandrel  are  divided  into 
vertical  segments,  incapable  (except  in  the  arch  ring)  of  exerting  other  than 
vertical  pressures.  The  theoretical  resistance  line,  thus  obtained,  may, 
especially  in  deep  arches,  pass  from  the  thickness  of  the  arch  ring  in  places ; 
so  that,  if  no  other  forces  were  acting,  the  arch  would  open  at  such  places ; 
on  the  intrados  when  the  resistance  line  cuts  the  extrados,  and  vice  versa; 


Fig:.  115. 

but  such  opening  is  usu^tlly  prevented  by  other  forces,  such  as  the  horizontal 
or  inclined  pressures  of  the  spandrels.  The  actual  resistance  line  is  thus 
confined  within  the  thickness  of  the  arch  ring.  In  general,  the  actual  resist- 
ance line.  Fig.  116,  approaches  the  extrados  at  the  crown,  and  the  intrados 
at  the  haunches,  so  that  the  arch  tends  to  sink  at  the  crown  (opening  there 
on  the  intrados),  and  to  rise  at  the  haunches  (opening  there  on  the  extrados), 
as  shown. 


Fig.  116. 


256.  In  order  to  avoid  any  tendency  of  the  joints  to  open  at  either  side, 
the  arch  should  be  so  designed  that  the  actual  resistance  line  shall  every- 
where be  within  the  middle  third  (see  tH  145,  etc.)  of  the  depth  of  the  arch 
ring. 

257.  In  general,  the  design  of  an  arch  is  reached  by  a  series  of  approxima- 
tions. Thus,  a  form  of  arch  and  spandrel  must  be  assumed  in  advance,  in 
order  to  find  their  common  center  of  gravity  for  the  purpose  of  determining 
the  horizontal  thrust,  H,  and  the  skewback  reaction,  R,  as  in  ^  252;  and,  if 
it  is  afterward  found  necessary  to  modify  the  form  first  assumed,  in  order  to 
satisfy  the  requirements  of  ^  256,  or  for  other  reasons,  we  may  have  to  re- 
compute H  and  R,  again  modifying  the  design,  and  so  on. 


432  STATICS. 

Practical  Considerations. 

358.  While  the  theoretical  thrust  and  resistance  lines,  based  upon  the 
foregoing  assumptions,  are  easily  found,  much  uncertainty  exists  as  to  the 
positions  of  the  actual  thrust  and  resistance  lines  in  a  masonry  arch. 

359.  In  the  first  place,  we  do  not  know  through  what  points  in  the  crown 
and  skewback,  respectively,  the  resultants,  H  and  R,  pass. 

260.  Again,  we  have  assumed  that  the  loads  on  the  arch,  like  those  on  the 
cord.  Figs.  Ill  to  113,  are  incapable  of  acting  otherwise  than  vertically; 
whereas  the  spandrel  walls  and  filling,  which  form  a  large  portion  of  the  load 
on  a  masonry  arch,  may  offer  resistances  acting  in  other  directions.  If  the 
loading  were  a  liquid,  like  water,  its  pressures  upon  the  arch  ring  would  be 
radial,  like  those  of  the  particles  of  steam,  in  a  boiler,  upon  the  boiler  tubes ; 
and  this  condition  is  probably  more  or  less  closely  approximated  in  the  case 
of  a  loading  of  clean  dry  sand;  and,  less  closely,  in  the  case  of  earth  filling. 
Hence,  although  the  determination  of  the  theoretical  thrust  and  resistance 
lines  in  an  arch  is  facilitated  by  the  assumption  that  the  arch  is  correctly 
represented  by  the  inverted  cord,  the  distinction  between  the  two  cases  must 
be  borne  in  mind  when  drawing  practial  conclusions  from  the  lines  so  found. 

361.  Thus,  in  many  cases,  the  theoretical  thrust  and  resistance  lines  cut 
the  intrados  or  the  extrados  in  places,  thus  passing  entirely  out  of  the  arch 
ring;  so  that  this  would  inevitably  fall  (see  ^  255),  were  it  not  for  horizontal 
or  inclined  resistances  exerted  by  the  upper  parts  of  the  abutments  through 
the  spandrel  walls  and  filling. 

363.  Hence,  in  order  to  determine  the  actual  resistance  line,  we  should 
not  only  have  to  know  through  what  points,  in  crown  and  in  skewback 
respectively,  the  resultants,  H  and  R,  pass,  but  we  should  also  have  to  ascer- 
tain and  take  into  account  the  possible  horizontal  and  inclined  resistances  of 
the  spandrel  walls  and  filling.  But,  as  this  is  ordinarily  impracticable,  we 
content  ourselves  either  with  determining  the  theoretical  thrust  and  resist- 
ance lines,  as  directed  above,  and  then  estimating,  as  well  as  may  be,  the 
resistances  of  the  spandrels,  or  with  reasoning  by  analogy  from  the  behavior 
of  actual  structures.     See  Stone  Bridges. 

363.  If  the  inverted  cord  correctly  represented  the  actual  thrust  line  in 
a  masonry  arch,  the  arch  stones,  in  elliptic  or  in  tieep  segmental  arches, 
twould  have  to  be  made  inordinately  deep,  in  order  that  the  resistance  line 
should  nowhere  leave  the  middle  third  of  their  depth  (see  ft  145,  etc.); 
and  it  might  therefore  appear  rational  to  make  the  profile  of  the  arch  corre- 
spond approximately  with  the  thrust  line,  which  usually  approaches  a  para- 
bola. But,  owing  to  the  spandrel  resistances,  the  actual  thrust  line,  even  in 
semicircular  arches,  probably  seldom  greatly  oversteps  the  middle  third. 

364.  With  a  wall  or  a  deep  continuous  filling,  over  an  arch,  if  the  arch  were 
to  settle,  or  were  to  be  removed,  the  wall  and  the  filling  above  it  would  form 
an  arch,  as  indicated  by  the  broken  lines  in  Fig.  117;  and  only  that  portion 


Fig.  117. 

below  this  arch  would  fall  out.     Hence,  only  this  portion  can  properly  be 
regarded  as  pressing  upon  the  arch. 

365.  Neglecting  the  strength  of  the  mortar,  the  inclination  of  each  joint 
between  two  arch  stones  must  of  course  be  such  that  the  angle,  between 
the  thrust,  at  any  joint,  and  a  normal  to  that  joint,  shall  be  less  than  the 
angle  of  friction.     See  i^  183,  184. 


DAMS. 


433 


266.  It  is  often  the  case  that  the  spandrels  or  the  spandrel  filling  are  of 
less  specific  gravity  than  the  arch  ring.  In  such  cases,  in  order  to  facilitate 
the  finding  of  the  lines  of  gravity  of  the  segments,  we  may,  before  dividing 
the  half -arch  and  its  spandrels  into  vertical  segments  (^  252),  consider  the 
lighter  structure  of  the  spandrels  as  being  reduced  to  an  equivalent  depth 
of  material  having  equal  specific  gravity  with  the  arch.  The  areas  of  the 
several  segments,  as  seen  in  profile,  and  as  thus  reduced,  may  then  be  taken 
as  representing  their  weights.     Thus,  in  Fig.  118,  where  t  1 1  represents  the 


Fig.  118. 


top  of  the  spandrels,  the  curved  line  e  e  e  represents  the  top  of  a  filling  of 
equal  weight  per  foot  run  with  the  spandrels,  but  of  equal  specific  gravity 
with  the  arch  ring.  When,  as  in  Fig.  119,  the  spandrels  consist  of  a  series  of 
transverse  arches,  we  may  assume  that  the  main  arch  carries  a  series  of  loads 
concentrated  at  the  piers  of  these  transverse  arches. 


Fig.  119. 


The  Masonry  Dam. 

267.  A  dam  must  be  secure  against  sliding,  on  its  base  or  on  any  plane 
,  within  the  body  of  the  dam,  against  overturning,  and  against  crushing  of  the 

material  at  any  point  and  consequent  opening  of  a  seam  at  either  face  of  the 
dam. 

268.  The  dam  will  be  secure  against  sliding  if  the  resultant  of  all  the  pres- 
sures, upon  any  surface,  forms,  with  a  normal  to  that  surface,  an  angle 
less  than  the  angle  of  friction  of  the  surface.  See  ![•[[  183,  etc.  In 
practice,  the  base  of  the  dam  is  let  wpU  down  into  the  rock  foundation,  as 
indicated  in  Fig.  122  (a),  and  continuity  of  joints  is  avoided  by  making  all 
the  stones  break  joints.  The  angle  of  friction  thus  becomes,  in  effect,  90°, 
and  sliding  cannot  occur  without  shearing  the  stones  themselves. 

269.  If  the  material  is  sufficiently  strong  to  resist  crushing,  under  the 
maximum  unit  stresses  brought  upon  it,  and  if  the  resultant  of  all  the  forces 
acting  upon  any  section  falls  within  the  body  of  the  dam,  the  dam  will  be 
secure  against  overturning.     But  see  H  270. 

270.  For  a  given  total  pressure  upon  any  section,  the  maximum  unit 
pressure  in  the  section  would  be  least  when  the  resultant  cut  the  middle 
point  of  the  section:  See  Center  of  Pressure,  HH  133,  etc.  It  is  generally 
impracticable  to  secure  this ;  but  the  dam  must  be  so  designed  that,  under 
the  maximum  unit  pressure,  the  given  material  shall  not  be  taxed  beyond 
its  safe  crushing  strength.  If  this  is  done,  and  if,  under  all  conditions,  the 
center  of  pressure  is  kept  within  the  middle  third  (see  T[  150)  of  each  hori- 
zontal section  throughout  the  dam,  there  will  be  no  tendency  to  open  on 
either  face  of  the  dam. 

28 


434 


STATICS. 


271.  Let  Fig.  120  represent  a  stone  block,  resting  upon  a  solid  foundation 
and  intended  to  sustain  the  pressure,  p,  of  quiet  water  on  one  side.  Through 
the  center  of  gravity,  g,  of  the  block,  draw  o'  N  vertically,  to  represent  the 
weight,  W,  of  the  block.  Then  the  point,  8,  where  o'  N  meets  the  foundation, 
is  the  center  of  pressure  for  the  block  alone,  i.  e.,  when  the  water  is  removed. 

372.  Let  h  be  the  depth  of  water  back  of  the  block,  and  let  the  block  be 
one  foot  in  length,  measured  normally  to  the  paper.  Then  the  amount,  in 
pounds, of  the  water  pressure,  against  the  vertical  back,  a  6,  is  p=62.5  AX>2^ 
and  its  center  of  pressure  is  at  a  depth,  d  ^  %  A,  below  the  water  surface. 

273.  Combining  p  with  W  (^  35)  we  obtain  R  as  their  resultant,  and  r 
as  the  center  of  pressure  upon  the  foundation  when  the  block  is  sustaining 
the  water  pressure. 

274.  Let  Fig.  121  represent  several  such  blocks  superposed.     Let 
Qi  =  cen  of  grav,  pi  =  cen  of  water  pres,  for  block  1 ; 

02  =    "     "       "     P2  =    "     "       "         "       "    blocks  1  and  2  combined; 

03  =  "  "  "  Pa  =  "  "  "  "  "  *-  1.  2,  and  3  combined; 
etc. 

Then,  finding  ri  and  si,  r2  and  s^,  rs  and  S3,  etc.,  for  joints  1-2,  2-3,  3-4, 
etc.,  as  before,  we  have  the  points  rj,  ro,  rs,  etc.,  in  the  resistance  line  for  full 
dam,  and  the  points  si,  82,  S3,  etc.,  in  the  resistance  line  for  empty  dam. 


Fig.   121. 


275o  In  Fig.  122  (a)  the  curves,  u  u  and  d  d,  indicate  the  up-stream  and 
down-stream  limits,  respectively,  of  the  middle  third  of  the  plane  separating 
each  two  blocks,  assuming  the  profile  with  vertical  back ;  and  the  points 

ri.  .  .  .  rj  and  aj s-j  are  points  in  the  corresponding  resistance  lines  for 

full  dam  and  for  empty  dam  respectively. 

276.  While  theory  would  require  the  cross-section  of  the  dam  to  terminate 
in  a  sharp  angle  at  the  top,  it  is,  of  course,  always  made  heavier  in  practice, 
as  indicated  in  Fig.  122  (a). 

277.  For  joint  2-3  we  may  suppose  that  block  2  had  at  first  been  designed 
rectangular,  as  shown  by  dotted  line  c  a  (Fig.  122  (c),  showing  blocks  1  and 
2  enlarged) ;  but  this  makes  the  center  of  pressure,  r\  for  the  full  dam,  fall 
beyond  the  middle  third  of  the  narrow  base,  a  b.  We  therefore  try  the  trape- 
zoidal shape  cib,  with  its  wider  base,  i  b,  and.  find  that,  with  this,  the  center 
of  pressure,  ro,  although  further  down-stream  than  before,  falls  within  the 
middle  third  of  said  wider  base.  The  remainder  of  the  profile  is  determined 
by  similar  trials. 


DAMS. 


435 


278.  Graphic  Method.  Suppose  the  cross-section  to  be  divided,  by 
horizontal  sections,  into  numerous  blocks,  1,  2,  3.  4,  etc.,  of  a  depth  approxi- 
mately =  the  top  width  of  the  dam.  In  Fig.  122  (6),  draw  0-1,  1-2,  2-3,  3-4, 
etc.,  vertically,  to  represent,  by  scale,  the  weights  of  the  several  blocks, 
respeatively,  and  0-1',  l'-2',  2'-3',  3'-4',  etc.,  horizontally,  to  represent 
the  water  pressures  against  said  blocks  respectively;  and  draw  I'-l,  2'-2, 
3'-3,  4'-4,  etc.,  representing,  in  amount  and  in  direction,  the  total  inclined 
pressures  upon  joints  1-2,  2-3,  3-4,  etc.,  and  upon  the  base,  respectively. 
Thus,  2'-2  represents  the  resultant  of  the  water  pressure  (see  Hydrostatics) 
upon  blocks  1  and  2,  and  the  combined  weight  of  those  blocks.  From  the 
points,  oi,  02,  etc..  Fig.  122  (c),  where  these  two  forces  meet  in  each  case,  draw 
oi  ri,  02  Ti,  etc.,  parallel  respectively  to  I'-l,  2''-2,  etc..  Fig.  122  (6),  to  the 
corresponding  joint.  We  thus  obtain  points,  ri  .  .  .  Tq,  in  the  resistance 
line  for  the  case  where  the  dam  is  filled  to  the  assumed  depth. 

The  foregoing  refers  to  the  diagram  for  a  dam  already  completed  or  de- 
signed. In  designing  de  novo,  we  of  course  begin  at  the  top,  and  lay  off 
the  lines  0-1,  0-1'  and  I'-l  in  Fig.  (6)  for  the  first  block;  then  lines 
1-2,  l'-2'  and  2'-2,  for  the  second  block,  and  so  on;  making  necessary 
changes,  as  in  ^  277. 


01' 2'    3'       4'  («) 


(C) 


Ol 

1 

c 

1 

1 

1 

1*1 
/  * 

>2 

/ 

/ 

h 

1    c 
i 

ih  *r'  6 

\ 

279.  In  order  that  the  resistance  line,  n  .  .  .  ro,  for  the  full  dam,  may 
be  brought  well  within  the  middle  third,  it  may  sometimes  be  necessary  to 
adopt  a  somewhat  unwieldy  cross-section;  but,  in  view  of  the  imminent 
danger  involved  in  the  smallest  opening  on  the  up-stream  side  of  the  dam, 
(see  H  281),  it  is  well  here  to  err  on  the  safe  side. 

280.  As  this  process  is  carried  further  down,  the  angle  formed  between 
the  down-stream  face  and  the  vertical  becomes  considerable ;  and  the  middle 
third,  in  each  of  the  lower  joints,  is  thus  brought  further  down-stream. 
The  centers  of  pressure,  s\  .  .  .  se,  for  empty  dam,  may  then  fall  beyond 
the  middle  third,  on  the  up-stream  side,  as  indicated  at  joints  5-6  and  6-7. 
To  obviate  this,  the  up-stream  face  is  sometimes  given  a  curved  profile,  as 
at  mn. 


436  STATICS. 

Practical  Considerations. 

281.  The  assumption  of  ideal  conditions  is  particularly  danger- 
ous in  the  case  of  masonry  dams.  Thus,  any  compression  of  the  material 
at  the  down-stream  face  may  open  seams  on  the  up-stream  face;  and  water, 
entering  these  seams,  will  exert  a  wedge-like  action,  shifting  the  resistance 
line  further  down-stream,  thus  still  further  increasing  the  tendency  to  crush- 
ing on  the  down-stream  face  and  to  opening  on  the  up-stream  face.  Again, 
if  any  relatively  smooth  joints  have  been  left,  the  water,  thus  penetrating 
into  or  under  the  dam,  increases  the  tendency  to  slide,  not  only  by  diminish- 
ing the  effective  weight  of  the  upper  portions,  but  also  by  acting  as  a  lubri- 
cant upon  the  seam  where  it  penetrates. 

It  has  been  suggested  that  failures  of  dams  may  have  been  occasioned, 
in  part  at  least,  by  vacuum,  formed  in  front  of  the  down-stream  face,  by  the 
action  of  the  sheet  of  water  falling  in  front  of  that  face. 

283.  Theoretically,  the  deflections  of  arches,  dams  and  other  structures 
composed  of  blocks,  may  be  found  by  means  of  the  formulas  in  ^^  162-167 
of  Trusses;  but,  owing  to  uncertainty  as  to  the  values  of  the  moduli  of 
elasticity,  E,  of  building  stones  and  of  mortar,  and  to  the  relative  inaccu- 
racy of  finish  in  masonry  work,  the  formulas  are  of  but  little  practical 
value  in  such  cases. 

THE  SCREW. 

283.  The  screw  is  a  spiral  inclined  plane.  The  force  (or  "power")  de- 
scribes a  spiral,  at  the  end  of  a  lever  arm,  while  the  resistance  (or  "weight") 
moves  along  the  axis  of  the  screw.  During  the  time  in  which  the  force 
makes  one  revolution,  the  resistance  traverses  the  "pitch,"  or  distance  be- 
tween the  centers  of  two  adjacent  threads. 

284.  Hence,  if  P  =  power,  w  =  weight,  d  ==  pitch,  I  =  lever  arm,  v  = 
rectilinear  velocity  of  weight,  and  V  =  linear  (circular)  velocity  of  power, 
ive  have,  theoretically  :* 

^  =  Y.  =  l,""! 

P  V  d    ' 


*  Neglecting  friction,  which,  however,  very  greatly  modifies  the  result. 


EQUILIBRIUM   OF    BEAMS. 


437 


FORCES    ACTING    UPON    BEAMS    AND    TBUSSES. 


Conditions  of  Equilibrium. 

285.  In  beams  and  trusses,  for  equilibrium,  it  is  necessary  and  sufficient 
that  the  resisting  forces,  exerted  by  the  material  of  the  structure,  and  the 
moments  of  those  forces,  shall  balance  the  external  or  destructive  forces  and 
their  moments.  We  here  discuss  chiefly  the  destructive  forces.  For  the 
resisting  forces,  see  Stresses,  under  Trusses,  and  Beams  or  Transverse 
Strength,  under  Strength  of  Materials. 

286.  The  destructive  forces  are  (1)  the  loads  upon  the  structure,  includ- 
ing its  own  weight,  "live"  or  moving  loads,  wind,  etc.,  and  (2)  the  reactions 
of  the  supports.  We  shall  here  discuss  the  action  of  vertical  loads  only,  in- 
cluding (a)  the  dead  load,  or  the  weight  of  the  structure  itself,  together  with 
the  roadway,  etc.,  and  (b)  the  live,  moving  or  extraneous  load  of  vehicles, 
trains,  persons,  etc.  The  action  of  horizontal  loads  (wind,  centrifugal  force, 
etc.)  is  governed  by  similar  laws,  and  is  discussed  under  Stresses,  in  Trusses. 

287.  Let  Fig.  123  (a)  represent  a  cantilever,  resting  upon  a  support,  b, 
and  bearing  a  load,  W,  at  its  outer  end,  a.  The  cantilever  is  prevented  from 
turning  about  b,  by  the  tension,  T,  of  a  horizontal  chain,  and  by  the  compres- 
sion, C,  in  a  horizontal  strut.*      Neglecting  the  weight  of  'the  cantilever 


-J 

>}ooooop 


Fl^.  123. 


itself,  the  cantilever  is  acted  upon  by  four  external  forces,  forming  two 
couples;  one  couple  consisting  of  two  vertical  forces — viz.,  the  load,  W,  and 
the  reaction,  R',  of  the  support ;  the  other  couple  consisting  of  two  horizontal 
forces — viz.,  the  tension,  T,  near  the  top,  and  the  compression,  C,  near  the 
bottom.  Were  it  not  for  the  reaction,  R',  of  the  support,  b,  the  load,  W, 
would  pull  the  cantilever  downward,  as  indicated  in  Fig.  123  (6) . 
288.  In  Fig.  123  (a)  we  have: 

Algebraic  sum  of  vertical      forces  =  R'  —  W  =  0 ; 
"     "  horizontal      "      =  T  —  C  =  0; 
"  *t     u  moments,  about  any  point,  as  o, 


W.w 


R'.r     +     T.^     +     C.c 


0. 


*  In  Figs.  123  to  127,  inclusive,  and  Figs.  132  and  133,  showing  cantilevers, 
beams  and  part  beams,  acted  upon  by  loads,  by  reactions,  by  pulls  of  chains 
and  by  pushes  of  struts,  the  arrows  denote  forces  acting  upon  the  cantilever 
or  beam,  or  upon  its  segments,  and  not  forces  acting  upon  the  load,  the  sup- 
ports, or  the  connecting  chains  or  struts.  Thus,  the  tension  in  a  chain  tends 
to  draw  together  the  two  bodies  which  it  connects.  Hence,  in  these  cases,  the 
corresponding  arrows  point  toward  each  other.  On  the  other  hand,  the  com- 
pression in  a  strut  tends  to  separate  the  two  bodies  between  which  it  acts. 
Hence  its  two  arrows  point  away  from  each  other. 


438 


STATICS. 


389.  If,  as  in  Fig.  124,  the  horizontal  forces  are  exerted  at  the  end  farthest 
from  the  support,  and  at  the  same  distance  apart  as  before,  their  amounts 
and  senses  must  remain  respectively  the  same  as  before ;  but  we  now  have 
compression,  C,  at  the  top,  and  tension,  T,  below.  Or,  if  Fig,  123  be  in- 
verted, R'  acts  as  the  load,  and  W  as  the  upward  reaction ;  and  we  have,  as 
in  Fig.  124,  compression,  C,  at  top,  and  tension,  T,  below.  Thus,  Fig.  124 
is  practically  Fig.  123  inverted. 


390.  The  condition  described  in  ^  289,  Fig.  124,  represents  also  the  condi- 
tion in  each  segment.  A,  B,  of  a  beam.  Figs.  125  (a)  and  (6)  or  Figs.  126  (a) 
and  (6),  supported  at  both  ends  and  bearing  a  concentrated  load,  W  +  W, 
Fig.  125,  or  W  +  w,  Fig.  126. 


Figr.  125. 


391.  Suppose  the  beam.  Fig.  125  (a)  or  Fig.  126  (a),  to  be  divided 
into  two  cantilevers,  or  part  beams,  as  in  Fig.  125  (b)  or  Fig.  126  (b) ;  each 
part  sustaining,  at  its  end,  a  part  of  the  original  load.  (See  1  292.)  The 
stresses  in  the  strut  and  chain.  Figs.  (6),  take  the  place  of  stresses  in  the  ma- 
terial (situated  in  the  dotted  line)  of  the  truss  or  beam,  Figs.  (a).  In  a  truss, 
these  forces  are  exerted  by  the  chords :  in  a  beam,  by  the  particles  or  fibers 
throughout  the  section. 


jjr    CJ^, 


Fi^.  126. 


393.  If,  as  inFig.  125  (a),  the  load  is  at  the  center  of  the  span,  the  spans, 
X  and  y,  of  the  cantilevers.  Fig.  125  (6),  are  equal,  as  are  also  the  loads, 
W  =  W,  carried  by  them.  But  if,  as  in  Fig.  126  (a),  the  load,  W  +  w,  on 
the  beam,  is  not  at  the  center  of  the  span,  the  partial  loads,  W  and  w,  sup- 
posed to  be  supported  at  the  ends  of  the*  two  cantilevers,  or  part  beams,  re- 
spectively, Fig.  126  (6),  are  unequal,  and  inversely  proportional  to  their 
leverages  about  their  respective  supports.  Hence,  the  moments  of  the  two 
opposite  couples  are  equal.  The  reaction  of  each  support  is  equal  to  the 
weight  carried  by  the  cantilever  resting  upon  it. 


END   REACTIONS. 


439 


End    Reactions. 

393.  In  a  cantilever,  Fig.  127,  there  is  but  one  vertical  support ;  the  reac- 
tion, R',  of  that  support,  is  =  the  sum  of  all  the  loads,  including  the  weight 
of  the  cantilever  itself;  and  the  reaction  due  to  each  partial  load  is  =  suck 
partial  load.     Thus,  if  B  =  weight  of  cantilever, 
R'  =  W  +  m;  +  B. 


Q^a 


Fig.  127. 


Tig.  128. 


W 


294.  The  reaction,  R',  must  not  be  confounded  with  other  vertical  forces. 
Thus,  a  cantilever  is  often  supported  as  in  Fig.  128  (a).  The  couple,  com- 
posed of  two  horizontal  forces,  T  and  C,  Fig.  127,  is  then  replaced  by  a  couple 
composed  of  two  vertical  forces,  V  and  V,  Fig,  128  (b) ;  one  of  which,  V,  co- 
incides with  the  reaction,  R'.  Here,  R'  +  V',  acting  upward,  is  the  anti- 
resultant  of  W,  w,  B  and  V,  acting  downward. 

295.  In  a  beam,  Fig.  129,  the  sum  of  the  two  end  reactions  is  =  the  sum 
of  all  the  loads,  including  the  weight  of  the  beam  itself. 

296.  The  reaction,  R,  of  the  left  support,  a,  Fig.  129,  due  to  the  load, 
W,  alone,  is  R  =^W  .  ^  (see  ^   17),  and  the  reaction,  R',  of  the  right 

support,  6,  is  =  W  —  R  =  W  .  -y .     If  the  load  is  central,  T^~i^~2'  ^^^ 

W 
R  =  R'  =  |. 


i 


m 


Figr.  129. 


Tig.  130. 


297.  Graphically,  Fig.  156,  suppose  a  concentrated  load,  W  (not  shown), 
to  be  placed  on  the  beam  at  any  point,  as  c.  Draw  a'  a"  anid  b'  b",  vertical 
and  each  =  W.  Join  a"  b'-,  also  join  a'  b",  and  draw  g  h  vertically  through 
c'.  Then  the  ordinate,  c'  g,  to  the  upper  line,  a"  b',  and  the  ordinate,  c'  h, 
to  the  lower  line,  a'  b'\  give  the  left  and  the  right  end  reactions,  R  and  R', 
respectively. 

298.  Where,  as  in  Fig.  130,  there  are  two  or  more  loads  (in  which  the 
weight  of  the  beam  may  or  may  not  be  included),  the  reactions  due  to  each 
load  may  be  separately  obtained,  the  sums  of  these  reactions  giving  the  total 
reactions ;  or,  the  common  center  of  gravity,  G,  of  all  the  loads  may  first  be 
found  (see  ^i[  125,  etc.),  and  then  the  reactions  found  as  for  a  single  load, 
W,  Fig.  129;  the  combined  weight  of  the  loads,  whose  center  of  gravity  is  at 
G,  being  supposed  concentrated  there. 


440 


STATICS. 


299.  In  a  beam,  Fig.  131,  under  a  load,  W,  uniformly  distributed  over  any 
part  of  the  span,  let  G  be  the  center  of  gravity  of  the  load,  and  let  x  and  y  be 
the  segments  of  the  span,  Z,  to  the  left  and  right  of  G  respectively.     Then, 

neglecting  the  weight  of  the  beam,  R  =  W  y ;  and  R'  =  W  —  R  =  W  — . 


W 


J 1 

Fig.  131. 


300.  If  the  load  is  uniformly  distributed  over  the  entire  span,  its  center 
of  gravity  is  at  the  center  of  the  span,  and  we  have : 

-  »  f  -  -I    and   R  -  R'=  -^ . 

Moments  and  Shears. 

301.  In  order  to  determine  what  internal  stressed  are  required,  at  any 
point  in  the  span,  to  maintain  equilibrium,  we  may  suppose  the  cantilever 
or  beam  to  be  cut  in  two  by  a  section,  c  c.  Fig.  132  or  Fig.  133,  at  such  point, 
and  inquire  what  forces  must  be  applied,  in  the  section,  in  order  to  main- 
tain equilibrium  and  hold  in  position  the  two  segments,  E  and  F,  into  which 
the  section,  c  c,  divides  the  span,  Fig.  132,  or  that  part  of  the  span  between 
the  load  and  a  support,  Fig.  133.  The  forces,  so  ascertained,  are  evidently 
equivalent  to  those  actually  exerted,  for  the  same  purpose,  by  the  material 
of  the  beam  itself. 

302.  In  Figs.  132  and  133,  moments  of  loads  and  of  reactions,  or  extern 
nal  or  bending  moments,  are  indicated  by  arrows  below  the  cantilever 
and  beam  respectively ;  while  the  resisting  moments  of  the  internal  forces 
are  indicated  by  arrows  within  the  body  of  the  cantilever  or  beam  respec- 
tively. 

303.  In  the  cantilever.  Fig.  132,  the  load,  w,  =  4  lbs.,  distant  6  ft. 
from  the  section,  e  c,  produces  there  a  left-hand  or  negative  moment  of6w~ 
6  X  4  ==  24  ft. -lbs.  Hence,  for  equilibrium,  the  horizontal  strut  and 
chain,  at  c  c,  must  exert  a  right-hand  or  positive  resisting  moment  of  24 
ft.-tt)s. ;  and,  being  2  ft.  apart,  they  must  exert  a  tension,  T,  and  compression, 

24 
C,  of  —   =  12  lbs.  each.     At  the  support,  moment  of  load  =  9w  =  9X4i'= 

36  ft.-lbs.;  and  T'  -  C  =  ^®  =  18  lbs. 

304.  But,  considering  only  the  forces  thus  far  discussed,  we  should  find  the 
right  segment,  F,  acted  upon,  at  c  c,  by  a  left-hand  couple,  =  rf  X  T  == 
rfXC  =  2X12<=  24  ft.-lbs.;  and,  at  the  support,  by  a  right-hand  couple, 
=  rf  X  T'  =  d  X  C  =  2  X  18  =  36  ft.-lbs.  In  other  words,  there  would 
be  an  unbalanced  excess  of  right-hand  moment,  =36  —  24  =  3  R'  =  3  X 
4  =  12  ft.-lbs.,  acting  upon  F.  F  also  receives,  at  the  support,  the  upward 
reaction,  R',  =  4  lbs.,  of  that  support.  Similarly,  the  couple,  rf  X  T  = 
d  X  C,  at  c  c,  exerts,  upon  the  left  segment,  E,  an  apparently  unbalanced 
right-hand  moment  of2X  12  =  6iy  =  6X4==24  ft.-lbs.,  and  E  receives, 
from  the  load,  w,  a  downward  pull  =  4  lbs. 

305.  For  equiliorium,  therefore,  the  vertical  chain  at  c  c  must  exert  a 
tension  =  S  =  w?=R'  =  4  lbs.,  pulling  F  downward,  and  E  upward.  The 
downward  tension,  — S,  acting  on  F  at  c  c,  forms,  with  the  reaction,  R',  of 
the  support,  a  left-hand  couple  =  3R'  =  3X4=12  ft.-lbs.,  balancing 
the  excess  of  right-hand  moment  acting  upon  F;  while  the  upward  tension, 
4-  S,  acting  on  E  at  c  c,  forms,  with  the  weight,  w,  a  left-hand  couple,  =  6v> 
=  6  X  4  =  24  ft.-lbs.,  balancing  the  excess  of  right-hand  mom.  acting  on  E. 


MOMENTS   AND   SHEARS. 


441 


306.  Similarly,  if  we  suppose  the  cantilever  cut  through  by  a  section  at 
any  other  point,  we  shall  find  that  a  vertical  force,  =  8  =  w  ==  R\  acting 
upward  upon  the  left  segment  and  downward  upon  the  right  segment, 
is  required  in  order  to  maintain  equilibrium  and  to  transmit  the  load,  w^ 
to  the  support,  so  that  the  two  segments  may  act  unitedly  as  a  single 
cantilever.  This  force,  S,  is  called  a  shear.  See  m  325,  etc.  Without  it, 
section  E  would  fall,  as  in  Fig.  123  (6). 

307.  In  the  beam,  Fig.  133.  the  total  load  is  16  lbs. ;  and,  its  distances, 
3  ft.  and  9  ft.,  from  the  left  and  from  the  right  support  respectively,  being 
as  1  to  3,  the  end  reactions  (^H  293,  etc.)  are  as  3  to  1 ;  or  R  =  16  X  i 
=  12  lbs.;  R'  =  16  X  i  =  4  lbs.  We  therefore  regard  the  beam  as  be- 
ing cut  by  a  section  at  the  load  (as  well  as  at  c  c),  and  the  total  Joad  of 
16  lbs.  as  divided  into  two  portions;  one,  W  =  R  =  12  lbs.,  attached  to  the 
end  segment,  M;  and  the  other,  w  =  R'  =  4:  lbs.,  supported  by  the  mid- 
dle segment,  E.  Here,  as  in  Fig.  132,  segments  E  and  F  together  form 
a  cantilever,  9  ft.  long,  loaded  with  a  weight,  w,  of  4  lbs.,  at  its  end;  but, 


Fl§r.  132. 


Wig.  133. 


in  Fig.  133,  the  horizontal  resisting  forces,  T'  and  C,  by  which  the  entire 
cantilever  (E  +  F)  is  upheld,  are  exerted,  not  at  the  support,  as  in  Fig.  132, 
but  at  the  end  farthest  from  the  support. 

308. , We  have,  therefore,  in  Fig.  133  : — at  c  c. 

Bending  moment,  positive, 

Acting  onE  =  2T'  —  6w;  =  2X  18  —  6X4  =  36  —  24  =12  ft.-lbs. 

Acting  on  F  =  3R' =  3X4=12  ft.-lbs. 

Resisting  moment,  negative,  =2T  =  2C  =  2X6  =  12  ft.-lbs.  Hence, 
T  =  C  =  6  lbs.;  and  shear,  S  =  to  =  R'  =  4  lbs. 

309.  In  Fig.  132  or  in  Fig.  133,  considering  the  segment  extendiilg  from 
the  load  to  either  support  (in  Fig.  132  there  is  but  one  such  segment),  it  will 
be  seen  that,  at  the  free  end  of  any  such  segment,  the  horizontal  stresses  are 
zero,  and  that  they  increase  uniformly  to  a  maximum  at  the  other  end  of  the 
segment.  Thus,  in  Fig.  132,  they  increase  uniformly  from  0,  at  the  loaded 
or  free  end,  to  18  lbs.,  at  the  support;  while,  in  Fig.  133,  they  increase  uni- 
formly from  0,  at  each  support,  or  free  end,  to  18  lbs.,  at  the  load. 


442 


STATICS. 


Moments  in  Cantilevers. 

310.  In  a  cantilever,  Fig.  134,  each  load  exerts,  about  any  point  between 
itself  and  the  support,  a  moment  =  its  weight  X  the  horizontal  distance  of 
its  center  of  gravity  from  such  point ;  and  the  total  moment,  at  any  point, 
is  the  sum  of  the  moments  of  the  several  loads  about  that  point.  Thus, 
neglecting  the  weight  of  the  cantilever  itself,  we  have : 
aliout  b,  moment  =  A  .  x  -\-  B  .  y; 
"     c,  "         =  A  .  m  +  B  .  n; 

*'     d,         "         =  A  .  z; 
"     A,  or  any  point  beyond  A,  moment  =  0. 


rig.  134. 


Figr.  135. 


311.  In  a  cantilever.  Fig.  135,  the  maximum  leverage  of  any  load,  W, 
is  evidently  its  distance,  I,  from  the  support,  b.  Hence,  the  maximum  bend- 
ing moment  of  any  load  upon  a  cantilever  is  at  the  support,  and  is  =  W.l. 
From  this  maximum,  the  moment  diminishes  uniformly  to  zero,  at  the  load. 
See  H  309. 

312.  Draw  6'  m,  Fig.  135  (6),  to  represent  the  maximum  moment  by  scale, 
and  join  m  W.     Then,  for  any  point,  c, 

moment  =  ordinate   at  c'  to   line   m  W. 

313.  In  a  cantilever.  Fig.  136  (a),  with  two  or  more  concentrated  loads, 
W  and  w,  let 

6'  w.   Fig.  136  (6),  =  moment  of  W  at  the  support; 
6'  m',  Fig.  136  (6),  =  moment  of  w,  at  the  support. 

Then,  for  both  loads,  W  and  w,  neglecting  the  weight  of  the  beam, 
at  d,  moment  =  moment  of  W  alone,  =  ordinate  at  d'; 
at  c,  moment  =  sum  of  moments  of  W  and  w, 

=  sum  of  two  ordinates,  c'  n  and  c'  n',  at  c'. 


Fig.  136. 


Fig.  137. 


314.  In  a  cantilever,  Fig.  137  (a),  under  a  load,  W,  uniformly  distributed 
©ver  a  length,  Z,  beginning  at  the  supoort,  b.  the  maximum  moment,  at  the 

support,  b,  is  =  W  .  "2  . 


MOMENTS. 


443 


In  Fig.  137  (6),  make  6'  m  =  said  max  moment,  and  draw  a  semi-parabola 
m  k',  with  apex  at  k'.  Then,  at  any  other  section,  c,  the  moment  is  repre- 
sented by  the  ordinate,  c',  of  said  parabola,  and  is  =  to  .  — ,  where  w  =  the 

weight  of  that  portion  of  W  beyond  c,  and  x  =  the  length  of  that  portion. 

At  k,  or  at  any  point  beyond  k,  moment  =  0. 

315.  In  Fig,  138,  neglecting  the  weight  of  the  cantilever  itself,  let  W  repre- 
sent the  weight  of  the  whole  load,  and  w,  that  of  the  shaded  portion,  concen- 
trated at  their  respective  centers  of  gravity,  G  and  g.     Then, 

about  6,  moment  =  W  .  x; 
"      c,         "         =  W  ,  y; 

"      d,         "         =   w  .  v; 

"      k,  or  any  point  beyond  k,  moment  ■■ 

W 


0. 


n 


Fig.   138. 

Moments  in  Beams. 

316.  In  a  beam.  Fig.  129,  the  upward  reaction  of  each  abutment  exerts; 
about  any  point,  a  moment  =  reaction  X  distance  of  support  from  such 
point ;  but  any  load,  between  such  point  and  the  support,  exerts  a  contrary 
moment  =  load  X  distance  of  load  from  such  point.     Thus, 

about  c,  moment  =  R'  .  e  =  R  (Z  —  z)  —  W  (y  —  z). 
At  each  support,  the  moment  is  0. 

317.  In  a  beam.  Fig.  129,  carrying  a  single  concentrated  load,  W,  the 
moment,  R'.^;,  at  any  point,  c,  is  =  Wz  =  l^-j  .  s  =  R  (Z  —  z)  —  W  (y  —  z) 


I  ' 

-  z)  —  W  (2/  —  2) .     At  a  point,  as  c,  not  under  the  load,  the 

moment,  Wz,  is  evidently  less  than  the  moment,  R'.j/,  about  the  point,  o, 
under  the  load.  In  other  words,  the  maximum  moment  is  at  the  point,  o, 
under  the  load. 


=  w.f  a- 


IF 


*» 


?    r 


K — a> — >}< y > 

\ 1 


Fig.  129  (repeated). 


Fig.  139. 


318.  From  the  point,  o',  Fig.  139  (6),  corresponding  to  the  point,  o.  Fig. 
(a),  where  the  load  is  applied,  erect  an  ordinate,  o'  m,  equal  by  scale  to  the 
(maximum)  moment,  =  R'  .  2/  =  R  .  a;,  at  that  point.  Join  a'm  6'.  Then 
the  ordinate  to  a'm,  or  to  m  b\  at  any  point,  c',  d\  e',  etc.,  represents  by 
scale  the  moment  at  the  corresponding  point,  c,  d,  e,  etc.,  in  the  span. 


44 


STATICS. 


319.  When  the  load,  W,  is  at  the  center  of  the  span;  I,  Fig.  140,  each  end 

W 
action  is  ==  — .     Hence,  the  moment,  s',  at  any  point,  s,  distant  y  from  a 

ipport,  as  b,  is 

.       W 

moment  —  -x-  .y. 

At  the  center  of  the  span  (i.  e.,  at  the  point  under  the  central  load,  W) 
e  have: 

maximum  moment,  M,  =  —  .   -?r  =  — i — . 
2       2  4 


Fig:.  139  (repeated). 


Tig,  140. 


In  order  that  the  maximum  moment  (at  o,  Fig.  139)  due  to  an  eccen- 
ic  load,  W,  may  be  equal  to  the  maximum  moment  (at  center  of  span,  I) 
le  to  a  given  center  load,  C,  we  must  have 


wf.s,  =  c 


or  W  =  C 


=  c(iy 


xy 


d,  e.  Fig.  141. 
m',  m",  repre- 
Make  the  long 


330.  When  there  are  two  or  more  concentrated  loads, 
3at  each  load  as  in  Fig.  139,  making  each  short  ordinate,  w, 

nt  the  maximum  moment  of  its  single  load,  c,  d  or  e,  alone.     „ 

dinates,  M,  M'  and  M''  =  the  sums  of  the  separate  moments,  as  measured 
c\  at  d',  and  at  e',  respectively.  Then  the  ordinate  to  a'  M  M'  M"  h',  at 
ly  point,  represents  the  total  moment  at  that  point,  due  to  the  several 
ads  combined. 


Fig.  141. 


Fig.  142. 


331.  In  a  beam.  Fig.  142,  under  a  uniform  load,  W,  covering  the  span,  Z» 
le  maximum  moment  is  at  the  center  of  the  span,  and  is 


moment  =  ^■^~  ^-^ 


W  J^ 
22 


'  24 


W  J^ 
24 


W.Z 

8  • 


MOMENTS. 


445 


Make  o'M,  by  scale,  =  the  maximum  moment;  and  draw  the  parabola, 
a'  M  b\  with  vertex  at  M.  Then  the  moment,  at  any  section,  as  s,  is  repre- 
sented by  the  corresponding  ordinate,  s\  to  that  parabola. 

Let  w  and  v  —  the  portions  of  W  to  the  left  and  right  of  s,  respec- 
tively.    Then,  moment  a,ts=--y*  =  -^x*  =  ~   moment  due  to  whole 

load,   W,  concentrated  at  s.* 
At  either  support,  moment  =  0. 

In  Fig.  131,  at  a  point,  c,  under  the  center  ot  gravity  of  a  load,  W,  uni- 
formly distributed  over  a  portion,  s,  of  the  span,  neglecting  the  weight  of 
the  beam,  ♦ 

.        13  W     s  ^  W.s         „,  W.s 

moment  =  R.x  —  o"  '  T  ""    ^'^  —       s,      ^  ^  ^  —  . 

323.  Let  W  =  the  total  load,  whether  concentrated  or  uniform,  and  let 
I  =  the  span.     Then  the  maximum  moment,  M,  is  as  given  below: 

=  WZ; 

2  • 
WZ. 

4  • 
WJ. 

8  • 
W_Z 

8  ' 
W  Z 

12  • 

323.  In  the  inclined  beam.  Fig.  143,  the  inclined  distances  may  be  used, 
instead  of  the  horizontal  distances,  in  finding  the  reactions.     Thus, 


Cantilever. 

Load, 

W,  at  end. 

M  at  support                   M 

" 

*'    uniform. 

"    "       "                          M 

Supported  beam.f 

- 

•*    at  center. 

"   at  center                     M 

.. 

" 

••    uniform. 

"    "       "                         M 

Fixed  beam.  J 

•• 

"    at  center. 

"    "       "     or  support  M 

♦• 

•• 

"    uniform. 

'*     "  support                  M 

reaction  R' 


w.f  =  w.-. 


But,  in  finding  moments  of  vertical  forces,  we  must  of  course  use  the 
horizontal,  not  the  inclined,  distances.    Thus,  at  c,  moment  R'c;  not  R'c'. 


Fig.    143. 


y       W 

*  Moment  ats    =   R'  y  —  v  -^  =  -^   y 

_,  X         W  w  W  —  w 

=  Rx-w-  =  -a:--^x: 2~  ""  ^ 

With  W  concentrated  at  «, 
moment  at  s 


W  -J-  y  =  wy  ■■ 


W 


t  Beam  supported  at  each  end,  but  not  fixed. 
I  Beam  fixed  at  each  end. 


446 


STATICS. 


324.  In  curved  beams,  the  same  principles  apply  as  in  straight  beams. 
Thus,  Fig,  144,  at  s,  moment  =  W  .  I.     Again,  in  Fig.  145  (a),  reaction  R' 

=  — y-  ,  and  at  s,  moment  =  R\y.     Or,  as  in  Fig.  145  (6),  from  o,  where 

the  load  is  applied,  draw  o  a  and  o  b,  to  the  two  supports  respectively,  and, 
by  means  of  the  force  parallelogram,  find  the  components,  p  and  q,  of  W. 
Then,  at  a, 

moment  =  n  .  n. 


^      Cb) 


Tig.  145. 


Shear. 

325.  In  the  beam,  a  b.  Fig.  146  (a),  consider  the  segments,  a  c  and  c  b,  to 
the  left  and  to  the  right  respectively  of  the  plane  n  n.  Besides  the  horizontal 
forces  acting  across  the  plane  n  n,  we  have  seen  (T[  305)  that  we  require  also, 
for  equilibrium,  a  vertical  force,  =  the  left  end  reaction.  R,  acting  down- 
ward upon  the  left  segment,  a  c,  and  forming  a  couple  with  R;  and,  at  the 
same  time,  acting  upward  on  the  right  segment,  c  6,  being  =  the  load,  W, 
minus  the  right  end  reaction,  R'.  This  force  is  called  the  shear,  S,  in  the 
section  n  n.  It  may  be  regarded  as  the  transmission  of  the  vertical  forces 
from  loads  to  supports  or  vice  versa. 

326.  The  two  segments,  a  c  and  c  b,  thus  tend  to  slide  vertically  past  each 
other,  the  right  segment,  c  b,  tending  downward,  owing  to  the  preponder- 
ance of  the  load,  W,  over  the  right  end  reaction,  R';  and  this  tendency  is 
resisted  by  the  shear,  S,  which  is  =  the  left  end  reaction,  R.  The  same  ten- 
dency exists  uniformly  between  W  and  a,  and  is  resisted  throughout  by  a 
shear  =  S  =  R. 

327.  Between  the  load,  W,  and  the  right  support,  6,  also,  a  uniform  shear 
exists;  but  here  the  shear,  S',  is  =  the  right  end  reaction,  R',  =  R  —  W; 
and,  whereas  the  shear,  S,  to  the  left  of  the  load  was  right-handed  or  clockwise 
(the  portion  to  the  right  of  any  section,  n  n,  receiving  the  downward  force), 
and  is  called  positive,  or  +,  the  shear  on  the  right  of  the  load  is  Ze/^-handed  or 
counterclockwise  (the  portion  to  the  left  of  any  section  receiving  the  down" 
ward  force),  and  is  called  negative,  or  — . 

328.  The  shears,  S  and  S',  to  the  left  and  to  the  right  of  the  load,  W,  are 
Vepresented  by  the  diagrams  in  Fig.  146  (6) ;  that,  S,  on  the  left  of  the  load 

being  drawn  above  the  zero  line,  a'  b\  to  indicate  a  positive  shear,  and  vice 
versa. 

329.  Comparing  Figs.  146,  147  and  148,  notice  that,  between  the  left  sup- 
port, a,  and  the  load,  W,  Fig.  146,  we  have  positive  shears,  S  =  90,  Fig.  146, 
and  8  =  15,  Fig.  147;  so  that,  in  Fig.  148,  where  both  loads,  W  and  w,  are 
placed  upon  the  same  beam,  we  have,  between  a  and  W,  a  total  positive 
shear  of  S  +  s  =  90  4-  15  =  105.  Between  the  right  support,  b,  and  the 
load,  ly.  Fig.  147,  we  have  negative  shears,  S'  =  —  30,  Fig.  146,  and  s'  = 
—  45,  Fig.  147 ;  so  that,  in  Fig-.  148,  between  b  and  w,  we  have  a  total  negative 
shear  =  S'  +  s'  =  —  30  —  45  =  —  75.  But,  between  the  points  of  appli- 
cation of  W  and  of  w,  we  have  S'  =  —  30,  Fig.  146,  and  s  =  -f  15,  Fig.  147; 
leaving,  between  W  and  w,  Fig.  148.  s  +  S'  =  15  —  30  =  —  15.  If  the 
total  right  end  reaction,  R'  -f  r',  exceeds  w,  as  we  here  suppose,  the  shear,  at 
any  point  between  the  two  loads,  W  and  w,  Fig.  148,  is  negative,  as  indi- 
cated ;  and  vice  versa. 


SHEAR. 


447 


330.  In  any  section,  the  shear  is  =  the  reaction  at  either  end,  minus  any 
loads  between  that  end  and  the  given  section. 

331.  If,  as  in  Fig.  149,  the  right  end  reaction,  R'  +  r',  is  =  the  load,  w, 
then  the  left  end  reaction,  R  +  r,  is  =  the  load,  W ;  and  there  is  no  shear  at 
any  point  between  the  two  loads.  In  other  words,  if  the  beam  be  cut  by  a 
section  at  any  point  between  W  and  w,  horizontal  forces  alone  will  pre- 
serve equilibrium,  no  vertical  forces  being  required,  since  the  two  segments 
have  no  tendency  to  slide  vertically  past  each  other. 

332.  A  similar  condition  exists  in  any  section  where  the  sign  of  the  shear 
changes  from  -f  to  —  or  vice  versa.  Thus,  if  the  beam  be  cut  by  a  section 
immediately  under  W,  Fig.  146  or  148,  or  under  w.  Fig.  147,  horizontal  forces 
equivalent  to  the  fiber  stresses  in  the  beam,  will  suffice  to  preserve  equilib- 
rium, without  a  vertical  force,  or  shear ;  there  being  no  tendency  of  the  two 
segments  to  slide  past  each  other.  Also,  when,  as  in  Fig.  149,  under  W 
and  under  w,  the  shear  changes,  in  amount,  from  any  value,  on  one  side  of 
a  section,  to  0,  on  the  other  side,  the  shear  in  the  section  itself  is  =  0. 


120 


L«^ 


^ 


fy,^^9&^ 


90 


m 


M-301 


Cb) 


\w60 


w 

6| 

'■mr^is 

r-4sW 

15 

1 

8=- 

-45 

Fig.  146. 


Fig.  147. 


12o( 


60  ( 


1 

W 

5= 

90 

8+S'= 

"15 

-4S 

S^ 

-30 

Fif?.  148. 


Figr.  149. 


333.  But  in  the  section  under  w,  Fig.  148,  where  the  shear  changes  in 
amount,  although  not  changing  sign  from  +  to  —  or  vice  versa,  there  is  a 
shear  =  the  less  of  the  two  shears  on  the  opposite  sides  of  the  section,  for 
this  is  the  amount  of  the  shear  transferred  through  the  section,  or  is  the 
tendency  of  either  segment  to  slide  past  the  other. 

334.  With  any  number  of  loads,  if  that  portion  of  the  total  load  to  the 
left  of  any  section  be  called  X,  and  that  portion  to  the  right  of  the  same  sec- 
tion be  called  Y,  it  will  be  found  that  the  shear  in  the  section  is  equal  to  the 
difference  between  that  part  of  X  which  goes  to  the  right  support,  h,  and  that 
part  of  Y  which  goes  to  the  left  support,  a. 

335.  With  a  load,  W,  Fig.  150  (a),  uniformly  distributed  over  the  entire 

W 
span,  the  maximum  shear,  =  R  =  R'  =  -g,  is  at  each  support,  a  and  6. 

The  minimum  shear,  =  0,  is  at  the  center,  c,  of  the  span,  which  is  also  the 
point  of  maximum  bending  moment,  see  ^  321  and  Fig.  142.     At  any  point. 


448 


STATICS. 


d,  the  shear  is  given  by  the  corresponding  ordinate,  d',  Fig.  150   (6).        See 
Relation  between  Moment  and  Shear,  til  359,  etc. 

336.  With  a  load,  W,  uniformly  distributed  over  any  part,  y,  of  the  span, 
Fig.  151  (a),  find  the  end  reactions,  R  and  R',  as  in  ^  299.     Then 
between  a  and  d,  shear  =  S    =  R ; 
"         eand  h,      "       =  S'  =  R'; 
at  c,  "       =  0. 

J  R  R' 
d  c  =  y  .  ^-j^j\    z  =  y--x  =  ce  =  y.. 


W 


•  W 


Fig.  150. 


Figr.  151. 


< 

U V 

i9f{|i||i||jj| 

I«            I 

I    c 

'"""'' '""'^|W 

■""%n            , 

— 1 — 
1 

1 

s\ 

It 

\at 

"\ 

^!^. 

i^) 

r 

V 

^ 

Fig.  152. 


Fig.  153. 


837.  When  the  loaded  portion,  y,  of  the  span,  begins  at  one  of  the  sup- 
ports, b,  Fig.  152  (a),  then  since  R  =  W-^  =  ^'oJ'  ^®  ^^^^ 


W^ 


=  d  c 


■■  y . 


ry_ 
R  _  ^^  _  y_^  y^ 
'  •  W  ^  W  ^2Z  2Z* 
338,  When  a  concentrated  load,  W,  Fig.  153,  is  added  to  a  load  uniformly 
distributed  over  the  entire  span,  or  over  a  part  of  it,  each  load  produces  the 
same  shears  as  if  it  alone  were  upon  the  span.  Those  due  to  W  are  repre- 
sented in  Fig.  153  (6),  while  those  due  to  the  uniform  load  are  represented 
in  Fig.  153  (c).  The  resultant  shear,  due  to  both  loads  combined,  is  repre- 
sented in  Fig.  153  (d).  Note  that,  between  v  and  r,  the  addition  of  W,  with 
its  positive  shear,  reduces  the  negative  shear  due  to  the  uniform  load,  and 
that,  between  r  and  z,  the  addition  of  W  reverses  the  negative  shear;  also 
that  it  shifts  the  zero  point  from  z  to  r. 

For  Continuous  Beams,  see  Beams,  under  Strength  of  Materials. 


INFLUENCE   DIAGRAMS. 


449 


Influence  Diagrams. 

339.  The  end  reactions,  due  to  a  given  load,  and  consequently  the 
moments,  shears  and  stresses,  produced,  at  any  given  point  in  a  span,  by 
such  load,  vary  as  the  position  of  the  load,  relatively  to  the  supports,  is 
changed.  A  diagram.  Figs.  154  (6),  155  (6),  156  (6),  showing  the  changes 
thus  produced  at  any  given  point,  is  called  an  influence  diagram,  or  influ- 
ence line.* 

Influence  Diagram  for  Moments. 

340.  Thus,  in  Fig.  154  (6),  a'Mb'  is  the  moment  influence  diagram  for  the 
point,  c,  under  a  single  concentrated  load,  W.f 


Fig.  154. 


341.  In  Fig.  154,  let  I  be  the  span,  x  the  variable  distance  of  the  load,  W,t 
from  the  right  support,  h,  and  y  the  constant  distance,  a  c,  of  a  given  point, 
c,  from  the  left  support,  a.  Then,  for  any  position  of  W,  the  left  end  reac- 
tion, R,  is  =  W  .  y ;  and  the  moment  of  that   reaction  about  c,  =  R  .  j/, 

=  W  .  y  .  2/.    The  right  end  reaction  is  R'  =  W  — j — ,  and  its  moment,  about 

c,  is  R'  a  —  y)  =  W^-^  (l  —  y). 

So  long  as  W  is  between  b  and  c,  the  moment  at  c  is  =  R  .  j/  =  W.  y- .  y. 

342.  Since  W,  y  and  I  are  constant,  the  moment,  at  c,  while  W  is  between 
b  and  c,  is  proportional  to  the  variable  distance,  x,  of  the  load  from  b.  It 
therefore  increases  uniformly,  from  0,  when  W  is  at  b,  to  its  maximum  value, 
M,  when  W  is  at  c.  See  ^  317.  Hence,  if  the  ordinate,  c'  M,  be  made 
equal,  by  scale,  to  the  maximum  moment,  M,  then  the  moment,  at  c,  for  any 
position,  d,  e  or  /,  of  W,  between  c  and  6,  is  given  by  the  corresponding  ordi- 
nate, d',  e'  or  /',  to  the  line  b'  M.  Similarly,  the  moments,  at  c,  for  any 
positions  of  W  between  c  and  a,  are  given  by  the  ordinates  to  the  line  a'  M. 

343.  For  the  moment,  at  c,  for  any  number  of  loads,  in  any  positions,  find 
the  moment,  at  c,  for  each  load  separately,  as  above,  and  take  their  sum. 

344.  It  is  customary  to  construct  the  moment  influence  diagram  for  the 
moments  of  a  load,  W,  =  unity  (1  ton,  1  pound,  1  thousand  kilograms,  etc.). 
Each  ordinate  must  then  be  multiplied  by  its  corresponding  load,  measured 
in  the  corresponding  unit,  in  order  to  obtain  the  required  moment. 

345.  When  W  is  at  the  point,  c,  we  have  x  =  I  —  y.  Hence,  ordinate, 
c'  M,  =  maximum  moment,  =  W  .  — j—  .  y;  or,  if  W  =  1,  c'  M  =  — ^  .  y. 


The  area  of  the  diagram,  a'Mb',  is  ■■ 


c'M 


1  ^ziy 

2  •    I 


-.2/ 


I 
(I  —  y)  y 


*  See  "Calculation  of  the  Stresses  in  Bridges  for  Actual  Concentrated 
Loads,"  by  Prof.  Geo.  F.  Swain,  "Trans.  Am.  Soc.  C.  E.,"  vol.  xvii,  July, 

1887. 

t  Inasmuch  as  the  load,  in  this  discussion,  occupies  different  positions  at 
different  times,  it  is  not  shown  in  Fig.  154. 


29 


450 


STATICS. 


346.  If  a  load,  —  1,  be  distributed  over  a  length,  =  1,  at  c.  Fig.  155  (a), 
the  resulting  moment,  at  c,  may  be  represented  by  the  area  of  the  rectangle 
standing  on  c^  Fig.  (6),  the  height  of  said  rectangle  being  the  ordinate,  c'  M, 
and  its  length  =  1.  Similarly,  the  moment,  at  c,  due  to  a  uniformly  dis- 
tributed load,  e  /,  of  1  per  unit  length.  Fig.  (a),  may  be  represented  by  the 
sum  of  the  areas  of  the  rectangles  between  e'  and  f\  Fig.  (b) ;  and,  if  we  sup- 
pose the  load,  e  /,  Fig.  (a),  of  1  per  unit  length,  to  be  divided  into  a  very 
large  number  of  very  narrow  vertical  strips,  the  resulting  moment,  at  c,  may 
be  taken  as  represented  by  the  area  of  the  shaded  trapezoid  over  e'  /',  Fig. 
155  (6) .  The  moment,  at  c,  due  to  a  load  of  p  (lbs.,  tons,  etc.)  per  unit  length, 
and  occupying  the  same  length,  e/,  is  =  p  X  area  of  trapezoid  over  e'  /',  Fig. 
155  (6). 

347.  Hence,  the  maximum  moment,  at  c,  due  to  a  uniform  load  of  p  (lbs., 
tons,  etc.)  per  unit  of  length,  occurs  when  that  load  covers  the  entire  span. 


This  maximum  moment  is  =  p  X  area  a'M6',  =  p 


jl  —  y)  y 


See  H  345. 


f       b' 


Fig.  155. 


Fii:.  156. 


Influence  Diagram  for  Shear. 

348.  Under  a  single  concentrated  load,  the  shear,  at  any  point  between 
the  load  and  either  support,  is  '=  the  reaction  of  that  support.  See  m  326 
and  327. 

349.  In  the  shear  influence  diagram.  Fig.  156,  as  in  the  moment  influence 
diagram.  Fig.  154,  let  I  be  the  span ;  x  the  variable  distance  of  the  load,  W, 
from  the  right  support,  6,  and  y  the  constant  distance,  a  c,  of  a  given  point, 
c,  from  the  left  support,  a.  Then,  for  any  position  of  W,  the  left  end  reac- 
tion, R,  or  the  shear,  S,  at  any  point  between  the  load  and  the  left  support,  is 

=  W  .  -7- ;  and  the  right  reaction,  R',  or  the  shear,  S',  at  any  point  between 

l  —  x 


I 


the  load  and  the  right  support,  is  =  W  . 


I 


350.  The  influence  line  for  shear,  like  that  for  moments,  1[  344,  is  usually 

constructed  for  a  load   =   unity,  so   that    S    =    R   =   -y;  and  S'  =  R'  = 

I X 

— -J — .     Each  ordinate  of  the  shear  diagram  must  then  be  multiplied  by  W, 

in  order  to  obtain  the  required  shear, 

351.  Since  W  (  =  1)  and  I  are  constant,  R  and  S  vary  directly  (and  R'  and 
S'  inversely)  with  x.  Thus,  when  W  (  =  1)  is  at  fc,  we  have  x  =  0 ;  S  =  R 
=  0,  and  S'  =  R'  =  W  =  1.  When  W  (  =  1)  is  at  a,  we  have  x  =  Z;  S  =  R 
=  1,  and  S'  =  R'  =  0.  Draw  a'  a"  and  h'  h'\  each  =  W  (  =  1),  and  join 
a"  b'  and  a'  6".  Then,  with  W  at  c,  the  (positive)  shear,  S,  at  each  point,  as 
/,  between  c  and  a,  is  given  by  the  ordinate,  c'  g,  to  the  line  a''  6' ;  while  the 
ordinate,  c'  h,  to  the  line,  a'  b",  gives  the  (negative)  shear  at  each  point,  as  c, 
between  c  and  6. 

353.  Similarly,  with  W  at  e,  the  ordinate,  e'  t,  gives  the  (positive)  shear  at 
each  point,  as  c,  between  e  and  a;  while  e^  p  gives  the  (negative)  shear  at 
each  point  between  e  and  b. 


INFLUENCE   DIAGRAMS. 


451 


353.  It  will  be  noticed  that,  as  the  load,  W,  passes  from  one  side  to  the 
other  of  any  point,  as  c,  the  shear  at  that  point  is  reversed,  tl^p  total  change 
in  shear  being  =  hc'-\-c'g  =  hg  =  the  load,  W. 

354.  With  a  load,  W  (  =  1),  at  h,  the  shear  at  c  is  =  0.  See  H  351.  As 
the  load  advances  from  h  toward  a,  the  positive  shear,  S  =  R  =  -y-i  **  ^'  ^^* 
creases  in  proportion  to  the  ordinates  to  the  line  6'  g,  becoming  =  c^  g  — 


l  —  y 
I 


when  W  is  just  to  the  right  of  c.     With  W  just  to  the  left  of  c,  we  have, 


</  A  =  -T-,     But  as  W  proceeds  from  c  to  a, 


negative  shear  at  c  =  S'  = 

this  negative  shear,  at  c,  decreases  in  proportion  to  the  ordinates  to  the  line 
h  a',  becoming  0  when  W  reaches  a.  Thus,  a'hgb'  is  the  shear  influence 
diagram  for  the  point,  c.  Similarly,  a'pth'  is  the  shear  influence  diagram 
for  the  point,  e,  etc. 

355.  If  a  series  of  nearly  uniform  and  equidistant  concentrated  loads, 
such  as  the  wheel  loads  of  a  locomotive  and*  train,  come  upon  the  span,  at  the 
support,  h,  and  advance  toward  a,  the  shear  at  c  evidently  increases  until  the 
first  load,  reaches  c.  It  is  then  suddenly  diminished,  by  an  amount  =  the 
first  load,  as  that  load  passes  c.  It  then  continues  to  diminish,  as  each 
wheel  passes  over  c,  but  more  slowly,  until  the  first  load  reaches  a.  See  If 
358. 

356.  With  a  uniformly  distributed  load,  of  unity  per  unit  length,  moving 
as  in  ^  355,  the  shears  at  c  (see  1[  346)  are  represented  by  the  areas  of  those 
portions  of  the  diagram,  a'hgb',  successively  covered  by  the  load,  portions 
of  the  diagram  below  the  zero  line,  a'  b',  being  taken  as  negative.  Thus, 
when  the  head  of  the  load  reaches  e,  the  (positive)  shear  at  c  is  given  by  the 
area  of  the  triangle,  b'  e'  t.  With  head  of  load  at  c,  the  shear  at  c  reaches  its 
maximum,  and  is  gi.ven  by  the  area  of  triangle,  b'  c'  g.  With  head  of  load  at 
/,  the  shear  at  c  is  =  area  b'  c'  g  —  area  /'  c'  h  n. 

357.  Similarly,  the  shears  at  e  are  given  by  the  areas  of  portions  of  the  dia- 
gram, a'  pt  b'. 

358.  Fig.  157*  shows  the  influence  diagram,  0  dz\^  16,  for  the  shears  at 


pomt  6,  for  a  given  uniformly  distributed  load,  at  least  as  long  as  the  span, 
coming  upon  the  span  at  point  0,  passing  across  it,  and  leaving  it  at  point  8; 
also  the  corresponding  diagram,  0  e  16,  for  the  left  support,  8;  and  that, 
0  £^  16,  for  the  right  support,  0. 

For  the  action  of  internal  resisting  forces  in  beams  and  trusses,  see 
Transverse  Strength,  under  Strength  of  Materials,  and  Stresses,  under 
Trusses. 


*  First  published  in  our  9th  Edition,  1885. 


452 


STATICS. 


Belation  between  Moment  and  Shear. 

359.  The  shear,  at  any  point  in  the  span,  is  simply  the  rate  at  which 
the  bending  moment  is  changing  at  that  point. 

360.  Thus,  in  Fig.  158,  the  moment,  M,  Fig.  (6),  at  the  support,  b,  due  to 
the  concentrated  load,  W,  of  6  lbs.,  is  =  WZ  =  6X4  =  24  ft.-lbs.;  but, 
between  the  support  and  the  load,  the  moment  is  decreasing  at  the  uniform 
rate  of  6  ft. -lbs.  for  each  foot  of  x,  or  6  ft. -lbs.  per  foot  =  6  lbs. ;  and  this  6  lbs. 
is  the  uniform  shear,  V,  Fig.  (c),  throughout  the  beam.  Hence  the  shear 
diagram.  Fig.  (c),  is  a  horizontal  line ;  i.  e.,  its  ordinates  are  of  equal  length. 

361.  Again,  in  Fig.  159,  the  shear  diagram  ordinates  between  a"  and  o''. 
Fig.  (c),  are  positive;  showing  the  (algebraic)  increase  of  the  bending  moment, 
M,  Fig.  (6),  as  we  proceed  from  the  left  support,  a,  toward  the  center,  o,  of 
the  span;  while  the  negative  shear  diagram  ordinates,  between  o''  and  b'\ 
show  the  (algebraic)  decrease  of  the  bending  moment  as  we  proceed  from 
the  center,  o,  to  the  right  support,  b.  At  the  center,  o,  the  rate  of  change  of 
bending  moment  is  zero,  as  is  also  the  vertical  shear. 


-«(<?) 


Figr.  158. 


Fig:.  159. 


362.  Both  in  Figs.  158  and  159,  the  bending  moment,  M,  is  constantly 
changing;  but  in  Fig.  158  its  rate  of  change  (=  6  ft.-lbs.  per  ft.  of  span)  is 
constant.  Hence,  the  moment  diagrani  is  a  straight  inclined  line,  and  the 
shear  diagram  is  a  horizontal  line;  whereas  in  Fig.  159  the  rate  of  change  of 
bending  moment  is  constantly  varying,  being  =12  ft.-lbs.  per  foot  of  span 
(shear  =12  lbs.)  at  the  support,  and  diminishing  to  zero  at  the  center,  a,  of 
the  span.  Hence,  in  Fig.  159,  the  moment  diagram.  Fig.  (6),  is  no  longer 
straight,  but  curved;  and  the  shear  diagram.  Fig.  (c),  is  no  longer  horizontal, 
but  inclined. 

363.  But,  in  Fig.  159  (c),  the  shear,  V,  or  the  rate  of  change  of  the  bending 
moment  (although  no  longer  conston^,  as  it  was  in  Fig.  158  (c)),  nevertheless 
diminishes  uniformly,  as  we  proceed  from  a  toward  6.  Thus,  at  the  point,  1, 
Fig.  159,  midway  between  a  and  o,  the  bending  moment  is  changing  at  the 
rate  of  6  ft.-lbs.  per  foot,  or  half  as  fast  as  at  a.  Hence,  the  shear  diagram, 
although  no  longer  a  horizontal  line,  is  still  a  straight  line ;  and  the  uniform 
decrease  ( =  6  ft.-lbs.  per  foot  per  foot)  in  the  rate  of  change  of  the  bending 
moment,  or  the  uniform  decrease  ( =  6  lbs.  per  foot)  in  shear,  is  indicatea  by 
the  horizontal  diagram  in  Fig.  159  (d). 

364.  In  either  Fig.,  let  a  straight  line  be  drawn,  tangential  to  the  moment 
diagram  (6),  at  any  point,  c',  and  forming,  with  the  horizontal  zero  line,  a'  b\ 

■  an  angle,  A.  (In  Fig.  158,  this  line  coincides  with  the  moment  diagram.) 
Then  the  tangent  of  A  is  given  by  the  shear  diagram  ordinate,  c'\  corre- 
sponding to  the  point,  c;  or,  for  any  point,  V  =  tan  A. 


MOMENT   AND"  SHEAR.  45S 

365.  In  Fig.  158,  where  this  angle,  A,  Fig.  (6),  is  constant,  the  shear  ordi- 
nates.  Fig.  (c),  are  of  constant  length.  In  other  words,  the  ^ear  diagram. 
Fig.  158,  is  a  horizontal  line.  w 

366.  Since  the  shear  diagram  ordinates  represent  forces  (as  in  lbs.,  etc.) 
and  the  abscissas  represent  dista,nces  (as  in  ft.,  etc.)  the  product  of  the  dis- 
tance between  any  two  shear  ordinates,  multiplied  by  the  mean  of  those  ordi- 
nates, is  an  area  representing  a  moment  (in  ft.-lbs,,  etc.).  This  moment  is  = 
the  difference  between  the  two  moments  represented  by  the  corresponding 
ordinates  in  the  moment  diagram. 

367.  Thus,  in  Fig.  158  (6),  the  increase  in  (negative)  bending  moment,, 
between  points  1  and  3,  is  =  18  —  6  =  12  ft.-fts. ;  and,  in  Fig.  158  (c),  the 
moment  represented  by  the  (shaded)  area,  between  the  same  two  points,  is 
=  2  ft.  X  6  lbs.  =  12  ft.-lbs.  In  Fig.  159,  the  (algebraic)  increase  in  bend- 
ing moment.  Fig.  {h),  between  the  left  support,  a,  and  the  center,  o,  of  the 
span,  is  =  8  +  4  =  12  ft.-lbs.;  and  the  moment,  represented  by  the  shear 
diagram  area  (triangle)  between  the  same  two  points,  Fig.  (c),  is 

1  -,  Z         12  X  2        ,o*^   ^ 
=  "2^2" 2-^  =12  ft.-lbs. 

368.  Again,  in  Fig.  159,  at  any  two  points  equally  distant  from  the  center,. 
o,  of  the  span,  the  moments  are  equal;  or  difference  of  moments  =  zero; 
and,  since  shear  ordinates  helow  the  zero  line,  a"  h".  Fig.  (c),  are  considered 
as  negative,  the  algebraic  sum  of  the  two  corresponding  shear  triangles,  Fig. 
(c),  is  also  =  zero. 

Similarly,  areas  in  Fig.  159  (d)  correspond  to  differences  of  ordinates  in 
Fig.  139  (c). 


454 


STRENGTH   OF   MATERIALS. 


.STRENGTH  OF  MATERIALS. 


GENERAIi  PRINCIPL.es. 
Art.  1  (a)  Stress  or  Strain*  occurs  yvhen  force  acts  upon  a  body  in  such 
a  way  that  its  particles  tend  to  move  (at  the  same  time)  with  diflferent  velocities 
or  in  diflferent  directions ;  to  do  which  they  must  either  separate  from  each  other 
or  come  closer  together.  This  occurs,  for  instance,  when  a  body  is  so  placed 
as  to  oppose  the  relative  motion  of  two  other  bodies ;  as  when  a  block  is  placed 
between  a  weight  and  a  horizontal  table.  In  this  case,  each  of  the  two  bodies 
(the  weight  and  the  table)  imparts  a  force  to  the  opposing  body  (the  block) ;  and 
the  stress  is  the  opposition  of  these  equal  forces.  The  tendency  of  the  particles 
of  the  block  to  separate  or  to  come  closer  together  calls  into  action  the  inberent 
forces  of  its  material,  and  these  act  between  the  particles  and  tend  to  keep 
them  in  their  original  relative  positions. 

(b)  Compression  and  Tension.  If  two  opposite  forces  are  simul- 
taneously imparted  to  a  body  in  the  same  straight  line,  the  stress  is  either  com- 
pressive (when  the  forces  act  toward  each  other)  or  tensile  (when  they  act /rom 
each  other). 

Compressive  stress  tends  to  push  the  particles  closer  together.  Tensile  stress 
tends  to  pull  th&xxi  farther  apart.f 

(c)  If  t-wo  imparted  forces,  as  a  o,  &  o,  meet  at  an  ang-le,  as  at  o ; 
then  two  equal  and  opposite  components,  a"  o  and  b"  o,  will  cause  compressive 
or  tensile  stress  in  the  body,  while  the  other  two,  a'  o  and  b'  o,  unite  to  form  the 


„    (6) 


resultant,  c  o,  which,  unless  balanced  by  other  forces,  moves  the  body  in  its  own 
direction,  and,  in  doing  so,  produces  another  stress  among  the  particles  of  the 
body.    See  (a),  above. 

(d)  If  tbe  two  forces  are  parallel,  forming  a  "  couple,"  as  in  a  punch 
and  die,  the  stress  is  a  sbear  (tending  to  slide  some  of  the  particles  over  the 
others,  see  p.  499),  and  is  accompanied  also  by  a  transverse  stress  (causing  a 
tensile  stress  in  some  of  the  particles  and  a  compressive  stress  in  others)  as  in  the 
case  of  a  beam.  The  transverse  stress  is  proportional  to  the  distance  between  the 
two  forces  (i.  e.,  to  the  arm  or  leverage  of  the  couple),  so  that^  when  they  are  very 
close  together,  as  in  a  pair  of  shears,  the  transverse  stress  is  very  small  and  is 
neglected,  and  the  shearing  stress  alone  is  considered. 

(e)  If  two  contrary  couples,  in  different  planes,  act  upon  a 
body,  the  stress  is  called  torsion  or  twisting.  See  p.  499.  Thus,  torsion  takes 
place  in  a  brake  axle  when  we  try  to  turn  it  while  its  lower  end  is  held  fast  by 
the  brake  chain. 

(f)  But  the  ultimate  tendency  of  any  of  these  forms  of  stress  is  either  to 
separate  certain  particles  or  to  drive  them  closer  together,  as  in  cases  (tensile  or 
compressive  stress)  where  the  two  forces  are  in  one  line.f  We  shall,  therefore, 
in  these  introductory  articles,  consider  only  this  simple  and  fundamental  form 
of  stress,  assuming  that  it  is  caused  by  the  action  of  two  opposite  imparted 
forces,  acting  in  the  same  straight  line  so  that  they  are  entirely  employed  in 
causing  the  stress. 

(g")  A  stress  may  be  stated  in  any  unit  of  weight,  as  in  pounds,  and 
is  equal  to  one  of  the  two  opposite  forces. 

*  For  another  use  of  the  word  "strain,"  see  Art.  2  (d). 

f  Indeed,  even  in  cases  of  compressive  stress,  it  is  only  by  the  separation  of  the 
particles  that  the  structure  of  the  body  and  its  inherent  forces  can  be  destroyed. 


STRENGTH    OF    MATERIALS.  455 

Art.  3  (a)  Stretcli  and  Rupture.  It  appears  from  experiment  that  the 
inherent  cohesive  forces  called  into  action  by  the  first  application  of  any  stress 
are  always  less  than  that  stress,  however  small  it  may  be.  In  other  words  ;  any 
stress,  however  slight,  is  believed  to  produce  some  derangement  of  the  particles. 
But  the  inherent  forces  increase  with  this  derangement  (up  to  a  certain  point) " 
and  thus,  in  many  cases,  they  become  equal  to  the  stress  and  so  prevent  further 
derangement.  When  the  stress  exceeds  the  greatest  inherent  force  which  the 
body  can  exert,  the  particles  separate  to  such  an  extent  that  the  inherent  forces 
cease  to  act.    The  body  is  then  said  to  be  broken,  or  ruptured. 

(to)  Different  materials  toeliave  very  differently  when  under  stress. 
Brittle  ones  seem  to  resist  almost  perfectly  up  to  a  certain  point,  allowing  no 
perceptible  derangement  of  the  particles ;  and  then  yield  suddenly  and  entirely. 
In  ductile  materials,  on  the  contrary,  considerable  derangement  takes  place 
before  the  inherent  resisting  forces  finally  yield. 

(cl  The  ultimate  stress  of  a  body  is  that  which  is  just  sufficient  to  break 
it  or  crush  it,  or,  in  short,  to  destroy  its  structure  so  that  it  can  no  longer  resist. 
In  other  words,  a  stress  just  less  than  the  ultimate  is  the  greatest  stress  to  which 
the  body  can  be  subjected. 

Caution.  In  brittle  materials,  such  as  brick,  stone,  cement,  glass,  cast-iron, 
etc.,  especially  when  subjected  to  tension,  the  point  of  rupture  is  clearly  marked, 
and  hence  the  ultimate  strength  may  in  such  cases  be  stated  with  precision. 
But  with  ductile  or  malleable  materials,  such  as  copper,  lead  and  wrought  iron, 
especially  when  under  compression,  it  is  often  difficult  or  impossible  to  state  the 
ultimate  strength  definitely.  For  instance,  a  cube  of  lead  may  be  gradually 
crushed  into  a  thin  flat  sheet  without  rupture.  In  other  words,  there  is 
practically  no  load  which  can  break  it  by  crushing.  In  such  cases,  we  may 
arbitrarily  assume  some  given  amount  of  distortion  as  marking  the  point  of 
ultimate  stress.  Thus,  by  the  *'  ultimate "  load  of  a  rolled  iron  beam 
we  mean  "that  one  which  so  cripples  the  beam  that  it  continues  to  yield 
indefinitely  without  increase  of  load."  Such  assumptions,  however,  necessarily 
give  rise  to  some  ambiguity,  and  care  should  therefore  always  be  taken  to  define 
or  to  ascertain  clearly  in  what  sense  the  term  "  ultimate  stress  "  is  employed. 

The  ultimate  strengtli  of  a  material  (or,  more  briefly,  its  strength)  is 
the  greatest  inherent  force  which  its  particles  can  exert  in  opposition  to  a 
stress.  In  other  words,  it  is  that  inherent  resistance  which  is  just  equal  to  the 
ultimate  stress.  Hence  strength,  like  stress,  is  stated  in  units  of  weight,  and 
we  may  use  the  terms  " ultimate  strength  "  and  "ultimate  stress "  indifferently, 
as  denoting  practically  the  same  thing. 

(d)  For  want  of  a  convenient  and  appropriate  name  for  the  cbange  of 
iiliape  caused  by  stress;  modern  writers  have,  rather  unfortunately,  given  to 
it  the  name  of  strain,*  which,  in  ordinary  language,  is  used  to  signify  stresSy 
as  above  defined.  We  prefer  to  use  the  word  "  stretcli "  1  jr  change  of  shape, 
in  inches,  etc.  {rGga.vdiu\g  compression  as  negative  "stretch"),  and  ** strain"  or 
"stress"  for  the  action  of  the  two  opposing  forces,  in  pounds,  etc. 

(e)  By  the  **lengtli."  of  a  body,  we  mean  its  dimension  measured  in  the 
line  of  the  stress ;  and,  by  **  area,"  the  area  of  the  resisting  cross  section  at 
right  angles  to  that  line.  Thus,  if  a  slab  of  iron,  2  inches  thick  and  10  in(?hes 
square,  be  laid  flat  upon  a  smooth  and  horizontal  surface,  and  if  a  load  be  placed 
upon  it  so  as  to  be  uniformly  distributed  over  its  upper  flat  surface,  the  "  length  " 
is  2  inches,  and  the  "  area,"  10  x  10  =  100  square  inches.f 

(f )  Units  adopted.  Unless  otherwise  stated,  we  shall  understand  the 
stress  in  any  case  to  be  given  in  pounds,  the  stretch  and  the  length  in  inches, 
and  the  area  in  square  inches. 

*  The  word  "  strain "  is  not  thus  defined,  even  as  a  scientific  term,  in  either 
Webster's  or  Worcester's  dictionary. 

t  Under  stresses  approaching  the  ultimate  stress,  the  area  of  the  cross  section 
generally  increases  under  compression,  and  diminishes  under  tension,  to  difler- 
«nt  extents  in  different  materiab ;  but  we  are  here  concerned  only  with  cases 
within  the  limit  of  elasticity,  (Art.  4  a,  p.  458)  and  in  such  cases  the  change  of 
area  is  generally  very  slight  and  may  be  neglected. 


456  STRENGTH   OF   MATERIALS. 

Art.  3  (a).    If  the  total  stress  (in  lbs.,  etc.)  upon  a  body  be  divided  by  the 

area  of  the  resisting  surface  (in  square  inches,  etc.)  the  quotient,  ?l£^  ^  is  the 

area 
mean  stress  per  unit  of  area,  or  (as  it  is  sometimes  called)  the  intensity  oi 
the  stress.    Or, 

Stress  per  unit  of  area  =    total  stress  ^ 

area 
Thuf ,  if  a  bar  of  iron,  2  inches  wide  by  1  inch  thick,  (having  therefore  2  square 
inches  of  area  of  cross  section)  and  10  feet  long,  be  subjected  to  a  total  tensile 
stress  of  20,000  lbs.  in  the  direction  of  its  (10  ft.)  length,  we  have 

mean  stress     )        total  stress         20,000  lbs.  m  nnn  iko  ^«-c^„„,.«  i««i. 

per  unit  of  area) ^e^.—  "  a^SSiTF.  "  '"'"^  ""  P'^"'!''"^  !•»=''• 

Caution.  Strictly  speaking,  the  stress  on  a  surface  is  seldom  distributed 
uniformly  over  it.  Thus,  in  the  case  of  the  bar  just  referred  to,  if  the  stress  is 
applied  by  means  of  grips,  clamping  the  sides  and  edges  of  the  bar,  the  stress 
per  square  inch  near  those  sides  and  edges  is  probably  greater  than  that  near 
the  center  of  the  bar,  because  the  stress  is  not  perfectly  and  uniformly  trans- 
mitted from  the  outer  to  the  inner  fibres.  And,  in  cases  of  compression,  the 
load,  instead  of  being  uniformly  distributed  over  the  surface,  as  it  appears  to  be, 
is  often  in  fact  supported  by  a  few  projecting  portions  of  it.  In  practice,  these 
considerations  are  often  of  the  greatest  importance,  but  in  studying  the. 
principles  of  resistance,  we  may,  for  convenience,  temporarily  neglect  them,  and) 
assume  the  stresses  to  be  uniformly  distributed  over  their  respective  areas. 

(b)  if  the  total  stretch  of  a  body  (in  inches,  etc.),  under  any  given  stress,  be 
divided  by  the  original  length  of  the  body  (in  the  same  measure),  the  quotient 
is  the  stretcb  per  unit  of  len^tli.    Or, 

StretcH  per  unit  of  lengtli  =      total  stretch     ^ 

original  length 

Thus :  if  the  foregoing  bar,  10  feet  (or  120  inches)  long,  is  found  to  stretch 
.04  inch,  under  its  load  of  20,000  lbs.  total,  or  10,000  lbs.  per  square  inch,  we  have 

stretch  per  unit  of  length  _     total  stretch    _    M__  ^^^gg  ^^^^        ^^^^ 
under  said  load  original  length        120 

(c)  Tlie  Modulus  of  Elasticity.  In  materials  which  undergo  a  per- 
ceptible stretch  before  rupture,  it  has  been  found  by  experiment  that  up  to  a 
certain  degree  of  stress,  called  the  limit  of  elasticity  (Art.  4  a,  p.  458),    the  ratio, 

total  stress    ^  ^^  ^^^  given  body,   remains  very  nearly  constant.    In  other 
total  stretch 

words,  within  the  limit  mentioned,  equal  additions  of  stress  cause  practically 
equal  additional  stretches. 

In  order  to  compare  bodies  of  different  dimensions,  we  state  the  same  fact  by 

saying  that,  within  the  elastic  limit  the  quotient,      stress  per  unit  of  area 

stretch  per  unit  of  length 
remains  practically  constant. 

This  quotient,  as  found  by  experiment  with  any  given  material,  is  called  the 
modulus  of  Elasticity  of  that  material,  and  is  usually  denoted  by  the 
capital  letter  E.  It  is  of  course  expressed  in  the  same  unit'  as  the  stress  per 
unit  of  area,  as,  for  instance,  in  pounds  per  square  inch;  but  it  is  usually  stated 
simply  in  pounds,  the  words  "per  square  inch"  being  understood. 

(d)  It  will  be  noticed  that  the  greater  the  stress  required  to  produce  a  given 
stretch  in  a  body,  the  greater  is  its  modulus  of  elasticity.  Hence  the  modulus. 
K  is  a  measure  of  the  resistance  which  the  body  can  make  against  a  change  in 
shape.  This  resistance  we  call  the  "  elasticity "  of  the  body,  although  in 
every-day  language  (afid,  indeed,  often  in  a  scientific  sense  also)  we  apply  the 
term  "  elasticity  "  rather  to  the  ability  of  a  body  to  sustain  considerable  distor- 
tion without  losing  its  power  of  returning  to  its  original  shape. 


8TBENOTH  OP   MATERIALS.  467 


(•)  Since  total  stress 

Stress  per  unit  of  area  =  a^xcaa  ^ 


and  since 


•tretch  per  unit  of  length  =     total  stretch 
original  length 


we  may  find  the  modulus  of  elasticity  of  any  material,  from  experiment  apoa 
»ny  specimen  of  it,  thus : 

Modulu.  of  el..tlclty  -    total  stress    X  original  length  ^     _  _  (,, 
total  stretch  X  ^ea 

From  this  we  have  the  following  equations: 

Total  stress,  in  lbs.  modulus  of         total  stretch  ^         area 

lequired  for  a  given  =-  elasticity  in  lbs  X  in  inches  ^  in  square  ins  .  .  .  (Q 

total  stretch,  in  inches       per  square  inch       original  length  in  inches 

Stress  per  unit  of  area,  modulus  of  stretch  ner 

in  lbs.  per  square  inch,  required  — »  elasticity  in  lbs.  X  ^j.^  ^f  length    •••(?) 
for  a  given  stretch,  in  inches  per  square  inch  ^ 

Total  stretcliL,  in     ^ total  stress  v^      original  length,  in  inches 
inches,  under  any  stress  ^      in  lbs,     -^  modulus  of  elasticity,  ^  area,  in  •  •  •  (^ 

in  lbs.  per  square  inch  ^  sq.  ins 

^  ori^nal  length  ^         stress,  in  lbs,  per  square  inch  .  .  ,  .  (5i 

"■      in  inches  modulus  of  elasticity  in  lbs.  per  sq.  inch  *  *  *  ' 

BtretcH  per  luilt  of  length stress,  in  lbs,  per  square  inch^ ^gj 

modulus  of  elasticity,  in  lbs  per  sq.  m. 

The  modulus  of  elasticity  is  used  chiefly  in  connection  with  the  stiffhess  of 

beams.    In  any  beam,  supported  at  both  ends  and  loaded  at  the  center: 

iifn«iHiiis  n«         /load     ,     %  weight  of  clear  \  ^  cube  of  span 
elasticUy,^*    -  Un  lbs.  +  spanof1>eam,inlbs.j  X     in  inches       .  .  ,   p) 
lbs.  per  sq.  inch  40  v  deflection,  ^  moment  of  inertia 

'^  in  inches    ^ 

*'•  BS    _  (WJlMJ^*  ...   (8) 

48  Al 
If  the  beam  is  rectangular ^  this  becomes 
Modulna  of        (  ^^^   4-  H  weight  of  clear  \  ^  cube  of  span 
elasticity,  in    =-  Un  lbs.  -^  span  of  beam,  in  IhaJ  ^    in  inches        .  .  .  (i| 
lbs  per  square  inch         a  y  deflection,  ^  breadth,  ^  cube  of  depth 
*  '^  in  inches  ^    inches    ^      in  inches 

Corresponding  formulae  for  modulus  of  elasticity  in  beams  otherwise  sup- 
ported and  loaded,  may  be  readily  deduced  from  those  for  deflection 

(f )  If  equal  additions  of  stress  could  produce  equal  additional  stretches  in  a 
body  to  an  indefinite  extent,  both  within  and  beyond  the  elastic  limit,  then  a 
stress  equal  to  the  Modulus  of  Elasticity  would  double  the  lengWi  of  a  bar  when 
applied  to  it  in  tension,  or  would  shorten  it  to  zero  when  applied  in  cow»pr«s8to». 
In  other  words,  if  equation  (5), 

total  stretch.  =  original  length  ^  stress  per  square  inch 
in  inches       -^   modulus  of  elasticHy    ' 

held  good  beyond  the  elastic  limit,  as  it  does  (approximately)  within  that  Umlt| 
and  if  we  could  make  the  stress  per  square  inch  equal  to  the  rsodulas  of 
elasticity,  we  should  have  total  stretch  =  original  length. 


458  STRENGTH   OF   MATERIALS. 

For  example,  a  one-inch  square  bar  of  wrought  iron  will,  within  the  limit  ol 

elasticity,  stretch  or  shorten,  on  an  average,  about  t'^^jtu  of  its  length  undei 
each  additional  load  of  2240  lbs.  If  it  could  continue  to  stretch  or  shorten 
indefinitely  at  this  rate,  it  is  evident  that  12000  times  2240  lbs.,  or  26  880  000  Ibs^ 
(which  is  about  the  average  modulus  of  elasticity  for  such  bars)  could  either 
stretch  the  bar  to  double  its  length  or  reduce  it  to  zero. 

If  equal  additional  stresses  applied  to  a  bar  could  indefinitely  produce 
stretches,  each  bearing  a  constant  proportion  to  the  increased  length  of  the  bar^  if 
in  tension;  or  to  the  diminished  length,  if  in  compression;  then  the  same  load 
which  would  double  the  length  of  the  bar  if  applied  in  tension,  would  reduce 
it  to  half  its  length,  if  applied  in  compression. 

is)  ^6  gi^6  below  a  table  of  average  Moduli  of  Blastlclty,  in  round 
numbers,  for  a  few  materials ;  remarking,  by  way  of  caution,  that,  even  in  the 
case  of  ductile  materials,  the  stretches  produced  by  stresses  within  the  elastic 
limit  are  so  small,  and  (owing  to  difierences  in  the  character  of  the  material)  so 
irregular,  that  a  satisfactory  average  can  be  arrived  at  only  by  comparing  many 
experiments ;  while,  in  the  case  of  materials,  such  as  stone,  brick,  etc.,  where 
almost  no  perceptible  stretch  takes  place  before  rupture,  it  is  scarcely  worth 
while  to  give  any  values  as  representing  the  actual  moduli.  Thus,  eighteen 
experiments  upon  a  single  brand  of  neat  cement  for  the  St.  Louis  bridge,  indi* 
cated  a  Modulus  varying  from  800  000  to  6  930  000  (!)  pounds  per  square  inch 
in  tension,  and  from  500  000  to  1  600  000  in  compression. 

(h.)  Owing  to  the  fact  that  the  stretches  within  the  elastic  limit  are  seldom, 
if  ever,  exactly  proportional  to  the  stresses,  but  only  approximately  so,  the 
modulus  of  elasticity,  as  found  by  experiment  for  a  given  material,  will 
generally  vary  somewhat  with  the  stress  at  which  the  stretch  is  taken. 

Art.  4:  (a)  The  stress  beyond  which  the  stretches  in  any  body  increase  per* 
ccptibly  faster  than  the  stresses^  is  called  the  limit  of  elasticity  of  that 
body.  Owing  to  the  irregularity  in  the  behavior  of  different  specimens  of  the 
same  material,  and  to  the  extreme  smallness  of  the  distortions  caused  in  most 
materials  by  moderate  loads,  and  because  we  often  cannot  decide  just  when  the 
stretch  begins  to  increase  faster  than  the  load,  the  elastic  limit  is  seldom,  if 
fever,  determinable  with  exactness  and  certainty.*  But  by  means  of  a  large 
number  of  experiments  upon  a  given  material  we  may  obtain  useful  average 
or  minimum  values  for  it,  and  should  in  all  cases  of  practice  keep  the  stresses 
well  within  such  values ;  since,  if  the  elastic  limit  be  exceeded  (through  mis- 
calculation, or  through  subsequent  increase  in  the  stress  or  decrease  in  the 
Strength  of  the  material)  the  structure  rapidly  fails.  The  table,  below,  gives 
approximate  average  elastic  limits  for  a  few  materials. 

(1»)  Brittle  materials,  such  as  stones,  cements,  bricks,  etc.,  can  scarcely  be  said 
to  have  an  elastic  limit ;  or,  if  they  have,  it  is  almost  impossible  to  determins 
it ;  since  rupture,  in  such  bodies,  takes  place  before  any  stretch  can  be  sati» 
factorily  measured.  Thus,  in  the  18  specimens  of  one  brand  of  cement^ 
referred  to  in  Art.  Sg  above,  the  experiments  indicated  an  elastic  limit  varyinij 
between  16  and  104  (!)  pounds  per  square  inch  in  tension,  and  from  424  to  15(« 
in  comprtssion, 

(c)  Experiments  show  that  a  small  permanent  **set'*  (stretch)  probably 
takes  place  in  all  cases  of  stress  even  under  very  moderate  loads ;  but  ordinarilt 
it  first  becomes  noticeable  at  about  the  time  when  the  elastic  limit  is  exceeded, 
Many  writers  define  the  elastic  limit  as  that  stress  at  which  the  first  marked 
permanent  set  appears. 

(d)  The  elastic  ratio  of  a  material  is  the  quotient, elastic  limit j| 

ultimate  strength 
1$  usually  expressed  as  a  decimal  fraction. 

♦  The  U.  S.  Board  appointed  to  test  Iron,  Steel,  Ac.,  found  a  variation  of  nearly 
4000  lbs.  per  square  inch  in  the  elastic  limit  of  bars  of  one  make  of  rolled  iroi^ 
prepared  with  great  care  and  having  very  uniform  tensile  strength ;  and,  in 
another  very  carefully  made  iron,  a  difference  of  over  30  per  cent,  between  twe 
bars  of  the  same  sise.    Report,  1881,  Vol.  1,  p.  31. 


STRENGTH   OF   MATERIALS. 


(e)  Table  of  Moduli  of  Klastlclty  and  of  Blastlc  Limits   for 
dlffereitt  materials. 

The  values  here  given  are  approximate  averages  compiled  from  many  sources. 
Authorities  differ  considerably  in  their  data  on  this  subject.  See  Art.  3  (g)  and 
(h),  and  Art.  4  (a)  and  (6). 


Modulus 
or  Coeff 


Elasticity. 


Stretch  or  Compression 

in  a  length  of  10  ft, 

.under  a 

load  of 

Approx  elas 

1000  fts  per 

1  ton  per 

limit. 

sq  in. 

sq  in. 

Ins. 

Ins. 

B>s  per  sqin. 

.075 

.168 

4500 

.092 

.207 

4000 

.086 

.192 

5000 

.013 

.029 

6000 

.009 

.019 

16000 

.120 

.269 

4600 

.007 

.015 

6300 

.007 

.015 

10000 

.120 

.269 

2000 

.015 

.034 

8200 

.010 

.022 

4600 

to 

to 

to 

.005 

.012 

8000 

.007 

.015 

6250 

.006 

.015 

20000 

to 

to 

to 

.003 

.007 

40000 

.004 

.009 

30000 

.005 

.010 

27000 

.008 

.018 

13000 

.109 

.244 

2300 

.167 

1100 

.120 

1100 

.086 

.192 

2700 

.120 

.269 
to 

to 

.060 

.134 
.179 

.080 

3S00 

.075 

.168 

S300 

.008 

.018 

3700 

.075 

.168 

S300 

.004 

.009 

34000 

to 

to 

to 

.003 

.006 

44000 

.003 

.007 

3900O 

.120 

.269 

4000 

.060 

.134 

5000 

.026 

1500 

Ash 

Beech 

Birch 

Brass,  cast 

*'        wire 

Chestnut 

Copper,  cast 

"        wire 

Elm 

Olass  

Iron,  cast 

'*      "    average 

"    wrought,  in  either  t 

bars,  sheets  or  plates < 

"  "  "    average 

•'  wire,  hard 

"  wire  ropes 

Larch 

Lead,  sheet 

*•      wire 

Mahogany 

Oak 

' '    average 

Pine,  white  or  yellow , 

Slate 

Spruoe 

Steel  bars 

"        "    average , 

Sycamore 

Teak , 

Tin,  cast •• 


R)s  per  sq  in. 
1  600  000 
1  300  000 
1  400  000 
9  200  000 

14  200  000 
1  000  000 

18  000  000 

18  000  000 
1  000  000 
8  000  000 

12  000  000 
to 

23  000  000 

17  500  000 

18  000  000 

to 
40  000  000 
29  000  000 
26  000  000 

15  000  000 
1  100  000 

720  000 
1  000  000 
1  400  000 

1  000  000 

to 

2  000  000 
1  500  000 
1  600  000 

14  500  000 
1  600  000 
29  000  000 

to 
42  000  000 
35  500  000 

1  000  000 

2  000  000 
4  600  000 


(f)  Yield  point.  In  testing  specimens  of  iron  and  steel,  it  is  commonly 
found  that,  after  passing  the  elastic  limit,  see  (Art  4  a),  a  point  is  reached  where 
the  ratio  between  the  stretch  and  the  stress  suddenly  begins  to  increase  more 
rapidly.    This  is  commonly  called  the  yield  point.    See  Fig.  D. 


460  STRENGTH   OF   MATERIALS. 

Art.  5  (»)  Resilience  is  a  name  given  to  the  work  (as  in  inch-pounds) 
which  must  be  done  in  order  to  produce  a  certain  stretch  in  a  given  body. 
This  work  is  equal  to 

resilience  _  said  stretch  y  mean  stress  in  pounds  employed  in  producing 
in  inch-pounds         in  inches    ^  the  stretch. 

The  total  resilience  is  the  work  done  in  causing  rapture.  The  elastic  resilience 
(frequently  called,  simply,  the  resilience)  is  that  done  in  causing  the  greatest 
stretch  possible  within  the  elastic  limit, 

(b)  Suddenly  applied  loads.  Place  a  weight  of  4  lbs.  in  a  spring 
balance,  but  let  it  be  upheld  by  a  string  fastened  to  a  firm  support  in  such  a 
way  that  the  scale  of  the  balance  shall  show  only  1  lb.  By  now  cutting  thia 
string  with  a  pair  of  scissors,  we  suddenly  apply  4  —  1=3  lbs. ;  and  the  weight 
will  descend  rapidly,  until,  for  an  instant,  the  scale  shows  about  1  +  twice 
3  =  7  lbs.  In  other  words,  the  load  of  3  lbs.  applied  suddenly  (but  without  jar 
or  shock)  has  produced  nearly  twice  the  stretch  that  it  could  produce  if  addsd 
grain  by  grain,  as  in  the  shape  of  sand. 

For,  when  the  load  is  first  applied,  the  inherent  forces,  as  noticed  in  Art.  2  (a), 
are  insufl&cient  to  counteract  its  stress.  Hence  the  load  begins  to  stretch  the 
spring.  The  work  thus  done  is  equal  to  the  product,  suddenly  applied  weight 
of  3  lbs.  X  the  stretch  of  the  spring ;  and  it  has  been  expended  (except  a  small 
portion  required  to  counteract  friction)  in  bringing  the  resisting  forces  into 
action,  thus  storing  in  the  spring  potential  energy  nearly 

sufficient  to  do  the  same  work ;  i.  e.,  to  lift  the  weight  (3  lbs.)  to  the  point  (1  lb. 
on  the  scale)  from  which  it  started.  But  a  portion  of  this  energy  has  to  work 
against  friction  and  the  resistance  of  the  air.  Therefore  the  weight  does  not 
rise  quite  to  its  original  height. 

The  shortening  of  the  spring  nearly  to  its  original  length  has  now  reduced 
its  inherent  forces  almost  to  zero ;  and  the  weight  again  falls,  but  not  so  far  aa 
before.  It  thus  vibrates  through  a  less  and  less  distance  each  time,  and  finally 
comes  to  rest  at  a  point  (4  lbs.  on  the  scale)  midway  between  its  highest  and 
lowest  positions  (1  lb.  and  7  lbs.)  Thus,  within  the  limit  of  elasticity,  a  load 
applied  suddenly  (though  without  shock)  produces  temporarily  a 
•tretcli  nearly  equal  to  t\>rlce  tlutt  wliicl&  It  could  produce  It 
applied  gradually ;  i.  c,  twice  that  which  it  can  maintain  after  it  comes 
to  rest. 

Renkark.  If  the  load  is  added  in  small  instalments,  each  applied  suddenly^ 
then  each  instalment  produces  a  small  temporary  stretch  and  afterward  main- 
tains a  stretch  half  as  great.  Thus,  under  the  last  small  instalment,  the  bar 
Stretches  temporarily  to  a  length  greater  than  that  which  the  total  load  can 
maintain,  by  an  amount  equal  to  half  the  small  temporary  stretch  produced  by 
the  sudden  application  of  the  last  small  instalment. 

(c)  The  Modulus  of  E21astlc  Resilience  (often  called,  simply,  th# 
Modulus  of  Resilience)  of  a  material,  is  the  work  done  upon  one  cubic  inch  of  it 
by  a  gradually  applied  load  equal  to  the  elastic  limit.    Or, 

Modulus  stretch  in  inches  mean  stress 

ivf  TwtriioTino  "=  P*^  ^^<^^  ^f  l^i^gif^  X  in  lbs.  per  square  inch 
oi  resuience       ^^  ^^^  elastic  limit        causing  that  stretch 

If^  as  is  usually  done,  we  assume  this  mean  stress  to  be  >^  the  elastic  limilk 
then,  by  formula  (6) 

Modulus  elastic  limit  ^  ly  rf„«*,v  KmM 

Of  resiUence  -  ^.odulus  of  elasticity  ^  ^  ^^^^'^  "^ 

^^  ly  square  of  elastic  limit 
*  modulus  of  elasticity 

The  elastic  resilience  of  any  piece  is  then 

Eesilience  =  modulus  of  resilience  X  volume  of  piece  in  cubic  incheisr 


STRENGTH   OF   MATERIALS.  461 

The  modulus  of  resilience  of  a  material  is  a  measure  of  its  capacity  for  resist- 
ing shocks  or  blows. 

Elastic  Ratio.  The  elastic  ratio  of  a  material  is  the  ratio  between  its 
elastic  limit  (Art.  4  a,  and  its  ultimate  strength  (Art.  2  c. 

Thus,  if  the  ultimate  tensile  strength  of  a  steel  bar  be  70,000  pounds  per 
square  inch,  and  its  elastic  limit  in  tension  39,900  pounds  per  square  inch,  its 
elastic  ratio  is 

39,900      ^  ^^ 

■70:000  =  ^•^^•.        - 

Inasmuch  as  it  is  now  generally  conceded  that  the  permissible  working  load 
of  a  material  should  be  determined  by  its  elastic  limit  rather  than  by  its  ulti- 
mate strength,  it  follows  that,  other  things  being  equal,  a  high  elastic  ratio  is 
in  general  a  desirable  qualification ;  but,  on  the  other  hand,  it  is  possible,  by 
modifying  the  process  of  manufacture,  to  obtain  material  of  high  elastic  ratio, 
but  deficient  in  "  body  "  or  in  resilience — i.  e.,  in  capacity  to  resist  the  etfect  of 
blows  or  shocks,  or  of  sudden  application  or  fluctuation  of  stress. 

In  the  manufacture  of  steel  it  is  found  that  the  elastic  ratio  is  increased  by 
increasing  the  reduction  of  area  in  hammering  or  rolling,  and  that  the  rate  of 
increase  of  elastic  ratio  with  reduction  of  area  increases  rapidly  as  the  reduc- 
tion becomes  very  great.  This  is  indicated  by  the  following  experiments  by 
Kirkaldy  on  steel  plates  :* 


Plates  1  inch  thick,  mean  elastic  ratio  0.53 

'      "'  "  "  "        0.53 

"       0.54 

"        0.61 


^ 


*  Annual  Report  of  the  Secretary  of  the  Navy,  Washington,  1885,  Vol.  I.  p. 
499;  and  Merchant  Shipping  Experiments  on  Steel,  Parliamentary  Papar,  d 
2897,  London,  1881. 


462 


STRENGTH   OF   MATERIALS. 


Art.  6.  A  section  may  be  weakened  by  increasing  its 
\¥idtli.  On  pp  400,  etc.,  we  cousidered  the  case  where  the  width  o^  ...he  base 
is  fixed  and  where  the  point  of  application  of  the  resultant  of  the  forces  acting 
upon  it  is  shifted  to  different  positions  along  the  base.  We  will  now  notice  the 
case  where  the  resultant  is  applied  at  a  constant  distance  from  one  end  of  the 
base,  but  where  the  base  is  of  varying  width,  so  that  this  constant  distance  may 
be  equal  to,  or  greater  or  less  than,  the  half  width  of  the  base. 

Let  Fig.  E  represent  a  side  view  of  a  bar  of  uniform  thickness  =  1,*  but  (as 


Scale  of  Unit  Pressures 


m   i 

g    a 

C 

e 

1.75 
1.50 

\ 

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F 

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p. 

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1.00 
0.75 
0.50 

V] 

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K 

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Ujdt 

prcj 

T»PP 

ere 

Ige- 

O.ftO 

1 
i 

»sur< 

.sat 

0.25  0.50  0.7$  1.00  1.25  1.50  2.00  2.50  S.OO  i.00 

Widths  (ab=-l) 

shown)  of  varying  width,  and  subjected  to  pressures,  the  resultant  P  of  which 
is  =  1,*  and  passes  through  the  center  of  that  section  ab  whose  width  is  1.*  f 


*  We  adopt  the  value  1  for  the  pressure  P,  the  width  a  b,  and  the  thickness 
merely  in  order  to  facilitate  the  explanation.  It  is  not  essential  to  the  applica- 
tion of  the  principle. 

t  We  here  suppose  ourselves  dealing  with  a  perfectly  rigid  and  homogeneous 
material.  In  practice,  these  values  would  be  more  or  less  modified  by  yielding 
of  the  particles  under  stress,  by  unevennesses  in  the  surfaces  of  the  supposed 
cross  sections,  etc.  Nevertheless,  the  general  principles  here  laid  down  hold 
good. 


STRENGTH   OF   MATERIALS. 


463 


The  pressure  per  unit  of  area  of  cross  section  (or  "unit  stress")  in  the  section 
j>  p 

a 6  is  then  ~rTT^  =  --=  P=  1*,  and  may  be  assumed     to  be  uniformly  dis- 
tributed over  it. 

But  at  other  sections  of  the  bar  the  resultant  is  nearer  to  one  edge  than  to  the 
other,  and  the  unit  stress  can  then  no  longer  be  assumed  to  be  uniformly  dis- 
tributed over  the  cross  section,  but,  as  explained  on  pp.  400  to  403  is  a'max- 
imum  at  the  edge  nearest  to  the  resultant,  and  diminishes  gradually  and  uni- 
formly    to  a  minimum  at  the  farther  edge. 

This  is  indicated  by  the  shaded" triangles,  etc.,  in  Fig.  E  and'  by  the  curves  in 
Fig.  F,  which  show,  for  the  several  sections  in  Fig.  E,  the  mean  unit  stress* 
and  the  unit  stresses  at  the  upper  and  lower  edges  respectivelv,  calculated  by  the 
rules  on  pp.  231  a  and  231  b. 

These  stresses  are  also  given  in  the  following  table: 

P 
IJiiit  {Stresses  in  Fig.  E ;  the  unit  stress  --  in  section  a  b  being  taken  as  l.f 

ab  ° 


Stress  per  unit  of  area  of 

cross  section. 

Width 

Mean. 

At  lower  edge  m  e. 

At  upper  edge  nf. 

4.00 

0.25 

0.8125 

—  0.3125  t 

ef 

3.00 

Vb 

1.00 

-Vz 

2.50 

0.40 

1.12 

-0^32 

2.00 

0.50 

1.25 

—  0.25 

cd 

1.50 

% 

1^ 

0 

1.25 

0.80 

1.28 

0.32 

ab 

1.00 

1.00 

1.00 

1.00 

gh 

0.75 

1^ 

0 

2% 

ikt 

0.50 

2.00 

-   n 

8.00 

mn^ 

0.25 

4.00 

—  32  J 

40.00 

It  is  important  to  notice  that  for  a  given  force  P,  and  for  widths  less 
than  3  a  6,  the  strongest  section  of  this  bar  is  not  the  widest  one,  but  that  (a  6)  at 
which  the  resultant  P  passes  through  the  center  of  the  section.  In  other  words, 
a  bar  may  be  weakened  by  additions  to  its  cross  section  if 

those  additions  are  such  as  to  cause  the  resultant  of  the  pressures  to  pass  else- 
where than  through  the  center  of  any  cross  section.  This  fact  is  entirely  inde- 
pendent of  the  weight  of  the  added  portion. 

Among  the  sections  wider  than  ab,  the  weakest  is  that  {cd)  whose  width 
is  =  1.5 a 6.    At  that  section  the  lower  edge  we  has  its  maximum  unit  stress 

(  =  1/^  X  — 7  )  while  at  d  in  the  upper  edge  there  is  no  pressure.    Beyond  c  d 

the  upper  edge  w/is  in  tensionX  and  the  unit  pressure  along  me  decreases,  be- 

p 
coming  again  =  —r  at  ef,  where  the  width  e/  is  =  3  a  6,  and  decreasing  still 

further  with  further  increase  in  width. 


*  In  the  case  discussed  on  pp.   400  to  403,        the  mean  pressure  un=  — , 

uv 
remained  constant  so  long  as  the  entire  surface  uv  was  called  into  play.    Here, 
on  the  contrary,  the  area  of  the  section  varies.    Hence  the  mean  unit  pressure 
varies  also,  and  inversely  as  the  area. 

t  See  foot-note  *,  p.  462. 

X  In  the  present  discussion,  as  well  as  in  that  on  pp.  400  to  403,  we  have 

assumed  cases  of  coru,pression  for  illustration,  but  the  principle  involved  applies 
equally  to  cases  where  the  force  applied  is  tennile.  In  such  cases,  however,  the 
terms  "pressure  "  and  "  tension  "  are  of  course  reversed. 

^  The  unit  stresses  at  the  edges  in  section  i  k  are  too  great  to  be  shown  con- 
veniently in  either  figure ;  while  those  in  section  m  n  (as  the  table  shows)  far 

p 
exceed  the  limits  of  the  figures.    The  pressure  at  k  would  be  —  =  oc  (infinity) 

were  it  not  for  the  tensions  in  the  lower  part  of  the  section. 


464 


STRENGTH    OF    MATERIALS. 


When  the  width  becomes  less  than  that  at  ab,  as  at  g h,  etc.,  the  upper  edge  o 
the  bar  comes  nearer  to  the  resultant  than  the  lower  edge,  and  hence  receive 
the  maximum  pressure. 

When  the  width  =  %ab,  as  at  g h,  the  distance  of  the  resultant  from  th 
upper  edge  is  V^  the  width  of  the  section.    The  pressure  at  the  lower  edge  i 

PI  P 

then  =  0 ;  the  mean  pressure  in  a  A  is  — -  X  7rir~  —  1/^  X  ~t>  ^^^  ^^^  pressure  a 

ah      v.io  ao 

p 
the  upper  edge  is  twice  the  mean  pressure  in  g  A,  or  2%  X  ~^. 

When  the  width  becomes  less  than  %  a  &,  as  at  i  A;  and  m  n,  the  pressure  at  th 
lower  edge  me  becomes  negative  or  tensile.*  Thus,  when,  as  at  iA;,  the  widt; 
is  —  }/^ab,  and  the  resultant  passes  through  the  upper  edge,  the  unit  pressur 

P  P 

at  that  edge  is  =  S  x      ,  ,  while  the  lower  edge  sustains  a  tension  of  4  X  —  ;  anc 

ab  ab 

as  the  section  is  further  reduced,  these  stresses  are  still  further  and  very  rapidl 
increased. 

The  condition  of  those  sections  (such  as  m/i)  where  the  line  of  the  resultan 
passes  outside  the  section,  is  similar  to  that  of  the  section  mn  of  a  bent  hoo' 
sustaining  a  load,  as  in  Fig,  G. 


Figr.  o. 


Messrs.  William  Sellers  &  Co.,  of  Philadelphia,  had  occasion  to  test  a  numbc 
of  cast-iron  beams,  each  having  a  large  circular  opening,  as  in  the  annexe 
figure.  These  beams  broke,  not  at  the  smallest  section  directly  under  the  cente 
of  the  opening,  but  a  little  to  one  side,  where  the  section  was  deeper,  as  indicate 
in  Fig.  H . 


STRENGTH   OF    MATERIALS.  465 

Fatig^ue  of  Materials.  In  the  following  articles  on  Strength  of  Mate- 
rials, the  ultimate  or  breaking  load  is  that  which  will,  during  its  first  application, 
rupture  the  given  piece  within  a  short  time.  But  Wohler's  and  Spangenberg's 
experiments  show  that  a  piece  may  be  ruptured  by  repeated  appliea- 
tions  of  a  load  much  less  than  this ;  and  that  tbe  oftener  the  load  is  applied  the 
less  it  needs  to  be  in  order  to  produce  rupture.  Thus,  wrought  iron  which  re- 
quired a  tension  of  53000  lbs  per  sq  inch  to  break  it  in  800  applications,  broke 
with  35000  lbs  per  sq  inch  applied  about  10  million  times  ;  the  stress,  after  each 
application,  returning  to  zero  in  both  cases. 

The  di/f  between  the  maximum  and  minimum  tension  in  a  piece  subjected  to 
tension  only,  or  between  the  max  and  min  compression  in  a  piece  subjected  to 
eomp  only;  or  the  sum  of  the  max  tension  and  max  comp  in  a  piece  subjected 
alternately  to  tension  and  comp ;  is  called  the  ran^e  of  stress  in  the  piece. 
Stresses  alternating  between  0  and  any  point  within  the  elastic  limit  may  be 
repeated  many  million  times  without  producing  rupture.* 

For  a  given  number  of  applications,  the  load  required  for  rupture  is  least  when 
the  range  of  stress  is  greatest.  If  the  stress  is  alternately  comp  and  tension, 
rupture  takes  place  more  readily  than  if  it  is  always  comp  or  always  tension. 
That  is,  it  takes  place  with  a  less  range  of  stress  applied  a  given  number  of 
times,  or  with  a  less  number  of  applications  of  a  given  range  of  stress.  For  a 
given  range  of  stress  and  given  number  of  applications,  the  most  unfavorable 
condition  is  where  the  tension  and  comp  are  equal. 

The  above  facts  are  now  generally  taken  into  consideration  in  designing 
members  of  important  structures  subject  to  moving  loads.  For  instance,  Mr. 
Jos.  M.  Wilson,  C.  E.,  Mem.  Inst.  C.  E.  (London  Eng.),  Mem.  Am.  Soc.  C.  E.,  uses 
the  following  formulae  for  determining  the  "permissible  stress"  in  iron  bridges, 
in  lbs  per  sq  inch ;  in  order  to  provide  the  proper  area  of  cross  section  for  each 
member. , 

For  pieces  subject  to  one  hind  of  stress  only  (all  comp  or  all  tension) 


,  /,      mm  stress  in  the  piece\ 

k  =  wf  ( IH 1 ■    ^.       . —  I 

V        max  stress  m  tbe  piece/ 


piece/ 

For  a  piece  subject  alternately  to  comp  and  tension,  find  the  max  comp  and  th« 
max  tension  in  the  piece.  Call  the  lesser  oi  these  two  maxima  "max  lesser", 
and  the  other  or  greater  one,  "  max  greater  ".    Then 

,  [  ^         max  lesser    \ 
a  =»  uf  1 1 I 

\         2  max  greater/ 

For  a  piece  whose  max  comp  and  max  tension  are  equal,  this  becomes 

»-«t(.-4)-|- 

The  above  a  is  the  permissible  tensile  stress  in  lbs  per  sq  inch  on  any  mem- 
ber; but  the  permissible  compressive  stress  is  found  by  "  Gordon's  formula"  for 
pillars,  p  495,  using  a  (found  as  above)  as  the  numerator,  instead  of/.  For  a  in 
the  divisor  or  denominator  of  Gordon's  formula  (which  must  not  be  confounded 
with  the  a  of  the  foregoing  formulae)  Mr.  Wilson  uses  for  wrought  iron : 

when  both  ends  are  fixed 36000 

when  one  end  is  fixed  and  one  hinged 24000 

when  both  ends  are  hinged 18000 

Experiments  show  that  materials  may  fail  under  a  long:  continued 

stress  of  much  less  intensity  than  that  produced  by  the  ult  or  bkg  load. 

*  This  does  not  always  hold  in  cases  where  the  elastic  limit  has  been  artificially  raised 
by  jM-ocees  of  manufacture,  etc.  Oft-repeated  alternations  between  tension  and  compres- 
sion below  such  a  limit  reduce  it  to  the  natural  one.  A  slight  flaw  may  cause  rupture 
under  comparatively  few  applications  of  a  range  of  stress  but  little  greater,  or  even  less, 
than  the  elastic  Hmit.  Rest  between  stresses  increases  the  resisting  power  of  a  piece. 
In  many  cases,  stresses  a  little  beyond  the  elastic  limit,  even  if  oft- repeated,  raise  that 
limit  and  the  strength,  but  render  the  piece  brittle  and  thus  more  liable  to  rupture  from 
shocks ;  and  a  little  further  increase  of  stress  rapidly  lessens,  or  may  entirely  destroy, 
the  elasticity.  A  tensile  stress  above  the  elastic  limit  greatly  lowers,  or  may  even  destroy, 
the  compressive  elasticity,  and  vice  versa.  If  a  tensile  stress,  by  stretching  a  piece,  reduces 
its  resisting  area,  it  may  thus  reduce  its  total  strength,  even  though  the  strength  per 
$q  in  has  increased.  Mr.  B.  Baker  finds  that  hard  steel  fatigues  much  faster  under  re- 
peated loads  than  soft  steel  or  iron. 

t  n  =  6500  lbs  per  sq  inch  for  rolled  iron  in  compression 

=  7Q00  lbs         "  "  "    tension  (plates  or  shapes). 

«7500B>9        '*  for  double  rolled  iron  in  tension  (Hnks  or  rods). 

30 


466  STRENGTH   OF    MATERIALS. 

TRANSVERSE  STRENGTH. 

1.  In  Statics,  ^^  285,  etc.,  we  discuss  the  action  of  external  or  destnn 
tive  forces  upon  cantilevers,  beams  and  trusses.  We  here  discuss  the  reai 
tion  of  the  internal  or  resisting  forces  (stresses)  in  solid  cantilevers  ar 
beams,  in  order  to  determine  their  loads.     See  also  Trusses. 

2»  Unless  otherwise  stated  or  apparent,  we  assume  that  the  stresses  in  a 
parts  of  the  cantilever  or  beam  are  within  the  elastic  limit. 

Conditions  of  Equilibrium. 

3.  For  equilibrium,  the  internal  forces,  and  their  moments,  must  balani 
the  external  forces  and  their  moments.  In  other  words,  if  the  cantilever  < 
beam  be  supposed  cut  by  a  section  at  any  point,  we  must  have 

(1)  2  vertical  forces       =  0 

(2)  2  horizontal  forces  =  0 

(3)  2  moments  —  0 

Or: 

(1)  Algebraic  sum  of  the  internal  vertical  stresses  ==  algebraic  sum  of  tl: 
external  vertical  forces  on  either  side  of  the  section ; 

(2)  Sum  of  horizontal  tensile  stresses  =  sum  of  horizontal  compressv 
stresses;  and 

(3)  Algebraic  sum  of  the  moments  of  the  internal  stresses  =»  algebra 
sum  of  moments  of  external  forces  on  either  side  of  the  section. 

4.  Cantilevers  and  beams  of  uniform  cross-section  have  usually  a  supe 
abundance  of  strength  against  shearing,  and  fail  (if  at  all)  near  tl 
middle,  where  the  bending  moment  is  greatest.  Hence  the  discussion  ^ 
their  resistance  turns  principally  upon  equilibrium  of  moments.  For  the 
resistance  to  vertical  shear,  see  Statics,  ^%  325,  etc.,  and  p.  499.  See  ah 
Horizontal  Shear,  ^  If  51  to  53,  below. 


5.  For  equilibrium,  therefore,  the  resisting  moment,  R  (  =  the  sum  of  tl 
resisting  moments,  r,  of  all  the  particles  in  any  cross-section  of  the  cant 
lever  or  beam,  Fig.  1  or  2),  must  be  equal  to  the  bending  moment,  M,  or  algi 
braic  sum  of  the  moments  of  all  the  external  forces  on  either  side  of  ti 
section. 


Reactions  of  Fibers. 

6.  In  a  truss  or  framed  beam  (see  Trusses)  the  resistance  of  each  of  i1 
two  chords  is  regarded  as  acting  in  a  line  passing  through  the  centers  of  gra^ 
ity  of  the  cross-sections  of  the  chord;  but,  in  a  solid  cantilever.  Fig.  1,  c 
beam.  Fig.  2,  the  total  resisting  moment  is  the  sum  of  the  separate  resistin 
moments  of  the  several  fibers  throughout  the  cross-section. 

Neutral  Surface.     Neutral  Axis. 

7.  When  a  cantilever  (or  beam)  bends,  the  fibers  in  the  upper  (or  lowei 
part  of  each  cross-section  are  extended,  while  those  in  the  lower  (or  uppei 


TRANSVERSE  STRENGTH. 


467 


part  are  compressed  (see  Figs.  1  and  2) ;  the  extension  and  compression 
being  greatest  at  the  top  and  bottom  of  the  section,  and  thence  decreasing 
uniformly  inward  toward  a  surface,  n  n.  Figs,  (a),  near  the  center  of  the 
cross-section.  In  this  surface,  which  is  called  the  neutral  surface,  the 
fibers  are  neither  extended  nor  compressed.  The  line,  o  o.  Figs.  (6),  formed 
by  the  intersection  of  the  neutral  surface  with  any  cross-section  of  the  canti- 
lever or  beam,  is  called  the  neutral  axis  of  that  section. 

8.  In  order  that  the  algebraic  sum  of  all  the  horizontal  stresses  in  the 
cross-section  may  be  zero,  as  required  for  equilibrium,  the  neutral  axis  must 
pass  through  the  center  of  gravity  of  the  section.  Hence,  the  neutral  sur- 
face passes  through  the  centers  of  gravity  of  all  the  cross-sections. 

9.  The  neutral  axis  may  be  found  by  balancing  the  section  (cut  out  of 
cardboard)  over  a  knife-edge.  Or  see  Center  of  Gravity,  under  Statics,  |If  ^ 
125,  etc.  Every  section  has  an  indefinite  number  of  neutral  axes,  all  passing 
through  its  center  of  gravity  in  as  many  different  directions.  The  axis  re- 
quired, in  any  given  case,  is  that  one  which  is  normal  to  the  plane  of  the 
bending  moment  under  consideration. 

^  In  the  following  discussion,  we  assume  that  the  neutral  axis  of  the  sec- 
tion is  normal  to  the  line  of  action  (usually  vertical)  of  the  load",  as  it  gen- 
erally is.  For  other  cases,  as,  for  instance,  the  case  of  roof  purlins,  see 
*'  The  Determination  of  Unit  Stresses  in  the  General  Case  of  Flexure,*'  by 
Prof.  L.  J.  Johnson,  Boston  Soc.  of  Civil  Engineers,  in  Jour.  Ass'n  of 
Eng'ng  Socs.,  vol.  xxviii.  No.  5,  May,  1902. 

Resisting  Moment.    Unit  Stress. 

10.  It  is  assumed  that  the  extension  or  compression  of  each  fiber,  and 
therefore  the  resisting  force  actually  exerted  by  it,  is  proportional  to  its 
vertical  distance,  t,  above  or  below  the  neutral  axis. 


Fig.  3. 


In  Fig.  3.  let 
T  = 


IM 


the  distance  from  the  neutral  axis,  o  o,  to  the  fiber  farthest  from 

that  axis,  either  above  or  below  the  axis; 
the  unit  stress  in  said  farthest  fiber; 
the  distance  from  the  neutral  axis  to  any  given  fiber; 
the  unit  stress  in  said  given  fiber; 
the  area  of  said  given  fiber; 
the  total  stress  in  said  given  fiber; 

the  resisting  moment  of  said  given  fiber  about  the  neutral  axis ; 
bending  moment  at  the  cross-section  under  consideration; 
R    =   the  resisting  moment  of  the  entire  cross-section ; 

=   2  r  =  the  sum  of  the  resisting  moments  of  all  the  fibers ; 
I    =   the  moment  of  inertia  of  the  cross-section.     See  *|^    14,  etc.; 
=   2  f2  a  =  the  sum,  for  all  the  fibers,  of  t-  a; 
I        R 
X    =  the  section  modulus,  =  7=  =  -.     See  t^  25,  etc. 

Then  the  unit  stress,  in  any   given  fiber,  is  =  «  =  S  ;p  I  its  total  stress; 

t  ^ 

F,  is  =  a  s  =  S  a  7=;  and  its  resisting  moment,  r,  is  ==  F  <  ==  S  a  7^.    Henoe^ 


the  resisting  moment,  R,  of  the  entire  section,  is 


2^0    =   -J.I. 


468  STRENGTH   OF   MATERIALS. 


Hence,  also. 

^      MT                MT                          SI 

Since, 

Q              rp                                                      ^              « 

"  =  --,  it  follows  that  ^  =^  Y*  ^ 

and 


R    =    M=    |.I    =    y.L 


The  resisting  moment,  R,  is  =  S  X,  and  the  moment  of  inertia,  I,  =»  TX. 
When  beams  are  tested  to  destruction,  the  value  attained  by  S  is  called 
the  Modulus  of  Rupture. 

11.  It  will  be  noticed  that  the  strengths  of  similar  beams  of  any  shape, 
and  those  of  rectangular  beams,  whether  similar  or  not,  are  directly  propor- 
tional to  the  product,  width  X  square  of  depth.     See  %  63. 

12.  When  the  stress,  S,  upon  the  extreme  fibers,  is  =  the  elastic  limit  of 
the  material,  failure  is  imminent.  The  permissible  unit  stress  is  usually 
taken  as  not  more  than  half  the  elastic  limit,  and  the  safe  load  is  that  under 
which  S  does  not  exceed  the  permissible  unit  stress. 

13.  The  same  quantity  of  material  that  composes  a  solid  beam,  Fig.  2, 
would  present  greater  resistance  to  bending  or  breaking  if  it  were  cut  in  two 
lengthwise  along  the  neutral  surface,  n  n,  and  converted  into  top  and  bot- 
tom chords  of  a  truss;  because,  first,  the  leverage  with  which  the  resistance 
acts  is  thus  greatly  increased ;  and,  second,  the  depths  of  the  chords  are  so 
small,  compared  with  their  distances  from  the  neutral  axis,  that  their  fibers 
may  be  assumed  to  act  unitedly  and  equally.  Hence,  practically,  all  the 
fibers  in  the  upper  chord  must  be  crushed,  or  all  those  in  the  lower  pulled 
apart,  at  the  same  instant,  before  the  truss  can  give  way;  whereas,  in  the  solid 
beam,  the  extreme  upper  or  lower  fibers  yield  first ;  then  those  next  to  them, 
and  so  on,  one  after  the  other. 

Moment  of  Inertia. 

14.  Unlike  the  moment  of  a  force,  which  is  the  product  of  a  force  and  a 
distance,  the  moment  of  inertia,  being  the  sum  of  the  products  of  areas  of 
fibers  by  the  squares  of  their  distances  from  the  neutral  axis,  is  a  purely 
geometrical  quantity.  Thus,  the  moment  of  inertia  of  a  given  section  de- 
pends solely  upon  the  dimensions  and  shape  of  that  section,  and  is  inde- 
pendent of  the  material  and  the  span  of  the  beam  and  of  the  manner  in 
which  it  is  supported  or  loaded. 

Unit  of  Moment  of  Inertia.  The  moment  of  inertia  of  a  figure 
being  the  product  of  an  area  by  the  square  of  a  distance,  its  unit  is  the 
fourth  power  of  a  unit  of  length.     Thus,  in  a  rectangle  3  ins.  wide  and  4 

ins.  deep,  I  ==  -j-—  =  — ~ — ■  =  -—^  =  16  biquadratic  inches  =  16  inch*. 

1  X  6  3 
In  a  rectangle  1  inch  wide  and  6  ins.  deep,  I  =  —  ^ —  =  18  inch*. 

15.  Comparing  similar  sections  of  any  shape,  their  moments  of  inertia 
(like  their  strengths,  see  1[  11)  are  proportional  to  the  product,  breadth  X 
cube  of  depth. 

16.  The  following  illustrated  table,  pp.  469-471,  gives,  for  several 
figures  of  frequent  occurrence, 

(1)  I  =  the  moment  of  inertia  =  ^  t-  a; 

(2)  T  =  the  distance  from  the  neutral  axis  to  the  farthest  fiber; 

(3)  X  ==  the  section  modulus  =  -_,  =  — =-    ==  "5^  =  a"J 

(4)  A  =  the  area  of  the  cross-section. 

17.  In  sections  where  the  distance  from  the  neutral  axis  to  the  lower- 
most fiber,  and  the  corresponding  section  modulus,  differ  from  those  (T 

Jmd  X)  pertaining  to  the  uppermost  fiber,   those   corresponding  to  the 
owermost  fiber  are  distinguished  as  T'  and  X'  respectively. 

18.  In  each  figure  the  neutral  axis  is  indicated  by  a  horizontal  line 
«t)ssing  the  section. 


MOMENTS    OF    INERTIA. 


469 


Moments  of  Inertia^  etc. 


I 

Moment 

of 

inertia. 


Distance  frora 

neutral  axis 

to  farthest 

fiber 


T 

Section, 
modulus. 


■Til 


BD3 
12 


6 


^~ 


^ 


1 


B  (D3_d3) 


B  (D^-d^) 


U—n-J 


-11 

12 


B3 


12 


B 


^B« 
12 

=  0.118  B  ^ 


U 


i^^ 


B4_6* 
12 


B^-&^ 
GB 


B^b^ 
12 


BPS 
36 


V2 


3 


12 


B 

B^-b-^ 


BD- 

21 


470 


STRENGTH  OF   MATERIALS. 


eT* 

c 

1 

g" 

Q 

«L 

"11 

II 

li 

t=l 

^ 

t= 

^ 

"^ 

§ 

"^ 

eo 

1 

Q 

eo 

«• 

g- 

Q 

S 

«           « 

CO 

t: 

00            *~ 

k 

"Ml 

J 

1 

5     o 

s; 

M    «    3 

t5 

II       « 

'^ 

f 

® 

0 

a 

s 

t 

1.^^ 

«        P3 

2  g« 

«|o« 

o|« 

§        1 

i 

i 

"ill 

« 

1 

1 

i    S 

Erf          H 

fil« 

o 

« 

e 

■« 

"« 

5 

««               M< 

■* 

00 

«              Q 

1:^ 

o 

§ 

1 

i     1 

O                     o 

b 

II 

1; 

t 

il          II 

;« 

^[^ 

-.n 

1» 

1 

w— S^-> 

^ 

Ij               fi                i' 

r      *N      >, 
1                1 

^! 

s 

1 

\*-^-^ 

h-hJ 

!*5.^h>l 

k^H-*!*^* 

00 

c 

Id 

s 

; 

-1 

MOMENTS    OF    INERTIA. 


471 


O       o 


II  5   ^ 

II      o    "^ 


P  '53  '^ 


S     O     0) 


ftl" 


k-h-W 


h-^^— >i 


RS3        I   »]0  B$l 


^^ 


l^ts-» 


472 


STRENGTH    OF   MATERIALS. 


19.  The  moment  of  inertia  of  any  figure,  about  its  neutral  axis,  is  the  sum 
of  the  moments  of  inertia  of  its  several  parts,  about  that  same  axis. 

30.   Let    I   =  the  moment  of  inertia  of  the  entire  figure  about  its  neutral 
axis,  o  o; 
i  =    the  moment  of  inertia  of  any  part,  about  the  neutral  axis, 
o  o,  of  the  entire  figure; 
m   ==  the  moment  of  inertia  of  that  part,  about  its  own  neutral 

axis; 
'a  =  the  area  of  that  part ; 

t   =  the  distance  of  its  center  of  gravity  from  the  neutral  axis, 
o  o,  of  the  entire  figure. 

Then  1  =  2  i;  and  i   =    m  +at^, 
21.  Thus,  in  Fig.  4, 

ti  =  ^  +  B  D  h^t 


and    I  =  1*1  +  ^. 


12 


*-&-^ 


^ 


".rz.vi*?.. 


[<         -B ^ 


IC 


Tig.  4. 


Figr.  5. 


22.  Hence,  in  any  hollow  section,  as  in  the  hollow  rectangle,  Fig.  5,  let 
I'  =■  the  moment  of  inertia  of  the  whole  figure  (including  both  the  shaded 
and  the  unshaded  rectangles),  i  =  that  of  the  missing  or  unshaded  rectangle, 
and  I  =  that  of  the  shaded  portion;  all  referred  to  the  neutral  axis,  o  o,  of 
the  shaded  portion.    Then  1  =  1'  —  i. 


Figr. 


23.  In  the  case  of  an  irregular  section,  as  Fig.  6,  let  the  section  be  divided 
into  numerous  strips,  parallel  to  the  neutral  axis  and  narrow  enough  to  be 
considered  as  rectangular;  and  proceed  as  in  m[  19  to  21. 


TRANSVERSE  STRENGTH.  473 

34.  The  narrower  the  strips  are  taken;  the  less  m  becoiies.  If  the  strips 
be  taken  so  narrow  (relatively  to  the  depth  of  the  section)  that  m  may  be 
neglected,  then  I  =  2  i'-^  a,  as  in  H  10.  The  strips  need  not  be  of  uniform 
width. 

The  Section   Modulus. 

S 
35«  Definition.     If  the  resisting  moment,  R  =  -^ .  2  <2  ^^   be   divided 

by  the  unit  stress,  S,  in  the  extreme  fibers,  the  quotient,  X   =  -^  =      „ 

=  7p,  is  called  the  Section  Modulus.      This,  like  the  moment  of  inertia,  ^^ 

14,  etc.,  is  a  purely  geometrical  quantitj>  depending  solely  upon  the  dimen- 
sions and  shape  of  the  section,  and  beLig  independent  of  the  material,  of  the 
span,  and  of  the  manner  of  loading. 

36.  Having  the  section  modulus,  X,  we  have  orjy  tc  multiply  it  by  the 
unit  stress,  S,  in  the  extreme  fibers,  in  order  ta  obtain  the  resisting  moment, 
R;   or  R  =-  S  X. 

37.  Multiplying  the  section  modulus,  X,  by  the  distance,^  T,  from  the 
neutral  axis  to  the  farthest  fibers,  we  obtain  the  moment  of  inertia,  I;  or, 
I  =TX. 

38»  The  section  modulus  is  usually  given  in  tables  of  rolled  beams,  chan- 
nels and  shapes.     See  tables  of  Carnegie  Beams,  etc. 

Lioi.ding.    Strength. 

39.  The  following  illustrated  tablt  gives  the  maximum  moment,  M, 
corresponding  to  a  given  load,  W,  and  th"  load,  W,*  corresponding  to  a 
given  imit  stress,  S,  for  different  conditions  of  support  and  of  loading.  In 
this  table, 

M   —   maximum  bending  moment ; 

W   =  the  total  extraneous  load  *  on  the  beam,  whether  concentrated' 
at  one  point  (as  shown)  or  uniformly  distributed  over  the  span ; 
I  =a  the  span; 
S   =  the  unit  stress,  in  the  fibers  farthest  from  the  neutral  axis,  due  to 

the  extraneous  load,  W;  * 
T   =«  the  distance  from  the  neutral  axis  to  the  farthest  fibers; 
I   =  the  moment  of  inertia. 

In  rectangular  beams, 
h  =  breadth; 
d  =  depth; 

I   ==  moment  of  inertia  =»  -y^" » 

""     "    S6d2- 

Of  the  two  diagrams  under  each  loading,  the  first  represents  the  mo- 
ments, and  the  second  the  shears,  in  the  several  parts  ^f  the  span. 

30.  If  S  =  the  permissible  unit  fiber  stress,  then,  in  the  foregoing  formulas, 
W  =  the  permissible  extraneous  load.* 

31.  It  will  be  noticed  that  the  strengths  of  similar  beams  are  proportional 

to  their  values  of  — y- ;  i.  e.,  the  strengths  of  beams  of  similar  cross-sections 

are  directly  proportional  to  their  breadths  and  to  the  squares  of  their  depths, 
and  inversely  proportional  to  their  spans. 


*  The  beam  is  here  supposed  to  be  without  weight.     See  ^f  42,  etc. 


474 


STRENGTH   OF   MATERIALS. 


For  symbols,  see 

opposite  page. 


Maximum  bendlnff 
moment. 


General 


[u 

rectangular 
beams 


1                                     T/ 

'■'/// 

^^^-^^iSiill 

1 

iiiiiiiiiiijiiiiininiiin 

1 

At 
Support. 


I 


r 


i 


At     jj.._WZ 
Support.       2 


W=2-y- 


^  ~ 


4 


b_df 
bd^ 


A 


M-^ 


_2S  .  Ad^ 


TF^ 


"^^^i^ 


At 
Center. 


I !5a_ 


^^ 


iiiiiiiiiiiiiiiiii 


i 


At 
Center 
and  at 
Support. 


ly—s 


2S     bd^ 

3    *  Ti 

4S      bd 

3         { 


1 


*^-«iiiinu[llUI« 


Support.  12 


bd' 

11 

bd' 

'    I 


TRANSVERSE   STRENGTH.  475 

Symbols  in  Table  Opposite. 

S   =»   unit  stress,  due  to  W,  in  the  extreme  fibers; 
T   =    distance  from  the  neutral  axis  to  the  extreme  fibers; 
I    =   moment  of  inertia;  M   =   maximum  bending  moment; 

W   ==  load;  .  I  =  span. 

In  rectangular  beams, 

b  =  breadth,  „  _  _W_L .    '  W  ~n^^ 

d  -  depth,  ^       S6d2'  w-7ib^. 

Beam   1  Inch   Square,  1  Foot  Span. 

32.  In  a  beam,  1  inch  square  and  1  foot  (12  inches)  span,  supported  at 
both  ends,  we  have,  for  the  extraneous  center  load :  * 

W    =    S.-3^ ^—    =    S.^3^    =    -;     and    S    =    18  W. 

33.  For  any  other  rectangular  beam,  let  L  =  the  span  in  feet.  Then  the 
extraneous  center  load,*  W,  required  to  produce  the  same  unit  stress,  S,  in 
the  extreme  fibers,  is 

34.  Thus,  for  yellow  pine,  let  S  —  the  permissible  unit  stress  «  ~  the 

3240 
elastic  limit  == — ^  =  1620  lbs.  per  sq.  in.     Then,  for  a  beam  1  inch  square, 

1  foot  span,  supported  at  both  ends  and  loaded  at  center,  the  permissible 
load,*  W,  is 

^,       nS        S         1620       -_^ 

^   ==-12"r8'=-T8    =^^^^" 
and,  for  a  joist,  3  X  12  ins.,  20  ft.  span;  the  permissible  extraneous  ♦  center 
load  is 

W  =  W'^  =  90  X  ^  ^Q ^^  «  19441bs.; 

and  the  permissible  extraneous  uniform  load  is  =  2  W  =  2  X  1944  =  3888 
lbs. 

35.  If  the  load,  W.  the  span,  L,  and  the  coefficient,  W,  are  given,  we 

W  L 
have  h  (P  ^  -^n--     Thus,  in  the  case  of  the  yellow  pine  joist,  mentioned 

W 
in  ^  34,  of  20  ft  span,  with  a  uniform  extraneous  *  load,  where  2  W  = 

W  T    1 Q44  y  20 

2  X  1944  =  3888  lbs.,- we  have  h  d^  =  -^  -  s,?^-^  =  432. 

w  yo 

36.  Then,  if  either  6  or  d  is  given,  the  other  is  easily  found.  If  not, 
assign,  to  either  of  them,  an  arbitrary  value.     Thus,  if  6  =  6,  we  have  <P  = 

^  =  72;  and  6/  =  |/  72  =  say  8J.     With  b  =^  3,  d^  =  144,  and  d  =  12. 

37.  With  the  slide  rule,  in  the  foregoing  example,  place  the  runner  at  432; 
and,  assuming  6  =  6,  place  1  (or  10)  on  the  slide,  opposite  6  on  the  rule. 
Then,  in  the  scale  of  square  roots,  on  the  slide,  opposite  432  on  the  rule,  will 
be  found  838  and  265.  The  former  of  these  represents  the  desired  root,  and 
we  take  8.5  as  a  sufficient  approximation. 

38.  If  the  relation,  S  =  18  W,  held  beyond  the  elastic  limit,  and  if  W'  = 
the  center  breaking  load,  in  lbs.,  on  ^  beam  of  any  material,  1  inch  square, 
1  ft.  span,  supported  at  each  end ;  then,  for  any  other  beam  of  the  same  mate- 
rial, and  of  breadth  b  ins.,  depth  d  ins.,  and  span  L  ft.,  the  center  breaking 

bd'^ 
load  would  be  W  =  W'  -j — . 

39.  Notwithstanding  the  defective  basis  of  this  method,  as  applied  to 
loads  beyond  the  elastic  limit,  its  simplicity  renders  it  very  convenient,  and 
it  is  much  in  use.  See  the  following  table  of  values  of  W',  and  example,  ^  40. 

*  The  beam  is  here  supposed  to  be  without  weight.  For  the  weight  of  the 
beam  itself,  see  1f1[  42,  etc. 


476 


STRENGTH   OF   MATERIALS. 


Center  Breaking  ILoads,  W, 

in  Pounds,  for  Beams  1  Inch 

Square,  1 

Foot  Span, 

Supported  at  Each  End. 

WOODS. 

W 

but  at  about  the  average  of  2250 

W 

Ash,  English 

w 

650 

lbs  its  elas  limit  is  reached. 

"    Amer  White  (Author).     B 

650 

Steel,  hammered  or  rolled;  elas 

"    Swamp 

^ 

400 

destroyed  by  3000  to  7000.. 

6000 

•'    Black. 

600 

Under  heavy  loads  hard  steel 

Arbor  VitiB,  Amer 

250 

snaps   like  cast  iron,   and    soft 
steel  bends  like  wrought  iron. 

Balsam,  Canada 

....     ft- 

350 

Beech,                

850 
550 

STONES,  ETC. 

BlvA  stone  flagging,  Hudson  River 
Brick,  common,  10  to  30. .average 

"      Amer            

Se  S 

o  C 

Birch,  Amer  Black 

^  <t 

125 

"       Amer  Yellow  ...  . 

S'2 

850 

20 

Cedar,  Bermuda 

W'HD 

400 

"       good  Amer  pressed,  30  to 

"       Guadaloupe 

....  »  * 

600 

50 average 

40 

*•       Amer  White   

250 

Caen  StonA 

25 

or  Arbor  Vitae... 

Chestnut 

.....  §  g 

450 

Elm,  Amer  White 

650 
800 
500 

*'    Rock  Canada 

r.'g's 

Hemlock 

O^D. 

Hickory,  Amer.. . 

3S- 

800 

"    Bitternul 

800 

Iron  Wood,  Canada 

-     0 

600 

Locust 

^d 

700 

Lignum  Vitx 

...   .  •     % 

650 

Concrete,  see  article  on  Cen- 

Larch 

cr 

400 

crete. 

Mahogany 

a 

750 

Mangrove,  White..   

650 

Black 

1 

550 

Maple,  Black 

-«* 

750 

"      Soft 

S" 

760 

Oak,  English 

550 

"    Amer  White    (by  Author). 

600 

"        "      Red.  Black,  Basket... 

850 

"    Live    

600 
450 

Pine,  Amer  White... (by  Author). 

"      Yellow     " 

" 

500 

Granite,  50  to  150 average 

100 

"        "      Pitch        " 

it 

560 

"       Quincy         

100 

"    Georgia    

850 

Glass,  Millville,  N.  Jersey,  thick 

Poplar 

550 

flooring  ....(by  Author). 

Mortar,  of  lime  alone,  60  days  old 

"      1  measure  of  slacked  lime 

170 

Poon 

700 

10 

Spruce (by  Author). 

460 

"      Black 

650 

in  powder,  1  sand  

"      1  measure  of  slacked  lime 

8 

Sycamore 

500 

Tamarack 

400 

in  powder,  2  sand  

7 

Teak 

750 

Marble,  Italian,  White  (Author) 
"  Manchester,  Vt,  "           *' 
"  East  Dorset,  Vt,  "           " 
"  Lee,  Mass,            "           " 

116 

Walnut 

650 
360 

96 

Willow 

111 

86 

METALS. 

"   Montg'y  Co,  Pa,  Gray      " 

103 

Brass 

850 
2100 

"      '*          "      Clouded       " 
"  Rutland, Vt,  Gray            " 

142 

Iron,  cast,  1500  to  2700... 

average 

70 

"        "    common  pig .. 

2000 

"  Glenn'sFalls.N.Y.Black  " 

156 

"        "    castings  from 

pig 

2300 

"  Baltimore,  Md,  white, 

"        "    employed    in 

coarse " 

102 

2025 

Oolites,  20  to  50 

35 

•*        "    for  castings  2^4  or  3 

Sandstones,  20  to  70 average 

45 

ins  thick.... 

1800? 

Iron,  wrought,  1900  to  2600 av 

2250 

New  Jersey 

45 

Wrought  iron  does  not  break ; 

Slaf^.,  laid  on  its  bed,  200  to  450,  av 

325 

TRANSVERSE  STRENGTH.  477 

40.  Example.    In  the  yellow  pine  joist  of  H^  34  and  36,  3  X  12  ins., 
20  ft.  span,  we  have,  from  the  table,  W  =  say  500  lbs.     Hence 

Center  breaking  load  *  W  =  W  —  =  500  X  3  X  144  _  ^^  g^^  ^^^  ^^ 

Li  ZO 

about  5.5  times  the  permissible  load,  found,  by  means  of  the  permissible  unit 
stress,  in  ^  34. 


Dimensions. 

-,  and  ^ 

'  W  I  W  L 


41.  SinceW  =  nS-~-  -  W -^.  and  W  =  ^,  we  have  * 

ft  Li  IZ 


Breadth   =b-     ^  g^^   -      ,^^. 

Depth       ^d^  ^^j-g^  =  ^^^; 

where  W    =  extraneous  load  *  required; 

W   =*  extraneous  load  on  beam  1  inch  square,  1  ft.  span; 

W  I 
n    =  coeflScient  from  last  column  of  table  =*  g.  .      *, 

S    =  unit  fiber  stress; 
I    =  span  in  inches; 
L    =  span  in  feet. 

Weight  of  Beam  Considered  as  Load. 

43.  For  simplicity  we  have  hitherto  regarded  our  cantilevers  and  beams 
as  having  no  weight  of  their  own ;  and,  in  beams  of  the  moderate  dimensions 
usually  employed  in  buildings,  their  own  weight,  w,  is  so  small,  in  comparison 
with  their  loads,  W,  that  it  may  often  be  safely  neglected;  but  in  larger 
beams  it  must  generally  be  taken  into  account.  The  loads,  found  as  above, 
with  S  =  greatest  permissible  unit  stress,  must  then  be  regarded  as  including 
not  only  the  extraneous  load,  W,  but  also  the  weight,  w,  of  the  beam  itself, 
for  a  length  =  span. 

43.  If  the  beam  is  prismatic, — i.  e.,  of  uniform  cross-section, — its  weight, 
w,  acts  as  a  uniformly  distributed  load,  and  we  have,  for  the  extraneous  load- 
W,  in  the  case  of  a  concentrated  center  load  on  a  beam,  or  of  a  concentrated 
load  at  the  end  of  the  span,  Z,  in  cantilevers, 

W  =  whole  load  —  -^  • 
in  the  case  of  a  uniform  load, 

W  =  whole  load  —  w. 

44.  In  finding  the  breadth  or  the  depth  of  a  rectangular  beam, 
required  to  carry  a  given  load  with  a  given  span  and  given  unit  stress,  we 
may  provide  for  the  weight  by  successive  approximations.     Thus, 

45.  To  find  the  breadth,  6,  required  for  a  beam  of  given  depth,  d. 
Neglecting  the  weight,  w,  of  the  beam,  find  the  first  approximate  breadth,  6, 
by  the  formulas  in  ^  41,  for  the  extraneous  load,  W.  Next,  calculate  the 
weight,  w,  of  a  beam  with  width,  b,  treat  said  weight  as  a  uniform  load ;  and, 
by  the  same  formulas,  find  the  additional  breadth,  b',  required  to  carry  this 
additional  load,  w.  Then  6  +  6'  =  a  second  approximate  breadth.  If  nec- 
essary, find  the  weight,  w',  of  a  beam  of  breadth,  b',  and,  from  this,  a  second 
additional  breadth,  6",  required  to  carry  it.  Then  6  +  6'  +  6"  =  a  third 
approximate  breadth,  and  so  on. 

46.  To  find  the  depth,  d,  required  for  a  beam  of  given  breadth,  6; 
find  a  first  approximate  deptn,  d,  by  the  formula,  If  41,  for  the  extraneous 
load,  W.     Find  the  weight,  w,  of  a  beam  of  that  depth;  and  again  apply  the 

formula,  using  (in  place  of  W)  W  +  ly  if  W  is  a  uniform  load,  or  W  +  ^  if  W 

is  a  concentrated  load.  The  depth,  rf',  so  found,  is  a  second  approximation. 
We  may  again  apply  the  formula,  as  before,  using  the  weight,  w\  of  beam  of 
depth  d';  or,  more  simply,  increase  the  breadth,  as  in  If  45. 

♦The  beam  is  here  supposed  to  be  without  weight.     See  %^  42,  etc 


478  STRENGTH  OF  MATERIALS. 

47.  In  practice,  beams  of  rectangular  section  are  almost  always  of  timber] 
and  such  beams  are  economically  obtainable  only  in  certain  commercial 
sizes.     Hence,  the  second  approximation  will  usually  be  ail  that  is  required. 

Strengths  and  Weights  of   Similar  Beams  of  DiflPerent  Dimen» 
sions.     Comparison  between  Models  and  Actual  Structures. 

48.  In  any  given  beam,  let  Wi  =    the  load  causing  any  given  unit  stress, 

S.     Then,  Wi.  =  -, (for  n,  see  table,  p.  474) ;  and,  in  any  similar  beam, 

of  o  times  the  breadth,  depth  and  span,  the  corresponding  load,   W  = 

i .   Hence,  the  ratio  of  their  loads  is  -^^  =  a-;  or  W  =  a^  Wi! 

a  I  Wi  * 

but  the  ratio  of  their  weights  is  —  =  ",    ,  ,     =  a^\  or  w  =  a^  w\. 

wi  b  d  I 

49.  In  other  words,  comparing  one  beam  with  another,  of  a  times  its 
breadth,  depth  and  span,  their  strengths  are  as  the  squares  of  their  respective 
dimensions;  but  their  weights  are  as  the  cubes  of  th-ose  dimensions, 

50.  Hence,  if  a  model  of  a  beam  will  just  break  under  a  uniform  load 
(including  its  own  weight,  w)  =  2,  3  or  4,  etc.,  times  its  own  weight,  then  a 
beam  of  similar  cross-section,  but  of  2,  3  or  4,  etc.,  times  its  breadth,  depth 
and  span,  will  just  break  under  its  own  weight  alone. 

Horizontal    Shear. 

51.  When  (Figs.  7  and  8)  deflection  occurs  in  a  cantilever  or  beam  com- 
posed of  separate  horizontal  layers,  like  a  pile  of  loose  boards,  the  several 
layers  slide  upon  each  other;  but,  if  they  are  firmly  joined  together,  or  other- 
wise prevented  from  sliding,  they  exert,  upon  each  other,  a  horizontal  shear- 
ing force.  In  any  section,  this  force  diminishes  uniformly  from  a  maxi- 
mum, at  the  neutral  surface,  n  n,  to  zero,  at  the  top  and  bottom  of  the  section. 


52.  In  any  section  of  a  rectangular  beam,  the  maximum  horizontal 
shear,  per  unit  of  neutral  surface,  is 

2hd' 
where  V  =  the  vertical  shear  in  the  section,  and  6  and  d  —  the  breadth  and 
the  depth  of  the  section,  respectively. 

In  words,  the  unit  horizontal  shear,  at  any  point,  is  directly  pro- 
portional to  the  vertical  shear  at  that  point.  Hence,  the  horizontal 
shear  diagram  is  similar,  in  character,  to  the  vertical  shear  diagram; 
but  is  opposite  in  sense,  positive  vertical  shear  corresponding  to  nega- 
tive horizontal  shear. 

53.  If  the  horizontal  shear  is  resisted  by  a  fastening  applied  at  only 
one  point,  said  fastening  must  be  made  sufficiently  strong  to  resist  the 
8um  of  all  the  horizontal  shears  between  such  point  and  that  where  the 
shear  is  =  0. 

54.  In  Fig.  9,  diagrams  (b)  and  (c)  show  respectively  the  moments 
and  the  vertical  shears  due  to  concentrated  and  distributed  loads  on  a 
beam  as  shown;  and,  in  Fig.  (.d),  each  ordinate  represents  the  force 
which  must  be  applied,  at  the  corresponding  point,  in  order  to  resist 
the  sum  of  all  the  horizontal  shears  between  that  point  and  the  point 
of  zero  shear.  Ordinates  above  a  zero  line  indicate  positive  moments  or 
shears,  and  vice  versa.     In  positive  moments,  the  segment  to  the  left  of  a 


HORIZONTAL   SHEAR. 


479 


section  tends  to  turn  clockwise.  In  positive  shears,  the  Ze/<-hand  segment 
tends  to  slide  upward  or  the  upper  segment  to  slide  toward  the  right. 
Between  a  and  c,  between  c  and  d,  between  g  and  h,  and  between  h  and 
6,  all  the  diagrams  are  straight  lines,  Figs,  (6)  and  {d)  being  inclined, 
and  Fig.  (c)  horizontal.  At  c  and  at  h.  Figs.  (6)  and  (d)  change  their 
inclination,  and  Fig.  (c)  shifts  its  position..  Between  d  and  /,  and 
between  /  and  g  {i.  e.,  under  the  distributed  load),  Figs.  (&)  and  (d)  are 
parabolic  curves,  and  Fig,  (c)  shows  inclined  straight  lines.  At  /,  Figs. 
(6)  and  {d)  change  curvature,  and  Fig.  (c)  shifts  its  position.  At  e,  tne 
point  of  maximum  moment,  Figs,  (c)  and  (cf)  change  signs.  See  Relation 
between  Moment  and  Shear,  Statics,  *^\  359  to  368. 


}r%zontal  Shears 


Fig.  9. 

55.  Inasmuch  as  the  horizontal  shear  is  a  resistance  to  bending,  its  neglect, 
in  the  common  theory  of  beams,  as  heretofore  explained,  is  in  general  on  the 
side  of  safety.  But,  in  beams  composed  of  horizontal  layers,  means  must 
be  provided  for  its  transmission  from  one  layer  to  the  next. 


!©1 


i©l 

L i. 


\@\ 


-^ 


-r#- 


-^T 


-r^-- 


-^^ 


Fig.  10. 


56.  Thus,  deep  wooden  beams.  Fig.  10,  are  frequently  built  up  of  two  or 
more  timbers,  one  above  the  other.  In  order  to  prevent  deflection,  due  to 
the  sliding  of  these  timbers  upon  each  other,  blocks  are  inserted  between 
them  at  intervals,  as  shown,  or  the  adjacent  sides  of  the  timbers  are  so 
notched  as  to  interlock.  In  either  case,  the  timbers  are  tightly  bound 
together.  The  blocks  or  notches  then  serve  to  transmit  the  horizontal  shear 
from  one  timber  to  the  other.  In  Fig.  10  the  blocks  are  more  numerous  near 
the  ends  of  the  span,  as  required  by  the  diagram  of  horizontal  shear,  Fig.  9  id). 


480 

0 


STRENGTH    OF   MATERIALS. 


pf;  hs> 


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+ 

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£2.i 

+  . 
t^ 

a 


-^    Hs 


-E 


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Hiao  -^15 


I'tsi 


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ft 


DEFLECTIONS.  481 


I>eflectionfs. 


57.  The  opposite  table  gives  the  deflections  within  the  elastic  limit, 
of  Rnj  prismatic  beam  (beam  of  uniform  cross  section  throughout)  under  dif- 
ferent arrang-ements  of  support  and  of  load;  also  (in  the  last 
column)  the  eztraneotis  load  which  will  produce  a  ^iven  deflection, 

without  assistance  from  the  weight  of  the  beam  itself.  All  the  formulae  are  based 
upon  the  assumption  that  the  increase  of  deflection  is  proportional  to  increase 
of  load. 

The  letters  signify  as  follows : 

d  =  deflection  of  beam,  in  inches  (see  Figs). 
W  =  weight  of  extraneous  load,  in  pounds. 
w  =       "       "    clear  span  of  beam,  in  pounds. 
I  =  clear  span  of  beam,  in  inches  (see  Figs). 

E  =  modulus  of  elasticity  of  the  material  of  the  beam,  in  lbs  per  sq  inch. 
I  =»  moment  of  inertia  of  the  cross  section  of  the  beam,  in  inches. 

From  the  principles  embodied  in  the  opposite  table,  w©  find  that  in  beams  of 
similar  cross  section  and  of  the  same  material,  and  within  the  elastic  limit,  the  load, 
and  deflections  (neglecting  the  weight  of  the  beam  itself)  are  as  follows : 


With  the  same 

The  deflections  imder  a  giren  extraneous  load  are 

span 
"        and  breadth 
«    depth 
breadth  «        " 

inversely  as  the  breadths  and  as  the  cubes  of  the  depths 

**           «       breadths 
directly       "       cubes  of  the  spans 

With  the  same 

The  extraneous  loads  for  a  given  deflection  are 

span 

"        and  breadth 
"           "   depth 

breadth    "        " 

directly  as  the  breadths  and  as  the  cubes  of  the  depths 

"            "      breadths 
inversely    "      cubes  of  the  spans 

Deflection  In  Terms  of  Extreme  Fiber  Stress.     In  table,  p.  474, 

the  load  W  =    JcS  ^y-^;  where  Jb  =  a  coefficient,  as  below;  S  =  unit  stress 

in  extreme  fibers;  I  =  moment  of  inertia;  T  =  distance  from  neutral  axis 
to  extreme  fiber,  and  I  =  span.     From  the  table  opposite,  we  have : 

W  =  7?i  — j5 — ;   where  7n  =  a  coefficient,  as   below;  d  =  deflection,  and 

E  =  modulus  of  elasticity.    Hence,  k  S  ?p-,  =  fn  — -pr-  ;  and  d  =  —  .  =^7= 

1  *  i  m     hi  1, 

l^S  ^  m 

=  E-T-c'^^^^^"=  *• 

In  a  cantilever,  loaded  at  end,  w=       3;  fc=     l;c=    3. 

'*  "  "        uniformly,  m  =       8;  *  =     2;  c  =     4. 

In  abeam,  supported,  and  loaded  at  center,    m  =    48;  k  =    4;  c  =  12. 

'*        "  '*  "        "        uniformly,  m  =     76.8;  k=    8;  c  =    9.6. 

"         "      fixed,  "        "        at  center,    m  =  192;  *  =  12;  c  =  16. 

uniformly,  w  =  384;  yfc  =  12;  c  «  32. 

31 


482  STRENGTH    OF    MATERIALS. 

Elastic  liimit. 

58.  Under  moderate  loads,  the  deflections  are  practically  proportional  to 
the  load.  When  they  begin  to  increase  perceptibly  faster  than  the  load,  the  latter  ii 
said  to  have  reached  the  elastic  limit,  or  limit  of  elasticity.  It  is  generally  at  this 
point  that  tho  **  permanent  set  "  first  becomes  noticeable ;  i.  e.,  after  remoTal 
of  the  load,  the  beam  fails  to  return  to  its  original  unstrained  condition,  and  remain! 
more  or  less  bent.  Th*^  deflections  then  also  begin  to  increase  irregularly  ;  and  to 
continue  indefinitely  without  further  increase  of  load.  In  short,  the  beam  is  in 
danger.  Hence,  the  actual  load  must  never  exceed  the  elastic  limit;  and  should  not 
exceed  from  one-third  to  two-thirds  of  it,  according  to  circumstances. 

Tlie  limit  of  elasticity  of  a  beam  of  any  particular  form,  or  material,  is 
determined  by  exp«urimeitt  with  a  similar  beam,  as  in  the  case  of  constants 
for  breaking  loads,  Ac.  Thus,  load  a  beam  at  the  center,  by  the  careful  gradual 
addition  of  small  equal  loads;  carefully  note  down  the  deflection  that  takes  place 
within  some  minutes  (the  more  the  better)  after  each  load  has  been  applied;  in  order 
to  ascertain  when  the  deflections  begin  to  increase  more  rapidly  than  the  loads ;  for 
when  this  takes  place,  the  load  for  elastic  limit  has  been  reached.* 

It  is  not  the  deflections  of  the  whole  beam  that  are  to  be  noted,  but  those  of  its 
clear  span  only.  Several  beams  should  be  tried,  in  order  to  get  an  average  constant, 
for  even  in  rolled  iron  beams  of  the  same  pattern,  and  same  iron,  there  is  a  very 
appreciable  difference  of  strengths  and  deflections. 

Then,  to  get  the  constant,  using  the  total  load  applied  during  the  equal  deflections, 
Including  half  the  weight  of  the  beam  itself, 

Constant  for  elastic  limit  =.  ^      ,  ,^P^°  ^^/^"*  >< '^^*^^  ^°ff  ^°  ?>^-  . . 

Breadth  in  inches  X  Square  of  depth  m  inches 

The  constant,  for  -vrooden  beams,  may  be  had,  near  enough  for  common 
practice,  by  taking  one  third  of  the  breaking  constants  in  the  tabU,  page  476. 

Said  constant,  thus  calculated,  is  the  elastic  limit  of  a  beam  of  the  given  shape 
and  material,  1  inch  broad,  1  inch  deep,  and  of  1  foot  span,  supported  at  both  ends 
and  loaded  at  the  center.  To  obtain  from  it  the  elastic  limit  of  any  other  beam  of 
the  same  design  J  and  the  same  material,  similarly  supported  and  loaded,  but  of  other 
dimensions, 

Clastic  ^  constant  X  ^^^'^^^  ^^  inches  X  square  of  depth  in  inches    ; 
**"***  span  in  feet 

Multiply  the 

If  the  beam  is  result  by 

supported  at  both  ends  and  loaded  at  center,  1 

«          it      II      «        ti        a       uniformly,  2 

fixed  t        "      "      "        "        "       at  center,  2 

"              "     "      "        "        "       uniformly,  3 

"              **     one  end     "        **       at  other  end,  JiT 

"              *'        "      **       "        "       uniformly,  3^ 

♦  Of  course,  in  practice,  it  is  frequently  difficult  to  ascertain  with  precision,  when, 
or  under  what  load,  the  deflections  actually  do  begin  to  increase  more  rapidly  than  the 
successive  loads.  For  although  hy  theory  the  deflections  are  practically  equal  for 
equal  loads,  until  the  elastic  limit  is  reached,  yet  in  fact  they  are  subject  to 
more  or  less  irregularity;  for  no  material  composing  a  Ijeam  is  perfectly  uniform 
throughout  in  texture  and  strength.  Hence,  instead  of  regular  increase  of  deflec- 
tion, we  shall  have  an  alternation  of  larger  and  smaller  ones.  Therefore,  Home  judg- 
ment is  required  to  determine  the  final  point;  in  doing  which,  it  is  better,  in  case 
otf  doubt,  to  lean  to  the  side  of  safety.  It  is  assumed  always  that  the  load  is  not 
subject  to  jars  or  vibrations.     These  would  increase  the  deflections. 

t  A  beam  is  said  to  be  "fixed "  at  either  end  when  the  tangent  to  the  longitudinal 
axis  of  the  deflected  beam  at  that  end  remains  always  horizontal. 

X  The  shapes  of  the  two  beams  need  not  be  similar.  For  instance,  the  constant 
deduced  from  experiments  upon  any  rectangular  beam  is  applicable  to  any  other 
rectangular  beam,  whether  square  or  oblong. 


DEFLECTIONS. 


483 


The  Elastic  Curve. 

59.  When  a  cantilever.  Fig.  1,  9r  a  beam,  Fig 
any  manner,  bends,  under  the  action  of  any  load 
forms  a  curve,  such  that,  at  any  section, 

E_I  ^  I^ 
M        M  k' 


2,  supported  or  fixed  in 
the  neutral  surface,  n  n. 


where 
R 
M 
I 


the  radius  of  curvature,  at  the  section ; 

the  bending  moment,  at  the  section; 

the  moment  of  inertia  of  the  section : 

q 

the  elasticity  coefficient  of  the  material,  =  -77 ; 

any  unit  stress  within  the  elastic  limit ; 

the  unit  "stretch"  (elongation  or  compression)  produced  by  S  in 
the  given  material. 


(a)  (b) 

Fig.  1  (repeated). 


Fig.  3  (repeated). 


The  Deflection  Coefiicient. 

60.  Definition.  The  deflection  coefficient,  for  any  given  material,  is  the 
deflection,  in  inches,  of  a  beam,  of  that  material,  1  inch  square  and  of  1  foot 
span,  supported  at  each  end,  and  carrying,  at  its  center,  an  extraneous  load 
of  1  lb.  —  f  w',  where  w'  =  weight  of  clear  span  of  beam  alone,  in  lbs. 

61.  Let  y  =  the  deflection  coefficient  for  any  given  material.  Then,  in 
any  rectangular  beam,  of  the  same  material,  with  center  load  or  uniform 
load,  let 

b   =   the  breadth,  in  inches; 
d   =   the  depth,  in  inches; 
L    =   the  span,  in  feet; 

w   =   the  weight,  in  lbs.,  of  the  clear  span  of  the  beam  itself; 
W    =   center  load  +  i  w; 
=   f  (uniform  load  +  w). 
Then,  in  the  given  beam, 

W    L3 
Deflection  =  Y    =      y  -     ' 


Breadth*    =    6   =  W. 


Load 


W 


Depth  * 


ds.Y* 

3/wT. 

'^l'bY 


63.  The  deflection  coefficient,  y,  for  any  given  material,  is  obtained  by 
experiment,  thus:  At  the  center  of  any  rectangular  beam,  of  the  given  mate- 
rial, placed  horizontally  upon  two  supports,  at  any  convenient  and  known 
distance  apart,  place  any  load  that  is  within  the  elastic  limit,  and  measure 
the  resulting  deflection,  Y,  Let  W  =  the  extraneous  center  load  +  ^  w, 
where  w  =  the  weight,  in  lbs.,  of  the  clear  span  of  the  beam  itself.  Then 
the  deflection  coefficient  is 

2/    =   Y  .  ;j 


•  W  .  L3' 

where  b  and  d  =  the  breadth  and  depth,  in  inches,  and  L 
feet,  of  the  experimental  beam. 


■  the  span,  in 


*  In  calculating  the  breadth  or  the  depth,  if  it  is  necessary  to  provide  for 
the  weight  of  the  beam  itself,  we  first  let  W  =  the  extraneous  load  only, 
and  then  proceed  by  successive  approximations,  as  in  ^1i[  45  and  46,  remem- 
bering, however,  that  in  the  case  of  deflections,  5-8  of  the  weight  of  each 
additional  section  is  to  be  taken  as  equivalent  center  load,  and  not  1-2  as  in 
the  case  of  strengths. 


484  STRENGTH    OF    MATERIALS. 

63.  The  ratio  between  any  two  homologous  lines,  in  any  two  similar 
figures,  is  constant.  Hence,  in  determining  or  using  coefficients,  whether 
for  strength  or  for  deflection,  by  comparing  beams  of  similar  sections  but  of 
different  sizes,  we  may  use  any  two  homologous  lines  in  place  of  the  two 
breadths,  or  in  place  of  the  two  depths,  or  the  same  line  may  be  taken  in 
place  of  both  breadth  and  depth.     Thus,  in  Figs.  11, 


(a) 

Fig.  11. 

B  _D  _R  _ 
T  ~  d~  T   ~  ^' 
Hence, 

B  D2  _  384  _  _  _  _3  _  R  R2     _  1000  _  ^  _  «,, 
Td^  -1^  ~^~  ^    -  Tt^-'^   -  -125'  -  8  -  2*. 
Also, 

B3  D        1728  _  ,.  _  .4  _  R3  R  _  10,000  _  i^  _  04 

64.  Deflection  coefficients,  being  the  deflections,  y,  in  inches,  at  the 
centers,  of  beams  1  inch  square  and  of  1  foot  span,  supported  at  each  end  and 
loaded  at  center  with  extraneous  loads  of  1  lb.  —  5-8  the  weight  of  the  clear 
span  of  the  beam  itself. 

Average 

Cast  iron,  0.000018  to  0.000036 0.000027 

Rolled  bar  iron,    0.000012  to  0.000024 0.000018 

Rolled  tool  steel,  0.000010  to  0.000020 0.000015 

White  oak,  well  seasoned 0.00023 

Best  Southern  pitch  pine,  well  seasoned,    \  q  oo027 

White  ash,  well  seasoned,  J 

Hickory,  well  seasoned, 0.00016 

White  pine,  wpII  seasoned,  > 

Ordinary  yellow  pine,  well  seasoned. 

Spruce,  well  seasoned,  I  ^  000^2 

Good,  straight-grained  hemlock,  well  sea-   j ^.yi\}\3<i^ 

soned, 
Ordinary  oaks,  well  seasoned,  '' 

65.  Caution.  The  deflections  of  timber  of  the  same  kind  vary  greatly 
with  the  degree  of  seasoning,  the  age  of  the  tree,  the  part  from  which  the 
beam  is  cut,  etc.  The  coefficients  given  above  are  averages  deduced  from 
our  own  experiments  on  good  pieces,  well  seasoned,  on  which  the  loads  were 
allowed  to  remain  for  months.  In  all  kinds,  less  than  2  per  cent,  of  the 
breaking  load  produced  a  permanent  set  in  a  few  months.  Several  of  the 
sticks  bore  their  breaking  loads  for  months  before  actually  giving  way.  The 
vibrations  and  jars,  to  which  all  structures  are  exposed,  in  time  increase  the 
deflections. 

66.  Eccentric  Concentrated  Loads.  Let  Y,  Fig.  12  (a),  be  the  de- 
flection, at  the  center  of  the  span  {%.  e.,  at  the  point  of  application  of  the  load) 
of  a  beam  supported  at  each  end,  due  to  a  load,  W,  within  the  elastic  limit, 
at  the  center  of  the  span.  Then,  if  the  same  load,  W,  be  placed  eccentrically 
upon  the  same  beam,  as  in  Fig.  12  (6),  the  deflection,  Y',  at  the  point,  c,  of 
application  of  the  load,  and  due  to  the  load,  W,  is 

where 

I    =  the  span; 

m  and  n   =   the  segments  into  which  the  load  divides  the  span. 


DEFLECTIONS. 


485 


67.  Uniform  liOads.  Let  Y  be  the  deflection,  due  to  any  central  ex- 
traneous load  (within  the  elastic  limit),  on  a  beam  supported  at  each  end. 
Then  the  deflection,  Y',  of  the  same  beam,  due  to  the  same  load  uniformly 
distributed  over  the  span,  is 

y'  =  |y.      . 

68.  Inclined  Beams.  If  the  beam  is  inclined,  use  the  horizontal  pro- 
jection of  its  span,  in  place  of  the  span,  I,  in  determining  its  deflections. 

69.  Cylindrical  Beams.  Let  Y  be  the  deflection  of  a  square  beam 
under  any  given  load.  Then,  for  a  cylindrical  beam  whose  diameter  =  sid« 
of  the  square,  the  deflection,  under  the  same  load,  is  =  1.698  Y. 


Stiffest 


Fig.  12. 


Fig.  13. 


70.  Figs,  13  (a)  and  (6)  show,  respectively,  the  strongest  and  the  stififest 
rectangular  sections  which  can  be  cut  from  a  given  cylindrical  log,  of  diame- 
ter, D.     In  the  strongest  section  Fig.  (a),  00  =  -^,  and  6  =  A/^  D^.    In  the 

D  Is 

stiffest  section  Fig.  (6),  ac  =  —,  andrf  =  A/ 4"  ^^• 

71.  Maximum  Permissible  Deflection.  Under  even  a  perfectly 
safe  load,  a  beam  may  bend  too  much  for  certain  purposes.  Thus,  to  pre- 
vent the  cracking  of  the  plaster  of  ceilings,  it  is  usual  to  limit  the  deflection 

of  beams  to  -777:77  =  ^^  inch  per  foot  of  span  =  35^  ins.  per  100  ft.     In  long 
3oU         6\} 


lines  of  shafting,  for  machinery,  the  deflection  is  usually  limited  to 
1  inch  per  100  feet  of  span ;  in  highway  bridges  to  '^ 


1200 


640    =  16  inch  in  10  ft.; 

in  railroad  bridges  to  ^—-:  =  —  inch  per  100  feet. 
loUU         4 
the  maximum  permissible  deflection,  in  inches  per  foot  of 

span,  in  any  given  case; 
the  deflection  coefficient,  ^^_  60,  etc. 
the  clear  span  of  the  beam,  in  feet; 
the  weight  of  the  clear  span  of  the  beam,  in  lbs.  ? 
the  center  load  +  f  w; 
=   4  (uniform  load  +  w). 

■■  the  deflection,  in  inches,  for  the  whole  span,  L,  and  we  have, 
for  the  permissible  load,  W,  and  the  required  breadth,  b,  and  depth,  d,  for  a 
rectangular  beam  (see  IT  61) : 

T       ^  TXT        YL5d3        Ybd^ 

Load  =   W 


73.  Let    Y 

I 

w 
W 

Then,  Y  L 


Breadth*    =  &    = 
Depth*        =  d    = 


W.- 


'd^Y* 

W   ^ 
bY' 


U^y 


*  See  foot-note  to  1[  61. 


486  STRENGTH    OF   MATERIALS. 

Suddenly   Applied   Loads. 

73.  Suppose  a  load  to  be  applied  to  a  flexible  beam  suddenly,  thoug! 
without  falling  or  jarring;  as,  for  instance,  if  it  be  supported  by  a  cord  whic' 
allows  it  just  to  touch  the  beam  without  bearing  upon  it,  and  the  cord  b 
then  suddenly  cut  in  two.  The  deflection  of  the  beam,  in  such  a  case,  i 
theoretically  twice  as  great  as  when  the  same  load  is  applied  gradually,  as  b; 
very  slowly  relaxing  the  cord,  or  by  dividing  the  weight  into  small  fragment 
and  applying  them  at  intervals,  one  by  one.  See  Art.  5  (6),  unde 
Strength  of  Materials.  Hence  the  strength  of  the  beam  (within  th 
elastic  limit)  is  much  more  severely  taxed  in  the  former  than  in  the  latte 
case.  A  heavy  train,  coming  very  rapidly  upon  a  bridge,  presents  a  con 
dition  intermediate  between  the  two. 

Cantilevers  and  Beams  of  Uniform  Strength. 

74.  For  equilibrium,  the  resisting  moment,  R,  of  any  section,  must  bal 
ance  the  bending  moment,  M,  at  that  section.     Or, 

-|.I  =  M;  or,  S  =  M.y; 

where  S    =   unit  stress  in  extreme  fibers ; 

T    =   distance  from  neutral  axis  to  extreme  fibers ; 
I    =   moment  of  inertia  of  section. 

75.  In  a  beam  of  uniform  cross-section,  therefore,  since  T  and  I  are  un: 
form  throughout  the  span,  the  unit  stress,  S,  on  the  extreme  fibers,  varie 
with  the  bending  moment,  M.     For  uniform  strength  against  bending  mc 

T 
ments,  the  cross-section  must  so  vary  that  y  shall  be  inversely  proportion? 

to  M,  in  order  that  S  may  remain  constant. 

76.  The  following  table  shows,  in  elevation  and  in  plan,  the  theoretics 
shapes  of  rectangular  cantilevers  and  beams  of  uniform  strength  agains 
bending  moments,  under  concentrated  and  uniform  loads.  In  practice 
some  of  these  shapes  would  of  course  have  to  be  made  stronger  near  thei 
ends,  in  order  to  provide  a  sufficient  section  to  resist  shear. 

77.  Notwithstanding  the  reduction  in  material  which  would  be  effectec 
by  using  beams  of  uniform  strength,  their  use  is  seldom  economical,  excej 
in  the  case  of  cast  iron.  In  timber,  the  material  removed  would  not  b 
eaved;  and,  in  steel,  the  saving  in  material  would  often  be  offset  by  the  co£ 
of  additional  labor. 

Moreover,  it  will  be  noticed  that  the  deflections  of  beams  of  unifon 
strength,  under  a  given  loading,  are  considerably  greater  than  those  of  beam 
of  uniform  cross-section. 

In  the  table, 

W   —  concentrated  load; 

w   —  uniform  load  per  unit  of  span; 

I  =  span; 

X   =  distance  from  a  support  to  any  given  sectioHj 

d  =  depth  of  beam  at  that  section; 

h   —  breadth  of  beam  at  that  section; 

D   =  maximum  depth  of  beam; 

B    =  maximum  breadth  of  beam; 

S    =  unit  stress  in  extreme  fibers; 

Ti  t     X.  'x  m  '     ^         unit  stress 

E   —  elasticity  coefficient  =    —^ — - — -—r ; 
unit  stretch 

Y'   =   deflection,  due  to  extraneous  load,  in  beam  of  uniform  strength 

Y    =   deflection,  due  to  extraneous  load,  in  beam  of  uniform  cross-sec 

tion  =  maximum  cross-section  of  beam  of  uniform  strengtl 


UNIFORM   STRENGTH. 


487 


Cantilevers    of    Rectangular    Cross-section    and    of    Uniform 
Strength.     Profiles,    Plans    and   Deflections. 

For  symbols,  see  If  77. 


Concentrated  Load,  =  TT,  at  end. 


Breadth,  b,  constant. 
Profile,   parabola,  with  vertex  at  load. 
8Wi3 


VdWx 


EbD3 
D  =  Maximum  depth . 


Depth,  d,  constant. 
Plan,  triangle. 


Sd2     '  EBd3  2 


Uniform  Load,  =  to    per  unit  of  span. 


Breadth,  b,  constant. 
Profile,  triangle. 


Depth,  d,  constant. 

Plan,  two  parabolic  curves,  with  vertices 
at  free  end. 


Y    =- 7-   =  2  Y. 


Sd2 


488 


STRENGTH    OF   MATERIALS. 


Beams  of  Rectangular  Cross-section  and  of  Uniform  Strength. 
Profiles,  Plans  and  Deflections. 

For  symbols,  see  opposite  page. 


Concentrated  I,oad,=  TF,  at  center. 


Breadth,  b,  constant. 

Profile,  two  parabolic  curves,  with  vertices 
at  supports. 


d 
y'  == 


V3Wx 


2E6DS 


D  at  center  of  span. 
«=     2  Y. 


Depth,  d,  constant.      B  ==  maxinuim  width 
Plan,  two  triangles 


b    = 


ZWx 


8EBd3 


Uniform  Load,  =tv  per  unit  of  span. 


K\\\\\\-p\\\\\\\\\\\\\\\\W 


Breadth,  h,  constant. 
Profile,  ellipse  or  semi-ellipse . 


"-Vi^T-^ 


Depth,  d,  constant 

Plan,  parabolas  with  vertices  at  center  of  span 

"Sd2    ' 


CONTINUOUS    BEAMS. 


489 


Symbols  in  table  opposite: 

W   =    concentrated  load ;  w  =  uniform  load  per  unit  of  span ; 

I   =   span ;  x  =  dist  from  a  support  to  given  sec ; 

d   =    depth  of  beam  at  that  sec;      b  =  breadth  of  beam  at  that  sec; 

D   =   maximum  depth  of  beam ;      B  =  maximum  breadth  of  beam ; 

S   =   unit  stress  in  extreme  fibers; 

T7,  1     X-  -x  ai  •     4.         unit  stress 

E   =   elasticity  coefficient  =  — tt — - — --:  ; 
unit  stretch 
Y'   =    deflection  due  to  extraneous  load  in  beamxjf  uniform  strength; 
Y   =    deflection,  due  to  extraneous  load,  in  beam  of  uniform  cross-sec- 
tion =  max  cross-section  of  beam  of  uniform  strength. 

Continuous   Beams. 

78.  A  continuous  beam  is  one  which  rests  upon  more  than  two  supports. 

79.  The  resistances  and  deflections  of  continuous  beams,  like  those  of 
beams  with  fixed  ends,  are  determined  by  means  of  the  elastic  curve,  using 
the  calculus.  The  more  important  facts,  thus  deduced,  are  indicated  in 
Fig.  14  and  illustrated  table,  If  89. 

80.  Fig.  14  represents  the  general  character  of  the  deflections,  and  the 
variations  of  the  moments  and  of  the  shears,  in  uniformly  loaded  continuous 
beams. 

81.  Moments.  Fig.  14  (6).  Ordinates  drawn  above  the  zero  line,  a'  6", 
represent  positive  moments,  or  those  where  the  segment  of  the  beam,  to  the 
left  of  any  section,  tends  to  revolve  clockwise;  and  vice  versa. 

83.  At  each  end  of  the  beam,  at  one  point,  i  (called  the  inflection  point,  or 
point  of  contrary  flexure)  in  each  end  span,  and  at  two  such  points  in  each 
remaining  span,  the  moment  is  zero. 

83.  At  another  point,  m,  in  each  span,  the  positive  moment  reaches  a 
maximum  for  that  span ;  while  the  negative  moments  reach  their  maxima 
at  the  supports.  Both  the  positive  and  the  negative  moments  vary  in  the 
different  spans;  but,  if  the  spans  are  equal,  then  the  moments,  at  any  two 
points  equidistant  from  the  center  of  the  whole  beam,  are  equal. 


(C) 
Shears 


Figr.  14. 


84.  The  moment  diagram,  between  each  support  and  the  point,  w,  of 
maximum  positive  moment  on  either  side  of  it,  is  a  semi-parabola,  with  its 
apex  at  m. 

85.  Shears.  Fig.  14  (c).  Ordinates  drawn  above  the  zero  line,  a"  h", 
represent  positive  shears,  or  those  in  which  the  Ze/^-hand  segment,  at  any 
section,  tends  tp  slide  upward  past  the  right-Yia,nd  segment ;  and  vice  versa. 

86.  At  the  point,  m,  of  maximum  moment,  in  each  span,  the  shear  is  zero. 
Between  each  such  point  and  the  next  support  on  the  left,  the  shear  is  posi- 
tive, and  vice  versa. 

87.  At  each  support  the  shear  suddenly  changes,  by  an  amount  =  the  re- 
action of  the  support. 

88.  The  shear  diagram  is  a  series  of  straight  lines. 


490 


STRENGTH   OF  MATERIALS. 


Continuous  Beams. 


i 

I 


4-  ^ 


^<=>- 


II  n  a  n 

g     O    «    5. 


f 

I 


CQ       4^ 

S  5  e 

la!      . 
1 1  !  .1 


o    S    p    g  'S 

i  .•  i  I  i  ^  ^ 
I  1 1  i|  =•§ ! 

iS    -^     -a    ^H    -5      O    9    i 

«<l    t    II 


II 


8    8 


II    II    11 
e  «  » 


I 


+ 


o  ?  t= 


r 


II    K   II    II 

g    «  H    » 


J 


d^ 


CONTINUOUS    BEAMS.  491 

89.  The  illustrated  table  opposite  represents  the  conditions  theoreti- 
cally existing  in  uniformly  loaded  continuous  beams  of  from  two  to  five  equal 
spans.  Only  the  left  half  of  each  such  beam  is  shown,  the  right  half  being 
symmetrical  with  it. 

90.  The  Figs,  show  the  amount  of  the  maximum  positive  moment  in 
each  span,  that  of  the  negative  moment  at  each  support,  and  the  shear  on 
each  side  of  each  support. 

91.  The  Figs,  show,  also,  the  coefficient,  a,  for  the  distance,  a  I,  from  the 
left  support  of  each  span  to  the  point  of  maximum  moment  in  that  span ; 
and  the  coefficient,  x,  for  the  distance  or  distances,  x  I,  from  the  same  sup- 
port to  the  inflection  point  or  points  in  that  span.  In  each  central  span,  the 
sum  of  the  two  values  of  a;  is  =  1.     In  each  end  span,  x  =  2  a. 

93.  In  each  central  span,  the  point  of  maximum  positive  moment  is  at 
the  center  of  the  span.  In  other  words,  the  deflection  in  that  span  is  sym- 
metrical, or  a  =  0.5. 

93.  The  numerical  sum  of  the  two  shears,  one  on  each  side  of  a  support, 
is  ="  the  reaction  of  that  support.  At  each  central  support,  the  shears,  on  its 
two  sides,  are  equal. 

In  the  Figs., 

w   =   load  per  unit  of  span ; 
I   =  span; 
m   =   the  coefficient  for  moment; 
mwP    =   moment ; 

V   =  the  coefficient  for  shear; 
vwl   —  shear; 

a   =   the  coefficient  for  distance  to  point  of  maximum  moment ; 
al   =  distance  from  left  support  of  any  span  to  point  of  maximum 
positive  moment  in  that  span ; 
X   =  the  coefficient  for  distance  to  inflection  point ; 
xl   =   distance  from  left  support  of  any  span  to  either  inflection 
point  in  that  span. 

94.  Fig.  15  shows  the  values  of  m  and  of  v  in  a  uniformly  loaded  non-con- 
tinuous beam.  Comparing  these  with  the  corresponding  values  in  con- 
tinuous beams,  as  shown  in  the  illustrated  table,  opposite,  we  see  that  the 
continuous  beam  has  considerable  theoretical  advantage.    But  see  ^  95. 


0.5 


95.  Certain  practical  considerations,   however,   materially  reduce  these 
advantages  in  many  cases.     Thus,  in  a  continuous  railroad  bridge  of  100  ft. 

spans,  so  designed  that  the  maximum  deflection  shall  not  exceed  -j-  inch,  a 

3 

settlement  of  —  inch,  in  an  intermediate  pier,  would  deprive  the  bridge  of 

the  support  of  such  pier,  and  thus  practically  throw  two  adjacent  spans 
into  one,  bringing  upon  their  members  stresses  far  in  excess  of  those  for 
which  they  were  designed.  Again,  with  moving  loads,  the  theoretical  ad- 
vantage may  at  times  be  much  less  than  that  due  to  a  stationary  load  and 
indicated  in  the  illustrated  table. 


492 


STRENGTH   OF   MATERIALS. 


Cross-shaped  Beam.* 

96.  In  a  cross-shaped  beam,  Fig,  16,  of  homogeneous  material,  loaded  at 
center,  let 


W  = 

E  = 

Y  = 

L,  I  = 

T>,d  = 

T,   t  = 

I,  i  = 

S,  s  = 

P,  P  = 


the  load; 

the  elasticity  coefficient 


unit  stress  ^ 
unit  stretch' 


the  deflection  at  center; 

the  spans  of 

the  depths  of 

the  half  depths  of 

the  moments  of  inertia  of 

the  unit  stresses  in   the  extreme 

fibers  of 
the  portions  of  W  borne  by 


the  two  branches  respec- 
tively. 


Fii^.  16. 


Then  (see  illustrated  table, 
same  for  both  branches, 


p.  480),  since  the  deflection  is  necessarily  the 


and,  since  P  =  4 


L8JP 

EI 

P 

V 
,  and  p  =  4 

S 


Z3  p 

E-r  " 

t.L3' 


(see  table,  p.  474),  we  have 


97.  In  other  words,  in  order  that  both  branches  may  be  equally  strong, 
their  depths  (independently  of  their  breadths)  must  be  inversely  as  the 
squares  of  their  spans,  or  their  spans  inversely  as  the  square  roots  of  their 
depths. 

Resistance  of  Plates. 

98.  The  laws  governing  the  resistance  of  plates,  to  pressures  normal  to 
their  surfaces,  are  but  imperfectly  understood;  and  formulas  respecting 
them  must  be  used  with  caution  and  as  probable  approximations. 

99o  Rectangular  plate,  with  central  load,  W. 

t   =   the  thickness  of  the  plate ; 
L   =   its  longer  span ; 

I   =   its  shorter  span; 

S  =   the  maximum  unit  fiber  stress ; 
C   =   a  coefficient.     See  p.  493. 

1    W 
-  Z2    <2  ' 
For  a  square  plate,  L  =  Z,  and 

3  ^W 


a  _    3  L. 


W 


(Ls  +  Z^  )  ^ 
•       CLZ 


S  = 


4C>' 


w 


*See  foot-note  (t),  p.  493. 


STRENGTH    OF    PLATES.  493 

100.  Rectangular   plate,  uniformly  loaded.     Let 

w   =   the  UDiform  load  per  unit  of  surface;    other  letters  as  in  ^  99. 
Then,  according  to  Grashof,* 


1               L2Z2           W, 

For  a  square  plate,  h  =  I.     Hence, 

101.  Value  of  C. 

For  uni-  For  cen- 

If  the  plate  is                              form  load.  tral  load. 

merely  supported  along  its  four  edges,     C  =  1.125  C  =  2.00 

firmly  secured  along  its  four  edges,            C  =  0.75  C  =  1.75 

103.  Circular  plate,   uniformly  loaded.     Let 
w   =  the  load  per  unit  of  surface ; 
S   =   the  unit  fiber  stress  in  the  material ; 

E   =   its  elasticity  coeflBcient  =  — r- — - — — -:  ; 
unit  stretch 
r   =   the  radius  of  the  unsupported  portion  of  the  plate; 
t   =   the  thickness  of  the  plate ; 
Y    =   the  deflection  at  the  center. 


Then,  according  to  F.  Reuleaux.t  if  the  plate  is  merely  supported, 
If  the  plate  is  firmly  secured, 


V  —  ^  ^  '^ 
'  S*  6E<3* 


i^/<\2  J2     w         ^        wr* 


3  , 

For  strengths  of  cylinders,  pipes,  etc.,  see  Hydrostatics,  Art.  17. 


TRANSVERSE     AND    LONGITUDINAL    STRESS    COMBINED. 

103.  Although  the  combination  of  longitudinal  and  transverse  stress  in 
the  same  piece  is  objectionable,  it  is  often  unavoidable.  Thus,  in  a  timber 
roof,  the  rafters  generally  act  both  as  columns  and  as  beams. 

In  such  cases,  the  total  unit  stress,  S,  in  the  extreme  fibers,  is  the  sura 
of  the  uniform  stress,  Sc,  due- to  direct  compression  or  tension,  and  the 
extreme  fiber  stress,  Sb,  due  to  bending  moments  only,  under  the  action 
of  the  transverse  and  longitudinal  loads  combined.     Or  S  =  Sc  +  Sb- 

Let  Mb  =  the  bending  moment  due  to  the  transverse  load;  Mc  =  the 
bending  moment  due  to  longitudinal  load,  P;  and  M  =  the  total  resultant 
bending  moment,  =  Mb  —  Mc  when  the  longitudinal  load  is  tensile; 
=  Mb  +  Mc  when  the  longitudinal  load  is  compressive. 

But  Mc  =  P  d,  where  P  =  the  longitudinal  load,  and  d  =  its  leverage,  = 

the  deflection  of  the  beam,  due  to  all  causes  ;  and  (see  ^  57)    d  =  ^— rr.-  J 

ill   L  C 

where  I  =  span,  Sb  =  unit  stress  in  extreme  fibers,  due  to  bending; 
E  =  modulus  of  elasticity;  T  =  distance  from  neutral  axis  to  extreme  fibers, 
and  c  =  a  coefficient,  whose  values,  for  different  cases,  are  given  in  ^57. 

Hence,  Mc  =  P  J^^  ;  and  resultant  moment  M  -  Mb  +  P  ^-^ .     The 
h,  L  c  —       hi  L  c 

resisting  moment,  R  (see  ^  10),  is  =  Sb  ^;  and,   for  equilibrium,   R  =  M. 

I  Z^  S 

Hence,  Sb.  ws  =  Mb  +  P  --,  ^^  ;  whence  we  derive,  for  the  extreme  fiber 
1  — ■     hi  i.  c 

stress,  Sb.  due  to  bending  only,  under  the  action  of  the  transverse  and  longi- 
tudinal loads  combined, 

*  "Theorie  der  Elasticitat  und  Festigkeit,"  Berlin,  1878. 
t  "Der  Konstrukteur."  Braunschweig,  1889.     "The  Constructor,"  trans- 
lated by  H.  H.  Suplee,  Philadelphia,  1893. 


494  STRENGTH   OF   MATERIALS. 

S    _         ^         I  where  the  longi-    I      Sb  =         ^  |  where  the  longi- 

p  p  >■  tudinal  stress  is    I  P  Z^  f  tudinal  stress  is 

I  +  -pT"  I  tensile  I  I  —  .= —  i  compressive 

Besides  this  we  have  the  unit  stress,  Sc,  due  directly  to  the  longitudinal 

p 
load,  P,  and  =  -r,  where  A  is  the  area  of  cross-section  of  the  beam.    Hence, 

A 
for  the  total  unit  stress,  S,  in  the  extreme  fibers,  we  have 

P  MbT 

S     =    Sc  +  Sb     =    -r  +  . 

M  T 
When  the  deflection,  d,  is  negligible,  M  =  0 ;  Sb  =  — y- ,  as  in  1  10 ;  and 

P       M  T 

^  =  A  +  ~r- 

It  is  often  assumed  that  the  resultant  unit  stress,  S,  in  the  extreme  fibers, 
is  equal  to  the  sum  of  the  longitudinal  and  transverse  unit  stresses,  and  the 
piece  is  then  so  designed  that  the  resultant  unit  stress,  so  obtained,  shall 
not  exceed  the  permissible  unit  stresg. 


STRENGTH    OF   PILLARS.  495 

STRENOTH   OF  PII^IiARIS. 

The  foregoing  remarks  on  crushing  or  compressive  strength  refer  to  that  of 
pieces  so  short  as  to  be  incapable  of  yielding  except  by  crushing  proper.  Pieces 
longer  in  proportion  to  their  diameter  of  cross  section  are  liable  to  yield  by 
bending  sideways.    They  are  called  pillars  or  columns. 

The  law  governing  the  strength  of  pillars  is  but  imperfectly  understood;  and 
the  best  formulae  are  rendered  only  approximate  by  slight  unavoidable  and  un- 
suspected defects  in  the  material,  straightness  and  setting  of  the  column.  A 
very  slight  obliquity  between  the  axis  of  a  pillar  and  the  line  of  pressure  may 
reduce  the  strength  as  much  as  50  per  cent;  and  differences  of  10  per  cent  or 
more  in  the  bkg  load  may  occur  between  two  pillars  which  to  all  appearances 
are  precisely  similar  and  tested  under  the  same  conditions.  Hence  a  liberal 
factor  of  safety  should  be  employed  in  using  any  formulae  or  tables  for  pillars. 

In  our  following  remarks  on  this  subject,  the  pillars  are  supposed  to  sustain 
a  constant  load;  and  the  ultimate  or  breaking  load  referred  to  is  that  one  which 
would,  during  its  first  application,  cripple  or  rupture  the  pillar  in  a  short  time. 
But  struts  in  bridges  etc  often  have  to  endure  stresses  which  vary  greatly  in 
amount  from  time  to  time.    Their  ultimate  load  is  then  less. 

Long  pillars  with  rounded  ends,  as  in  Fig  1,  have  less  strength  than 
those  with  flat  ends,  whether  free  or  firmly  fixed. 


In  steel  bridges  and  roofs,  the  ends  of  the  struts  are  frequently 
sustained  by  means  of  pins  or  bolts  passing  through  (across) 
them,  at  either  one  or  both  ends.     These  we  will  call  liing-ed 
Figr.  1.  ends.   Pillars  so  fixed  are  about  intermediate  in  strength  between 
'  those  with  flat  and  those  with  round  ends.    There  is  much  uncer- 

tainty about  this  and  all  such  matters.    The  strength  of  a  given 
■n-mm  hinged-end  pillar  is  increased  to  an  important  extent  by  increas- 

^  '^^  ing  the  diameter  of  the  pin. 

W/%^  The  formula  in  most  general  use  for  the  strength  of  pillars, 

is  that  attributed  to  Prof.  I^ewis  Oordon  of  Glasgow,  and 
called  by  his  name.    With  the  use  of  the  proper  coeflScients  for  the  given  case, 
it  gives  results  agreeing  approximately  with  averages  obtained  in  practice  with 
pillars  of  such  lengths  (say  from  10  to  40  diams)  as  are  commonly  used. 
It  is  as  follows 

Breaking  load  in  fts  per  sqinch  „  / 

of  area  of  cross  section  of  pillar  /2 

in  which  r^a 

f  is  a  coefficient  depending  upon  the  nature  of  the  material  and  (to 
some  extent)  upon  the  shape  of  cross  section  of  the  pillar.  It  is  often  taken, 
approximately  enough,  as  being  the  ult  crushing  strength  of  short  blocks  of  the 
given  material.  For  good  American  wrought  iron,  such  as  is  used  for  pillars, 
40000  is  generally  used  ;  for  cast  iron  80000.  Mr.  Cleeman*  found  for  mild  steel 
(.15  per  cent  carbon)  52000 ;  and  for  hard  steel  (.36  per  cent  carbon)  83000  lbs. 
Mr,  C.  Shaler  Smith  gives  5000  for  Pine. 
a,  for  wrought  iron,  is  usually  taken  as  follows:  a  = 

when  both  ends  of  the  pillar  are  flat  or  fixed 36000  to  40000 

when  both  ends  of  the  pillar  are  hinged 18000  to  20000 

when  one  end  is  flat  or  fixed,  and  the  other  hinged...  24000  to  30000 

For  cast  iron  about  one  eighth  of  these  figures  is  generally  used ;  and  for  pine 
about  one  twelfth. 

1        is  the  length  of  the  pillar.    If  the  pillar  has,  between  its  ends,  supports 
which  prevent  it  from  yielding  side-ways,  the  length  is  to  be  measured 
between  such  supports. 
r       is  the  least  radius  of  gyration   of  the  cross  section  of  the  pillar.  I  and  / 
must  be  in  the  same  unit ;  as  both  in  feet,  or  both  in  inches. 

*  Proceedings  Engineers'  Club  of  Phila,  Nov  1884. 


496 


STRENGTH    OF    PILLARS. 


Radius  of  g^yration.  Suppose  a  body  free  to  revolve  around  an  axis  which 
passes  through  it  in  any  direction ;  or  to  oscillate  like  a  pendulum  hung  from  a  point 
of  suspension.  Then  suppose  in  either  case,  a  certain  given  amount  of  force  to  be 
applied  to  the  body,  at  a  certain  given  dist  from  the  axis,  or  from  the  point  of  sus- 
pension, so  as  to  impart  to  the  body  an  angular  vel ;  or  in  other  words,  to  cause  it 
to  describe  a  number  of  degrees  per  sec.  Now,  there  will  be  a  certain  point  in  the 
body,  such  that  if  the  entire  wt  of  the  body  were  there  concentrated,  then  the  same 
force  as  before,  applied  at  the  same  dist  from  the  axis,  or  from  the  point  of  suspen- 
sion as  before,  would  impart  to  the  body  the  same  angular  motion  as  before.  This 
point  is  the  center  of  gyration  ;  and  its  dist  from  the  axis,  or  from  the  point  of  sus- 
pension, is  the  Radius  of  gyration,  of  the  body.  In  the  case  of  areas,  as  of  cross* 
sections  of  pillars  or  beams,  the  surface  is  supposed  to  revolve  about  an  imaginan^ 
axis ;  and,  unless  otherwise  stated,  this  axis  is  the  neutral  axis  of  t^e  area,  whick 
passes  through  its  center  of  gravity.    Then 

Radius  of  g'y ration  =  l/Mon^ent  of  inertia  -7-  Area 

Square  of  radius  of  g'y ration  =  Moment  of  inertia  t-  Area 


In  a  circle,  the  radius  of  gyration  remains  the  same,  no  matter  in  what  direc- 
tion the  neutral  axis  may  be  drawn.  In  other  figures  its  length  is  diflerent  for 
the  different  neutral  axes  about  which  the  figure  may  be  supposed  to  be  capable 
of  revolving.  Thus,  in  the  I  beam,  page  893,  the  radius  of  gyration  about  the 
neutral  axis  X  Y  is  much  greater  than  that  about  the  longer  neutral  axis  A  B. 
In  rules  for  pillars  the  least  radius  of  gyration  must  be  used. 

The  following  formulae  enable  us  to  find  the  least  radius  of  gyration,  and  the 
square  of  the  least  radius  of  gyration,  for  such  shapes  as  are  commonly  used  for 
pillars. 


Shape  of  cross  section 
of  pillar 


Solid  square 


liCast  radius 


Square  of 
M^v  o.^»a«iAn    least  radius 
of  gyration     of  gyration 

side2 


Hollow  square  of   uniform 
thickness 


Solid  rectangle 


side 
1/12* 


least  side 
1/~I2~* 


12 


D2  +  < 


least  side2 


r — A- 

l^2^22z^2^T  Hollow  rectangle  of  uniform 

thickness  \  12  (C  A  —  c a)    12  (C  A  —  e  a) 


v; 


C8A  — c3a 


C3A  — c3a 


Solid  circle 


diameter 


diameterj^ 
16 


J.-D-H 


Hollow   circle   of  uniform 
thickness 


~l6~ 


D2  +  (j2 

16 


*  >/l2  =  about  3.4641. 


STRENGTH   OF   PILLARS.  497 

The  followiiig^  are  oikly  approximate : 

Shape  of  cross  section  I^ea«t  radius    ,  Square  of 

of  pillar  of  gyration    least  radius 

of  gyration 


-F-^- 


Phoenix  column. 
Carnegie  Z-bar  column. 


I  beam. 


D  X  0.3636  D2  X  0.1322 

B  X  0.590  B2  X  0.348 


F 

4.58 


F2 
21 


Channel. 


F 

3.54 


F2 
12;5 


Deck  beam. 


F 
6 


F2 
36.5 


Angle,  with  equal  legs 


F 
5 


F2 

25 


^.    Angle,  -with  unequal  legs  Fj 


F2/2 


2.6  (F+/)  13  (F2  +  /2) 


f 


h^id 


32 


T,  with  F  =  / 


Cross,  with  F  —  / 


F 
4.74 


F2 
22.5 


498 


STRENGTH    OF    IRON    PILLARS. 


The  young*  engineer  must  bear  in  mind  that  the  breakg  and  the 
safe  loads  per  sq  inch,  of  pillars  of  any  given  material,  are  not  constant  quantities ; 
but  diminish  as  the  piece  becomes  longer  in  proportion  to  its  diam.  If  a  very  long 
piece  or  pillar  be  so  braced  at  intervals  as  to  prevent  its  bending  at  those  points, 
then  its  length  becomes  virtually  diminished,  and  its  strength  increased.  Thus,  if  a 
pillar  100  ft  long  be  sufficiently  braced  at  intervals  of  20  ft,  then  the  load  sustaine*' 
may  be  that  due  to  a  pillar  only  20  ft  long.  Therefore,  very  long  pillars  used  ^.c. 
bridge  piers,  &c,  are  thus  braced;  as  are  also  long  horizontal  or  inclined  pieces, 
exposed  to  compression  in  the  form  of  upper  chords  of  bridges ;  or  as  struts  of  any 
kind  in  bridges,  roofs,  or  other  structures. 

Mistakes  are  sometimes  made  by  assuming,  say  5  or  6  tons  per  sq  inch,  as  the  safe 
compressing  load  for  cast  iron ;  4  tons  for  wrought;  1000  pounds  for  timber;  without 
any  regard  to  the  length  of  the  piece. 

But  although  the  final  crushing  loads,  as  given  in  tables  of  strengths  of  materials, 
are  usually  those  for  pieces  not  more  than  about  2  diams  high,  they  will  not  be  much 
less  for  pieces  not  exceeding  4  or  5  diams. 

Cautions.  Remember  a  heavily  loaded  cast-iron  pillar  is  easily  broken  by  a 
side  blow.  Cast-iron  ones  are  subject  to  hidden  voids.  All  are  subject  to  jars  and 
vibrations  from  moving  loads.  It  very  rarely  happens  that  the  pres  is  equally  dis- 
tributed over  the  whole  area  of  the  pillar;  or  that  the  top  and  bottom  ends  have  per- 
fect bearing  at  every  part,  as  they  had  in  the  experimental  pillars.f  Cast  pillars  are 
seldom  perfectly  straight,  and  hence  are  weakened. 

Hollow  pillars  intended  to  bear  beavy  loads  should  not  be  cast 
with  such  mouldings  as  a  a ;  or  with  very 
projecting  bases  or  caps  such  as  g.  Fig  19. 
It  is  plain  that  these  are  weak,  and  would 
break  off  under  a  much  less  load  than 
would  injure  the  shaft  of  the  piliar.  When 
such  projecting  ornaments  are  required, 
they  should  be  cast  separately,  and  be  at- 
tached to  a  prolongation  of  the  shaft,  as 
cd,  by  iron  pins  or  rivets. 

Ordinarily,  it  is  better  to  adopt  a  more 
simple  style  of  base  and  cap,  which,  as  at 
6,  can  be  cast  in  one  piece  with  the  pillar, 
without  weakening  it. 


Fi^.  19. 


Fig.  20. 

When  a  flat-ended  pillar,  Fig.  20,  is  so  irregularly  fixed,  that  the  pressure 
upon  it  passes  along  its  diagonal  a  a,  it  loses  much  of  its  strength.  Hence  the 
necessity  for  equalizing,  as  far  as  possible,  the  pressure  over  every  part  of  the 
top  and  bottom  of  the  pillar ;  a  point  very  diflacult  to  secure  in  practice. 


t  In  important  cases  both  ends  should  be  planed  perfectly  true. 


SHEARING    AND    TORSION. 


499 


ISHEARINO  STRENGTH. 

Shearing  or  detrnsion  occurs  when  a  body  is  acted  upon  by  two 
opposite  forces  in  parallel  and  closely  adjacent  planes,  tending  to  slide  some  of 
the  particles  over  the  others.  In  Fig  1,  the  twa  forces  are  (1)  the  downward 
pressure  of  the  weight,  W,  and  (2)  the  upward  reaction  of  the  support,  A. 

In  sing'le  shear.  Fig  1,  the  shearing  area,  a,  =  the  section  ^r^.  In  double 
sbear.  Fig  1,  a  =  gg^oo  =  1y^gg.  In  Fig  3,  a  =  6  X  cross  section  of 
piece.  In  Fig  4  (single  shear),  a  =  section  c  c.  In  punching  rivet  holes, 
a  =i circumference  of  hole  X  thickness  of  plate. 

In  any  case,  if  S  =  the  ultimate  unit  shearing  stress,  Shearing  strength  =  S  a. 


Fig.  1 


Fig.  % 


Fig.  4 


Ultimate  miit  shearing  stress,  S,  in  ibs  per  sq  inch.  The  following 
figures  indicate  the  range  of  values  of  S  in  metals  and  in  timber. 

Metals.  Wrought  iron,  35,000  to  55,000;  cast  iron,  20,000  to.  30,000;  steel, 
45,000  to  75,000 ;  copper,  33,000.  „r-.u  .r.  •      t^.    .       * 

With  the  grain.  Fig  4.     Across  the  grain. 
"  Spruce  250  to  500  3,250 

White  pine  "  2,500 

Timber :  )   Hemlock  '  *  *  * 

From  our  experiments : '^    Yellow  pine  4,300  to  5,600 

Oak  400  to  700  

White  oak  4,400 

TORSIONAIi  JSTRENOTH. 

Torsion  occurs  when  a  body  is  acted  upon  by  two  couples  or  moments  of 
contrary  sense  and  in  different  planes.  Thus,  torsion  takes  place  in  a  brake 
axle  when  we  try  to  turn  it  while  its  lower  end  is  held  fast  by  the  brake  chain  ; 
and  in  shafting,  when  it  transmits  the  motive  power  of  an  engine  to  tools.  Sup- 
pose such  a  body  to  be  divided,  by  cross  sections,  into  layers.  Then  each  layer 
tends  to  shear  across  from  those  next  to  it.  Hence,  in  order  to  maintain  equi- 
librium, each  two  adjacent  layers  must  exert,  in  the  cross  section  between  them, 
an  internal  resisting  moment  equal  to  one  of  the  two  external  and  contrary 
torsional  moments. 
Resisting  inomeiit  in  a  circular  cross  section  of  a  cylindrical  shaft.    Let 

P  =  the  torsional  force  of  one  of  the  two  external  moments in  pounds ; 

I  =  its  leverage,  =  its  distance  from  the  axis  of  the  shaft in  inches  ; 

M  =  P  Z  =  external  or  torsional  moment in  inch-pounds; 

T  =  distance  from  axis  to  farthest  fibers,  =  radius  of  shaft in  inches,* 

D  =  diameter  of  shaft,  =  2  T in  inches; 

S  =  unit  shearing  stress  in  farthest  fibers in  pounds  per  sq  inch  ; 

t  =  distance  from  axis  to  any  given  fiber in  inches  ; 

s  =  unit  stress  in  said  given  fiber in  pounds  per  square  inch  ; 

a  =  area  of  said  given  fiber in  square  inches ; 

F  =  total  stress  in  said  fiber in  pounds ; 

r  =  resisting  moment  of  said  fiber  about  the  axis in  inch-pounds  ; 

R  =  internal  resisting  moment  of  the  entire  cross  section 

=^  2  r  =  sum  of  resisting  moments  of  all  the  fibers in  inch-pounds; 

Ip=  polar  moment  of  inertia*  of  the  cross  section 
=  moment  of  inertia  of  cross  section  about  the  axis  of  the  shaft 
=  2  ^2  a^  ^  the  sum,  for  all  the  fibers,  of  f^  a in  inches. 

*Tn  any  figure,  the  polar  moment  of  inertia,  Ip,  is  =  the  sum  of  the  greatest 
and  the  least  moments  of  inertia  of  the  same  figure,  about  two  axes  lying  in  the 
figure  and  intersecting  in  its  center.  In  a  solid  circle,  each  of  these  is  a  moment 
oif  inertia  about  a  diameter,  and  is  =  tt  T*  -i-  4.  Hence,  in  such  a  circle, 
I    =  TT  T4  ^  2. 


500  STRENGTH   OF    MATERIALS. 

Then  the  unit  stress,  in  any  given  fiber,  is  5  =  S  /  -i-  T ;  its  total  stress,  F,  is 
=  as  =  Sat-7-T;  and  its  resisting  moment,-  r,  is  ==  F  t  ^=  S  a  t^  -r-  T.  Fof 
equilibrium,  the  internal  resisting  moment,  R,  of  the  entire  section,  must  be  =s 
the  external  torsional  moment,  M. 

Hence,  for  the  Internal  Keisisting'  moment,  R,  we  have : 

R  =  M  ==  2  r  =-  2  S  a  r^  --  T  =  (S  --  T)  ^  t^  a  =  {S -r- T)  Ip. 

Hence,  also,  S  =  M  T  ^  Ip ;    M  =  S  Ip  -^  T ;  and  P  =  M  ^  Z  =  S  Ip  -4-  (T  Z). 

In  a  solid  circle,  Ip  =  tt  T*  -r-  2.  Hence,  S  =  2MT-=-(7rT4)=i 
2  M  -V-  (tt  T3) ;    M  =  S  TT  T3  --  2 ;    R  =  S  tt  T3  -^  (2  /) ;     and 


Diameter,  D, 


--xV^^'=V'-^  =  x.,.Vf. 


For  approximate  ultimate  values  of  S,  for  torsion,  use  the 

values  for  shearing,  p  499,  with  safety  factors  from  5  to  10. 

Horse  power  of  shafting-.  In  one  revolution,  the  force,  P  Bbs,  de- 
scribing a  circle  with  radius  =  I  ins,  does  a  work  =  2  tt  Z  P  inch-ibs,  and,  in  n 
revolutions,  work  =  2  ir  I  F  n  inch- lbs.  If  n  be  the  number  of  revolutions  per 
mimcle,  the  horse  power  is : 

H  =  27rZPri-=-(12x  33,000)  ==  2  tt  M  w  -f-  (12  X  33,000) ; 

or,  since  F  I  =  M  =  R  ==  S  Ip  -i-  T,  we  have : 

II  ==  S  TT  71  Ip  -^  (12  X  16,500  T)  ;    and  S  =  12  X  16,500  T  II  -4-  (ttw  Ip). 

In  a  solid  cylindrical  shaft,  Ip  =  tt  T^  ^  2.    Hence, 
H  =  S  TT  n  TT  T4  -4-  (12  X  33,000  T)  =  S  n'^n  T3  ^  (12  X  33,000) ; 
S  =  12  X  33,000  H  --  (7r2  71  T3)  =  12  X  3,843  H  ^  (w  T3)  ; 
n  =  321,000  H  -r-  (D3  S)  :  and 


l>lame*«r,  D  =  2  T  =  2  x  -  !'1^<M^  -  -  i^Al^OOOH 


*  /12  X  3,348  H         ^  1321,000  H        ^„   "*  /  H 


Tlie  liiglier  the  speed,  the  less  is  the  force,  and  hence  the  less 
is  the  strength  of  shaft,  required  in  order  to  transmit  a  given  horse-power ;  but 
if  the  speed  is  increased  by  increasing  the  torsional  force,  the  horse-power 
transmitted  is  thereby  increased  also. 

Example.  Given,  a  wrought  iron  shaft;  let  S  =  6,000  Bbs  per  sq  inch; 
P  =  7,500  flos;  Z  =  10  ins;  M  =  75,000  inch-lbs.  Required  the  diameter,  D. 
Here,  D  =  1.72  X  fWTS  =  1.72  X  f  75,000  --  6,000  =  1.72  X  f  12:5  =» 
1.72  X  2.32  =  4  ins.  Let  the  horse-power,  H,  =  25.  Then  n  =  321,000  H  -f- 
(D3  S)  =  321,000  X  25  -^  (43  X  6,000)  =-  21  revolutions  per  minute.  Checking, 
D  =  68  X  f  H  -r-  (Sw)  =  68  X  f  25  ^  (6,000  X  21)  =  68  X  0.058  =  say  4  inches. 

Rectang-ular  Sections.  The  foregoing  equations  are  based  upon  the 
assumption  that  the  stress  increases  uniformly  from  the  axis  of  the  shaft  out- 
ward. It  has  been  shown  (notably  by  St.  Venant*)  that  this  assumption  is  not 
applicable  to  square  and  rectangular  sections.  In  a  rectangle,  let  B  =  the  longer, 
b  =  the  shorter  side,  and  c  =  6  -f-  B.  Then  S  =  M  (3  +  1.8  c)  -=-  (B  62)  j  and 
P  =  S  B  62  _L.  [(3  _f-  1.8  c)  I].  In  a  square,  with  side  =  b,  this  becomes :  S  = 
4.8  M  -^  b3 :     M  -  S  63  ^  4.8.     P  =  S  63  -^  (4.8  I). 

The  angle  of  torsion  is  that  described  by  one  of  the  external  torsional 
moments,  relatively  to  the  other.  Within  the  elastic  limit,  this  angle  is  pro- 
portional to  the  torsional  moment,  M,  and  (assuming  I  constant)  proportional 
to  the  force,  P.  Other  things  being  equal,  the  angle  is  proportional  to  the  dis- 
tance between  the  planes  of  the  two  contrary  external  moments,  and,  in  a  solid 
cylindrical  shaft,  is  inversely  proportional  to  D^.  It  is  inadvisable  to  allow 
the  angle  of  torsion  to  exceed  1°  in  a  length  =  20  diameters,  in  shafts  revolving 
in  one  direction.     In  reciprocating  shafts  allow  still  less.     See  Fatigue,  p  465. 

Practical  Considerations.  In  many  cases  the  diameter  of  the  shaft 
must  be  made  greater  than  that  required  by  the  foregoing  formulas  ;  as  in  a  long 
shaft,  in  order  to  keep  the  angle  of  torsion  within  permissible  limits;  in  fly- 
wheel and  other  shafts,  carrying  considerable  bending  loads  in  addition  to  the 
torsion ;  and,  in  most  cases,  to  allow  for  additional  moments  due  to  alternate 
acceleration  and  retardation. 

*See  Treatise  on  Natural  Philosophy,  by  Sir  William  Thomson  and  Peter 
Guthrie  Tait,  Part  II,  New  Edition,  Cambridge,  1890,  pp  236,  etc. 


HYDROSTATICS. 


601 


HYDROSTATICS. 


Art.  1.    Hydrostatics  treats  of  the  pressure  of  water  and  of  other 
fluids  at  rest. 
At  any  given  point  within  a  fluid  tlie   pressure   is  equal  in  all 

directions;   and  the  pressure  against  any  point  of  any  surface,  whether 
plane  or  curved,  is  normal  to  the  pressed  surface  (or  to  a  plane  tangent . 
to  that  surface)  at  that  point. 

The  intensity  of  the  pressure  is  proportional  to  the  depth  of  the 
point  below  the  water  surface. 

Pressure  against  any  plane  surface. 
Let 

a  =  the  area  of  the  pressed  surface ; 

h  =  the  vertical  depth  of  the  center  of  gravity  of  the  pressed  surface  below 
the  free  surface  of  the  fluid  ; 

H  =  the  total  depth  of  the  fluid  ; 

w  —  the  weight  of  a  unit  volume  of  the  fluid  ;* 

p   =  the  mean  unit  pressure  on  tht^  pressed  surface; 

P  =  the  total  pressure  on  the  pressed  surface ; 


Then  the  mean  unit  pressure,  p,  is  equal  to  the  weight  of  a  prism  of 
the  fluid,  whose  base  is  =  1,  and  whose  length  is  =  A ;  or 

p  ~  h  w; 
and  the  total  pressure,  P,  is  equal  to  the  weight  of  a  prism  of  the  fluid, 
whose  base  is  =  a,  and  whose  length  is  =  h.     Or 
F  =  a  h  w  ==  a  p. 

In  the  diag-rams  of  Figs.  1  and  2,  the  ordinates  (supposed  to  be  drawn 
from,  and  normally  to,  the  pressed  surfaces  respectively)  represent  the  unit 
pressures  (as  in  lbs.  per  sq.  inch,  kilograms  per  sq.  centimeter,  etc.),  and  the 
area,  opposite  any  given  surface,  represents  the  total  pressure  on  that  surface. 
Thus,  in  Fig.  1  (a),  the  unit  pressures  at  n,  at  o',  at  c  and  at  o,  are  represented 
by  n  (=  0),  by  o'  q',  by  c  c'  and  hj  o  q  respectively;  and  the  total  pressure 
on  wo  is  represented  by  the  area  n  o  q,  that  on  ii  o'  by  the  area  n  o'  q',  that  on 
o'  0  by  the  area  o  q\  and  that  on  o'  c  by  the  area  c  q' . 

The  center  of  pressure  upon  any  surface  is  opposite  the  center  of 
gravity  of  the  area  representing  the  total  pressure  upon  that  surface.  Thus,  in 
Figs.  1,  the  center  of  pressure  on  no,  is  opposite  the  center  of  gravity  of  the 

2 
triangle,  n  o  q,  or  at  a  depth,  d,  =  — -  H,  below  the  water  surface.    See  The  Center 

of  Pressure,  Arts  8,  etc.,  also  ^^  133  etc.,  of  Statics. 

The  hydrostatic  paradox.  For  a  given  depth,  h,  both  the  mean  unit 
pressure,  p,  and  (for  a  given  area,  a)  the  total  pressure,  P,  are  independent 
of  the  quantity  of  water.  Thus,  in  P'igs.  1,  the  walls  sustain  as  great  a  pressure 
from  a  vertical  film  of  water  only  an  inch  thick,  as  if  the  water  extended  back 

♦For  water,  w  =  about  62.*^  ft)s.  per  cub.  ft.,  =  about  0.0362  lbs.  per  cub.  inch. 


502 


HYDROSTATICS. 


for  miles.  In  Fig.  2  (6),  the  excess  of  weight  of  water,  over  that  in  Fig.  2  (a), 
is  carried  by  the  lower  sloping  wall  of  the  vessel.  The  total  pressures,  F,  upon 
the  equal  bases,  a  b  and  a'  b',  Figs.  2  (c)  and  (d),  are  equal.  In  Fig.  2  (c),  the 
total  pressure  upon  the  base  is  greater,  and  in  Fig.  2  {d)  less,  than  the  weight  of 
the  water ;  but,  in  either  case,  the  algebraic  sum  of  all  the  vertical  pressures  in 
the  vessel  {upward  pressures  taken  as  negative)  is  =  the  weight  of  all  the  water 
in  the  vessel ;  and  the  algebraic  sum  of  all  the  horizontal  pressures  is  =  0. 

Thus,  let  the  lower  part  of  Fig.  2  (c)  represent  a  cubical  box,  8  feet  on  a  side, 
and  filled  with  water.  Now  let  the  tube,  n  o,  36  ft.  high  and  of  0.287  inch  bore, 
be  filled  with  water.  The  water  in  the  tube  alone,  although  weighing  only  about 
1  pound,  will  cause  an  additional  bursting  pressure  of  2250  fts.  per  square  foot, 
or  say  384  tons  total,  to  be  exerted  upon  the  top,  bottom  and  sides  of  the  box. 


yAir\Ar 

} 

y 

^A 

^ 

(a) 


Air  pressure  on  water  snrface.  In  addition  to  the  pressure  of  the 
water  itself,  the  free  surface  of  any  body  of  water  sustains  also  the  pressure  of 
the  air,  =  about  14.7  lbs.  per  sq.  inch.  This  pressure  (transmitted  through  the 
water  to  the  walls  of  the  vessel)  is  indicated  by  the  diagrams  (parallelograms) 
marked  "air"  in  Fig.  2  (6).  In  most  cases,  the  pressure  against  a  surface,  due 
to  the  air  pressure  on  the  water  surface,  is  counter-balanced  by  an  equal  pressure 
of  the  air  in  the  opposite  direction,  as  against  the  outer  sides  of  the  walls  of  the 
vessels  in  Figs.  2,  and  against  the  down-stream  sides  of  the  dams  in  Figs.  1. 

Strictly  speaking,  the  air  pressure,  exerted  directly  against  the  walls.  Fig.  2, 
or  against  the  dams.  Figs.  1,  being  exerted  practically  at  the  centers  of  gravity 
of  those  surfaces,  and  therefore  at  lower  elevations  than  the  opposite  pressure 
due  to  the  air  pressure  on  the  water  surface,  is  a  very  little  greater  than  the 
latter.  In  a  dam  100  ft  deep,  this  difference  would  amount  to  about  0.027  lb  per 
sq  inch,  =  0.0018  atmosphere. 

Braces  for  dams.  The  water  pressure  being  normal  to  the  pressed  sur- 
face, the  posts,  Fig.  3,  must  also  be  normal  to  the  surface,  D,  if  they  are  to  receive 


that  pressure  longitudinally  and  thus  avoid  bending  moments;  but  other  con- 
siderations may  forbid  their  being  so  placed.  Thus,  if  the  face,  D,  of  the  dam, 
is  nearly  vertical,  the  posts  would  have  to  be  made  inordinately  long;  and 
their  consequent  weakness,  as  pillars  (unless  made  inconveniently  thick)  might 
more  than  offset  the  advantage  due  to  directness  of  pressure.  Moreover,  the  feet 
of  the  down-stream  posts  would  project  beyond  the  crest  of  the  dam,  and  would 
thus  be  liable  to  injury  by  ice  or  logs,  etc.,  tumbling  over  the  dam. 

Inasnuich  as  the  pressure  increases  uniformly  from  the  water  surface  down- 
ward, the  posts  are  usually  placed  closer  together  near  the  bottom  than  near  the 
top,  although  the  shortness  of  the  lower  ones  renders  them  stronger  as  pillars. 
Similarly,  the  hoops  on  tanks,  if  of  uniform  strength,  are  placed  closer  together 
near  the  bottom. 


HYDROSTATICS. 


503 


Horizontal  and  vertical  components.  In  Figs.  1  (6)  and  (c),  the 
force  triangle  (Statics, ^^46,  etc.)  gives  us  the  horizontal  and  vertical  components, 
L  and  V,  of  the  total  normal  pressure,  P.  Or,  if  n  o  be  taken,  in  each  case,  as 
representing  the  total  normal  pressure,  P,  by  scale,  then  H  =  the  total  hori- 
zontal pressure.  In  Fig.  1  {a),  with  pressure  against  a  vertical  surface, 
L  -=  P,    and    V  =  0. 

In  Fig.  (6),  the  vertical  component,  V,  presses  the  wall  downward  against  its 
base;  but  in  Fig.  (c)  it  tends  to  uplift  and  overturn  it. 

The  depth,  H,  being  the  same  in  each  of  the  three  figures,  Figs.  1  a,  b  and  c, 
the  vertical  projections  of  the  three  submerged  surfaces  are  equal,  and  hence  the 
total  horizontal  pressures  are  equal  in  the  three  cases  ;  but  the  horizontal  projec- 
tion, and  consequently,  the  total  vertical  ])ressure,  vary  with  the  inclination  of 
the  surface.  Thus,  in  Fig.  1  (a),  the  horizontal  projection  and  the  vertical 
pressure  are  each  —  0. 

Pressures  in  cubical  and  otber  vessels,  full  of  water.    Let 

F 
F  =  the  weight  of  water  contained  in  a  prismatic  vessel;  and  /  =  -  =  the 

weight  of  that  in  a  conical  or  pyramidal  vessel  of  the  same  base  and  height. 
In  a  cubical  vessel,  we  have 

pressure  on  base =  F ; 

_  F 
~  2  ' 


one  side  . 


:  F  +  4  ^  =  3  F. 


"         "  base  and  four  sides  together  = 

In  a  conical  or  pyramidal  vessel,  we  have 

pressure  on  base =  3/=  F. 

In  a  spherical  vessel,  we  have 

total  pressure  , =  3  X  weight  of  water. 

Art.  2.    Unequal  pressures  iii  opposite  directions.    In  Figs. 
4,  let  a  =  the  area  of  that  portion,  n'  o,  which  is  subjected  to  pressure  on  both 


Fig.  4. 

sides;  and  let  H  and  h  =  the  vertical  depths  of  the  center  of  gravity  of  n'o 
below  the  two  water  surfaces,  n  and  w'  respectively.  Then,  in  each  Fig.,  the 
large  triangle,  n  o  q,  represents  the  sum  of  the  pressures  of  the  deeper  water  on 
the  left  against  the  entire  wall,  n  o;  the  trapezoid,  m  o,  represents  the  sum  = 
a  H  If,  of  the  pressures  of  the  water  on  the  left  against  the  portion,  n'  o ;  and  the 
sraaller  triangle,  n'  o  q\  represents  the  sura  —  a  hw,  of  the  pressures  of  the  shal- 
lower water  on  the  right  against  the  portion,  ??/  o.  Then  the  parallelogram,  n'  q, 
represents  the  excess  of  pressure,  from  the  left,  against  the  portion,  n' o.  This 
excess,  due  to  the  difference,  K-h,  between  the  two  levels,  is  imv/orw/?/ distributed 
over  n'o,  the  uniform  excess  unit  pressure  being  represented  by  the  ordinate 
n'  m  =  q"  q. 

The  pressure  coming  from  the  right  against  the  portion  n'  a,  and  represented 
by  the  triangle  n'o  q',  is  balanced  by  an  equal  portion  (represented  by  the 
triangle  n' o  5'")  of  the  total  pressure,  mo,  from  the  left  against  the  portion 
n'o;  and  the  centers  of  these  two  pressures,  each  being  at  a  depth  =  %  n' 0 
below  w',  are  opposite.  Hence  these  two  pressures  are  in  equilibrium.  But  the 
center  of  the  excess  pressure,  w'  q,  from  the  left,  is  opposite  the  center  of  gravity 
of  the  parallelogram,  n'  q,  or  at  the  center  of  gravity  of  w'  0.  Hence  the  portion 
n'o,  considered  independently  of  n' n,  is  acted  upon  by  the  unbalanced  force 
n' q,  ooraing  from  the  left,  acting  through  its  center  of  gravity  and  therefore 
tending  to  move  it  bodily  toward  the  right,  without  rotation. 


504 


HYDROSTATICS. 


a 


%r 


Mq  O 


This  will  be  understood  by  means  of  Pig  5,  which  may  represent  flTe 
planks,  1,  2,  3,  4,  and  5,  forming  a  dam.  and  seen  endwise;  each  one  1  ft 
in  depth,  and  say  20  ft  long  hor ;  making  the  area  of  each  surf  pressed, 
equal  to  20  sq  ft.  The  pres  in  lbs  against  each  separate  20  sq  ft  of  area, 
calculated  by  the  rule  in  Art  1,  is  shown  in  the  fig.  Now,  the  outward 
pres  against  the  upper  immersed  20  ft  area,  or  that  of  plank  3.  is  3125  fts ; 
while  the  counter-pres  against  it  from  the  other  side  is  625  fts  ;  making 
the  excess  of  outward  pres  equal  to  3125  —  625  =  2500  fts.  Again,  at  the 
lowest  plank,  number  5,  the  outward  pres  exceeds  the  inward  one  by 
5625  —  3125  =  2500  fts,  the  same  as  in  the  upper  one.  And  so  of  any  other 
equal  area  of  surf,  at  any  depth  whatever ;  the  excess  depending  upon  the 
vert  height  of  m  n,  will  be  equally  distributed  over  a  b.  It  only  remains 
to  show  that  the  total  excess  of  outward  pres  against  a  b,  is  equal  in 
amount  to  the  wt  of  a  uniform  column  of  water  with  a  base  equal  in  area 
to  a  6,  and  with  a  height  equal  to  mn.  Th'us,  we  have  seen  that  in  the 
instance  before  us,  the  excess  amounts  to  3  times  2500  fts,  or  to  7500  fts. 
Now,  the  wt  of  the  column  of  Tater  will  be  60  (or  area  of  a  bi  X  m  n(or 
2  ft)  X  62.5  fts  =  7500  fts ;  or  the  same  as  the  excess  pres  on  o  6. 

The  excess  of  pres  against  the  entire  side  s  b,  over  that  against  n  o,  is 
evidently  the  dilf  between  those  two  pressures  calculated  respectively  by  the  rule  in  Art  1. 

Art.  .3.  Surfaces,  vert,  ^s  b  m  c  s,  n  n  o  t,  Fig  6,  or  otherwise,  of 
equal  widths,  b  m,  a  n;  comineiiciiig-  at  the  level,  b  a  n  m,  ^f 
the  water,  but  extending  to  diflf  <lepths,  m  c,  n  o,  measured 
vert;  and  having*  the  same  inclination  to  the  surf  of  the 
water;  sustain  total  pressures  proportional  to  the  squares 
of  those  <lepths. 

In  Fig  6,  let  the  two  vert  sides,  a  not,  and  6  »»  c  s,  of  a  vessd, 
have  the  sami.  width  a  n,  and  b  m ;  then  if  the  depth  m  c,  be  2,  3, 

4,  5,  &c,  times  greater  than  the  depth  n  o,  the  pres  against  the  surf 

\V  hOv  bm.cs,  will  be  4,  9,  16,  25,  &c,  times  greater  than  that  against  on  o*. 

>X.      (^     nN.  This  will  be  seen  by  referring  to  the  pressures  figured  on  the  left  side 

>X,,^;^  ^^^  ^^  ^^S  ^'  where,  as  stated  in  Art  2,  the  surf  of  plank  1,  exposed  to 

\S"ri^V  "■  >.  the  pres  on  the  left  side,  is  20  sq  ft ;  that  of  planks  1  and  2,  40  sq  ft; 

^  that  of  planks  1,  2,  and  3,  60  sq  ft,  &c.     All  these  surfs  commence  at 

the  level  of  the  water;  and  all  of  them  being  vert,  are  of  course  at 
the  same  inclination  with  the  water  surf;  but  their  depths  are  re- 
spectively 1,  2,  and  3  ft.  The  pres  against  the  surf  of  1,  is  625  fts; 
that  against  the  surf  of  1,  2,  is  6254-1875  =  2500;  and  that  against 
the  surf  of  1,  2,  3,  is  625  +  1875  +  3120=5625.  But  2500  is/oMrtimes 
625 ;  and  5625  is  nine  times  625.  And  the  pres  against  the  entire 
surf  8  5,  (which  is  5  times  as  deep  as  plank  1,)  is  25  times  as  great  as  that  against  plank  1 ;  or 
625  X  25  =  15625  fts  =  the  sum  of  all  the  pressures  marked  on  the  left  side  of  Pig  5. 

This  follows,  from  the  Rule  in  Art  1 ;  for  twice  the  area  of  surf,  mult  by  twice  the  vert  depth  of  the 
cen  of  grav  below  the  surf,  must  give  4  times  the  pres ;  three  times  the  area,  by  three  times  the  depth, 
must  give  9  times  the  pres,  &c. 

It  follows,  also,  that  at  any  particular  point,  or  against  any  given  area  placed  at  various  depths,  the 
pres  will  increase  simply  as  the  vert  depth  :  thus,  if  there  be  three  areas,  each  one  sq  ft,  placed  in 
the  same  positions,  but  with  their  centers  of  grav  respectively  8,  16,  and  24  ft  below  the  surf,  the  pres 
against  them  will  be  respectively  as  8,  16,  and  24;  or  as  1,  2,  and  3. 

Art.  4.  The  pressure  of  quiet  w^ater,  in  any  one  g-iven  di- 
rection, against  any  given  plane  surface,  whether  vertical,  horizontal,  or  inclined, 

is  equal  to  the  weight  of  a  prismatic  column  of  water,  the  area  of  whose  section,  parallel  to  its  base, 
is  equal  to  the  area  of  the  projection  of  the  given  surface  taken  at  right  angles  to  the  given  direction, 
and  whose  height  is  equal  to  the  vertical  depth  of  the  center  of  gravity  of  the  given  surface  below 
the  upper  surface  of  the  water.     Hence  the 

Rule.  To  find  the  pres  in  lbs,  mult  together  the  area 
in  sq  ft  of  the  projection  taken  at  right  angles  to  the  given  direction ;  the 
vert  depth  in  ft  of  the  cen  of  grav  of  the  pressed  surf  below  the  upper  surf 
of  the  water ;  and  the  constant  62.5  fts  wt  of  a  cub  ft  of  water. 

Ex.  Let  m,  c  s  n.  Fig  7,  be  an  inclined  surf,  sustaining  the  pres  of  water 
which  is  level  with  its  top  m  c.  Then  the  total  pres  against  m  c  s  n,  and  at 
right  angles  to  it,  as  found  by  the  rule  in  Art  1,  is  an  illustration  of  the  pres- 
ent rule ;  because  the  projection  of  mcsn,  taken  at  right  angles  to  the  give* 
direction,  or  parallel  to  m  c  »  n,  is  in  fact  mcsn  itself,  or  equal  to  it.  Henee 
the  rule  in  Art  1  is  merely  a  simple  modification  of  the  present  one,  appli- 
cable to  the  case  of  total  pres  against  any  surf. 

But  if  it  be  reqd  to  find  only  the  vert  or  downward  pres 
against  mcsn,  in  pounds,  mult  together  the  area  of  the  hor  projection  aocm 
in  sq  ft;  the  vert  depth  in  ft  of  the  cen  of  grav  of  m  es  n  below* the  surf;  and  62.5.  Or  if  only  the 
Jior  pres  against  mcsn  be  sought,  mult  together  the  area  of  the  vert  projection  aosn;  the  vert 
depth  of  the  cen  of  grav  of  to  c  «  n;  and  62.5. 

In  Pig  8  also,  the  total  pres  against  efghis  found  by  rule  in  Art  1 ;  while 
the  hor  and  vert  pressures  against  it  are  found  as  in  Pig  7,  by  using  the  projec- 
tions e/ki,  and  ki  g  h.  In  Pig  7  the  vert  pres  is  downward ;  while  in  Pig  8 
it  is  upward ;  but  this  circumstance  in  no  respect  affects  the  rule. 

Rkm.  1.  At  any  given  dppth,  the  pres,  perp  to  anv  given  surf,  is  the  same 

In  all  directions;  but  Pigs  7  and  8  show  that  the  total  pres  oblique  to  a  given 

nni.        Q~  surf  will  be  less  than  the  perp  one  at  the  same  depth  ;  because  an  oblique  pro- 

j:1C1  O  jection  of  a  surf  must  be  less  than  the  surf  itself,  which  last  is  the  projectioo 

D  when  the  pres  is   perp  to  it.     Thus,  in  a  reservoir,  the  total  pres  perp  to  a 

sloping  side,  asmns  c.  Pig  7,  is  greater  than  either  the  vert  or  the  hor  pres  upon  it. 


HYDROSTATICS. 


505 


E^9 


a 


f  ^^ 

\o 

\ 

> 

N^ 

\ 

Jl 

m. 

^\"i 

k 

V 

\^ 

EolO 


;  -with  the  same  depth  ;  and  62.5 


Again,  let  Fig  9  represent  a  conical  vessel  full  of  ivater; 

Ma  base  6  c,  2  ft  diam ;  its  vert  height  a  n,  3  ft ;  then  the  circumf  of  the  base  will  be 
8.2832  ft;  the  area  of  the  base  3.1416  sq  ft;  the  length  of  its  slant  side  a  b  or  a  c,  3.16 

ft;  the  area  of  its  curved  slanting  sides  will  be : —  rr  9.93  sq  ft;   and  the 

Tert  depth  of  the  cen  of  grav  of  the  slanting  sides  will  be  at  two-thirds  of  the  vert 
height  a  n  from  the  apex  a,  or  2  ft. 

Here,  to  find  the  total  pres  against  the  base,  we  have  by  rule  in  Art  I,  3.1416  X  3 
X  62.5  =  589.05  lbs.  For  the  total  pres  against  the  slant  sides,  by  the  same  rule, 
9.93  X  2  X  62.5  =  1241.25  fi)s.  For  the  vert  pres  upward  against  the  entire  area  of  the 
slant  sides,  we  have  given  the  area  of  the  base  (which  is  here  the  hor  projection  of 
the  slant  sides)  r:  3.1416;  and  the  vert  depth  of  the  cen  of  grav  of  the  slant  sides,  2  ft.  Therefore, 
3.1416  X  2  X  62.5  —  392.7  tt)s,  the  upward  vert  pres. 

Finally,  for  the  hor  pres  in  any  given  direction  against  the  slant  sides  of  one  half  of  the  cone,  we 
have  the  vert  projection  of  that  half,  represented  by  the  triangle  a  b  c,  with  its  base  2  ft,  and  its  perp 
height  3  ft ;  and  consequently,  with  an  arta  of  3  sq'ft.  The  depth  of  its  cen  of  grav  is  2  ft ;  therefore, 
3  X  2  X  62.5  =  375  »)s,  the  reqd  hor  pre«.* 

In  Fig  10,  which  represents  a  vessel  full  of  water,  the  total  pres 
against  the  semi-cylindrical  surf  avemdk,  and  perp  to  it,  must  be 
also  hor,  because  the  surf  is  vert;  but  inasmuch  as  the  surf  is  curved, 
this  total  pres,  as  found  by  rule  in  Art  1,  acts  against  it  in  many  di- 
rections, which  might  be  represented  by  an  infinite  number  of  radii 
drawn  from  o  as  a  center.  But  let  it  be  reqd  to  find  the  hor  pres  in 
B)8,  in  one  direction  only,  say  parallel  to  o  e,  or  perp  to  a  d;  which 
■would  be  the  force  tending  to  tear  the  curved  surf  away  from  the  flat 
sides  a  b  nv,  and  d  c  t  k,  hj  producing  fractures  along  the  lines  a  v 
and  d  k;  or  which  would  tend  to  burst  a  pipe  or  other  cylinder.  In 
this  case,  mult  together  the  area  of  the  vert  projection  ad  kv  in  sq 
ft  J  the  depth  of  the  cen  of  grav  of  the  curved  surf  in  ft ;  (which,  in 
the  semi-cylinder  would  be  half  of  e  m,  or  of  o  i ;)  and  62.5.  Since 
the  resulting  pres  is  resisted  equally  by  the  strength  of  the  vessel 
along  the  two  lines  a  v  and  d  k,  it  is  plain  that  each  single  thickness 
along  those  lines  need  only  be  sufficient  to  resist  safely  one  half  of  it ; 
and  so  in  the  case  of  pipes,  or  other  cylinders,  such  as'hooped  cisterns 
or  tanks.    See  Art  17. 

Should  the  pres  against  only  one  half  of  the  curved  surf,  &s  edmk 
be  sought,  and  in  a  direction  parallel  to  o  d,  tending  to  produce  frac- 
tures along  the  lines  e  m,  and  d  k,  then  use  the  vert  projection  oen  ' 
as  before. 

It  follows,  that  if  the  face  of  a  metallic  piston  be  made  oonoave  or  convex,  no  more  pres  will  be  reqd 
to  force  the  piston  through  any  dist,  than  if  it  were  flat ;  for  the  pres  against  the  face  of  the  piston, 
in  the  direction  in  which  it  moves,  must  be  measured  by  the  area  of  a  projection  of  that  face,  taken 
at  right  angles  to  said  direction ;  and  the  area  of  said  projection  will  be  the  same  in  all  three  cases. 

Rem.  2.    If  a  bridg^e  pier,  or  otlier  construction. 
Fig  10  3^,  be  founded  on  sand  or  g^ravel,  or  on  any  kind  of 

foundation  through  which  water  may  find  its  way  underneath,  even  in  a  very  thin 
sheet,  then  the  upward  pres  of  the  water  will  take  effect  upon  the  pier ;  and  will  tend 
to  lift  it,  with  a  force  equal  to  the  wt  of  the  water  displaced  by  the  pier;  (Arts  18, 
19.  In  other  words,  the  effective  wt  of  the  submerged  portion  of  the  pier, 

will  be  reduced  623.^  lbs  per  cub  ft;  or  nearly  the  half  of  the  ordinary  wt  of  masonry. 

But  if  the  foundation  be  on  rock,  covered  with  a  layer 

of  cement  to  prevent  the  infiltration  of  water  beneath  the  masonry,  no  such  effect 

will  be  produced ;  but  on  the  contrarv,  the  vert  pres  downward,  afforded  by  the  bat- 

tering  sides  of  the  pier,  and  bv  its  offsets,  will  tend  to  hold  it  down,  and  thus  increase  its  stability ; 

which,  in  quiet  water,  will  then  actually  be  greater  than  on  land. 

Art.  5.  To  divide  a  rectangular  surf, 
ivlietlier  vert  an  ab  c  d,  or  inclined  as 
mnop,  Fig  11,  whose  top  ab  or  mti  is 
level  with  the  surf  of  the  water,  by  a 
hor  line  x  2,  such  that  the  total  pres 
against  the  part  above  .said  hor  line, 
shall  equal  that  against  the  part  be- 
low it. 

Rule.  Mult  one  half  cf  the  length  of  b  c,  or  mp,  as  the  case 
may  be,  by  the  constant  number  1.4142;  the  prod  will  be  b  2, 
or  mx. 

Ex.  Let  &  c  =  12  ft.  Then  6  X  1.4142  =  8.4852  ft ;  or  6  2. 
Let  mp  =  16  ft.     Then  8  X  1.4142  =  11.3136  ft,  or  m  x. 

Bkm.  The  line  x  2,  thus  found,  must  not  be  confounded  with 
the  cen  of  pres,  which  is  entirely  diff.    See  Art  8. 

Art  6.  In  a  rectangular  surf,  whether  vert  as  abed,  or  in- 
clined n»  mnop,  Fig  11,  whose  top  a  b  or  mn  coincides  with 
the  surf  of  the  water,  to  find  any  number  of  points,  as  1,2,  «tc, 
through  which  if  hor  lines,  as  1 «,  2a;,  Ac,  be  drawn,  they  will 
divide  the  given  surf  into  smaller  rectangles,  all  of  which 
shall  sustain  equal  pressures. 

EuLK.  First  fix  on  the  number  of  small  rectangles  reqd.  Then  for  point  1  from  the  top.  mult  l^e 
number  1 ,  by  this  number  of  rectangle-s.   Take  the  sq  rt  of  the  prod.  Mult  this  sq  rt  by  the  entire  length 


Firfiol 


*  In  a  sphere  filled  with  a  fluid  the  total  inside  pres  =  3  times  wt  of  fluid. 


506 


HYDROSTATICS. 


h  COT  mp,  aa  the  case  may  be.  Div  the  prod  by  the  uumber  of  rectangles.  The  quot  will  be  the  dist 
6  1.  orn  1,  as  the  case  may  be. 

For  the  dist  b  2,  or  n  2,  proceed  in  precisely  tiie  same  wayj  only  instead  of  the  number  1,  use  the 
number  2  to  be  mult  by  the  number  of  rectangles:  and  so  use  successively  the  numbers  3,  4,  5,  &c, 
if  it  be  reqd  to  find  that  number  of  points. 

Ex.  Let  6  c  =  10  ft ;  and  let  it  be  reqd  to  find  2  points,  1  and  2,  for  dividing  the  rectangular  surf 
abed  into  3  rectangular  parts,  which  shall  sustain  equal  pressures.    Here  we  have  for  point  1, 

1X3  =  3.     Thesqrt  of  3  =  1.732.     And  1.732  X  10  (or  6  c)  =  17.32.     And   _-H^'L__rr5.773ft=:JL. 


For  point  2,  we  have 


3  rectangles 
24.49 


2X3  =  6.   Thesqrt  of  6=2.449.   And  2.449  X  10  (or  6  c)  =  24.49.   And — =  8.163  ft  =  6  2. 

3  rectangles 
And  so  for  any  number  of  points. 

Rem.  1.  This  rule  will  be  found  useful  in  spacing-  tbe  cross- 
bars of  loek-gates;  tbe  boops  around  cylin<lrieal  cisterns  | 
and  tbe  props  to  a  structure,  like  Fig-  3. 

Rem.  2.  For  dividing^  any  surf,  asobcd^  Fig  12,  wbich  is  not 
rectangular,  in   tbe  same    manner, 

with  an  accuracy  sufficient  for  most  practical  purposes,  per- 
haps the  following  method  is  as  convenient  as  any. 

Rule.  First  div  the  surf,  as  in  Fig  12,  into  several  small 
hor  parts,  equal  or  not,  at  pleasure.  Then  by  Rule  in  Art  1, 
find  the  pres  on  each  part  separately,  as  is  supposed  to  be 
done  in  the  numbers  on  the  left  hand  of  the  fig.  The  sum  of 
these  (in  this  case  15510)  is  the  total  pres  against  the  entire 
surf  0  b  e  d.  Now  suppose  we  wish  to  div  this  surf  in  4  parts 
bearing  equal  pres;  first  div  15510  by  4  =  3878.  Then  begin- 
ning at  the  top,  add  together  a  number  of  the  separate 
pressures  sufficient  to  amount  to  3878;  by  this  means  find 
point  1.  Then  proceed  with  the  addition  until  the  snm 
amounts  to  twice  3878,  or  7756,  which  will  indicate  point  2; 
and  in  the  same  manner  find  point  3,  by  adding  up  to  three 
times  3878,  or  11634.  Then  the  hor  dotted  lines  ruled  through 
points  1.  2,  and  3,  will  give  the  reqd  divisions  approximately. 
In  this  manner  the  hoops  of  conical,  and  other  shaped  ves- 
sels, may  be  spaced  nearly  enough  for  practical  purposes. 


Total  =  15510 


a 


Figr.13. 


Art.  7.  Tbe  transmission  of  pressure  tbrougta  Trater.  TFa- 
ter,  in  common  i¥itb  otber  fluids,  possesses  tbe  important 
property  of  transmitting-  pres  equally  in  all  directions.  Thus, 

suppose  the  vessel,  Fig  13,  to  be  entirely  closed,  and  filled  with  water; 
and  suppose  the  transverse  area  of  T,C,  D,and  E,  to  be  each  equal  to  one 
sq  inch.  Then,  if  by  means  of  a  piston,  or  otherwise,  a  pres  of  1  ft,  1 
ton,  or  any  other  amount,  be  applied  to  the  one  sq  inch  of  area  of  T,  C, 
D,  or  E,  every  sq  inch  of  the  inner  surf  of  the  vessel,  and  of  the  pipe  a, 
will  instantly  receive,  at  right  angles  to  itself,  an  equal  pres  of  1  ft,  or 
1  ton,  <fec;  in  addition  to  the  pres  which  it  before  sustained  from  the 
water  itself;  and  this  will  occur  if  the  vessel  consist  of  parts  even  miles 
asunder ;  as,  for  instance,  if  T  were  miles  distant  from  E  ;  and  united 
to  it  by  a  long  series  of  tubes.  If  the  vessel  were  a  strong  steam  boiler 
full  of  water,  a  single  pres  of  a  few  hundred  pounds  at  T,  C,  &c,  would 
burst  it.    See  also  Fig  2  (c)  and  paragraph  above  it. 

Tbe  bydrostatic  press  acts  on  this  prin- 
ciple.    Any  body,  within  the  vessel,  would  also  receive 
an  equal  additional  pres  on  each  sq  inch  of  its  surf. 

If  the  top  of  T  be  open,  the  air  will  press  upon  the  sq  inch  of  the  exposed  surf  of  water  to  the  extent 
©f  nearly  15  fts  ;  and  the  same  degree  of  pres  will  also  be  transmitted  to  every  sq  inch  of  the  interior 
surf  of  the  vessel,  and  its  connecting  tubes ;  but  no  danger  of  bursting  will  result  from  this  atmo- 
spheric pres,  because  the  air  also  presses  every  sq  inch  of  the  outside  of  the  vessel  to  the  same  extent. 

Air,  and  other  gaseous  fluids,  transmit  pres  equally  in  all 
directions,  like  liquids ;  but  not  as  rapidly. 

Art.  8.     The  center  of  pressure.    Let  Fig  14 

represent  a  vessel  full  of  water,  and  suppose  the  side  P  to  be  perfectly 
loose,  so  as  to  be  thrown  outward  by  the  slightest  pres  of  the  water  from 
within.  Now,  there  is  but  one  single  point,  P,  in  every  surf  so  pressed, 
no  matter  what  its  shape  may  be,  to  which  if  we  apply  a  force  equal  to 
the  pres  of  the  water,  and  in  a  direction  opposite  to  said  pres,  the  side  P 
will  be  thereby  prevented  from  yielding.  Such  point  is  called  the  cen- 
ter of  pressure.  It  must  not  "be  understood  by  thiM  that  the  actual 
amount  of  pres  of  the  water  against  that  part  of  the  surface  which  is 
above  the  hor  dotted  line  passing  through  P,  is  equal  to  that  of  the  water 
below  said  line  ;  but  that  the  sum  of  the  products  of  the  several  pressures 
above  it,  mult  by  their  several  leverages,  or  vert  dists  from  P,  is  equal 
to  the  sum  of  the  products  of  the  pressures  below,  mult  by  their  lever- 
ages ;  or,  in  other  words,  that  tlie  sum  of  the  moments  around  the  point 
P,  of  the  pressures  above  the  line,  is  equal  to  the  sum  of  the  moments 
Pig.  14.  of  those  beiow  it ;   so  that  if  a  hor  iron  rod  b  b  were  passed  entirely 

through  the  side  P,  at  the  same  level  as  the  dotted  line,  as  shown  in  the 
fig.  so  as  to  serve  as  a  hinge  for  the  side  P  to  turn  on,  the  side  would  have  no  tendenny  to  turn. 


HYDROSTATICS. 


507 


Art.  9.  To  find  tlie  cen  of  pres  of  a  qaiet 
fluid,  ag-aiiist  a  plane  surface.    Fig  15. 

1.  The  center  of  pressure  of  a  quiet  fluid  against  auy  plane  surface 
whose  width  is  uniform  throughout  its  depth,  whether  said  surface  be 
vertical,  as  e  o,  or  inclined,  as  c  a,  (or  inclined  in  the  opposite  direction  :) 
and  whose  top  c,  or  e,  coincides  with  the  hor  water  surf;  is  distant  vert 
below  the  water  surf,  two-thirds  of  the  vert  depth,  s  x,  from  said  water 
surf  to  the  bottom  of  the  plane;  as  at  n  and  ».  Inasmuch  as  a  hor  line 
at  %  of  the  depth  of  sx,  intersects  both  ca  and  eo  at  %  of  their  lengths 
respectively,  we  might  say  at  once  that  the  center  of  pres  against  a  plane 
of  uniform  width  is  at  two-thirds  of  its  length  below  the  water  surface. 

Throughout  Art  9  any  measure,  as  yard,  foot,  or  inch 
&c,  may  be  used. 

2.  But  if  the  hor  top  a,  or  o,  Pig  16,  of  the  rectangular  plane  ag,  •r 
oft,  oe  covered  to  some  depth  with  water,  then  the  vert  depth  aim,  of  the 
oeu  of  pres  d,  or  e,  below  the  surf  of  the  water,  will  be  eqaal  to 

9     „      cube  of  so  —  cube  of  sw 

I  of ■ 

*  square  of  »  c  —  square  of  s  to 
where  «  c  is  the  vert  depth  of  the  bottom,  and  a  w  the  vert  depth  of  the 
top,  of  the  pressed  surf,  below  the  water  surf.  Or,  in  words :  From  the 
cube  of  «c,  take  the  cube  of  sw;  and  call  the  rem  a.  Then,  from  the 
square  of  s  c,  take  the  square  of  sw;  and  call  the  rem  5.  Div  a  by  6, 
and  take  two-thirds  of  the  quot  for  s  m. 


Fig.  16. 


3.  When  a  plane  surf  of  any  shape  whatever,  whether 
rectangular,  triangular,  or  circular,  &c ;  whether  vert  as 
op,  Fig  17,  or  inclined  as  mn,  is  entirely  immersed,  so  as  to 
be  pressed  over  the  entire  area  of  both  sides :  but  by  diff 
depths  of  water  on  its  two  sides  ;  then  the  cen  of  pres  coin- 
cides with  the  cen  of  grav  of  the  pressed  surf. 

In  the  3  foregoing  figures  the  supposed  surfaces  are  shown 
edgewise,  so  that  their  widths  do  not  appear. 


Fig.  17. 


4.  In  any  triangular  plane  surf,  whether  right-angled,  or 
otherwise,  as  ab  c,  Fig  18;  whether  vert,  or  inclined;  the  base 
ah  of  which  coincides  with  the  hor  surf  of  the  water;  the  cen 
of  pres  0,  will  be  in  the  center  of  the  line  c  r,  which  bisects  the 
base  a  b. 

5.  But  if  the  triangle,  as  as  c,  vert,  or  inclined,  have  its 
apex,  a,  at  the  surf  of  the  water ;  and  its  base  s  c,  hor;  then  the 
een  of  pres  x,  will  also  be  in  the  line  am  which  bisects  the  base; 
biit  ax  will  be  %  of  am. 


e.    If  any  plane  triangle  ahc,  Fig  19,  base  up,  and  hor;  have  its  base 
a  b  covered"  to  some  depth  n  d,  with  water ;  then  the  cen  of  pres  o,  will 
be  in  the  line  cs  which  bisects  the  base  ;  and  no  will  be  equal  to 
»»x2  4-  (2mx  X  ma)  -\-  3ma^ 

(TOX  +  2ma)  X  2.  X 


7.  The  center  of  pres  against  any 
plane  rectangular  surface.  Fig  20, 
whether  vert  as  m  n,  or  inclined  as 
po,OT  wx;  having  its  top  coinciding 
with  the  surf  of  the  water;  and 
pressed  by  diff  depths  of  water  on 
its  opposite  sides,  as  shown  in  the 
fig ;  will  be  vert  below  the  upper 
Water  surf,  a  dist  equal  to 


Fig.  19. 


Fig.  20. 


(sqofvert\       /               j,       j       sqofven\       /  ver«         area  of          ^^^     \ 
area  of  surf  m  n^^depth  a  b  \_/     <"•««  of  surf    y^depth  vh\-l  depth  X  surfcL  X  «^<5'«Arb  | . 
orpo,  orwx    ^ ~— )      \^cn,  or  eo,  or  sx     ^ /      \^    ^^       oreo,orax     T"  / 


f      area  of  surf       „  ha7fBfa.h)  —  Z'     "' 
Vmn,  or  po,  or  wx  ^  '*"*•/ •■'*V        Vcn, 


area  of  surf 
or  e  o,  or  s  X 


X  half  of  rb). 


508 


HYDROSTATICS. 


8.  To  find  the  center  of  pressure  against  either  a  circular  or  an  elliptic  sur- 
face, pressed  on  one  side  only  ;  whether  vertical  or  inclined  ;  and  having  its  top 
either  coinciding  with  the  surface  of  the  water  or  below  it. 
Let  h  =  the  vertical  depth  of  the  center  of  pressure  below  the  water  surface. 
r  =  the  vertical  or  inclined  5e?7it-diameter  of  the  surface. 
d  =  the  vertical  distance  of  the  center  of  the  pressed  surface,  below  the 
water  surface. 
Then 

r2 

In  a  vertical  circle,  with  top  at  water  purface,  h  =  l^X  radius. 

Art.  10.  Walls  for  resisting  the  pressure  of  quiet  water. 

A  study  of  our  remarks  on  retaining-walls  for  earth,  pp.  603,  etc.,  will  be  of  use  in 
this  connection.  It  is  of  course  assumed  that  the  water  does  not  find  its  way 
under  the  wall ;  and  that  the  wall  cannot  slide.  In  making  calculations  for  walls 
to  resist  the  pressure  of  either  earth  or  water,  it  is  convenient  to  assume  the 
wall  to  be  but  one  foot  in  length  (not  height,  or  thickness);  for  then  the  num- 
ber of  cubic  feet  contained  in  it,  is  equal  to  that  of  the  square  feet  of  area  of  its 
cross-section,  or  profile;  so  that  these  square  feet,  when  multiplied  by  the  weight 
of  a  cubic  foot  of  the  masonry,  give  the  weight  of  the  wall.  In  ordinary  cases, 
it  is  well,  for  safety,  to  assume  that  the  water  extends  down  to  the  very  bottom 
line  of  the  wall. 

Now,  by  Art  1,  the  total  pressure  of  quiet  water,  against  the  rectilineal  back  of 
a  wall,  whether  vertical  or  sloping,  is  found  in  ft)s,  by  multiplying  together  the 
area  in  square  feet  of  the  part  actually  pressed,  (or  in  contact  with  the  water;) 
half  the  vertical  depth  of  the  water,  in  feet,  (being  the  vertical  depth  of  the 
center  of  gravity  of  a  rectilineal  back,  below  the  surface);  and  the  constant 
62.5  lbs;  and  this  total  pressure  is  aXwajs perpendicular  to  the  pressed  area. 


When  the  back  of  the  wall  is  vertical,  as  in  Fig  20>^,  this  pressure  p  is  of 
course  less  than  when  it  is  battered;  and  is  also  horizoyital;  and  it  tends  to  over- 
throw the  wall,  by  making  it  revolve  around  its  outer  toe,  or  edge  t.  The  center 
of  pressure  is  at  c;  c  s  being  3^  the  vertical  depth  on;  in  other  words,  the  entire 
pressure  of  the  water,  so  far  as  regards  overthrowing  the  wall  as  one  mass,  may 
be  considered  as  concentrated  at  the  point  c;  where  it  acts  with  an  overthrowing 
leverage  1 1.  The  pressure  in  lbs,  multiplied  by  this  leverage  in  feet,  gives  the 
moment  in  foot-ft)s  of  the  overturning  force.  The  wall,  on  the  other  hand,  resists 
in  a  vertical  direction  g  a,  with  a  moment  equal  to  its  weight  (supposed  to  be 
concentrated  at  its  center  of  gravity  g),  multiplied  by  the  horizontal  distance 
a  i,  which  constitutes  the  leverage  of  the  weight  with  respect  to  the  point  ^  as  a 
fulcrum.  If  the  moment  of  the  water  is  greater  than  that  of  the  wall,  the  latter 
will  be  overthrown  ;  but  if  less,  it  will  stand. 

In  Fig  21  the  overturning  moment  of  the  water  is  equal  to  its  calculated  pres- 
sure p  X  its  leverage  1 1 ;  while  the  moment  of  stability  of  the  wall  is  equal  to 
its  weight  X  its  leverage  a  t.  By  aid  of  a  drawing  to  a  scale,  we  may  on  this 
principle  ascertain  whether  any  proposed  wall  will  stand.  For  we  have  only  to 
calculate  the  pressure  jo,  then  apply  it  at  c,  and  at  right  angles  to  the  back  ;  pro- 
.  long  it  to  Z ;  measure  1 1  by  the  same  scale.  Then  calculate  the  weight  of  wall ; 
find  its  center  of  gravity  g ;  draw  g  a  vertical,  and  measure  the  leverage  a  t.  We 
then  have  the  data  for  calculating  the  two  moments. 

If  the  water,  instead  of  being  quiet,  is  liable  to  waves,  the  wall  should  be 
made  thicker. 


HYDROSTATICS. 


509 


Fig.  24. 


Art.  11.  To  find  tbe  tbickness  at  base  of  a  wall  required  to  b< 
safe  against,  overturning  under  tiie  pres  of  quiet  water  level  with  its  top,  and 
pressing  against  its  entire  vert  back.    Caution.    See  Art.  13. 


(1st)  Vertical  wall.  Fig  22. 


Thickness 
in  feet     ' 


Height 
'  in  feel  ^ 


V 


Factor  of  safety  * 
3  X  sp  grav  of  wall 


Height      the  proper  decimal 
'  in  feet  X  in  following  table 


(2d)  Rigrbt  angeled  triangular  wall.  Fig  23. 


Thickness 
at  base     = 
in  feet 


Height  f 

Mn  feet  X  -^ 


Factor  of  safety  * 


2  X  sp  grav  of  wall 
thickness,  m  o,  of  vertical  wall  X  1.225. 


Height      the  proper  decimal 
in  feet  X  in  following  table 


Notwitlistanding  their  greater  thickness  at  base,  such  triangular  walls  con- 
tain, as  seen  by  the  fig,  not  much  more  than  half  the  quantity  of  masonry  reqd 
for  vert  ones  of  equal  stability.  This  is  owing  to  the  fact  that  their  cent  of 
grav  is  thrown  farther  back;  thus  increasing  the  leverage  by  which  the  wtof 
the  wall  resists  overthrow. 


(3d)  Wall  witb  vertical  back  and  sloping:  face.  Fig  24. 

/  (Ht2,  ft  X  factor  of  safety  *)  +  (batter  A  n^,  ft  X  sp  grav  of  wall) 


Thickness 
at  base 
in  feet 


3  X  specific  gravity  of  wall 
=  Height  in  feet  X  th«  proper  decimal  in  the  following  table. 


Fig.  22. 

Sp.  Gr. 

Lbs  per 
Cub  Ft. 

Resist  =  1.5  pres. 

Resist  =  2  prei. 

Resist  =  3  pres. 

Dressed  Granite. . . 
Dressed  Sandstone 

Mortar  Rubble 

Brickwork 

Fig.  23. 
Dressed  Granite... 
Dressed  Sandstone 

Mortar  Rubbl<3 

Brickwork . .. 

'2.5 
2.2 
2. 
1.8 

2.5 
2.2 

156 
137 
125 
112 

156 
137 
125 
112 

.447 

.477 
.500 
.527 

.548 
.584 
.613 
.646 

.51« 
.550 
.578 
.609 

.633 
.675 
.707 
.746 

.633 
.674 
.707 
.746 

.775 
.82tt 
.866 
.913 

Resist  =  1.5  pres. 

Resist  =  2  pres. 

Fig.  24. 

Batter 
1  in.  to 
a  foot. 

Batter 
2  ins.  to 
a  foot. 

Batter 
4  ins.  to 
a  foot. 

Batter 
6  ins.  to 
a  foot. 

Batter 
1  in.  to 
a  foot. 

Batter 
2  ins.  to 
a  foot. 

Batter 
4  ins.  to 
a  foot. 

"7551" 
.583 
.609 
.640 

Batt«r 
6  ins.  t* 
a  foot. 

Dressed  Granite... 
Dressed  Sandstone 

Mortar  Rubble 

Brickwork 

2.5 

2.2 
2. 
1.8 

156 
137 
125 
112 

.449 

.480 
.502 
.530 

.458 
.488 
.510 
.539 

.487 
.515 
.536 
.562 

.532 
.558 
.578 
.602 

.519 
.552 
.571 
.610 

.526 
.560 
.586 
.618 

.593 
.622 
.646 
.674 

•  Factor  of  safety  : 


.  Required  moment  of  stability  of  wall 
overturning  moment  of  water 


510 


HYDROSTATICS. 


Art.  12.    Table  showing  Iiow  tlie  stability  of  a  wall  sustain- 
ing-  water  is  affected  by  a  cbange  in  tbe  form  of  tbe  w^all ; 

the  quantity  of  masonry  remaining  the  same.  Rem,  When  the  base  of  a  tri- 
angular wall,  of  sp  grav  2,  is  less  than  ^  the  ht,  the  stability  is  greatest  when 
the  water  presses  the  vert  side ;  but  if  the  base  exceeds  ^  the  ht,  the  stability 
Is  greatest  with  the  water  on  the  battered  side.    Caution.    See  Art.  13. 


All  these  ivalls  contain  precisely  the  same 
quantity  of  masonry.  The  masonry  is  supposed 
to  be  mortar  rubble,  weighing  125  lbs  per  cubic  foot;  or  twice  as  much 
as  water;  or  about  the  same  as  ordinary  rough  mortar  rubble.  If 
the  sp  gr  of  the  masonry  is  actually  greater  or  less  than  this,  the 
safety  also  will  be  greater  or  less,  in  precisely  the  same  proportion. 

Vertical  wall 

Face  vertical :  back  batters  one-tenth  height 

"  "  "  "        one-fifth         "      

"  "  "  *•        one-fourth     "      

"  "  "  "        one-third       "       

"  "  "  "        four-tenths   "      

"  "  "         "        one-half        "      

Back  vertical ;  face  batters  one- tenth  height 

"  "  "  *'        one-fifth        "      

"  "  "  "        one-fourth    "      

"  "  "  "        one- third       " 

"  "  "  "        four-tenths   "       

"  "  "  "        one-half         "      

Back  and  face,  each  batter  one-tenth  height 

"  "  "        one-fifth        "      

"  ••  "  "        oue-fourth    *'      

'*  •'  "  "        one-third       "       

"  "  "  "        four- tenths   "      


Base  in 

r^eXt°5 

parts  of 

height. 

wall. 

.5 

1.5 

.55 

1.8 

.6 

2.2 

.625 

2.6 

.667 

3.5 

.7 

4.9 

.75 

14.0 

.55 

1.8 

.6 

2.1 

.625 

2.2 

.667 

2.4 

.7 

2.6 

.75 

2.9 

.6 

2.2 

.7 

3.4 

.75 

4.6 

.833 

9.0 

.9 

36.0 

rbx 


Art.  13.  Liiability  of  wall  or  foundation  to  crusli  under 
unequal  distribution  of  pressure.  Arts  11  and  12  apply  only  to 
the  stability  of  a  rigid  wall  resting  upon  a  rt^'/d  base,  and  therefore  incapable 
of  failure  except  by  overturning  as  a  ^vhole.  They  show  that  the  stability  is 
greatest  when  the  water  presses  against  the  sloping  side.  But  in  practice  the 
point  where  the  resultant  of  all  the  pressures  on  the  base  of  the  wall  cuts  the 
base,  must  not  be  so  near  to  either  toe  as  to  endanger  a  crushing  of  wall  or 
of  foundation.  This  consideration  often  makes  it  best  to  let  the  water  press 
against  the  vert  back,  notwithstanding  the  consequent  loss  in  stability. 
Art.  14.  Fig.  25  shows,  to  scale,  a  dam  wall  at  Poona,  India,  designed'  by  Mr. 
Fife,  C.  E.,  of  England.  It  is  of  mortar  rubble,  of  150 
lbs  per  cub  ft.  Its  total, vert  height  is  100  ft;  thickness 
w  v  at  base,  60  ft  9  ins ;  at  top,  rx,  13  ft  9  ins.  The  front 
ru  slopes  42  ft  in  100  ft;  and  the  back  xv,  5  ft  in  100  ft. 
Its  foundation  is  7  feet  deep;  but  we  here  assume  that 
the  water  presses  against  its  ew/ire  back  xv.  Through 
tl)e  cen  of  grav  G  draw  G*  vert.  From  c,  where  the 
direction  of  the  pres  P  of  the  water  strikes  G*,  lay  oflT 
en  by  scale  =  139.6  tons  (of  2240  lbs)  water  pres  against  1 
ft  in  length  of  xv;  and  c/  =  249.4  tons  wt  of  1  ft  length 
of  wall.  Complete  the  parallelogram  cnmt  of  forces. 
^^  Its  diag  c  m  represents  the  resultant  of  all  the  pressures 
^  upon  the  base  uv,and  cuts  the  base  at  a, 20  ft  back  from 
the  toe  u.  Doing  the  same  with  the  151.4  tons  pres  p 
against  ru,  we  get  the  resultant  oy,  which  is  greater 
than  cm,  and  cuts  the  base  (at  i)  only  12.7  ft  from  the 
toe  V,  or  7.3  ft  less  than  a  is  from  u. 

Hence,  when  the  water  presses  against  xv  the  wall  is 
less  liable  to  fracture  or  crushing,  and  the  earth  foun- 
dation w  V  is  more  evenly  loaded,  and  hence  less  liable  to 
yield  unequally  so  as  to  cause  cracks  in  the  wall.  On 
this  account  xv  is  made  the  back  of  the  wall,  although 
the  moment  of  stability  of  the  wall  is  then  only  2.2  (calling  the  overturning 
moment  of  the  water  1),  while  if  the  water  pressed  against  rw  it  would  be  3,  or 
36  per  cent  greater. 


HYDROSTATICS. 


511 


Art.  15.    The  points  a  and  i,  Fig  25,  are  called  centeri^  of  pressure 

upon  the  base,  or  centers  of  resistance  of  the  base.  If  similar  points,  as 
d  and  z,  be  found  in  the  same  way  for  other  lines,  as  f  h,  by  treating  a  part  (a8 
rxhf)  of  the  wall  as  if  it  were  an  entire  wall;  a  slightly  curved  line  joining 
these  points  is  called  the  line  of  pressure.  Thus,  &a  is  the  line  of  pres- 
sure when  the  water  presses  against  xv.  Each  point,  as  d,  in  6a,  shows  where 
any  joint,  as,  fh,  drawn  through  that  point,  is  cut  by  the  resultant  of  all  the 
forces  acting  upon  said  joint,  bi  is  the  line  of  pres  when  the  water  presses 
against  ru.  These  lines  do  not  show  the  direction  of  the  resultants.  Thus,  at  a, 
the  latter  is  em,  not  ha.  The  angle  between  the  direction  of  the  resultant  and  a 
line  at  right  angles  to  the  bed  or  joint,  must  be  less  than  the  angle  of  friction 
of  the  materials  forming  the  joint. 

If  from  the  end  m  or  y  of  the  resultant  of  the  pressures  upon  any  joint,  we 
draw  m2  or  yZ  hor,  then  c2ot  ol  (as  the  case  may  be)  measures  the  entire  vert 
pres  on  that  joint;  and  m2  ovyl  measures  the  hor  pres  against  the  back  of  the 
wall,  which  tends  to  cause  sliding  at  the  same  joint.  If  the  direction  of  the  re- 
sultant comes  within  the  limit  stated  in  the  preceding  paragraph,  m  2  or  yl  will 
be  less  than  the  frictional  resistance  to  sliding,  which  last  is  =  c2  (or  o/)  X  the 
coeflf  of  friction  for  the  surfaces  forming  the  joint.  Hence  sliding  cannot  take 
place.  Sliding  never  occurs  in  the  masonnj  of  walls  of  ordinary  forms.^  Good 
mortar,  well  set  aids  to  prevent  sliding,  but  it  is  better  not  to  rely  upon  it.  But 
entir  '  walls  have  sliddeu  on  slippery  foundations. 

Art.  16.  In  California  is  this  dam  of  a  mining  reservoir,  built  of 
rough  stone  without  mortar,  founded  on  rock.  Height,  70  feet;  base,  50;  top, 6; 
water-slope,  30  feet;  outer-slope,  14.  To  prevent  leaking  the 
water-slope  is  only  covered  with  3-inch  plank  bolted  horizon- 
tally to  12  by  12  inch  strings,  built  into  the  stone-work.  All 
laid  with  some  care  by  hand,  except  a  core  of  about  one-fifth  of 
the  mass,  wiiich  was  roughlyHhrown  in.  Cost  about  S3  per  cubic 
yard.    It  has  been  in  use  since  1860. 

Rem.  If  a  dam  is  compactly  baclied  with  earth 
at  its  natural  slope,  and  in  sufficient  quantity  to  prevent  the 
water  from  reaching  the  dam,  the  pressure  against  the  dam  will 
not  be  increased. 

Art.  17.  To  find  the  thickness  of  a  cylinder  to  resist  safely  the 
pressure  of  water,  steam,  &c,  against  its  interior.    If  riveted,  see  next  page. 

liVhere  the  thicliness  is  less  than  one-thirtieth  of  the 
radius,  as  it  is  in  most  cases,  the  usual  formula 

Thickness  pressure       ^^      ,.     ^ 

(1>  "'inches     -  safe  strength    Xradins* 

Is  employed.  It  regards  the  material  as  being  subjected  only  to  a  direct  tensile 
strain,  which  is  sufficiently  correct  in  such  thin  shells. 

For  somewhat  g-reater  pressures  and  thicknesses.  Professor 
F.  Keuleaux  (Der  Konstrukteur,  p  52)  gives 

Thickness  _      pressure       (^   , pressure         \  ..     ^ 

(2)  in  inches      -  ^e  strength   I    "^  2  X  safe  strength/  ^  ^       '^* 

For  very  great  pressures  and  thicknesses,  as  in  hydraulic 
presses,  cannons,  Ac,  Professor  Reuleaux  (Konstrukteur,  p  53)  gives  Lamp's 
formula : 


Thickness 

(^)  in  inches      • 


(V^ 


X  radius.* 


I  safe  strength  -}-  pressure 
safe  strength  —  pressure 
The  three  formulae  give  results  as  follows,  pressures  and  strengths  in  lbs  per 
square  inch : 


-) 


Diameter. 

Radius. 

Pressure. 

Safe 

tensile 

strength. 

Thickness,  inches. 

Formula  (1). 

Formula  (2). 

Formula  (3). 

20  inches. 

10  inches. 

50 

500 
5000 

10000 

.05 

.50 
5.00 

.050125 
.5125 

6.25 

.05 

.513 

7.33 

The  thicknesses  given  by  the  formulas  appropriate  to  the  several  pressures  are 

Erinted  in  heavy  type.    It  will  be  seen  that  in  these  cases  the  results  differ 
ut  slightly,  except  for  very  great  pressures. 

*  In  all  three  formulae  take  the  radius  in  inches,  and  the  pressure  and  strength 
in  pounds  per  square  inch. 


512 


HYDROSTATICS. 


Rem.  2.  Want  of  uniformity  in  tbe  coolings  of  thick  castings  make!> 

them  proportionally  weaker  than  thin  ones,  so  that  in  order  to  reduce  thickness  in  important  cases 
we  should  use  only  best  iron  remeited  3  or  4  times,  by  which  means  an  ult  cohesion  of  about  30000 
lbs  per  sq  inch  may  be  secured.  But  even  with  this  precaution  no  rule  will 
apply  safely  in  practice  to  cast  cylinders  whose  thickness  exceeds  either 

about  8  to  10  ins,  or  the  inner  rad  however  small. 

Under  a  pres  of  8000  lbs  per  sq  inch,  water  will   ooze  tbrong^li  cast 
iron  8  or  10  inS  thick ;  and  under  but  250  lbs  per  sq  inch,  through  .5  inch. 
Table  of  tliiekn esses  of  sing-le-riveted  wrought  iron  pipes, 

tanks,  standpipes,  &c,  by  the  above  rule,  to  bear  with  a  safety  of  6  a  quiet  pressure  of  1000  ft  head 
of  water,  or  434  lbs  ye-  sq  inch  ;  the  ult  coh  of  fair  quality  plate  iron  being  taken  at  48000  fiis  per  sq 
inch,  or  at  8000  lbs  for  a  safety  of  6 ;  which  is  farther  reduced  to  8000  X  .56  =  4480  fts,  to  allow  for 

weakening  by  rivet  holes;  for  sing-le-riveted  cyls  have  but  about  ,56  of  the 
strength  of  the  solid  sheet;  and  double- riveted  ones  about  .7.  With  the 
above  pres  and  other  data,  the  rule  here  leads  to  thickness  =  .1016  X  inner  rad  in  ins. 


Dl. 

Ths. 

Di. 

Ths. 

Di. 

Ths. 

Di. 

Ths. 

Di. 

Ths. 

Di. 

Di. 

Ths. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

;Ft. 

Ins. 

.5 

.025 

5 

.254 

16 

.813 

30 

1.52 

60 

3.05 

120 

10 

•     6.09 

1.0 

.051 

6 

.305 

18 

.914 

33 

168 

66 

3.35 

132 

11 

6.70 

1.5 

.076 

8 

.406 

20 

1.016 

36 

1.83 

72 

3.66 

144 

12 

7.31 

'2.0 

.102 

10 

.508 

22 

1.117 

42 

2.13 

84 

4.27 

192 

16 

9.75 

3.0 

.152 

12 

.609 

24 

1.219 

48 

2.44 

96 

4.88 

240 

20 

12.19 

4.0 

.203 

14 

.711 

27 

L371 

54 

2.74 

108 

5.49 

288 

24 

14.63 

For  a  less  head  or  pressure,  or  for  any  safety  less  than  6,  it  is  safe  axA 

near  enough  in  practice,  to  reduce  the  thickness  of  wrought  iron  cyls  in  the  same  proportion  as  s^d 
head,  pres,  or  safety  is  less  than  the  tabular  one. 

Double-riveted  cylinders,  Fairbairn  says,  are  about  1.25  times  as  strong 
as  single-riveted.    Hence  they  may  be  one-fifth  part  thinner.    Liap-welded 

ones  are  nearly  1.8  times  as  strong  as  single-riveted;  and  hence  may  be  only 

.56  as  thick. 

Many  continuous  miles  of  double- riveted  pipes  in  California  have 

been  in  use  for  years  with  safetys  of  but  2  to  2.6.  In  one  case  the  head  is  1720  ft,  with  a  pres  of  746  fts 
per  sq  inch  ;  diam  11.5  ins  ;  thickness,  .34  inch. 

Cast  iron  city  water  pipes  must  be  thicker  than  required  by  formula 
(1),  in  order  to  endure  rough  handling  and  the  effects  of  "water-ram" 

^due  to  sudden  stoppage  of  flow,  see  second  Rem,  p513),  and  to  provide  against 
irregularity  of  casting  and  the  air  bubbles  or  voids  to  which  all  castings  are 
more  or  less  liable.  In  the  following  table  the  ultimate  tensile  strength  of  cast 
iron  is  taken  at  18,000  Bbs  per  square  inch.  Column  A  gives  thicknesses  by  Mr. 
J.  T.  Fanning's  formula  (Hydraulic  Engineering,  p  454). 

-100)X  bore,  ins  /        bore,  insX 

.4  X  ultimate  tensile  strength  \  100      / 

These  correspond  with  average  practice.  The  addition  of  100  ft)s  to  the  pres  is 
made  in  order  to  allow  for  water-ram.  Column  B  gives  thicknesses  by  formula 
(1),  taking  coefficient  of  safety  =  8  (thus  making  safe  tensile  strain  =  2250 

ft)s  per  square  inch)  angl  adding  three-tenths  of  an  inch  to  each  thickness  given 
by  the  formula: 


Thickness )  _^  (pres,  R)s  per  sq  in  - 
in  inches  J  ™  ^~-  ~ 


Head  in  feet    50 


100 


200 


300 


500 


1000 


Pressure, 

21.7 

43  4 

86.8 

130 

217 

434 

K)s  per  sq  in 

Bore,  ins. 

Thickness  of  pipe,  in  Indies. 

A 

B 

A 

B 

A     B 

A    B 

A 

B 

A 

B 

2 

.36 

.31 

.37 

.32 

.38    .34 

.39    .36 

.42 

.40 

.48 

.51 

3 

.37 

.31 

.38 

.33 

.40    .35 

.42    .40 

.45 

.45 

.54 

.60 

4 

.39 

.32 

.40 

.34 

.42    .38 

.45    .42 

.50 

.50 

.61 

.72 

6 

.41 

.33 

.43 

.36 

.47    .42 

.50    .48 

.57 

.60 

.75 

.94 

8 

.45 

.34 

.47 

.38 

.52    .47 

.57    .55 

.66 

.70 

.90 

1.15 

10 

.47 

.35 

.50 

.40 

.56    .50 

.62    .60 

.74 

.81 

1.04 

1.35 

12 

,49 

.36 

.53 

.42 

.60    .54 

.67    .66 

.82 

.91 

1.18 

1.57 

16 

.55 

.38 

.60 

.46 

.70     .62 

.79    .77 

.98 

1.10 

1.46 

2.00 

18 

.57 

.39 

.63 

.48 

.74    .65 

.85    .84 

1.06 

1.21 

1.60 

2.20 

20 

.61 

.40 

.67 

.50 

.79    .68 

.91     .90 

1.15 

1.31 

1.75 

2.50 

24 

.66 

.42 

.73 

.53 

.87     .77 

1.02  1.01 

1.30 

1.51 

2.03 

2.84 

30 

.74 

.82 

.45 
.47 

.83 
.93 

.59 
.65 

1.01     .89 
1.15  1.01 

1.19  1.19 

1,55 

1.82 
2.12 

2.46 

2.88 

3.47 

36 

1.36  1.37 

1.80 

4.11 

48 

.98 

.53 

1.13 

.77 

1.42  t.24 

1.70  1.73 

2.28 

2.73 

3.73 

5.38 

HYDROSTATICS. 


613 


Table  of  thickness  of  lead  pipe  to  bear  internal  pressures  with  a 

safety  of  6;  taking  the  ultimate  cohesion  of  lead  at  1400  Sts  per  sq  inch. 

Rem.  Although  these  thicknesses  are  safe  againstquiet  pressures,they  might  not 
resist  shocks  caused  by  too  sudden  closing  of  stop-cocks  against  running  water. 


Heads  in  Feet. 

Heads  in  Feet. 

a 

100        200        300    1    400     1     500 

1- 

a 
a 

100    1    200    1    300   1    400         500 

0 

Pres  in  lbs  per  sq  inch. 

Pres  in  lbs  per  sq  inch. 

1 

43.4   1  86.8        130        174     1     217 

43.4    1  86.8   1    130        174     1     217 

Thickness  in  Inches. 

Thickness  in  Inches. 

^ 

.026 

.055 

.089 

.128 

.171 

1 

.102       .221 

.357 

.511 

.682 

% 

.038 

.083 

.134 

.192 

.256 

1J< 

.127       .276 

.447 

.639 

.853 

^ 

.051 

.111 

.179 

.256 

.341 

1^ 

.153       .332 

.536 

.767 

1.02 

H 

.064 

.138 

.223 

.320 

.427 

m 

.178       .387 

.626 

.895 

1.20 

'       H 

.076 

.166 

.268 

.383 

.512 

2 

.204       .442 

.714 

1.02 

1.36 

K 

.089 

.193   1    .313 

.447 

.597 

Rem.  The  valves  of  water-pipes  must  be  closed  slowly,  and 

the  necessity  for  this  precaution  increases  with  their  diams.  Otherwise  the  sud- 
den arresting  of  the  momentum  of  the  running  water  will  create  a  great  pressure  against  the  pipes 
in  all  directions,  and  throughout  their  entire  length  behind  the  gate,  even  if  it  be  many  miles ;  thus 
endangering  their  bursting  at  any  point.     Hence  stop-gates  are  shut  by  screws. 


Fig. 


Art.  18.  Buoyancy.  When  a  body  is  placed  in  a  liquid,  whether  it  float 
or  sink,  it  evidently  displaces  a  bulk  of  the  liquid  equal  to  the  bulk  of  the  im- 
mersed portion  of  the  body  ;  and  the  body,  in  both  cases  and  at  any  depth,  and 
in  any  position  whatever,  is  buoyed  up  by  the  liquid  with  a  force  equal  to  the 
weight  of  the  liquid  so  displaced.  Thus,  if  we  immerse  entirely  in  water  a 
piece  of  cork,  c,  c,  Fig  26,  or  any  body  of  less  specific  gravity  than  water,  the 
cork  will,  by  its  weight,  or  force  of  gravity,  tend  to 
descend  still  deeper;  but  the  upward  buoyant  force  of 
the  water,  being  greater  than  the  downward  force  of 
gravity  of  the  cork,  will  compel  the  latter  to  rise.  In 
this  case,  the  cork  receives  a  total  downward  pressure 
equal  to  the  weight  of  the  vertical  column  of  water 
above  it,  shown  by  the  vertical  lines  in  vessel  1  ;  and 
a  total  upward  pressure  equal  to  the  weight  of  the 
column  shown  in  vessel  2.  The  difference  between 
these  two  columns  is  evidently  (from  the  figs)  equal  to 
the  bulk  of  the  cork  itself;  therefore  the  difference 
between  their  weights  or  pressures  (or,  in  other  words, 
the  buoyancy  of  the  water)  is  equal  to  the  weight  or 

pressure  of  the  water  which  would  have  occupied  the  place  of  the  cork;  or,  in 
other  words,  of  the  water  which  is  displaced  by  the  cork.  This  difference,  or 
buoyancy,  will  plainly  be  very  nearly  the  same  at  any  depth  whatever  of  entire 
immersion.  It  increases  slightly  with  the  depth,  owing  to  increase  in  the  density 
of  the  water;  but,  on  the  other  hand  it  is  diminished  by  compression  of  the 
cork.  Now  the  cork,  if  left  to  itself,  will  continue  to  rise  until  a  portion  of  it 
reaches  above  the  surface,  as  in  vessel  3.  The  downward  pressing  column 
then  ceases  to  exist ;  and  the  cork  is  pressed  downward  only  by  its  own  weight. 
But,  as  it  now  remains  stationary,  the  upward  pressure  of  the  water  must  be 
equal  to  the  weight  of  the  cork.  But  the  upward  pressure  of  the  water  arises 
only  from  the  sliaded  column  shown  in  vessel  3;  and  this  column  is  (as  in  the 
case  of  total  immersion)  equal  to  the  bulk  of  water  displaced.  Therefore,  in  all 
cases,  the  buoyancy  is  eqtial  to  the  weight  of  water  displaced ;  and  when  the 
body  floats  on  the'surface,  the  buoyancy,  or  the  weight  of  water  displaced,  is 
also  equal  to  the  weight  of  the  body  itself. 

If  the  body  be  of  a  substance  heavier  tlian  ivater,  its  weight  is  greater 
than  the  buoyancy  of  the  displaced  water,  and  the  body  therefore  sinks,  with  a 
force  equal  to  the  difference  between  the  two.  Thus,  a  cubic  foot  of  cast  iron 
weighs  450  lbs.,  and  a  cubic  foot  of  water  62.5  lbs.,  so  that  the  iron  sinks  with  a 
force  of  450  —  62.5  =  387.5  lbs. 

The  same  principle  applies  to  other  fluids.  Thus,  light  bodies,  such  as 
Bmoke,  a  balloon,  etc.,  in  air,  all  tend,  like  a  cork  in  water,  to  fall ;  but  the  air, 
being  heavier,  crowds  them  out  of  the  lower  positions  which  they  tend  to  assume, 
and  pushes  them  upward. 

Althougli  a  pound  of  lead  and  a  pound  of  feathers,  weighed  in  the  air,  balance 
each  other,  yet  in  a  vacuum  the  feathers  will  outweigh  the  lead,  by  as  much  as 
the  bulk  of  the  air  displaced  by  them  outweighs  that  displaced  by  the  iron. 


514         BUOYANCY,   FLOTATION,   METACENTER,   ETC. 


The  downwd  forc6  of  grar  may  be  regarded  as  concentrated  at  the  cen  of 

grav  G  of  a  floating  body.  The  upwd  pres,  or  buoyancy,!  of  the  water  may  similarly 
be  regarded  as  acting  at  the  cen  of  gr  W  of  the  displaced  water.*  W  is  also  called 
the  center  of  pressure,  or  of  buoyancy ,  of  the  water ;  and  a  vert  line 
drawn  through  it  is  called  the  axis,  or  vertical,  of  buoyancy,  or  of  flo* 
tation.  Ordinarily,!  W  shifts  its  position  with  every  change  in  that  of  the  body. 
Thus  in  L  it  is  at  the  cen  of  gr  of  the  rectangle  0066;  and  in  N  at  that  of  the  tii» 
angle  a  a  v. 

When     a     floating  t, 

body,  L,  P  or  R,  is  at 
rest,  and  undisturbed 
by  any  third  force, 
as  F,  it  is  said  to  be 
in  e^iuilibrium, 
and  G  and  W  are  then  ^ 
in  the  same  vert  line 
1 1  Figs  L  and  R,  or 
e  eFigP;  which  line 
is  called  the  axis, 
or  vertical,  of 
equilibrium.^ 

When  a  third  force,  as  F,  causes  the  axis  of  equilib  to  lean,  as  in  Figs  N,  0  and  S, 
then  if  a  vert  line  be  drawn  upwd  from  the  cen  W  of  buoy,  the  point  M  where  said 
line  cuts  said  axis,  is  called  the  metacenter  of  the  body.  |  G  and  W  are  then  no 
longer  in  the  same  vert  line; J  and  the  two  opp  and  vert  forces,  grav  and  buoy,  act- 
ing upon  those  points  respectively,  form  a  "couple"  and,  when  the 
third  force  F  is  removed,  they  no  longer  hold  the  body  in  equilib,  but  cause  it  to 
rotate.  If  (as  in  Figs  0  and  S)  the  positions  of  G  and  W  are  then  such  that  the 
metacenter  M  is  above  the  cen  of  gr  G,  this  rotation  will  tend  to  restore  the  body  to 
its  former  position,  and  the  body  is  said  to  have  been  (before  the  application  of  the 
third  force  F)  in  stable  equilibrium.^  But  if  (as  in  N)  M  is  below  G  the  direc- 
tion of  rotation  is  such  as  to  upset  the  body,  by  causing  it  to  depart  further  from  its 
former  position,  and  the  body  is  said  to  have  been  in  unstable  equilibrium.:]: 


The  tendency  ormomentin  ft-lbs  of  a  floating  body  either  to  upset  or  to  right  itself,  is, 


_  the  wt  of  the  body  (or  the  equal  ^  the  hor  dist  between  "W  M  and  G  H, 
upwd  pres  of  the  water)  in  lbs    ^  Figs  N,  O  and  S,  in  ft. 

The  third  force  P  may  of  course  be  so  great  as  to  overpower  the  tendency  of  the  body  to  right  it» 
self.  Thus,  a  ship  may  upset  in  a  hurricane,  although  judiciously  loaded  and  ballasted  for  ordlaarj 
Arinda.    A  hor  section  ef  a  body  at  water-line  is  called  its  plane  of  flotation. 

*  Tbe  body  is  in  fact  acted  upon  by  otber  forces,  such  as  the  hor 

pressures  of  the  water  against  its  immersed  portions  ;  but  as  all  of  these  in  any  one  given  direction 
are  balanced  by  equal  ones  in  the  opposite  direction,  they  have  no  effect  upon  the  forces  G  and  W. 
It  is  also  acted  upon  by  the  air,  which  presses  it  downwards  with  a  force  of  14.75  lbs  per  sq  inch ;  but 
this  is  balanced  by  an  equal  pres  of  the  surrounding  air  upon  the  surface  of  the  water. 

t  This  buoyancy  is  made  up  of  the  parallel  upward  pressures  of  the 
innumerable  vert  filaments  of  the  displaced  water  as  shown  by  Fig  26,  and 

the  axis  of  flotation  is  their  resultant,  as  in  the  case  of  paraUel  forces* 

}  The  shape  of  a  body  (as  that  of  a  sphere  or  cylinder  U)  may  be  such  that  the  position  of  Its  cen  of 
buoy  W,  relatively  to  that  of  its  cen  of  gr  G,  is  not  changed  by  the  rotation  of  the  body  about  a  given 
axis  (as  any  axis  of  the  sphere  or  the  longitudinal  axis  of  the  cyl),  but  remains  constantly  in  the 
same  vert  line  with  G,  so  that  the  body,  in  rotating,  remains  in  equilib.*  Such  a  body  is  said  to  be 
in  indifferent  equilibrium  about  said  axis.  But  if  a  cyl  U  be  made  to 
rotate  tibout  its  transverse  axis  xx,  It  plainly  comes  under  the  remarks  on  Figs  R  and  S,  and  may 
(before  rotating)  be  in  either  stable  or  unstable  equilib  about  that  axis  according  to  the  way  in  which 
Its  wt  is  distributed. 

II  This  metacenter  shifts  its  position  on  the  line  1 1  according  to  the  inclination  of  the  latter. 

g  Uneven  loading*,  instead  of  a  third  force,  may  cause  a  vessel  at  rest  to 
lean  as  at  P ;  and  yet  the  vessel  so  leaning  may  be  in  equilib ;  for  its  axis  e  e  of  equilib  may  be  vert, 
although  not  coinciding  with  the  axis  of  symmetry  of  the  vessel,  as  it  does  at 

ttiuL,. 

^  In  floating  bodies,  this  may  sometimes  (as  in  Figs  R  and  S)  be  the  case  even  when  the  cen  ^ 
huoy  W  (not  the  metacenter)  is  below  the  cen  of  gr  G ;  because,  when  the  body  is  forced  to  lean,  w 
moves  to  another  point  In  it,  and  this  point  maj  be  such  aa  to  bring  H  above  O.    W  is  always  below 

G  in  bodies  of  uniform  density,  floating  at  rest,  if  any  part  of  the  iaody  is  above  water.  When  such 
bodies  are  entirely  submerged,  W  and  G  coincide. 


HYDROSTATICS. 


515 


Art.  19.  A  body  ligrtiter  than  water,  if  placed  at 
tlie  bottom  of  a  vessel  coiitaining'  virater,  w^ill  not 
rise  unless  the  w^ater  can  g-et  under  it,  to  buoy  it, 
or  press  it  upward,  as  the  air  presses  a  balloon  of 
Sinoke  upward.  Thus,  if  one  side  of  a  block  of  light  wood, 
perfectly  flat  and  smooth,  be  placed  upon  the  similarly  flat  and  smooth  bottom  of  • 
vessel,  and  held  there  until  the  vessel  is  filled  with  water,  the  downward  pres  wiU 
keep  it  in  its  place,  until  water  insinuates  itself  beneath  through  the  pores  of  tha 
wood.  But  if  the  wood  be  smoothly  varnished,  to  exclude  water"  from  its  pores,  it 
will  remain  at  the  bottom. 

On  the  other  hand,  a  piece  of  metal  may  be  pre« 
vented  from  sinking:  in  water,  by  subjecting  it  to  a  suffi< 

cient  upward  pres  only,  while  the  downward  pres  is  excluded.  Thus,  if  the  bottonv 
of  an  open  glass  tube,  t,  Fig  27,  and  a  plate  of  iron  m,  be  made  smooth  enough  to  b« 
water-tight  when  placed  as  in  the  fig;  and  if  in  this  position  they  be  placed  in  « 
vessel  of  water  to  a  depth  greater  than  about  8  times  the  thickness  of  the  iron,  the 
upward  pres  of  the  water  will  hold  the  iron  in  its  place,  and  prevent  its  sinking; 
because  it  is  pressed  upward  by  a  column  of  water  heavier  than  both  the  column  of  air,  and  its  own 


Fin  27 


weight,  which  press  it  downward. 


Pid  28 


On  this  principle  iron  ships  float. 

Rem.  1.    A  retaining- wall,  as  in  Fig'  28, 

founded  on  piles,  may  be  strong  enough  to  re- 
sist the  pres  of  the  earth  e  behind  it,  in  case  water  does  not  find 
its  way  underneath ;  and  yet  may  be  overthrown  if  it  does ;  or 
even  if  the  earth  «  s  around  the  heads  of  the  piles  becomes  sata* . 
rated  with  water  so  as  to  form  a  fluid  mud.  In  either  case,  tha 
upward  pres  of  the  water  against  the  bottom  of  the  wall  will  Tir» 
tually  reduce  the  wt  of  all  such  parts  as  are  below  the  water  surf, 
to  the  extent  of  6'2}4  H>s  per  cub  ft ;  or  nearly  one-half  of  the  or- 
dinary wt  of  rubble  masonry  in  mortar. 
Rem.  2o  Although  the  piles  under  a  wall,  as  in  Fig  28,  may  be 
^::j  abundantly  sufficient  to  sustain  the  wt  of  the  wall ;  and  the  wall 

equally  strong  in  itself  to  resist  the  pres  of  the  backing  e;  yet  if 
the  soil  8  8  around  the  piles  be  soft,  both  they  and  the  wall  may  be  pushed  outward,  and  the  latter 
overthrown  by  the  pres  of  the  backing  e.  From  this  cause  the  wing-walls  of  bridges,  when  built 
on  piles  in  very  soft  soil,  are  frequently  bulged  outward  and  disfigured.  In  such  cases,  the  piling, 
and  the  wooden  platform  on  top  of  it,  should  extend  over  the  whole  space  between  the  walls;  or  else 
some  other  remedy  be  applied. 

Art.  20,  Draugrh  t  of  vessels.  Since  a  floating  body  displaces  a  wt  of  liquid 
equal  to  the  wt  of  the  body,  we  may  determine  the  wt  of  a  vessel  and  its  cargo,  by  ascertaining  how 
many  cub  ft  of  water  they  displace.  The  cub  ft,  mult  by  62%,  will  give  the  reqd  wt  in  lbs.  Suppose, 
for  instance,  a  flat-boat,  with  vert  sides,  60  ft  long,  15  ft  wide,  and  drawing  unloaded  6  ins,  or  .5  of 
a  ft.  In  this  case  it  displaces  60  X  15  X  .5  =  450  cub  ft  of  water ;  which  weighs  450  X  623^  =  28125 
fts ;  which  consequently  is  the  wt  of  the  boat  also.  If  the  cargo  then  be  put  in,  and  found  to  sink 
the  boat  2  ft  rruyre,  we  have  for  the  wt  of  water  displaced  by  the  cargo  alone,  60  X  15  X  2  X  62J^  == 
112500  B)s  ;  which  is  also  the  wt  of  the  cargo.  So  also,  knowing  beforehand  the  wt  of  the  boat  and 
•argo,  and  the  dimensions  of  the  boat,  we  can  find  what  the  draught  will  be.   Thus,  if  the  wt  as  befora 

140625 

be  140625  fts.  and  the  boat  60  X  15,  we  have  60  X  15  X  62J^  =  56250;  and :=  2.5  ft  the  required 

56250 
draught.    In  vessels  of  more  complex  shapes,  as  In  ordinary  sailing  vessels,  the  caloalation  of  the 
amount  of  displacement  becomes  more  tedious ;  but  the  principle  remains  the  same. 


516 


HYDRAULICS. 


HTDKAULIOS. 

Flow  of  Water  tbrodgb  Pipes. 

Much  of  the  theory  of  hydraulics  is  still  matter  of  dispute.  This,  and  the 
unavoidable  imperfections  of  actual  work,  render  it  advisable  to  use  liberal 
safety  factors  iu  applying  hydraulic  formulas.  Even  new  pipes  are  liable  to 
tuberculations,  which  materially  diminish  the  flow,  and  these  sometimes  greatly 
increase  uuder  the  action  of  chemicals  in  the  water.  Air  in  the  pipes  also 
diminishes  the  flow. 


The  term  HEAD  or  TOTAIi  HEAB  of  water,  as  applied  to  the  fiowage  of 
water  through  canals,  pipes,  or  openings  in  reservoirs,  &c,  means  the  vert  dist  it;  or  p  o,  Fig  1,  from 
the  level  surf,  mi,  of  the  water  in  the  reservoir,  or  source  of  supply,  to  the  center  (or  more  properly  t« 
thecen  of  grav)  o,  of  the  orifice  (whether  the  end  of  a  pipe,  r  o,to,  v  o,zo,  lo;  ov  any  other  kind  of 
opening)  through  which  the  disch  takes  place  freely,  into  the  air;  or  the  vert  dist  a  u,  or  fg,  from 
the  same  surf,  m  i,  to  the  level  surf,  g  u,  of  the  water  in  the  lower  reservoir ;  when  the  disch  takes 
place  under  water.  Thus,  in  the  case  of  disch  into  the  air,  the  vert  dist  i  v  or  po,  is  the  total  head 
for  either  of  the  pipes  ro,  t  o,  v  o,  z  a,  or  I  o;  and  ik  is  the  head  for  the  orifice,  k,  in  the  side  of  the 
reservoir.  And  for  disch  under  water,  au,  or  fg,  is  the  head  for  either  the  pipe  j,  or  the  opening  n; 
without  any  regard  whatever  to  their  depths  below  the  surf  of  the  lower  water ;  which,  according  t« 
the  older  authorities,  do  not  at  all  affect  their  disch. 

A  portion  of  a  pipe  may  have  a  head  greater  than  the  total  head  of  the  entire  pipe.  Thai  th« 
point  6  in  the  pipe  lo,  has  a  head  61;  while  the  eutire  pipe  has  only  the  he&d  p  o. 

Both  in  theory  and  in  praotice  it  is  immaterial  as  reg-ards 
the  vet,  and  tSae  quantity  ol"  water  disehar$sed.  whetlier  the 
pipe  is  inclined  downward,  as  ro.  Fig  1;  or  hor,  as  t^o;  or  in- 
clined upward,  as  ^o;  provided  the  total  head  po,  and  also 
the  leni^th  of  the  pipe,  remain  nnchang-ed.    If  one  pipe  is  longer 

than  another,  its  sides  will  evidently  present  more  friction  against  the  water,  and  thus  diminish  the 
•vel  and  the  quantity  of  disch.  The'  inclined  pipes,  r  o,  lo,  being  of  course  a  little  longer  than  the 
hor  one  vo,  will  therefore  each  disch  a  trifle  less  water;  but  if  the  hor  one  were  extended  slightly 
fceyond  o,  so  as  to  give  it  the  same  length  as  the  others,  then  each  of  the  three  would  disch  the  same 
quantity  iu  the  same  time. 

Art.  1  a.    nivisions  of  the  Total  Head.    In  any  pipe,  as  so.r  o, 

t  o,  V  o,  z  o,or  I  o.  Fig  1,  the  total  head  has  three  distinct  duties  to  perform;  1st,  to  overcome  the 
resistance  to  entry  at  s,  r,  t.  v,  z,  or  I;  '2d,  to  overcome  the  resistances  within  the  pipe;  and,  3d,  to 
^ive  to  the  water,  entering  the  pipe,  the  uniform  velocity  with  which  it  actually  flows. 

For  convenience,  we  regard  the  total  head  as  divided  into  three  portions,  corresponding  to  these 
duties;  namely,  1st,  the  entry  head;  2d,  the  resistance,  or  friction,  head;  and,  3d.  the  velocity  bead. 

Art.  1  6.  The  velocity  head  is  the  height  through  which  a  body  must 
fall,  in  vacuo,  to  acquire  the  vel  with  which  the  water  actually  flows  into  the  pipe.    It  is  th«refore  =: 

-^,  in  which  v  is  the  vel  in  ft  per  sec ;  and  g  is  the  acceleration  of  gravity,  or  32.2 

Art.  1  c.  Experiment  shows  that,  with  the  usual  sharp-edged  entry,  the  en- 
try head  is,  near  enough  for  practice,  =  half  the  vel  head.  If  the  entry  is  shaped 
Ake  Fig  7,  scarcely  any  entry  head  will  be  required.     But,  in   pipes  longer  than  about  1000 

diameters,  the  entry  head  bears  so  slight  a  proportion  to  the  total  head,  that  this  advantage  is  of  but 
little  importance.     It  becomes  more  apparent  in  shorter  pipes. 

Art.  I  d'.  In  Fig  1  we  will  assume  that  for  any  of  the  pipes,  is  representa 
the  sum  of  the  vel  and  entry  beads.  Then  the  remainder  s  v.  or  to  o,  of  the  total  head,  is  the 
JTriction  head;  or  the  head  which  is  just  suflBcient  to  balance  the  friction  and 
other  resistances  within  the  pipe ;  and,  since  the  entry  head  balances  the  resistance  at  the  entrance 
to  the  pipe,  the  velocity  head  has  only  to  give  velocity  to  the  water  iu  the  vessel,  causing  it  to  ent«r 


HYDRAULICS.  517 

the  pipe  as  rapidly  as  it  flows  through  it,  and  thus  Iceeping  the  pipe  supplied.  If,  by  shortening  the 
pipe,  or  by  smoothing  its  inner  surf,  we  diminish  the  total  friction,  then  a  less  friction  head  will  be 
required  ;  but  the  vel  will,  at  the  same  time,  be  increased,  and  this  will  require  a  greater  vd  head, 
and  entry  head,  so  that  the  three  together  make  up  the  total  head,  as  before.  Since  the  friction  is 
equal  to  the  force  or  head  reqd  to  overcome  it,  it  also  is  represented  by  wo. 

Art.  1  e»  The  friction  head  maj'  as  in  v  o, «  o,  and  Z  o,  Fig  1,  be  all  above  the  entrance 
to  the  pipe,  and  therefore  outside  of  the  pipe  ;  or,  as  in  a  pipe  laid  from  «  to  o,  it  may  be  all  helore 
the  entrance,  and  ivithin  the  pipe;  or,  as  in  ro  and  to,  it  may  be  partly  above,  and  partly  below,  the 
entrance;  and  therefore  partly  within,  and  partly  without,  the  pipe.  The  vel  and  disch,  after  the 
pipe  is  filled,  are  not  affected  by  this  difference  in  position  of  the  entry  end  ;  but  the  pressures  m  the 
pipe,  and  the  vels  while  the  water  is  filling  an  empty  pipe,  are  affected  by  it,  as  explained  in  Arts  1  2 
and  1  o. 

Art.  1/*.  But  it  is  necessary  that  the  entry  end  of  the  pipe 
should  be  placed  so  far  helO'W  the  surf  m  i,  that  thero  shall  be  left, 
above  the  cen  of  grav  of  the  entry  end,  at  least  a  head,  i  s,  sufficient  to  perform  the  duties  of  the  entry 
and  vel  heads.  If  the  entry  end  of  any  of  the  pipes  be  raised  above  6,  a  portion  of  the  vel  head  will 
be  m  the  pipe.  In  other  words,  the  head  in  the  pipe  will  be  more  than  sufficient  to  overcome  the 
resistances  in  the  pipe ;  and  the  surplus  will  act  as  vel  head,  and  will  give  greater  vel  to  the  water 
<n  the  pipe.  The  reduced  head  thus  left  above  the  entry  end  will  plainly  be  insufficient  to  maintain 
the  supply  for  the  greater  vel,  and  the  pipe  will  run  only  partly  full. 

In  ordinary  cases  of  pipes  of  considerable  length,  the  sum  of  the  entry  and  vel  heads  theoreticallj 
required,  is  but  a  small  portion  of  the  total  head,  and  i  arely  exceeds  a  foot.  Indeed,  in  a  pipe  of 
considerable  diameter,  the  upjter  half  of  its  cross  section  at  the  entry  end  may  often  be  more  than 
enough  to  provide  sufficient  entry  and  vel  heads  above  the  cen  of  grav  of  said  cross  section:  so  that 
the  top  of  the  entry  end  might,  so  far  as  these  considerations  alone  are  concerned,  project  above  the 
Burf  of  the  water  in  the  reservoir.  But  the  end  of  the  pipe  should  in  practice  always  be  entirely  be- 
low the  surf;  otherwise  air  and  floating  impurities  viU  be  drawn  into  it,  and  cause  obstructions. 
Moreover,  the  water  surf  of  reservoirs  is  always  liable  to  considerable  changes  of  height;  and  the 
entry  end  of  the  pipe  must  be  placed  at  such  a  depth  that  the  water  can  flow  into  it  with  sufficienJ 
Tel  when  at  its  lowest  stages.     As  before  stated,  this  will  cause  no  diminution  or  increase  of  discb. 

Art.  1  a.  To  find  the  friction  head  reqd  for  any  part  of 
a  pipe;  knowing  the  fric  head  reqd  for  the  whole  pipe  Since  the  friction,  in  a 
pipe  of  uniform  diam,  is  (other  things  being  equal)  in  proportion  to  its  length;  and  since  wo.  Fig  1, 
represents  the  total  friction,  or  reqd  friction  head,  we  have 

Total  length   ,   Length  of  the  »  •  „q  *  The  friction  head  reqd 
of  the  pipe    •    given  portion   •  •  •         for  that  portion, 

fir,  having  drawn  w  o  by  scale,  «  w  hor,  and  «  o ; 

Total  length  ,  Length  of  the  ...-,«   A  dist,  as  a  c,  to  be  laid 
of  the  pipe    •    given  portion  ••'*'•         off  from  son  so. 

*  *  sw  •  ■*  dist,  as  «  b,  to  be  laid  off 

•  •  •  from  s  on  « tTc 


Or 


Then  a  vert  line,  as  h  c,  drawn  from  6  or  c,  and  joining  «  w  and  s  o,  gives  by  scale  the  friction  head 
reqd. 

Art.  1  7l.«     If  the  pipe  is  straight,  as  r  o,  v  o^l  o,  the  friction  in  any  part  begin' 
ning  at  the  reservoir,  as  2  6  in  the  pipe  I  o,  may  be  found  at  once  by  drawing  a  line  6    i  vert  upward 
from  the  axis  of  the  pipe  at  6.     The  line  2    3  will  then  give' the  friction  in  1 6.    It  also  gives  the  fric- 
tion in  r  i,  or  in  that  part  ot  v  o  which  lies  between  v  and  the  dotted  line  1    6.     It  must  be  remem- 
bered that  all  the  pipes   in  Fig  1  are  supposed  to  be  of  the 
I  same  actual  length.   They  would  thus  end  at  different  points 

_    J/y 'T}  o,  and  strictly,  a  separate  diagram  must  be  drawn  for  each 

JT)\_~ -ito/j  pipe.     In  a  part  of  the  pipe  not  beginning  at  the  reservoir, 

t/'j  CU  US  io  r  o,  V  o.  or  I  o,  between  points  vertically  under  c  and 

I  X,  the  amount  ot  friction  is  given  by  the  line  d  x,  for  it  is 

j  plainlv  — ya;  —  5  c. 

Art.  1  j.     If  the  pipe  is  vert,  as  v  o. 


'St 


2 


•T  ,  Fig  1  A;  let  is  (on  its  axi,s  io)  represent,  as  before,  the  sum 

— /C  of  the  vel  and  entry  heads.     From  s,  v,  and  o,  respectively, 

draw  hor  lines  s  w,  v  k,  and  o  y,  making  o  y  ~  v  o.     Draw 

the  oblique  line  s  y.    Then,  to  find  the  friction  in  any  part, 

"^.^  as  V  q,  beginning  at  the  reservoir  ,•  from  q  lay  off  g  d  hor,  and 

N  equal  to  v  q,  and  draw  the  vert  line  a  d,  crossing  ay  aX  g, 

s  Then  h  g  will  give  the  friction  in  v  q. 

\      \  Art.  1  h.    If  the  pipe  is  curved,  and 

\         ^^  »f  the  curvature  is  uniformly  distributed  along  its  length,  or 

\        \  so  slight  that  it  maybe  neglected;  the  friction  heads  reqd. 

\       ^  for  the  several  portions  of  the  pipe,  may  be  found  in  the 

\      '^  same  way  as  for  straight  pipes,  as  in  Art  I  H.    Otherwise 

\    \  thev  must  be  found  by  proportion,  as  in  Art  1  G. 

M  Art.  1  I.       While   water  is    filling 

^y       an  empty  pipe,  the  excess  of  the  total  head 

■|^j^o*,X  A  above  the  requirements  of  friction,  &c,  gives  to  the  water* 

**'  g-reater  vel  than  it  has  after  the  pipe  is  filled; 

but  this  gradually  decreases  as  the  advancing  water  encoun* 
ters  the  friction  along  the  increased  lengths  of  pipe  filled;  and  finally  becomes  least  when  the  water 
fills  the  whole  length,  and  begins  to  flow  from  the  disch  end.  o.  But  if  only  the  vel  and  entry 
heads  are  left  above  the  entry  end,  as  in  a  pipe  laid  from  a  to  o,  there  will  plainly  be  no  such  excest 
)r  total  head,  and.  consequently,  no  such  change  of  vel  during  the  filling  of  the  pipe. 


518 


HYDRAULICS. 


Art.  1  m-r,*  Relation  between  clisebarg^e,  area,  velocity  and 
pressure.  In  Fig  1  B-D,  where  the  pipe,  b  F,  running  full,  receives  water 
from  an  unlimited  reservoir,  R,  at  &,  and  discliarges  through  an  orifice,  F;  the 
vohime  of  water,  passing  any  given  cross  section  of  h  F,  in  a  given  time,  is 
constant  and  equal  to  the  rate  of  discharge  at  F.  Thus  ; — if  the  rate  of  discharge, 
at  F,  be  Q  cubic  feet  per  second,  then  Q  cubic  feet  will  pass  each  cross  section  of 
the  pipe,  b  F,  per  second. 


JJi 


Tig.  1  B-I>. 


r > 


Let  a  =  the  area  of  cross  section,  and  F=?  the  velocity,  of  the  stream  issuing 
through  the  short  pipe  beyond  F.     Fis  called  the  velocity  of  efflux. 

Let  Ai,  Ao,  etc.,  be  the  different  areas  of  cross  section  of  b  F,  and  let  Vi,  v^,  etc., 
be  the  velocities  at  those  cross  sections  respectively.    Then  Q  =  a  V  =  AiVj^ 

=  Ai  Vq,  etc.:  or  F  =  — ,    v\  =  ~,  vo  =  ■-^,  etc".    In  other  words,  the  veloci- 
^    *"         '  a'  Ai  A^ 

ties  are  inversely  as  the  areas  of  cross  section.    Also,  a  =  %,  Ai  =  — ,  Ao  =  -, 

etc. 


The  losses  of  pressure,  due  to  the  velocities,  respectively,  are  di 


_vr 


.d2- 


t'22 


2g'    ^       2g' 

etc.;  as  represented  by  the  ordinates  between  the  line  o  o',  of  static  pressure,  and 
the  diagram,  ol23456i^,  of  actual  pressures.  The  difference,  due  to  velocity, 
between  the  pres  heads  at  any  two  points,  as  c\  and  c>i^  where  the  velocities  are 


2^       • 
static  head  in 


Vi  and  vo  respectively,  is  j92  —  i'l  =  "l  —  ^2  —  o ^  ' 

The  remaining  pressure  head,  pi,  jso*  etc.,  at  any  point,  is  = 
reservoir  —  velocity  head  at  the  point,"=  H —  d\,  H —  d^,  etc. 

The  loss  of  pressure  head,  at  F^  is  (6  i^)  =  j^a  =  IT  —  d^;  and  the  pressure 
drops  to  zero  ;  i.e.,  to  the  atmospheric  pressure. 

Art.  1  a.  Open  piezometers.  If  the  lower  ends  of  vertical  or  inclined 
tubes,  open  at  both  ends,  be  inserted  into  a  pipe,  b  F,  Fig.  1  J5  i>,  as  at  Cj,  C2, 
etc.,  the  water  surface,  in  these  tubes,  will  stand  at  heights,  7? i,  joo,  etc.,  corre- 
sponding to  the  pressure  heads  at  the  points  where  the  tubes  are  inserted.  Such 
tubes  are  called  open  piezometers.  In  order  that  the  water  level  may  be 
observed,  they  are  of  glass,  at  least  in  those  portions  where  that  level  is  likely  to 
be  found.  An  obstruction,  in  the  pipe,  between  Co  and  F,  would  raise  the  level 
in  a  piezometer  at  Cg ;  while  an  obstruction  between  b  and  c^  would  lower  it. 

*  In  Art.  1  m-r,  for  simplicity,  we  neglect  all  resistances,  including  those  due 
to  the  abrupt  enlargements  and  contractions  of  the  pipe. 


HYDRAULICS. 


519 


Fig.l  F 


Art.  1  f .  If  we  imagine  any  pipe,  full  of  water,  to  be  supplied  with  a  nurobM 
of  piezometers,  then  a  line,  joining  the  tops  of  the  cohimns  of  water  in  the  several 
piezometers,  is  called  the  bydraulic  grade  line. 

Art.  1  «*.   In  a  straight  tube  of  uniform  diara  tliroughout,  as  ro,  v  o,  or  I  o,  Fig 

1  runuing  full  and  discharging  freely  into  the  air,  the  hyd  grade  line  is  a  straight  line  drawn 

from  its  disch  end  o  to  a  point  s  immediately  over  the  entry  end  of  the  pipe,  and  at  a  depth  below 

the  surf  equal  to  the  sum  of  the  vel  and  entry  heads. 
If  the  orifice  at  o  be  contracted,  the  hyd  grade  line  must  be  drawn 

from  «  to  some  point,  as  e,  immediately  over  o,  and  depending,  for  its  height,  upon  the  amount  of 

contraction  at  o.  But  in  this  case 
the  point  s  will  also  be  higher  than 
before,  because  the  vel  in  the  pipe  is 
reduced  by  the  contraction  ;  and  the 
sum  i  s  of  the  vel  and  entry  head* 
will  be  less. 

If  the  disch  at  o  is 
nnder  ivater,  the  effect 
upon  the  position  of  the  grade  line 
will  be  the  same  as  that  of  a  con- 
traction of  the  orifice  at  o.  The 
point  e  will  be  on  the  surf  of  the 
lower  water,  and  immediately  over  o. 

Art.  1  V.     If  the  pipe,  of  uniform 

diam,  (whether  discharging  freely  or  through  a  con- 
tracted opening  at  o,  whether  into  the  air  or  under 

water),  is  bent  or  cnrved,  the  hyd  grade 
line  will  still   be  straight,  provided  the 

resistances  are  equal  in  each  equal  division  of  the  hor 
length  of  the  pipe,  as  in  Fig  1  E,  where  equal  divisions 
vw,to  X,  &c,  of  the  total  length,  correspond  with  equal 
divisions  v  a,  a  b,  &c,  of  the  hor  length. 

But  in  Fig  1  F,  the  hyd  graOe  line  will  take  the 
shape  8  ao.    For  if,  in  accordance  with  Art.  1  G,  we 
divide  s  o  into  two  equal   parts,  s  m,  m  o,  correspond- 
ing with  the  two  equal  parts  vr,ro,  of  the  length  of  the 
0    pipe,  we  obtain  m  c  =  a  e  for  the  head  consumed  in  the 
resistances  In  v  r,  leaving  only  r  a  for  the  pres  head  at  r. 
Art  Xw,    In  a  very  large  vessel,  the  total  head  upon  any  point  at  the  level 
of  the  entrance  I  to  a  pipe  I  o  o'  Fig  1  G,  is  represented  by  i  /,  as  already  ex- 
plained but  of  this  total  head  a  portion,  as  i  s,  is  required  to  act  as 
velocity  head  and  entry  head  for  the  entrance  at  I,  leaving  only  s  I  as  the  pres- 
sure head  upon  a  point  in  the  pipe,  immediately  to 
the  right  of  I.    Thus  while  the  pressure,  in  pounds 
per  square  inch,  in  the  vessel  at  I,  is 

p  =  ilX  0.434    . 
that  in  the  pipe  at  I  is 

p  =  slX  0.434. 
But  now  a  portion,  as  sv,  of  si,  is  expended  in 
Zo  in  balancing  or  "overcoming"  the  resistances 
throughout  that  portion  of  the  pipe ;  and,  in  doing 
this  work,  it  gradually  diminishes  from  sv  (at  /)  to 
nothing  (at  o)  as  indicated  by  the  dotted  line  se. 
Thus,  at  the  point  6,  a  portion  =  bc  has  already  been 
expended  in  overcoming  the  resistances  in  the  pipe 
between  I  and  6,  leaving  c  6  as  the  pressure  head  at 
6,  of  which  c  in  must  still  be  expended  against  resist- 
ances in  the  wide  pipe  between  6  and  o,  leaving 
in&  =  vl  =  eoas  the  pressure  head  for  a  point  just 
to  the  left  of  the  contraction  at  o.  The  pressure  in 
to  is  thus  gradually  diminished  from  si  (at  I)  to 
eo  =  vl  (at  o). 
Now  a  portion  es'  of  eo  is  required  to  act  as  ve- 
locity and  entry  head  for  the  entrance  o  to  the  narrower  portion  o  o'  of  the  pipe ; 
because  we  need  at  o  not  only  an  additional  entry  head  to  overcome  the  resist- 
ance due  to  the  square  shoulder  formed  by  the  contraction,  but  also  an  addi- 
tional velocity  head  to  give  the  increase  of  velocity  which  must  take  place  as  tlie 
water  passes  from  the  wide  pipe  lo  to  the  narrower  one  oo';  for,  so  long  as  a 
pipe  runs  ftcll  and  the  discharge  remains  constant,  the  velocity  in  each  part  of 
the  pipe  must  be  inversely  as  the  area  of  cross  section  of  that  part ;  because  in 
each  second  the  same  quantity  of  water  passes  each  point;  and  this  constant 
quantity  is  =  area  X  velocity.  Hence,  as  the  area  diminishes,  the  velocity 
increases. 

There  remains,  therefore,  s'  o  as  the  pressure  head  upon  a  point  in  the  narrow 
part  just  to  the  right  of  o;  and  this  m  turn  gradually  diminishes  to  nothing  at 


520 


HYDRAULICS. 


the  end  o'  of  the  pipe,  as  indicated  by  the  dotted  line  s  o\  being  all  expended  in 
overcoming  the  resistances  in  o  o'.  We  thus  have,  for  the  hydraulic  gradient  in 
Fig  1  G,  the  broken  line  ises' o'. 

When  tlie  pressure  is  thus  diminished  by  overcoming  resistances,  or  by  ac- 
celerating velocity,  the  diminution  is  called  los<S  of  head.  Thus  we  say  that 
is  is  lost  at  the  entrance  /, 5v  as  friction  head  in  lo,  es'  at  the  contraction  o,  and 
s'  0  as  friction  head  in  o  o', 

At  o\  all  the  available  head,  i  Z,  has  been  used  up.  The  water  flowing  out  at  o\ 
therefore  exerts  no  lateral  pressure,  so  that  the  stream  flows  out  in  parallel  lines, 
and  its  capacity  for  forward  pressure  is  due  entirely  to  its  kinetic  energy  (energy 
of  motion)  and  to  the  pressure  of  the  air  upon  the  surface  in  the  reservoir  at  i; 
but  this  last  is  of  course  balanced  by  the  air  pressure  from  without  against  the 
opening  o'.  Where  a  great  reduction  of  cross  sectional  area  in  a  pipe  is  followed 
(down-stream)  by  a  re-enlargement,  the  increase  of  velocity  may  (under  certain 
circumstances)  consume  not  only  the  entire  head  of  water,  but  also  a  portion  or 
all  of  the  atmospheric  pressure  on  the  surface  in  the  reservoir,  thus  causing  a 
partial  or  complete  vacuum  at  the  constriction.    See  the  Venturi  Meter. 

Tlie  syption,  or  siphon.  If  one  leg  a  6  of  a  bent  tube  or  pipe  ahCy 
Fig  M,  of  any  diaiii,  filled  with  water,  and  with  both  its  ends  stopped, 
be  placed  in  a  reservoir  of  water,  as  in  the  fig;  and  if  the  stoppers  be 
then  removed,  the  water  in  the  reservoir  will  begin  to  flow  out  at  c,  and 
will  continue  to  do  so  until  its  level  is  reduced  to  t,  which  is  the  same  as 
that  of  the  highest  end  c  of  the  pipe  or  syphon.  The  flow  will  then  stop. 
The  parts  a  b  and  6  c  are  called  the  legs  of  the  syphon,  b  being  its  high- 
est point ;  and  this  is  correct  so  far  as  relates  to  it  merely  as  a  piece  of 
tube ;  but  considering  it  purely  with  regard  to  its  character  as  a  hydrau- 
lic machine,  the  part  t  a  below  the  level  of  the  highest  end  c,  may  be  en, 
tirely  neglected;  for  the  water  in  the  reservoir  will  not  be  drawn  down 
below  the  level  of  the  highest  end,  whether  that  be  the  inner  or  the  outer 
one.  Therefore,  if  the  disch  end  be  above  the  water  in  the  reservoir,  as, 
for  instance,  at  w,  no  flow  will  take  place.  The  vert  height  b  o,  from  the 
highest  part  of  the  syphon,  to  the  lowest  level  t,  to  which  the  reservoir 
is  to  be  drawn  down,  must  not,  theoretically,  exceed  about  33  or  34  ft; 
or  that  at  which  the  pres  of  the  air  will  sustain  a  column  of  water. 
Practically  it  must  be  less,  to  allow  for  the  friction  of  the  flowing  water, 
and  for  air  which  forces  its  way  in.  And  still  less  at  places  far  above  sea 
level ;  for  at  such  the  reduced  weight  of  the  atmospheric  column  will  not 
balance  so  great  a  height  of  water.  In  order  readily  to  understand,  or 
at  any  time  to  recall  the  principle  on  which  the  syphon  acts,  bear  in 
mind  that  we  may  theoretically  consider  the  end  of  the  inner  leg  to  be 
not  actually  immersed  lelow  the  water  surf,  but  only  to  be  kept  precisely 
at  it,  as  the  surf  descends  while  the  water  is  flowing  out;  but  may  re- 
j?ard  the  vert  dist  6  a  as  the  length  of  the  outer  leg ;  and  a  varying  dist,  which  at  first  is  b  s,  and  finally 
b  o  (as  the  surf  of  the  reservoir  descends)  as  the  length  of  the  inner  leg;  and  that  the  flow  continues 
only  while  this  outer  leg  is  longer  than  this  inner  one.  The  books  are  wrong  in  saying  that  the  outer 
leg  6  c  must  be  longer  than  the  inner  one  b  a,  in  order  that  the  water  may  run  at  all.  The  principle 
then  is  simply  this:  that  both  these  legs  6  c,  and  bi,  being  first  filled  with  water,  (the  part  ia  being 
considered  at  first  as  a  portion  of  the  reservoir,  and  not  of  the  syphon,)  it  follows  that  when  the  stop- 

Eers  are  removed  from  the  ends  c  and  a,  the  air  presses  equally  against  these  ends ;  but  the  great  vert 
ead  of  water  boin  the  outer  leg  6  c,  presses  against  the  air  at  c,  with  more  force  than  the  small  head 
©f  water  6s  in  the  inner  leg  bi,  does  against  the  air  at  a  or  i.*  Consequently,  the  water  in  i  c  will 
tend  to  fall  out  more  rapidly  than  that  in  bi;  and  -as  it  commences  to  fall,  would  produce  a  vacuum  at 
b,  were  it  not  that  the  pres  of  the  air  against  the  other  end  a  or  t,  forces  the  water  up  i  6,  to  supply 
the  place  of  that  which  flows  out  at  c.  In  this  manner  the  flow  continues  until  the  surf  of  the  water 
in  the  reservoir  descends  to  t.  on  the  same  level  as  c.  The  pressures  of  the  vert  heads  bo,  bo,  in  the 
two  legs  be,  bt,  being  then  equal,  it  ceases. 

The  syphon  principle  may  b«  employed  for  draining  ponds  into  lower  ground  at  a  considerable  dist, 
even  though  an  elevation  of  several  feet  (in  practice  perhaps  not  exceeding  about  28  ft  above  the  level 
to  which  the  pond  is  to  be  reduced)  may  intervene.  In  such  c  case  an"  escape  must  be  provided 
at  the  summit  (or  summits,  if  there  are  more  than  one)  of  the  bends,  for  the  disch  of  free  air,  which 
will  inevitably  enter,  and  soon  stop  the  flow,  unless  this  precaution  be  t-'jicn.  The  air-valve 

will  not  answer  for  this,  because  as  soon  as  the  valve  v  opens,  the  syphon  becomes 
in  eftect  two  separate  tubes  open  at  top ;  and  the  water  will  fall  in  both.  An  ori- 
fice at  the  escape  will  be  needed  for  filling  the  syphon  at  the  start ;  and  to  pre- 
vent the  water  thus  introduced,  from  running  out.  stopcocks  must  be  provided  at 
the  ends,  and  kept  closed  until  the  filling  is  completed. 

The  greatest  pains  must  be  taken  to  make  all  the  joints  perfectly  air-tight. 

The  motive  power  or  head  which  causes  the  flow  in  a  syphon,  is  the 
vert  dist  s  o,  from  the  surf  of  the  reservoir,  to  the  disch  end  c;  or  in  other  words, 
it  is  the  diflT,  s  o,  between  the  theoretical  lengths  b  s  and  b  o,  of  the  two  legs.    Con- 


Fig.  M. 


•  Said  pressure  of  the  air  is  of  course  not  exerted  directly  at  a  or  < ;  but  is  transmitted  to  a  tbroogb 
toe  water  in  the  vessel ;  and  thence  upward  to  i  through  the  water  in  the  siphon. 


HYDRAULICS. 


521 


sequently,  the  farther  c  is  below  s  the  more  rapid  will  be  the  flow ;  and  it  is  plain 
that  as  the  surf  gradually  sinks  below  a-,  the  less  rapid  will  the  flow  become.  Hav- 
ing this  head,  the  entire  length  ahc  of  the  syphon,  and  its  diam,  all  in  ft,  the 
disch  may  be  tound  approximately  by  either  of  the  rules  given  in  Art  2  for  straight 
pipes.  These  rules  give  553^  galls  per  min,  instead  of  .the  433^  galls  actually  dischd 
by  Col  Crozet's  syphon,  with  a  head  of  20  ft.     See  below. 

In  a  true  »^yphoii,  agnyo  Pig  H,  free  from  air  inside,  and  running  full, 
tlie  total  head  />o  is  measured  vertically  from  the  surface  mi  in  the  reser- 
voir to  the  center  of  gravity  of  the  outlet  o,  as  in  Fig  1 ;  the  hydraulic 
gradient  (with  the  restriction  named  in  Art  1  v)  is,  as  before,  a  straight  line 
r  *ro  drawn  from  the  foot  * 

JE JiSM J?- — — jP      of  the  combined  entry  and 

'■  velocity  heads  to  the  end 
o;  and  the  velocity  and 
discharge  are  the  same  as 
they  would  be  if  all  parts 
of  the  pipe  were  brought 
below  sro.  But  see  cau- 
tions 1  and  2,  below. 

The  pressure  at 
any  point,  g,  n  or  y,  is 
then  given  by  a  vertical 
line,  gv,  nr  or  yv,  drawn 
from  the  point  in  question  to  sro\  but  for  points,  as  n,  situated  above  sro,  this 
pressure  is  negative  or  inuxird ;  while  at  points  where  sro  and  the  pipe  are  at  the 
same  level,  as  at  /and  e,  there  is  neither  pressure  nor  vacuum. 

Caution  1.  But  if  the  water  be  admitted  to  the  empty  pipe  at  a,  while  the 
end  0  is  open,  the  pipe  will  not  form  a  true  syphon.  The  part  agn  will  then  run 
full,  and  will  have  5 en  as  its  hydraulic  gradient;  but  upon  reaching,  at  w,  a 
portion  no  of  the  pipe  with  a  much  steeper  grade,  the  water  will  run  off,  in  no, 
with  a  velocity  greater  than  that  with  which  it  arrives  from  a  n.  Hence  the 
stream  in  no  will  have  a  less  area  of  cross  section  than  in  an,  and  therefore  can- 
not fill  no,  but  will  run  off  in  it  as  in  an  open  gutter. 

Caution  2.  The  tendency  to  vacuum  at  points  above  sro  causes  an  accu- 
mulation, at  n,  of  particles  of  air  that  have  been  carried  into  the  syphon  by  the 
water  or  have  found  their  way  in  through  imperfect  joints,  etc' ;  and  these 
bring  about  a  condition  approaching  that  described  in  Caution  1;  for  their 
expansive  force,  by  reducing  the  negative  pressure  or  vacuum  n  r  at  n,  diminishes 
the  total  head  //  r  of  the  part  agn,  while,  by  practically  reducing  the  cross-sec- 
tion of  the  syphon  at  n,  they  require  that  a  portion  of  the  remaining  head  he 
used  at  n,  as  entry  head  to  overcome  the  resistance  caused  by  the  contraction, 
and  as  velocity  head  to  give  the  increase  of  velocity  needed  for  passing  the  nar- 
rowed section  at  n.  Now  since  the  friction  head  required  for  the  part  agn  re- 
mains about  the  same,  the  velocity  head  in  the  reservoir  is  considerably  dimin- 
ished, and  the  water  arrives  at  n  too  slowly  to  keep  n  o  filled.  The  accumulation 
of  air  at  n  thus  retards  the  flow  and  disturbs  the  distribution  of  the  pressures, 
so  that  these  are  no  longer  correctly  indicated  by  vertical  lines  drawn  to  sro. 

At  Blue  Ridge  Tunnel,  Virginia,  Col.  C.  Crozet  constructed  a  drainage 
syphon  1792  ft  long  of  cast  iron  faucet  pipes  3  ins  bore,  9  ft  long.  Its  summit  was 
9  ft  above  the  surface  of  the  water  to  be  drained  ;  and  its  discharge  end  was  20  ft 
below  said  surface,  thus  giving  it  a  head  of  20  ft.  At  the  summit  570  ft  from  the 
inlet,  was  an  ordinary  cast  iron  air-vessel  with  a  chamber  3  ft  high  and  15  ins 
inner  diam.  In  the  stem  connecting  it  with  the  syphon  was  a  cut-ofT  stop- 
cock ;  and  at  its  top  was  an  opening  6  ins  diam,  closed  by  an  air  tight  screw  lid. 
At  each  end  of  the  syphon  was  a  stopcock.  To  start  the  flow  these  end 
cocks  are  closed,  and  the  entire  syphon  and  air-vessel  are  filled  with  water  through 
the  opening  at  top  of  air-vessel.  This  opening  is  then  closed  airtight,  and  the  two 
end  cocks  afterwards  opened ;  the  cut-off  cock  remaining  open.  The  flow  then 
begins,  and  theoretically  it  should  continue  without  diminution,  except  so 
far  as  the  head  diminishes  by  the  lowering  of  the  surface  level  of  the  pond.  But 
in  practice  with  very  long  syphons  this  is  not  the  case,  for  air  begins  at  once 
to  disengage  itself  froni  the  water,  and  to  travel  up  the  syphon  to  the  summit, 
where  it  enters  the  air-vessel,  and  rising  to  the  top  of  the  chamber  gradually 
drives  out  the  water.  If  this  is  allowed  to  continue  the  air  would  first  fill  the  en- 
tire chamber,  and  then  the  summit  of  the  syphon  itself,  where  it  would  act  as  a 
wad  completely  stopping  the  flow.  The  -water-level  in  the  ai  r  chamber 
can  be  detected  by  the  sound  made  by  tapping  against  the  outside  with  a  hammer. 


522 


HYDRAULICS. 


To  prevent  this  stoppag>e,  the  cut-off  at  the  foot  of  the  chamber  is 
closed  before  the  water  is  all  driven  out;  and  the  lid  on  top  being  removed  the 
chamber  is  refilled  with  water,  the  lid  replaced,  and  the  cut-oflf  again  opened. 
The  flow  in  the  meantime  continues  uninterjupted,  but  still  gradually  diminish- 
ing notwithstanding  the  refilling  of  the  chamber;  and  after  a  number  of  refill- 
ings  it  will  cease  altogether,  and  the  whole  operation  must  then  be  repeated  by 
filling  the  whole  syphon  and  air  chamber  with  water  as  at  the  start. 

At  Col.  Crozet's  syphon  at  first  owing  to  the  porosity  of  the  joint-caulking, 
which  was  nothing  but  oakum  and  pitch,  air  entered  the  pipes  so  rapidly  as  to 
drive  all  the  water  from  the  chamber  and  thus  require  it  to  be  refilled  every  5  or 
10  minutes;  but  still  in  two  hours  the  syphon  would  run  dry.  The  joints  were 
then  thoroughly  recaulked  with  lead,  and  protected  by  a  covering  of  white  and 
red  lead  made  into  a  putty  with  Japan  varnish  and  boiled  linseed  oil.  But  even 
then  the  chamber  had  to  be  refilled  with  water  about  every  two  hours ;  and  after 
six  hours  the  syphon  ran  dry,  and  the  whole  had  to  be  refilled.  In  this  way  it 
continued  to  work. 

Care  in  making  the  joints  air-tight,  and  an  outside  and  inside  coating  of  the 
pipes  and  air-vessel  with  coal  pitch  varnish  are  important  precautions. 

Art.  2.  Approximate  formulse  for  the  velocity  of  water  in 
straight,  smooth,  cylindrical  iron  pipes,  as  ro,  v  o,  lo,  Fig.  1.  Having  the  total 
headp  o,  and  the  length  and  diameter  of  the  pipe. 


Approx 
mean  vel 

in  ft  per  sec 


1        coefficient  I 

^  =       m       X  a/; 
J         as  below  >' 


diam  in  ft  X  total  head  in  ft 
total  length  in  ft  +  54  diams  in  ft 


Table  of  coefficients  "  m  ". 


Diam  of  pipe, 

m 

l>iam  of  pipe, 

m 

feet 

inches 

feet 

inches 

0.1 
0.2 
0.3 
0.4 
0.5 
0.6 
0.7 
0.8 
0.9 
1.0 

1.2 
2.4 
3.6 
4.8 
6.0 
7.2 
8.4 
9.6 
10.8 
12.0 

23 
30 
34 
37 
39 
42 
44 
46 
47 
48 

1.5 
2.0 
2.5 
3.0 
3.5 
4.0 
5.0 
6.0 
7.0 
10.0 

18 
24 
30 
36 
42 
48 
60 
72 
84 
120 

53 
57 
60 
62 
64 
66 
68 
70 
72 
77 

For  heads  not  less  than  4  feet  per  mile,  this  formula  gives  results  practically 
corresponding  with  those  by  Kutter's  formula  (p.  523)  with  coefficient  n  of 
roughness  =  0.012.  But  slight  differences,  as  to  roughness,  etc.,  may  cause  con- 
siderable variations  of  velocity,  especially  in  small  pipes ;  for,  in  such  pipes,  a 
given  roughness  of  surface  bears  a  greater  proportion  to  the  whole  area  of  surface 
than  in  a  pipe  of  large  diameter.  Extreme  accuracy  is  not  to  be  expected  in 
such  matters. 

As  in  a  river  the  velocity  half  way  across  it,  and  at  the  surface,  is  usually 
greater  than  at  the  bottom  and  sides,  so  in  a  pipe  the  velocity  is  greater  at  the 
center  of  its  cross  section  than  at  its  circumf.  The  niean  velocity 
referred  to  in  our  rules  is  an  assumed  uniform  one  which  would  give  the  same 
discharge  that  the  actual  ununiform  one  does. 

Hence 


Discharge    _^  Mean  velocity  w 

in  cub  ft  per  sec  '°"        in  ft  per  sec        "^ 


Area  of  cross  section 

of  pipe  in  sq  ft. 


1  cubic  foot      =  7.48052  U.  S.  gallons 

1  U.  S.  gallon  =    .13368  cubic  foot  =  231  cubic  inches. 


»  For  intermediate  diametera,  eto,  take  intermediate  coeffioieat«  from  the  table  by  simple  prv 
portion. 


HYDRAULICS.  523 

In  the  case  of  long  pipes  with  low  heads,  the  sum  of  the  velocity  and  entry 
heads  is  frequently  so  small  that  it  may  be  neglected.    Where 

this  is  the  case,  or  where  their  amount  can  be  approximately  ascertained,  Kut- 
ter's  formula,  although  designed  for  open  channels,  may  be  used.  This 
formula  is  the  joint  production  of  two  eminenl  Swiss  engineers,  E.  Ganguillet 
and  W.  R.  Kutter,  but  for  convenience  it  is  usually  called  by  the  name  of  the 
latter.* 

It  is,  properly  speaking,  a  formula  for  finding  the  coefficient  c  in  the  well 
known  formula, 

mean  velocity  =  c  "j/mean  radius  X  slope 


..V 


diameter 

X  slope 


According  to  Kutter, 

For  Sng^lisb  measnre.  For  metric  measure. 


C     = 


slope  n  slope        n 

77n       .00281  \  ~      '  7~~~m\b5  \ 


l/meau  rad  in  feet  x^mean  rad  in  metres 

See  also  tables  of  c,  pp  566  etc. 

Tlie  mean  radius  is  the  quotient,  in  feet  or  in  metres,  obtained  by  divid- 
ing the  area  of  wet  cross  section,  in  square  feet  or  in  square  metres,  by  the  wet 
perimeter  (see  below)  in  feet  or  in  metres.  In  pipes  running  full,  or  exactly  half 
full,  and  in  semicircular  open  channels  running  full,  it  is  equal  to  one-fourth  of 
the  inner  diameter. 

The  wet  perimeter  is  the  sum,  ab  co  Figs  28, 29,  30,  of  the  lengths, 

^b,bc,co,  in  feet  or  in  metres,  found  by  measuring  (at  right  angles  to  the  length 
!>f  the  channel)  such  parts  of  its  sides  and  bottom  as  are  in  contact,  with  the 
>ater.    In  pipes  running  full,  it  ia  of  course  equal  to  the  inner  circumference. 

The  slone  is  ^ JHcHonheBdwoF^ 

length  of  pipe  measured  in  a  straight  line  from  end  to  end. 

=  sine  of  angle  wso,  Fig  1. 

In  open  channels,  this  becomes 

,  __  fall  of  water  surface  in  any  portion  of  the  length  of  the  channel 

""  length  of  that  portion 

=  fall  of  water  surface  per  unit  of  length  of  channel 

=  sine  of  the  angle  formed  between  the  sloping  surface  and  the  horizon. 

The  number  indicating  the  slope  in  any  given  case  is  plainly  the  same  for 
English,  metric  and  all  other  measures. 

*'  n  "  is  a  "  coefficient  of  roughness  "  of  wet  perimeter,  and  of  course 
depends  chiefly  upon  the  character  of  the  inner  surface  of  the  pipe.  For  iron 
pipes  in  good  order  and  from  1  inch  to  4  feet  diameter,  n  may  be  taken  at  from 
.010  to  .012;  the  lower  figures  being  used  where  the  pipe  is  in  exceptionally  good 
condition. 

If  the  diameter,  or  the  mean  radius,  is  in  feet,  metres  etc,  the  velocity  will  be 
in  feet,  metres  etc,  per  second. 

.  *  See  "  Flow  of  Water,"  translated  from  Ganguillet  and  Kutter,  by  Rudolph  Hering 
and  John  C.  Trautwine,  Jr.,  New  York,  John  Wiley  &  Sons,  1889.    $4.00. 


524  HYDRAULICS. 

The  diameter  or  the  slope,  required  for  a  given  velocity, 

may  be  found  by  trial  as  follows:  assume  a  diameter,  or  a  slope,  as  the  case  may 
be  ;  take  the  corresponding  c  from  tables,  pp  566,  etc.    Then  say 

Approx  l>iain  required  ^  mean  ^  4  ^  /  velocity  \ 
for  the  given  vel  radius  ^  \  cT/sl6pe) 

Approx  Slope  required  _  /         velocity         y^  /    velocity    \2 
for  the  given  vel  Vci/raean  radius/       Ul/r^m5i/* 

With  the  approximate  diameter  (or  slope)  and  c,  thus  obtained,  say 
v' =  cv'raean  radius  X  slope.  If  •y'  is  near  enough  to  the  given  velocity,  the 
assumed  diameter  (or  slope)  is  the  proper  one.  If  not,  try  again,  assuming  a 
grecUer  di&meter  or  slope  than  before  if  v'  is  less  than  the  required  velocity,  and 
vice  versa. 


Curves  and  bends  do  not  greatly  affect  the  discharge,  so  long  as  the  total 
heads,  and  total  actual  lengths  of  the  pipes  remain  the  same  ;  provided  the  tops 
of  all  the  curves  be  kept  below  the  hydraulic  grade  line ;  and  provision  be  made 
for  the  escape  of  air  accumulating  at  the  tops  of  the  curves. 

Relation  between  area,  velocity,  and  discharge. 

Let  g  ^  rate  of  discharge  (as  in  cubic  feet  per  second), 
V  =  mean  velocity  (as  in  feet  per  second), 
a  =  area  of  cross  section  (as  in  square  feet). 

Then :      q  =  av;    v  =  -  ;    a  = -. 
a  V 

Relation  of  discharge  to  diameter  *_and  slope.  If  we  assume 
velocity  =  c  ^/mean  radius  X  slope,  or  v  =  c  }/  r  s  (page  523);  and  if  the  pipe 
be  of  circular  cross  section,  we  have,  for  the  rate,  Q,  of  discharge  through  a  pipe 
of  diameter,  d,  and  area.  A,  of  cross  section,  running  full: — 

Q  =  Av=  —  ,c-s^  =  g ; 

or :  Q  is  proportional  to  the  ^/g  power  (square  root  of  fifth  power)  of  the  diam- 
eter, and  to  the  ^  power  (square  root)  of  the  slope.  For  tables  of  fifth  powers, 
and  of  square  roots  of  fifth  powers,  see  pp  67-69. 

Effect  of  resistances. 
The  pressure  head  of  running  water,  upon  any  point  in  a  pipe  between 
the  orifice  and  the  reservoir,  is : 

r  +v,o  Vir.o/1  t^^  head  consumed       1 

(the  total    )  ^   PtnthP       *^^  in  overcoming  re- 

=  -l  head  on      V  minus  \   ^pI Vt  +  entry  +  sistances  in  the  pipe      V 

I  that  point)  Ju  V'     •   *       liead         between  the  reservoir 

l<t^a^POi°t  and  the  point.  J 

Thus,  at  the  point  6,  in  the  pipe,  1 0,  Fig  1,  the  pressure  head  is  A  =  (3  6)  =  (1  6) 
—  [(1  2)  -f  (2  3)]  ;  where  (1  2)  =  i  *  =  the  sum  of  the  velocity  and  entry  heads. 
At  4,  in  the  pipe  r  0,   h=  (3  4)  =  (14)  —  [(1  2)  +  (2  3)]. 

In  Fig  1,  let  the  straight  line,  s  0,  represent  the  actual  length  of  the  pipe, 
"Vhether  straight,   bent  or  curved,   etc.;   and  s  v  the  sum  of  the  resistances 
(supposed  to  be  uniformly  distributed)  within  the  pipe.     Then,  the  angle,  5  0  v, 
is  called  the  hydraulic  gradient,  and  sine  s  0  v  =^  s  v  -^  s  0. 
In  the  vertical  pipe,  v  0,  Fig  1  ^,  the  pressure,  at  q,  \s  =  g  d. 

*  Diameter  =  4  X  mean  radius,  or  d  =  4  r  (p  523). 


HYDRAULICS. 


525 


TABIiE    OF  l¥EIOHT  OF  1¥ATER  CONTAINED  Mf  ONE 
FOOT  1.ENOTH;  OF  PIPES  OF  DIFFERENT  BORES. 

(Original.) 

Water  at  maximum  density,  62.425  lbs.  per  cubic  foot  =  1  gram  per  cubic  centi- 
meter ;  corresponding  to  a  temperature  of  4°  Centigrade  =  39.2°  Fahrenheit. 
Weight  =  0.340475558  X  square  of  bore  in  inches. 


Bore. 

Water. 

Bore. 

Water. 

Bore. 

Water. 

Bore. 

Water. 

Ins. 

Lbs. 

Ins. 

Lbs. 

Ins. 

Lbs. 

Ins. 

Lbs. 

/4 

0.005320 

6 

12.25712 

23 

180.1116 

62 

1308.788 

0.021280 

&\i 

13.29983 

24 

196.1139 

63 

1351.347 

1 

0.047879 

6^ 

14.38509 

25 

212.7972 

64 

1394.588 

0.085119 

6% 

15.51292 

26 

230.1615 

65 

1438.509 

Ps 

0.132998 

7 

16.68330 

27 

248.2067 

66 

1483.112 

% 

0.191518 

7^4 

17.89625 

28 

266.9328 

67 

1528.395 

% 

0.2(50677 

?ll 

19.15175 

29 

286.3399 

68 

1574.359 

1 

0.340476 

20.44981 

30 

306.4280 

69 

1621.004 

ri 

0.430914 

8 

21.79044 

31 

327.1970 

70 

1668.330 

0.531993 

8K 

23.17362 

32 

348.6470 

71 

1716.337 

1-^ 

0.643712 

ti 

24.59936 

33 

370.7779 

72 

1765.025 

ik 

0.766070 

26.06766 

34 

393.5897 

73 

1814.394 

\% 

0.899068 

9 

27.57852 

35 

417.0826 

74 

1864.444 

iM 

1.042706 

9^4 

29.13194 

36 

441.2563 

75 

1915.175 

1% 

1.196984 

9j| 

30.72792 

37 

466.1110 

76 

1966.587 

2 

1.361902 

32.36646 

38 

491.6467 

77 

2018.680 

1 

1.537460 

10 

34.04756 

39 

517.8633 

78 

2071.453 

1.723658 

103^ 

37.53743 

40 

544.7609 

79 

2124.908 

1.920495 

11 

41.19754 

41 

572.3394 

80 

2179.044 

2.127972 

11^ 

45.02789 

42 

600.5989 

81 

2233.860 

2.346089 

12 

49.02848 

43 

629.5393 

82 

2289.358 

2.574846 

12J^ 

53.19931 

44 

659.1607 

83 

2345.536 

2j| 

2.814243 

13 

57.54037 

45 

689.4630 

84 

2402.396 

3 

3.064280 

133^ 

62.05167 

46 

720.4463 

85 

2459.936 

^/k 

3.324957 

14 

66.73321 

47 

752.1105 

86 

2518.157 

% 

3.596273 

143^ 

71.58499 

48 

784.4557 

87 

2577.060 

3.878229 

15^' 

76.60700 

49 

817.4818 

88 

2636.643 

4.170826 

15>^ 

81.79925 

50 

851.1889 

89 

2696.907 

^% 

4.474062 

16 

87.16174 

51 

885.5769 

90 

2757.852 

^% 

4.787938 

16^" 

92.69447 

52 

920.6459 

91 

2819.478 

3% 

5.112453 

17 

98.39744 

53 

95(5.3958 

92 

2881.785 

4 

5.447609 

17K 

104.27064 

54 

992.8267 

93 

2944.773 

^K 

6.149840 

18 

110.31408 

55 

1029.9386 

94 

3008.442 

4i| 

6.894630 

18M 

116.52776 

56 

1067.7314 

95 

3072.792 

7.681980 

19 

122.91168 

57 

1106.2051 

96 

3137.823 

5 

8.511889 

193^ 

129.46583 

58 

1145.3598 

97 

3203.535 

5^ 

9.384358 

20 

136.19022 

59 

1185.1954 

98 

3269.927 

^y% 

10.299386 

21 

150.14972 

60 

1225.7120 

99 

3337.001 

m 

11.256973 

22 

164.79017 

61 

1266.9096 

100 

3404.756 

Tlie  'we^glat  of  water  in  a  given  length  (as  one  foot)  of  any  pipe  or  other 
circular  cylinder  is  in  proportion   to  the  square  of  the  bore  or 

inner  diameter.  Hence  the  weight  of  water  in  1  foot  length  of  any  cylinder  of 
other  diameter  than  those  in  the  table  can  be  found  by  multiplying  that  for  a  1 
inch  pipe,  0.340475558,  by  the  square  of  the  inner  diameter  of  the  given  cylinder  in 
Inches.  Thus,  for  a  cylinder  120  inches  diameter :  diameters  =  120^  =  14400,  and 
weight  of  water  in  1  foot  depth  =  0.340475558  X  14400  =  4902.848  lbs.  Or,  weight 
for  120  ins.  diam.  =  100  X  weight  for  12  ins.  diam.  =  100  X  49.02848  =  4902.848  lbs. 
Similarly,  {^^)  2  _  _4^9_  _  0.191406,  and  0.340475558  X  0.191406  =  0.065169  lb.  = 
weight  in  1  foot  of  ^"g-  inch  pipe.  Here,  also,  -f-^  =  half  of  | ;  hence,  weight  for 
3?^  inch  =  one-fourth  of  weight  for  f  inch  =  one-fourth  of  0.260677  =  0.065169. 

ll'eight  of  one  square  inch  of  water  1  foot  hig^h,  at  62.425  fts. 
per  cubic  foot  =  62.425  -=-  144  =  0.433507  K). 


♦Actual.     See  nominal  and  actual  diameters,  foot  note,  p  526. 


526 


HYDRAULICS. 


TABIiE  3.    Areas  and  Contents  of  Pipes ;  and  square  roots 
of  ]>ianis.    (Original.)  Correct. 


* 

Area  in 

* 

Area  in 

* 

Area  in 

Diam. 

Diam. 

in 
Feet. 

sq  ft,  also 
cub  ft, 

Sq.  rt. 
of 

Diam. 

Diam. 

sq  ft,  also 
cub  ft. 

Sq.  rt. 
of 

Diam. 

Diam. 

sq  ft,  also 
cub  ft, 

Sq.  rt. 
of 

ll?8. 

in  1  foot 
length  of 

Diam. 
in  Ft. 

in 
Ins. 

in 

Feet. 

in  1  foot 
length  of 

Diam. 
in  Ft. 

in 
Ins. 

in 
Feet. 

in  1  foot 
length  of 

Diam. 
in  Ft. 

Pipe. 

Pipe. 

Pipe. 

M 

.0208 

.0003 

.145 

4. 

.3333 

.0873 

.579 

15. 

1.250 

1.227 

1.118 

5-16 

.0260 

.0005 

.161 

H 

.3438 

.0928 

.588 

H 

1.271 

1.268 

1.127 

Va 

.0313 

.0008 

.177 

k- 

.3542 

.0985 

.596 

^ 

1.292 

1.310 

1.136 

7-16 

.0365 

.0010 

.191 

.3646 

.1040 

.604 

H 

1.313 

1.353 

1.146 

H 

.0417 

.0014 

.204 

}4 

.3750 

.1104 

.612 

16. 

1.333 

1.396 

1.155 

9-16 

.0469 

.0017 

.217 

% 

.3854 

.1167 

.621 

K 

1.354 

1.440 

1.163 

% 

.0521 

.0021 

.228 

% 

.3958 

.1231 

.629 

34 

1.375 

1.485 

1.172 

11-16 

.0573 

.0026 

.2;)9 

% 

.4063 

.1296 

.637 

H 

1.396 

1.530 

1.181 

% 

.0625 

.0031 

.250 

5. 

.4167 

.1363 

.645 

17. 

1.417 

1.576 

1.190 

13-16 

.0677 

.0036 

.260 

M, 

.4271 

.1433 

.653 

1/ 

1.437 

1.623 

1.199 

% 

.0729 

.0042 

.270 

yi 

.4375 

.1503 

.660 

34 

1.458 

1.670 

1.207 

15-16 

.0781 

.0048 

.280 

% 

.4479 

.1576 

.669 

^ 

1.479 

1.718 

1.216 

1. 

.0833 

.0055 

.289 

34 

.4583 

.1650 

.677 

18. 

1.5 

1.767 

1.224 

1-16 

.0885 

.0062 

.297 

% 

.4688 

.1725 

.685 

H 

1.542 

1.867 

1.241 

% 

.0938 

.0069 

.305 

H 

.4792 

.1803 

.693 

19. 

1.583 

1.969 

1.258 

3-16 

.0990 

.0077 

.314 

% 

.4896 

.1878 

.700 

^ 

1.625 

2.074 

1.274 

M 

.1042 

.0085 

.322 

6. 

.5 

.1964 

.707 

20. 

1.667 

2.182 

1.291 

5-16 

.1094 

.0094 

.330 

H 

.5208 

.2131 

.722 

}i 

1.708 

2.292 

1.307 

% 

.1146 

.0103 

.338 

^ 

.5417 

.2304 

.736 

21. 

1.750 

2.405 

1.323 

7-16 

.1198 

.0113 

.346 

H 

.  .5625 

.2485 

.750 

34 

1.791 

2.521 

1.339 

14 

.1250 

.0123 

.354 

1. 

.5833 

.2673 

.764 

22. 

1.833 

2.640 

1.354 

9-16 

.1302 

.0133 

.361 

Va, 

.6042 

.2867 

.777 

34 

1.875 

2.761 

1.369 

% 

.1354 

.0144 

.368 

34 

.6250 

.3068 

.791 

23 

1.917 

2.885 

1.384 

11-16 

.1406 

.0155 

.375 

% 

.6458 

.3276 

.803 

^ 

1.958 

3.012 

1.399 

% 

.1458 

.0167 

.382 

8. 

.6667 

.3491 

.817 

24. 

2.000 

3.142 

1.414 

13-16 

.1510 

.0179 

.389 

M 

.6875 

.3712 

.829 

25. 

2.083 

3.409 

1.443 

K 

.1563 

.0192 

.395 

>i 

.7083 

.3941 

.841 

26. 

2.166 

3.687 

1.472 

15-16 

.1615 

.0205 

.402 

M 

.7292 

.4176 

.854 

27. 

2.250 

3.976 

1.500 

2. 

.1667 

.0218 

.408 

9. 

.75 

.4418 

.866 

28. 

.2  333 

4.276 

1.528 

1-16 

.1719 

.0232 

.414 

34 

.7708 

.4667 

.879 

29. 

2  416 

4.587 

1.555 

H 

.1771 

.0246 

.420 

3^ 

.7917 

.4922 

.890 

.30. 

2.500 

4.909 

1.581 

3-16 

.1823 

.0260 

.427 

K 

.8125 

.5185 

.902 

31. 

2.584 

5.241 

1.607 

M 

.1875 

.0276 

.433 

10. 

.8333 

.5454 

.913 

32. 

2.666 

5.585 

1.633 

5-16 

.1927 

.0291 

.440 

Va, 

.8542 

.5730 

.924 

33. 

2.750 

5.940 

1.658 

% 

.1979 

.0308 

.445 

.8750 

.6013 

.935 

34. 

2.834 

6.305 

1.683 

7-16 

.2031 

.0324 

.451 

H 

.8958 

.6303 

.946 

35. 

2.916 

6.681 

1.708 

H 

.2083 

.0341 

.457 

11. 

.9167 

.6600 

.957 

36. 

3.000 

7.069 

1.732 

9-16 

.2135 

.0358 

.462 

H 

.9375 

.6903 

.968 

38. 

3.166 

7.876 

1.779 

% 

.2188 

.0375 

.467 

34 

.9583 

.7213 

.979 

40. 

3.333 

8.727 

1.825 

11-16 

.2240 

.0394 

.473 

H 

.9792 

.7530 

.990 

42. 

3.500 

9.621 

1.871 

H 

.2292 

.0412 

.478 

12. 

1. 

.7854 

1.000 

44, 

3.666 

10.56 

1.914 

13-16 

.2344 

.0432 

.484 

H 

1.021 

.8184 

1.010 

48. 

4.000 

12.57 

2.000 

% 

.2396 

.0451 

.489 

H 

1.042 

.8522 

1.020 

54. 

4.500 

15.90 

2.121 

15-16 

.2448 

.0471 

.495 

H 

1.063 

.8866 

1.031 

60. 

5.000 

19.63 

2.236 

3. 

.2500 

.0491 

.500 

13. 

1.083 

.9218 

1.041 

66. 

5.500 

23.76 

2.345 

H 

.2604 

.0532 

.510 

H 

1.104 

.9576 

1.051 

72. 

6.000 

28.27 

2.449 

.2708 

.0576 

.    .520 

H 

1.125 

.9940 

1.060 

78. 

6.500 

33.18 

2.550 

% 

.2813 

.0621 

.530 

H 

1.146 

1.031 

1.070 

84. 

7.000 

38.48 

2.646 

^ 

.2917 

.0668 

.540 

14. 

1.167 

1.069 

1.080 

90. 

7.500 

44.18 

2.739 

% 

.3021 

.0716 

.550 

H 

1.187 

1.108 

1.090 

96. 

8.000 

50.27 

2.828 

H 

.3125 

.0767 

.560 

H 

1.208 

1.147 

1.099 

% 

.3229 

.0819 

.570 

H 

1.229 

1.18f7 

1.110 

*  Caution.  In  the  tables  on  pages  525  and  526,  the  diameters  or  bores  are 
the  actual  ones,  as  measured  in  inches.  Wrought-iron  steam,  gas,  and  water 
pipes  are  commonly  designated  by  fictitious  or  "  nominal"  diameters,  which  are 
mere  arbitrary  names  for  the  pipes.  In  the  smaller  sizes  especially,  the  use  of 
these  nominal  diameters  tends  to  mislead.  Thus,  the  pipe  whose  "  womma^ " 
inner  diameter  is  one-eighth  inch,  has  an  actual  inner  diameter  of  full  quarter  inch. 


HYDRAULICS.  527 

Art.  3.    To  findwtlie  total  bead  required  for  a  g^iven  velocity,  ot 

fiveii  discliarg-e,  through  a  straight,  smooth,  cylindrical  iroa  pipe  of 
nown  diam  and  length. 

If  the  discharge  is  given,  first  find    ' 


mean  velocity discharge  in  cubic  feet  per  second 

in  feet  per  second  ""  area  of  cross  section  of  pipe^iu  square  feet 


Then 


Vdiam  X  head      _  mean  velocity  in  feet  per  second 
length  +  54  diams  ~     the  proper  divisor  as  foUowa 


diam  of  pipe  in  ft    .05    .10    .50    1      1.5    2      3      4 
divisor  40     43     46    48     51    64    58    61 

(for  intermediate  diams,  take  intermediate  divisors  by  guess.) 

From  table  Art  2,  take  the   coefficient  m  corresponding  to  this  value  of 


i 


diam  X  head      ^  ^^^  ^^  ^^^  ^.^^^  ^.^^^    ^^^^^ 


length  +  54  diams' 


Total  _  Igfeet^'perler  X  (^"°g^^  ^°  ^^  +  ^^  ^'^"^^  ^°  ^^ 
in^ftft"  m^Xdiaminfeet 


To  find  the  Friction  bead.    Weisbach^s  formula. 

.01716      \      I^ength  Vel^in 


(01716  \       i^eDgin  vei-m 

.0144  + Iw  in  feet  ^  ft  per  sec 

^velinft  l-^    Diam  64.4 

per  sec  I      ,„  foot 


Friction  head  _ 
in  feet        " 

per  sec     f      in  feet 


For  tbe  total  bead,  we  have  only  to  add  together,  the  friction  bead 

so  found,  the  velocity  bead,  taken  from  the  next  table,  or  from  Table  10, 
opposite  the  given  velocity,  and  tbe  entry  bead  (=  say  half  the  velocity  head). 
The  sum  of  the  velocity  head  and  entry  head  rarely  amounts  to  a  foot. 

TABliE  4  Of  tbe  vel,  and  discbarge  of  water  through  straight,  smooth, 
cylindrical  cast-iron  pipes;  with  tbe  friction  bead  required  for  each  100  feet  in 
length;  and  also  tbe  velocity  bead.  Calculated  by  means  of  Weisbach's  formula,  by 
James  Thompson,  A  M ;  and  George  Fuller,  C  E,  Belfast,  Ireland.  Tbe  vel  bead 
remains  tbe  same  for  any  length  of  pipe ;  being  dependent  only  on  tbe  velocity  of  th« 
water  in  tbe  pipe. 

Tbe  entry  bead  is  equal  to  about  half  the  vel  bead. 


628 


HYDRAULICS. 


TABIiE  4 


Vel. 
head  in 
Feet. 

Diauk  in  Inches 

Yel.  in 
Feet 

3 

3H 

4 

4H 

5 

per  Sec. 

Prhead 
Ft  per 
100  ft. 

Cub  ft 
per  Min 

Frhead 
Ft  per 
100  ft. 

Cub  ft 
per  Min 

Frhead 
Ft  per 
100  ft. 

Cub  ft 
per  Min 

Prhead 
Ft  per 
100  ft. 

Cub  ft 
per  Min 

Prhead 
Ft  per 
100  ft. 

Cub  ft 
per  Min 

2.0 

.062 

.659 

5.89 

.565 

8.02 

.494 

10.4 

.439 

13.2 

.395 

16.3 

2.2 

.075 

.780 

6.48 

.669 

8.82 

.585. 

11.5 

.520 

14.6 

.468 

18.0 

2.4 

.090 

.911 

7.07 

.781 

9.62 

.683 

12.5 

.607 

15.9 

.547 

19.6 

2.6 

.105 

1.05 

7.65 

.901 

10.4 

.788 

13.6 

.701 

17.2 

.631 

21.3 

2.8 

.122 

1.20 

8.24 

1.03 

11.2 

.900 

14.6 

.800 

18.5 

.720 

22.9 

3.0 

.140 

1.35 

8.83 

1.16 

12.0 

1.02 

15.7 

.905 

19.8 

.815 

24.5 

3.2 

.160 

1.52 

9.42 

1.31 

12.8 

1.14 

16.7 

1.02 

21.2 

.915 

26.2 

3.4 

.180 

1.70 

10.0 

1.46 

13.6 

1.27 

17.8 

1.13 

22.5 

1.02 

27.8 

3.6 

.202 

1.89 

10.6 

1.62 

14.4 

1.41 

18.8 

1.26 

23.8 

1.13 

29.4 

3.8 

.225 

2.08 

11.2 

1.78 

15.2 

1.56 

19.9 

1.39 

25.2 

1.25 

31.0 

4.0 

.250 

2.28 

11.8 

1.96 

16.0 

1.71 

20.9 

1.52 

26.5 

1.37 

32.7 

4.2 

.275 

2.49 

12.3 

2.14 

16.8 

1.87 

22.0 

1.66 

27.8 

1.50 

34.3 

4.4 

.302 

2.71 

12.9 

2.33 

17.6 

2.03 

23.0 

1.81 

,29.1 

1.63 

36.0 

4.6 

.330 

2.94 

13.5 

2.52 

18.4 

2.21 

24.0 

1.96 

304 

1.76 

37.6 

4.8 

.360 

3.18 

14.1 

2.72 

19.2 

2.38 

25.1 

2.12 

31.8 

1.91 

39.2 

5.0 

.390 

3.43 

14.7 

2.94 

20.0 

?.57 

26.2 

2.28 

33.1 

2.05 

40.9 

5.2 

.422 

3.68 

15.3 

3.15 

20.8 

2.76 

27.2 

2.45 

34.4 

2.21 

42.5 

5.4 

.455 

3.94 

15.9 

3.38 

21.6 

2.96 

28.2 

2.63 

35.8 

2.37 

44.2 

5.6 

.490 

4.22 

16.5 

3.61 

22.4 

3.16 

29.3 

2.81 

37.1 

2.53 

45.8 

5.8 

.525 

4.50 

17.1 

3.85 

23.2 

3.37 

30.3 

3.00 

38.4 

2.70 

47.4 

6.0 

.562 

4.78 

17.7 

4.10 

24.0 

3.59 

31.4 

3.19 

39.7 

2.87 

49.1 

6.2 

.600 

6.08 

18.2 

4.36 

24.8 

3.81 

32.4 

3.39 

41.0 

3.05 

50.7 

6.4 

.640 

5.39 

18.8 

4.62 

25.6 

4.04 

33.5 

3.59 

42.4 

3.23 

52.3 

6.6 

.680 

5.70 

19.4 

4.89 

26.4 

4.28 

34.5 

3.80 

43.7 

3.42 

54.0 

6.8 

.722 

6.02 

20.0 

5.16 

27.3 

4.52 

35.6 

4.01 

45.0 

3.61 

55.6 

7.0 

.765 

6.35 

20.6 

5.45 

28.0 

4.77 

36.6 

4.24 

46.4 

3.81 

57.2 

Vel- 

head  in 

Feet. 

Diam.  in  Inches. 

Vel.  in 
Feet 

5 

7 

8 

9 

10 

per  Sec. 

Prhead 
Ft  per 
100  ft. 

Cub  ft 
per  Min 

Prhead 
Ft  per 
100  ft. 

Cub  ft 
per  Min 

Frhead 
Ft  per 
100  ft. 

Cub  ft 
per  Min 

Frhead 
Ft  per 
100  ft. 

Cub  ft 
per  Min 

Frhead 
Ft  per 
100  ft. 

Cub  ft 
pwMin 

2.0 

.062 

.329 

23.5 

.282 

32.0 

.247 

41.9 

.220 

53.0 

.198 

65.4 

2.2 

.075 

.390 

25.9 

.334 

35.3 

.293 

46.1 

.260 

58.3 

.234 

72.0 

2.4 

.090 

.456 

28.2 

.390 

38.5 

.342 

50.2 

.304 

63.6 

.273 

78.5 

2.6 

.105 

.526 

30.6 

.450 

41.7 

.394 

54.4 

.350 

68.9 

.315 

85.1 

2.8 

.122 

.600 

32.9 

.514 

44.9 

.450 

58.6 

.400 

74.2 

.360 

91.6 

3.0 

.140 

.679 

35.3 

.582 

48.1 

.509 

62.8 

.453 

79.5 

.407 

98.2 

3.2 

.160 

.763 

37.7 

.654 

51.3 

.572 

67.0 

.508 

84.8 

.458 

105 

3.4 

.180 

.851 

40  0 

.729 

54.5 

.638 

71.2 

.567 

90.1 

.510 

111 

3.6 

.202 

.943 

42.4 

.808 

57.7 

.707 

75.4 

.629 

95.4 

.566 

118 

3.8 

.225 

1.04 

44J 

.892 

60.9 

.780 

79.6 

.693 

101 

.624 

124 

4.0 

.250 

1.14 

47.1 

.979 

64.1 

.856 

83.7 

.761 

106 

.685 

131 

4.2 

.275 

1.25 

49.5 

1.07 

67.3 

.935 

87.9 

.832 

111 

.748 

137 

4.4 

.302 

1.35 

51.8 

1.16 

70.5 

1.02 

92.1 

.905 

116 

.814 

144 

4.6 

.330 

1.47 

54.1 

1.26 

73.7 

1.10 

96.3 

.981 

122 

.883 

150 

4.8 

.360 

1.59 

56.5 

1.36 

76.9 

1.19 

100 

1.06 

127 

.954 

157 

5.0 

.390 

1.71 

58.9 

1.47 

80.2 

1.28 

105 

1.14 

132 

1.03 

168 

5.2 

.422 

1.84 

61.2 

1.58 

83.3 

1.38 

109 

1.23 

138 

1.10 

170 

5.4 

.455 

1.97 

63.6 

1.69 

86.6 

1.48 

113 

1.31 

143 

1.18 

177 

5.6 

.490 

2.11 

65.9 

1.81 

89.8 

1.58 

117 

1.40 

148 

1.26 

183 

5.8 

.525 

2.25 

68.3 

1.93 

93.0 

1.68 

121 

1.50 

154 

1.35 

190 

6.0 

.562 

2.39 

70.7 

2.05 

96.2 

1.79 

125 

1.59 

159 

1.43 

196 

6.2 

.600 

2.54 

73.0 

2.18 

99.4 

1.90 

130 

1.69 

164 

1.52 

203 

6.4 

.640 

2.69 

75.4 

2.31 

102 

2.02 

134 

1.79 

169 

1.61 

209 

6.6 

.680 

2.85 

77.7 

2.44 

106 

2.14 

138 

1.90 

175 

1.71 

216 

6.8 

.722 

3.01 

80.1 

2.58 

109 

2.26 

142 

2.01 

180 

1.81 

222 

7.0 

.765 

3.18 

82.4 

2.72 

112 

2.38 

146 

2.12 

185 

1.90 

229 

HYDRAULICS. 


529 


TABI.E  4     - 

-  (Continued., 

Vel- 
head  in 
Feet. 

Diam.  in  Inches. 

Vel.  in 
Feet 

11 

12           1 

13 

14 

15 

per  Sec. 

Frhead 
Ft  per 
100  ft. 

Cub  ft 
per  iMin 

Frhead 
Ft  per 
100  ft. 

Cub  ft 
per  Min 

Frhead 
Ft  per 
100  ft. 

Cub  ft 
per  Min 

Frhead 
Ft  per 
100  ft. 

Cub  ft 
per  Min 

Frhead 
Ft  per 
100  ft. 

Cub  ft 
per  Min 

2.0 

.062 

.180 

79.2 

.165 

94.2 

.152 

110 

.141 

128 

.132 

147 

2.2 

.075 

.213 

87.1 

.195 

1(13 

.180 

121 

.167 

141 

.156 

162 

2.4 

.090 

.248 

95.0 

.228 

113 

.210 

133 

.195 

154 

.182 

176 

2.6 

.105 

.287 

103 

.263 

122 

.242 

144 

.225 

167 

.210 

191 

2.8 

.122 

.327 

111 

.300 

132 

277 

156 

.257 

179 

.240 

206 

3.0 

.140 

.370 

119 

.339 

141 

.313 

166 

.291 

192 

.271 

221 

3.2 

.160 

.416 

127 

.381 

151 

.352 

177 

.327 

205 

.305 

235 

3.4 

.180 

.464 

134 

.425 

160 

.393 

188 

.365 

218 

.340 

250 

3.6 

.202 

.514 

142 

.472 

169 

.435 

199 

.404 

231 

.377 

265 

3.8 

.225 

.567 

150 

.520 

179 

.480 

210 

.446 

243 

.416 

280 

4.0 

.250 

.623 

158 

.571 

188 

.527 

221 

.489 

256 

.457 

294 

4.2 

.275 

.680 

166 

.624 

198 

.576 

232 

.534 

269 

.499 

309 

4.4 

.302 

.740 

174 

.679 

207 

.626 

243 

.582 

282 

.543 

324 

4.6 

.330 

.803 

182 

.736 

217 

.679 

254 

.631 

295 

.589 

339 

4.8 

.300 

.867 

190 

.795 

226 

.734 

265 

.682 

308 

.636 

353 

5.0 

.390 

.935 

198 

.857 

235 

.791 

276 

.734 

321 

.685 

368 

5.2 

.422 

1.00 

206 

.920 

245 

.850 

287 

.789 

333 

.736 

383 

6.4 

.455 

1.07 

214 

.986 

254 

.910 

298 

.845 

346 

.789 

397 

5.6 

.490 

1.15 

222 

1.05 

264 

.973 

309 

.903 

359 

.843 

412 

6.8 

.525 

1.22 

229 

1.12 

273 

1.04 

321 

.964 

372 

.899 

427 

6.0 

.562 

1.30 

237 

1.19 

283 

1.10 

332 

1.02 

385 

.967 

442 

6.2 

.600 

1.38 

245 

1.27 

292 

1.17 

343 

1.09 

397 

1.01 

456 

6.4 

.640 

1.47 

253 

1.35 

301 

1.24 

354 

1.15 

410 

1.08 

471 

6.6 

.680 

1.55 

261 

1.42 

311 

1.31 

365 

1.22 

423 

1.14 

486 

6.8 

.722 

1.64 

269 

1.50 

320 

1.39 

376 

1.29 

436 

1.20 

500 

7.0 

.765 

1.73 

277 

1.59 

330 

1.46 

387 

1.36 

449 

1.27 

615 

Vel- 
head  in 
Feet. 

Diam.  in  Inches 

, 

Vel.  in 
Feet 

16 

17 

18 

19 

20 

perSec. 

Frhead 
Ft  per 
100  ft. 

Cub  ft 
per  Min 

foo^ft'iperMin 

Fr  head 
Ft  per 
100  ft. 

Cub  ft 
per  Min 

Frhead 

Ft  per 
100  ft. 

Cub  ft 
per  Min 

Frhead 
Ft  per 
100  ft. 

Cub  ft 
per  Min 

2.0 

.062 

.123 

167- 

.116 

189 

.110 

212 

.104 

236 

.099 

262 

2.2 

.075 

.146 

184 

.138 

208 

.130 

233 

.123 

260 

.117 

288 

2.4 

.090 

.171 

201 

.161 

227 

.152 

254 

.144 

283 

.137 

314 

2.6 

.105 

.197 

218 

.185 

246 

.175 

275 

.166 

307 

.158 

340 

2.8 

.122 

.225 

234 

.212 

265 

.200 

297 

.189 

331 

.180 

366 

3.0 

.140 

.255 

251 

.240 

2S4 

.226 

318 

.214 

354 

.204 

393 

3.2 

.160 

.286 

268 

.269 

302 

.254 

339 

.241 

378 

.229 

419 

3.4 

.180 

.319 

284 

.300 

321 

•  .283 

360 

.269 

401 

.255 

445 

3.6 

.202 

.354 

301 

.333 

340 

.314 

382 

.298 

425 

.283 

471 

3.8 

.225 

.390 

318 

.367 

.359 

.347 

403 

.328 

449 

.312 

497 

4.0 

.250 

.428 

335 

.403 

378 

.380 

424 

.360 

472 

.342 

623 

4.2 

.275 

.468 

352 

.440 

397 

.416 

445 

.394 

496 

.374 

660 

4.4 

.302 

.509 

368 

.479 

416 

.452 

466 

.429 

619 

.407 

676 

4.6 

.330 

.552 

385 

.519 

435 

.490 

48S 

.465 

643 

.441 

602 

4.8 

.360 

.596 

402 

.561 

454 

.^0 

509 

.502 

667 

.477 

628 

5.0 

.390 

.642 

419 

.605 

473 

.571 

530 

.541 

690 

.514 

664 

5.2 

.422 

.690 

435 

.650 

•492 

.614 

551 

.581 

614 

.562 

680 

5.4 

.455 

.740 

452 

.696 

511 

.657 

672 

.623 

638 

.592 

707 

6.6 

.490 

.791 

469 

.744 

529 

.703 

594 

.666 

661 

.632 

733 

6.8 

.525 

.843 

486 

.793 

548 

.749 

615 

.710 

685 

.674 

769 

6.0 

.562 

.897 

502 

.844 

567 

.798 

636 

.755 

709 

.718 

786 

6.2 

.600 

.953 

519 

.897 

586 

.847 

657 

.802 

732 

.762 

811 

6.4 

.640 

1.01 

636 

.951 

605 

.898 

678 

.851 

756 

.808 

838 

6.6 

.680 

1.07 

653 

1.01 

624 

.950 

700 

.900 

780 

.855 

864 

6.8 

.722 

1.13 

569 

1.06 

643 

1.00 

721 

.951 

803 

.904 

896 

7.0 

.765 

1.19 

586 

1.12 

662 

1.06 

74.'' 

1.00 

827 

.953  1  Plfi 

34 


530 


HYDRAULICS. 


TABLE  4 

—  (Continued.) 

Vel- 
head  in 
Feet. 

Diam^iu  laches. 

Vel.  in 

Feet 

22 

24 

26 

28 

30 

perSec. 

Prhead 
Ft  per 
100  ft. 

Cub  ft 
per  Min 

Prhead 
Ft  per 
100  ft. 

Cub  ft 
per  Min 

Frhead 
Ft  per 
100  ft. 

Cub  ft 
per  Min 

Frhead 
Ft  per 
100  ft. 

Cub  ft 
per  Min 

Frhead 
Ft  per 
100  ft. 

Cub  ft 
per  Mia 

2.0 

.062 

.090 

316 

.082 

377 

.076 

442 

.070 

513 

.066 

589 

2.2 

.075 

.106 

348 

.097 

414 

.090 

486 

.083 

564 

.078 

648 

2.4 

.090 

.124 

380 

.114 

452 

.105 

531 

.097 

616 

.091 

707 

2.6 

.105 

.143 

412 

.131 

490 

.121 

575 

.112 

667 

.105 

766 

2.8 

.122 

.164 

443 

.150 

528 

.138 

619 

.128 

718 

.120 

824 

3.0 

.140 

.185 

475 

.170 

565 

.157 

663 

.145 

770 

.136 

883 

3.2 

.160 

.208 

507 

.191 

603 

.176 

708 

.163 

821 

.152 

942 

3.4 

.180 

.232 

538 

.213 

641 

.196 

752 

.182 

872 

.170 

1001 

3.6 

.202 

.257 

570 

.236 

678 

.218 

796 

.202 

923 

.189 

1060 

3.8 

.225 

.284 

601 

.260 

716 

.240 

840 

.223 

974 

.208 

1119 

4.0 

.250 

.311 

633 

.285 

754 

.263 

885 

.244 

1026 

.228 

1178 

4.2 

.275 

.340 

665 

.312 

791 

.288 

929 

.267 

1077 

.249 

1237 

4.4 

.302 

.370 

697 

.339 

829 

.313 

973 

.290 

1129 

.271 

1296 

4.6 

.330 

.401 

728 

.368 

867 

.339 

1017 

.315 

1180 

.294 

1355 

4.8 

.360 

.434 

760 

.397 

905 

.367 

1062 

.341 

1231 

.318 

1414 

6.0 

.390 

.467 

792 

.428 

942 

.395 

1106 

.367 

1283 

.343 

1472 

5.2 

.422 

.502 

823 

.460 

980 

.425 

1150 

.394 

1334 

.368 

1531 

5.4 

.455 

.538 

855 

.493 

1018 

.455 

1194 

.423 

1385 

.394 

1590 

5.6 

.490 

.575 

887 

.527 

1055 

.486 

1239 

.452 

1437 

.422 

1649 

5.8 

.525 

.613 

918 

.562 

1093 

.519 

1283 

.482 

1488 

.450 

1708 

6.0 

.562 

.652 

950 

.598 

1131 

.552 

1327 

.513 

1539 

.478 

1767 

6.2 

.600 

.693 

982 

.635 

1168 

.586 

1371 

.544 

1590 

.508 

1826 

6.4 

.640 

.735 

1013 

.673 

1206 

;622 

1416 

.577 

164-2 

.539 

1885 

6.6 

.680 

.778 

1045 

.713 

1241 

.658 

1460 

.611 

1693 

.570 

1943 

6.8 

722 

.821 

1077 

.753 

1282 

.695 

1504 

.645 

1744 

.602 

2003 

7.0 

'.765 

.867 

1109 

.794 

1319 

.733 

1548 

.681 

1796 

.635 

2061 

HYDRAULICS.  531 

Art*  4         To  find  tbe  discbarg^e  throiig'h  a  compound  pipe 

r  o,  Fig  1  H,  composed  of  any  number  of  pipes,  a,  6,  c,  z,  of  different  diaiu* 
eterS)  which  decrease  from  the  reservoir  toward  the  outflow  o. 


' ^     '^  u  ^^       1 


Fio.iH 

First  find  what  part  of  the  total  head  H  is  employed  in  forcing  the  water  through 
the  last  pipe  z  alone^  thus : 

Let  La,  lib,  L  c,  and  Lz,  be  the  lengths  in  ft  of  the  pipes  a,  6,  c,  and  z,  respectively; 
D  a,  D  6,  D  c,  and  D  z  their  diameters  in  ft ;  and  A  a,  A  6,  A  c,  and  A  z  the  areas  of 
their  cross  sections  in  sq  ft.    Then 

The  head  in 
in  forSng^he  (     _  The  total  head  H  in  feet 


waterthrough  f  -         r   A  z^  X  ^  Z     ^    /La-h(54XDa)    ,    L  6-f  (54XD  fc)    .    L  o-f  (54XD  c)\  H 
thejast  pipe  ^       1+  ll  ^^^r,,^j,  ^^  X    (,  a  a3  X  D  a   +  ^aTT^TdT  +      Ac^XdTJA 

However  many  divisions  the  pipe  may  have,  proceed  in  tlie  same  way  as  above,  using 

Area*  X  Diam        -      ^,      .    ^                        i.  j.  •  •              j   Length -f  54 Diams 
for  the  last  or  narrowest  division;   and 


Length  +  64  Diams  *  Area^  x  Diam 

for  each  of  the  others. 

Then,  by  the  formulae.  Art.  2,  find  the  velocity  in  ft  per  second,  and  discharge  in 
cub  ft  per  second,  of  the  last  pipe  z,  using  its  actual  diameter,  length,  and  croee 
sectional  area,  and  the  head  just  found.  Said  discharge  is  evidently  the  discharge 
for  the  compound  pipe. 

For  the  velocity  in  any  portion^  as  6,  say 

area  of  cross  section     ,    area  of  cross    ,  ,    velocity    ,    velocity  in  the 
of  the  given  portion    •     section  of  a      •  •       iu  z         •    given  portion. 

For  the  above  rule  and  formula,  we  are  indebted  to  Mr,  Howard  Murphy,  C  B, 
of  Phila;  and  for  the  opportunity  of  testing  it  experimentally,  to  Messrs  Morris, 
Tasker  &  Co,  Limited,  Pascal  Iron  Works,  Philadelphia. 


532 


WATER-PIPES. 


Art.  4  a.*  The  Ventari  Meter  is  designed  for  the  measurement  of  the 
flow  of  liquids  in  pipes  of  large  dimensions,  running  full. 

Tlie  meter  proper,  patented  by  Clemens  Herschel,  consists  essentially 
of  a  mere  constriction  in  the  area  of  cross-section  of  the  pipe,  with  openings 
in  the  pipe  opposite  its  normal  and  its  constricted  diameters,  for  measuring,  by 
piezometers  or  pressure-gauges,  the  pressures  at  those  points;  while  the 
register,  patented  by  Messrs.  Frederick  N.  Connet  and  Walter  W.  Jackson, 
is  an  elaborate  mechanism,  provided  with  clock-work  and  dials. 

Theory .t    Let  Figs.  1   to    3 
represent  a  Venturi  meter  tube,  Fig.  1. 

with  three  piezometers  in  place, 
viz.:  No.  1,  over  the  tube  up-stream 
from  the  constriction  ;  No.  2,  over 
the  constriction  itself;  and  No.  3, 
over  the  tube  down-stream  from 
the  constriction.  Let  the  unshaded 
area  W  in  Figs.  1  to  3,  represent 
the  depths  at  which  the  water 
stands  above  any  assumed  hori- 
zontal datum  plane  0-0;  and  let 
the  shaded  area  A  represent  the 

uniform  pressure    of  the  atmos-       ^  " 

phere,  which,  for  convenience,  we 

may  suppose  to  be  converted  into  some  liquid  of  the  specific  gravity  of  water, 
but  distinguishable,  by  its  appearance,  from  the  water. 

The  vertical  distance,  between  the  upper  boundary  of  this  latter  area  and 
any  given  point  in  the  tube,  represents  the  combined  pressure  of  air  and  water 
at  such  point. 

The  velocities  in  the  meter  tube,  at  any  instant,  are  of  necessity  inversely 
proportional  to  the  areas  of  cross  section;  and,  as  the  heads  corresponding  to 
the  several  velocities  are  proportional  to  the  squares  of  those  velocities,  the 
remaining  or  pressure  heads  must  vary  also,  the  smallest  or  lowest  pressure 
head  standing  over  the  throat,  where  the  velocity  is  greatest. 

The    increase    of  velocity,   ac- 
FiG.  2.  quired    by  the   fluid  in    passing 

from  section  1  to  section  2,  is  again 
given  up  in  passing  from  section 
2  to  section  3;  and,  in  the  case  of 
a  perfect  fluid,  the  pressure  lost 
between  sections  1  and  2  would  be 
perfectly  restored  in  passing  from 
section  2  to  section  3.  In  practice, 
a  small  total  loss  occurs.  This  loss 
is  greater  with  high  than  with  low 
velocities. 

For  a  given  head  in  piezometer 
No.  1  and  given  diameter  of  pipe 
at  section  1,  the  expenditure  of 
head  in  velocity  between  sections 
1  and  2  increases  as  the  area  of  the  throat  is  diminished  and  as  the  throat 
Telocity  is  thereby  increased.^  In  Fig.  2  is  shown  the  case  where  all  of  the 
water  head  above  the  top  of  the  throat  is  required  to  maintain  the  velocity 
through  the  throat. 

In  Figs,  1  and  2  the  head,  If,  expended  in  the  increase  of  velocity  between 
sections  1  and  2  is  represented  by  the  difference  in  level  between  the  tops  of  the 
two  water  columns  1  and  2,  or  between  the  tops,  of  the  two  corresponding  air 
columns.  In  Fig.  2  this  difference  is  equal  to  the  io^a^  vertical  height  of  the 
water  column  at  section  1  above  the  top  of  the  throat  at  section  2. 

*  Abridged  from  a  description  prepared  by  the  writer  as  Chairman  of  a  Com- 
mittee of  the  Franklin  Institute.  Journal  of  the  Franklin  Institute,  February, 
1899. 

t  The  Venturi  meter,  apart  from  its  merits  as  a  measuring  device,  embodies 
important  hydraulic  principles.  Hence  its  theory  is  here  stated  niore  fully  than 
would  otherwise  be  necessary. 

X  In  a  given  Venturi  tube  the  pressure  and  velocity  at  the  throat  may  be 
varied  also  by  modifying  those  at  sectons  1  and  3,  as  by  regulating  the  openings 
of  the  valves  of  influx  to  and  of  efflux  from  the  meter  tube,  by  changing  the 
total  head  on  the  system,  etc. 


WATER-PIPES. 


533 


Fig.  3. 


If  now  (Fig.  3)  the  throat  sec- 
tion be  still  further  reduced 
(the  other  conditions  remain- 
ing as  before),  the  throat  veloc- 
ity will  thereby  be  still  further 
increased  ;  for  the  total  pressure 
available  for  increase  of  veloc- 
ity between  sections  1  and  2  con- 
sists not  merely  in  the^depth  of 
water  above  the  tube,  but  also 
in  the  atmospheric  pressure,  rep- 
resented by  the  shaded  area  A 
above  the  water  W. 

In  Fig.  3  all  the  water  has  dis- 
appeared from  piezometer  2,  and  q 

even    a  portion  of   the  liquid 

representing  the    air  has   also 

disappeared,  leaving  only  a  portion  of  the  latter  to  represent  Such  pressure  as 

now  remains  in  the  throat.    In  other  words,  the  pressure  within  the  throat  is 

now  less  than  the  atmospheric  pressure. 

In  Fig.  3,  the  loss  of  head,  due  to  increase  of  velocity  between  sections  1  and  2, 
is  H=  hw  ■\-ha  =  the  entire  available  head  of  water,  hw,  plus  a  portion,  ha,  of  the 
atmospheric  pressure.  The  latter  portion,  ha,  is  frequently  called  "the 
vacuum." 

The  top  of  the  water  column  having  now  disappeared  below  the  top  of  the 
throat,  it  is  no  longer  feasible  to  ascertain  the  loss  of  head  by  taking  the  differ- 
ence between  the  levels  of  the  water  surfaces  in  piezometers  1  and  2.    The 

degree    of    "  vacuum "    may    be 
Fig.  4.  found,  as    shown   in    Fig.  4,  by 

using,  in  place  of  the  piezometers, 
a  glass  tube  bent  over  and  led 
downward  into  an  open  vessel  con- 
taining water  or  mercury.  The 
height  to  which  the  water  (or  the 
mercury,  converted  into  feet  of 
water)  rises  in  this  tube,  shows 
the  extent  of  the  vacuum,  or  the 
portion,  A«,  of  the  air  pressure 
which  has  been  called  into  service 
in  producing  the  high  velocity 
through  the  throat.  By  adding 
this  to  hw,  we  obtain,  as  above, 
the  total  loss  of  head  H  between 
sections  1  and  2. 
When  the  reduction  of  area  at  the  throat  has  proceeded  so  far  that  the  entire 
available  pressure  of  water  and  air  at  section  1  is  required,  in  order  to  maiu- 

FiG.  5. 


tain  tlie  corresponding  velocity  through  the  throat  (t.  e.,  when  the  line  repre- 
senting the  upper  surface  of  the  air  falls  to  the  level  of  the  top  of  the  throat), 


534 


WATER-PIPES. 


no  further  increase  of  throat  velocity  can  be  secured  (with  a  given  total  head 
over  section  1)  by  still  further  narrowing  the  throat.  If  the  throat  is  further 
narrowed,  the  velocity  through  it  will  remain  the  same ;  and,  the  rate  of  dis- 
charge being  thus  diminished,  the  velocity  through  section  1  will  be  neces- 
sarily reduced.    In  other  words,  throttling  begins. 

Let  vi  be  the  velocity  in  section  1,  above  the  throat,  and  V2  the  "throat  veloc- 
ity," or  velocity  in  the  throat  or  section  2. 

Referring  to  Fig.  5,  the  velocity  head  at  section  1,  measured  from  an  assumed 
datum  represented  by  the  upper  horizontal  lines,  is 


and  that  at  section  2  is 


'^-t' 


2g 


Neglecting  resistances  to  flow,  the  loss  of  head,  between  sections  1  and  2, 
or  "  the  head  on  the  Venturi,"  is  equal  to  the  increase  in  the  velocity  head,  ot 
to  the  loss  in  pressure,  between  a^,  and  02,  or 

Hence,  h.^  = '^  =  B.  +  h-^ 


and  tbroat  velocity  =  vg 


^2^(H  +  AJ  =  ^j2g{H+^)^ 


In  other  words,  the  velocity  at  the  throat  is  that  corresponding  to  the  "  head 
H  on  the  Venturi,"  plus  the  head  corresponding  to  the  velocity  of  approach  Vi 
in  section  1. 

But,  since  the  velocities  are  inversely  as  the  areas  of  cross-section  a^  and  a^, 
■  ~        '■"         ""^  "^    ~  ai2 

^-     -  -  "-        -      2, 


and  throat  velocity  =  vg  =    .       ^       -V2g  H, 

The  ratio  — ^» 

between  the  area  a^  of  cross-section  at  the  throat,  and  that,  Oj,  at  the  upper  end 
of  the  up-stream  cone,  is  called  the  tbroat  ratio.    For  a  ratio  of  1 :  9  we  have 

•  «!  _         9         _  _9-  _      /si 

or  V2  =  1.0062)^2^7  i/. 

The  Tenturi  tube,  for  pipes  not  over  60  inches  in  diameter,  is  formed  of 
8«^veral  short  sections  of  cast  iron  pipe,  having  the  required  taper,  and  fur- 

*  By  Bernouilli's  theorem,  j?i  +  ^1  =  Pa  +  ^2- 


WATER-PIPES.  5e35 

nished  with  flanges,  by  means  of  which  the  sections  are  bolted  together  to  form 
the  two  truncated  cones  required. 

In  the  smaller  sizes,  the  shorter  cone  is  generally  in  one  section  and  the 
longer  cone  in  two  or  more  sections. 

The  throat  section  is  generally  made  in  a  separate  piece,  and  is  either  made 
of  bronze  or  lined  with  that  metal. 

The  ends  of  the  Venturi  tube  are  furnished  with  either  bell,  spigot,  or  flanged 
ends,  according  to  the  character  of  the  pipe  in  which  the  tube  is  to  be  used. 

For  still  iarjser  streams,  such  as  those  in  masonry  conduits  or  riveted 
flumes,  the  Venturi  tube  may  be  made  of  wooden  staves,'  sheet  steel,  cement 
concrete,  brick  or  other  suitable  material,  metal  being  used  for  the  throat  piece 
and  where  required  by  the  pressure. 

The  tliroat  piece  is  surrounded  by  an  annular  chamber  called  the  press- 
ure ctiamber.  which  communicates  with  the  interior  of  the  throat  by  means 
of  several  holes  drilled  radially  through  the  walls  of  the  latter  at  equal  or 
nearly  equal  distances  around  the  circumference. 

A  similar  pressure  chamber  is  provided  at  the  larger  end  of  the  short  cone  for 
observing  the  pressure  in  the  normal  section  up-stream  from  the  throat ;  and, 
if  it  is  desired  to  ascertain  the  final  loss  of  head  due  to  the  passage  of  the  water 
through  the  Venturi,  a  similar  chamber  must  be  provided  at  the  larger  end  of 
the  longer  or  down-stream  cone. 

In  designating-  tlie  size  of  the  meter,  the  diameter  of  the  pipe  of  which 
it  forms  a  part  is  used,  and  not  the  throat  diameter.  Thus,  a  meter  for  use  in  a 
6-inch  pipe  is  called  a  6-inch  meter. 

The  reg'ister  gives  periodic  registrations,  usually  every  ten  minutes,  in 
which  the  head  ff  =  h2— hi,  existing  &t  the  instant  of  registry,  is  recorded  in 
terms  of  the  total  discharge  in  cubic  feet  since  the  last  registry  and  as  an  in- 
crease in  the  total  number  of  cubic  feet  registered.  In  other  words,  the  registry 
involves  the  assumption  that  the  average  velocity,  during  the  period  between  two 
registrations,  is  equal  to  the  velocity  at  the  instant  of  the  following  registration. 

The  register  may  be  placed  at  a  considerable  distance  (not  exceeding,  say,  500 
feet)  from  the  Venturi  tube.  It  must  be  placed  at  such  a  depth  below  the 
hydraulic  grade  line  that  the  pressures  existing  in  the  Venturi  tube  shall  at  all 
times  be  transmitted  to  the  register. 

The  pipe  lines,  connecting  the  Venturi  with  the  register,  must  be  covered, and 
a  shelter  from  weather  and  frost  must  be  provided  for  the  register. 

The  size  and  cost  of  the  register  are  independent  of  the  size  of  the  Venturi. 

Behavior.  From  experiments  by  Mr.  Herschel,*  f  ^  by  the  Bureau  of 
Water,  Philadelphia,f  and  by  others,!  it  appears  that  the  Venturi  meter  may  ordi- 
narily be  depended  upon  to  give  results  within  3  per  cent,  of  the  true  discharge. 

With  a  48  inch  Venturi,  Mr.  Herschel  ^  found  a  total  loss  of  head,  due  to 
the  passage  of  the  water  through  the  Venturi  tube,  of  about  10.6  per  cent,  of  the 
head  H  on  the  Venturi.  With  two  54  inch  Venturis,  Professors  Marx,  Wing,  and 
Hoskinsgf  found  a  loss  of  14.9  per  cent.,  part  of  which,  no  doubt,  was  due  to 
the  presence  of  a  42  inch  gate  valve  in  the  down-stream  cone.  This  last  result 
would  add  about  1.12  feet  to  the  head  required  in  pumping  20,000,000  gallon^ 
daily  through  a  48  inch  main  and  a  Venturi  having  a  throat  ratio  of  1 : 9. 

The  Venturi  meter  has  been  found  to  give  perfectly  satisfactory  results  in 
measuring  the  flow  of  brine  and  very  hot  water. 

Venturi  tubes  are  made  with  throat  ratios  ranging  from  1:4^  (or  2: 9)  to 
1:  16.  The  former  are  adapted  to  high,  and  the  latter  to  low  velocities;  for, 
where  the  velocity  in  the  pipe  is  low,  it  is  necessary  to  accelerate  it  greatly  in 
the  throat  in  order  to  obtain  sutficient  loss  of  pressure  to  secure  reliable'  in- 
dications in  the  register.  These  cannot  be  obtained  where  the  throat  velocity 
is  less  than  about  3  feet  per  second.  With  a  throat  ratio  of  1 :  16,  this  would 
give  a  pipe  velocity  of  {^e  ^^^^  P^i*  second.  On  the  other  hand,  a  meter  with 
a  high  throat  ratio,  adapted  to  low  velocities,  would, with  high  velocities,  exceed 
the  upper  limit  of  the  register. 

Owing  to  its  unobstructed  channel,  free  from  moving  parts,  the  Venturi 
meter  is  far  less  liable  to  clogging  than  the  forms  of  meter  in  common  use. 

The  prices  of  the  principal  sizes  of  the  Venturi  meter  are  as  follows: — on 
board  cars  at  Providence,  R.  1. 

6  inch  ^600.00  24  inch  ^1,130.00  48  inch  $3,060.00 

12  inch    770.00  36  inch    1,680.00  60  inch    4,890.00 

These  prices  include  the  register,  which,  in  the  smaller  sizes,  constitutes  the 
principal  item  of  cost.    Discount,  1901,  10  per  cent. 

*  Trans.  Am.  Soc.  Civil  Engrs.,  Nov.,  1887,  Vol.  XVII.,  page  228. 
fJournal  of  the  Franklin  Institute.  Feb.,  1899. 

1[  Journal  New  England  Waterworks  Assn.,  Vol.  VIII.,  No.  1,  Sep.,  1893. 
2  Trans.  Am.  Soc.  Civil  Engrs,,  Vol.  XL.,  Dec,  1898,  pp.  471,  etc. 


536 


WATER-PIPES. 


Fig.  6. 


7i  =  ftt>» 


Art.  4  b.  Tlie  Ferris-Pitot  meter,  invented  and  patented  by  Mr. 
Walter  Ferris,  of  Philadelphia,  in  designed  to  measure  the  flow  of  liquids  in 
pipes  running  full.  It  consists  of  a  device  for  the  registration  of  the  results 
obtained  by  the  Pitot  tube,  described  on  pages 561  and  562,  and  of  special  devices 
to  prevent  the  clogging  of  the  tubes  and  to  permit  their  examination  while 
in  use. 

In  Fig.  6  let  P  represent  the  level  at  which  the  water  stands  in  the  straight 
Pitot  tube,  5.    Then  h^ki^,  or  the  difference  in  level 
between  the  columns  in  the  two  tubes,  is   the  head 

(theoretically  =—  )  due  to  the  velocity  of  the  water 

in  the  pipe  as  it  impinges  against  the  open  up-stream 
end  of  the  bent  tube,  c.  For  a  given  velocity,  v,  this 
ditference,  //,  is  constant,  and  is  Independent  of  the 
pressure  represented  by  P. 

The  Ferris  register,  like  that  of  the  Venturi  meter, 
records  the  velocity  (existing  ai  the  instant  of  registra- 
tion) in  terms  of  the  total  discharge  since  the  last  regis- 
try and  as  an  increase  in  the  total  number  of  cubic  feet 
registered.  The  registry  thus  involves  the  assumption 
that  the  average  velocity,  during  the  period  between 
registrations,  is  equal  to  the  velocity  at  the  end  of  that 
period.  In  the  Ferris  meter  the  registration  is  made 
every  two  minutes. 

Evidently  the  instrument  measures  the  velocity  at 
only  one  point  in  the  cross-section  of  the  pipe,  arid  it  may  thus  be  used  to  de- 
termine successively  the  velocities  at  any  number  of  such  points,  but  the  ve- 
locity at  such  a  point  may  or  may  not  be  equal  to  the  mean  velocity  in  the  entire 
cross-section.  The  instrument  is  therefore  usually  calibrated  by  reference  to 
some  accepted  standard,  and  the  coeflicient  or  coefficients  thus  obtained  are 
used  in  subsequent  observations. 

The  recording  mechanism  is  operated  by  a  small  hydraulic  motor,  driven  by 
means  of  the  flow  of  the  water  in  the  pipe  itself.  For  this  purpose  a  second 
pair  of  Pitot  tubes,  is  inserted  into  the  pipe;  and  the  current,  flowing  through 
these  tubes,  drives  the  motor  without  loss  of  water,  the  water  used  for  power 
being  returned  to  the  pipe.  If  the  velocity  in  the  pipe  is  less  than  3  feet  per 
second  it  must  be  increased  by  means  of  a  "  reducer." 

Experiments  made  by  Mr.  Ferris  and  by  the  Bureau  of  Water,  Philadelphia, 
indicate  that  the  Ferris-Pitot  meter  will  ordinarily  register  within  3  per  cent, 
of  the  true  discharge. 

In  general,  the  size  and  cost  of  the  registering  apparatus  are  independent  of 
the  size  of  the  pipe. 


HYDRAULICS. 


537 


Art.  5.    Resistance  of  curves  and  bends  in  water  pipes. 

Much  uncertainty  exists  respecting  these  matters.     Weisbach's  form- 
ula,* for  the  resistance  due  to  a  circular  curve,  Figs.  2  and  3,  is 


A  =  C 


A 

180 


il  =  [0.131+ 1.8.7  (^)  J] 


A 

180 


'^9 


•where 


h  =  head  in  feet  required  to  overcome  resistance  due  to  curve  or  bend^ 

C  =  experimental  coeflQicient, 

A  =  angle  of  deflection,  in  degrees, 

V  =  mean  velocity  of  flow  in  pipe,  in  feet  per  second, 

g  =  acceleration  of  gravity  =  32.2  ft  per  sec  per  sec, 


^9 


=  head  theoretically  due  to  velocity  v, 


D  =  inside  diameter  of  pipe,  in  feet, 
r  =  inside  radius  of  pipe,  in  feet, 
R  =  radius  of  axis  of  curve,  in  feet. 


IfrH-R=     0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

0:9 

1.0 

then  C  =  0.131 

0.138 

0.158 

0.206 

0.294 

0.440 

0.661 

0.977 

1.408 

1.978 

Fig.  a. 


Fig.  2. 


Figr.  4. 

(See  next  page.) 


According  to  this  formula,  the  resistance  due  to  curvature  decreases  rapidly 
as  R  increases  from  J^  D  to  2  D ;  and  but  little  further  decrease  occurs  beyond 
R  =  5  D  ;  but,  from  very  careful  and  elaborate  experiments  on  city  water  mains, 
from  12  to  30  ins  diameter,  in  Detroit,  Mich.,f  the  investigators  conclude  that  a 
line  of  pipe  with  a  curve  of  short  radius  R  (down  to  a  limit  of  R  =  2>^  D)  causes 
less  resistance  than  does  a  line  of  equal  length  and  equal  total  angle  A,  with  a 
curve  of  longer  radius  R.  Their  results  were  approximately  as  follows,  where 
H  =  resistance  due  to  a  section  of  80  diameters  in  length,  with  a  curve  of 

A  =  90°  at  mid-length, 
h  =  resistance  in  a  tangent  of  length  =  80  diameters. 
If  R  -f-  D  =    1  2         2.5         3  4  5  10  15         20         25 

then  B.-irh  =  1.35      1.14      1.13      1.14      1.18      1.24      1.50       1.66       1.80       1.95 

They  found  also  that  the  loss  of  bead,  due  to  a  curve,  occurs  not  only 
in  the  curve  itself,  but  that  head  continues  to  be  lost  in  the  following 
tangent,  for  some  distance  down  stream  from  the  curve. 

Their  experiments  led  to  the  inference  that  even  very  sliffbt  deflections, 
A,  in  the  line,  cause  material  losses  of  bead,  and  that»care  in  securing 
a  straight  alignment  is  therefore  highly  advisable.    For  bends,  see  next  page. 

*Der  Ingenieur,  pp.  444,  445. 

t  Paper  by  Gardner  S.  Williams,  Clarence  W.  Hubbell,  and  George  H.  Fenkell, 
Transactions,  American  Society  of  Civil  Engineers,  1901. 


538  HYDRAULICS. 

For  abrupt  ang'les.  Fig.  4,  Weisbach  gives :  Resistance,  in  feet  of  head  =— 


h  =  c^  ==  (0.95  sin2  J^  A  +,2.05  sin*  y^  A)  ^ 


\iy^K==  10° 

then  c       =  0.03 


20° 
0.14 


30° 
0.36 


40° 
0.74 


45° 
0.98 


50° 
1.26 


55° 
1.56 


60° 
1.86 


65° 
2.16 


2.43 


Figr.  4. 


In  addition  to  the  resistance  offered  to  flow,  curves  and  bends  in- 
volve additional  labor  and  expense  in  manufacture  and  in  laying;  and  vertical 
bends  and  curves  lead  to  the  formation  of  pockets  of  sediment  at  the  feet  of 
slopes,  and  of  air  cushions  at  their  summits. 


a  h 


Art.  6.  Although,  in  Fig.  5,  the  static  pressures  upon  the  equal  bases,  a  b 
and  a'  &',  of  the  two  pipes  are  equal  (see  Hydrostatics,  Art.  1)  ;  yet,  in  order  to 
pump  water  through  either  pipe,  at  a  given  velocity,  an  additional  force  is 
required,  in  order  to  overcome  resistances  to  flow  ;  and  these  resistances  and  the 
additional  force  required  in  order  to  overcome  them,  will  be  greater  in  the  longer 
than  in  the  shorter  pipe. 


HYDRAULICS. 


539 


Art.  7.     Flow  throngli  orifices.    Theoretically  the  velocity,  v,  of  a 

fluid  flowiug  through  a  small  orifice  iu  the  side  or  bottom  of  a  very  large  vessel, 
is  equal  to  that  acquired  by  a  body  falling  freely  in  vacuo  through  a  height 
equaUto  the  head,  h,  or  depth,  measured  vertically  from  the  level  surface  of  the 
fluid  iu  the  vessel,  to  the  center  of  gravity  of  the  orifice ;  or, 


^l/2gh   =  |/ 64.4  A   =  8.03  ^Z h\ 


and 


h  = 


2^' 


64.4 


=  0.0155  v\ 


This  law  applies  equally  to  all  flnids.  Thus,  theoretically,  mer- 
cury, water,  air,  etc.,  all  flow  with  equal  velocities  from  a  given  orifice  under  a 
given  head. 

For  deviations  in  practice  from  this  theoretical  law,  see  Art.  9,  etc. 

Table  10. 
Velocities  theoretically  due  to  given  heads. 


Head 

Vel. 

Head 

Vel. 

Head^Vel. 

Head 

Vel. 

Head 

Vel. 

Head 

Vel 

Head 

Vel. 

Feet. 

Ft  per 

Feet. 

Ft  per 
sec. 

Feet. 

Ft  per 
sec. 

Feet. 

Ft  per 
sec. 

Feet. 

Ft  per 

sec. 

Feet. 

Ft  per 
sec. 

Feet. 

Ft  per 
sec. 

.005 

.57 

.29 

4.32 

.77 

7.04 

1.50 

9.83 

7. 

21.2 

28 

42.5 

76 

69.9 

.010 

.80 

.30 

4.39 

.78 

7.09 

1.52 

9.90 

.2 

21.5 

29 

43.2 

77 

70.4 

.015 

.98 

.31 

4.47 

.79 

7.13 

1.54 

9.96 

.4 

21.8 

30 

43.9 

78 

70.9 

.020 

1.13 

.32 

4.54 

.80 

7.18 

1.56 

lO.O 

.6 

22.1 

31 

44.7 

79 

71.3 

.025 

1.27 

.33 

4.61 

.81 

7.22 

1.58 

10.1 

.8 

22.4 

32 

45.4 

80 

71.8 

.030 

1.39 

.34 

4.68 

.82 

7  26 

1.60 

10.2 

8. 

22.7 

S3 

4(5.1 

81 

72.2 

.035 

1.50 

..35 

4.75 

.83 

7.31 

1.65 

10.3 

.2 

23.0 

34 

46.7 

82 

72.« 

.040 

1.60 

.36 

4.81 

.84 

7.35 

1.70 

10.5 

.4 

23.3 

35 

47.4 

83 

73.1 

.045 

lw70 

.37 

4.87 

.85 

7.40 

1.75 

io:6 

.6 

23.5 

36 

48.1 

84 

73.5 

.050 

1.79 

.38 

4.94 

.86 

7.44 

1.80 

10.8 

.8 

23.8 

37 

48.8 

85 

74.0 

.055 

1.88 

.39 

5.01 

.87 

7.48 

1.85 

10.9 

9. 

24.1 

38 

49.5 

86 

74.4 

.060 

1.97 

.40 

5.07 

.88 

7.53 

1.90 

11.1 

.2 

24.3 

39 

50.1 

87 

74.8  . 

.065 

2.04 

.41 

5.14 

.89 

7.57 

1.95 

11.2 

.4 

24.6 

40 

50.7 

88 

75.3 

.070 

2.12 

.42 

5.20 

.90 

7.61 

2. 

11.4 

.6 

24  8 

41 

51.3 

89 

75.7 

.075 

2.20 

.43 

5.26 

.91 

7.65 

2.1 

:i.7 

.8 

25.1 

42 

52.0 

90 

76.1 

.080 

2.27 

.44 

5.32 

.92 

7.70 

2.2 

11.9 

10. 

25.4 

43 

52.6 

91 

76.5 

.085 

2.34 

.45 

5  38 

.93 

7.74 

2.3 

12.2 

.5 

26.0 

44 

53.2 

92 

76.9 

.090 

2.41 

.46 

5.44 

.94 

7.78 

2.4 

12.4 

11. 

26.6 

45 

53.8 

93 

77.4 

.095 

2.47 

.47 

5.50 

.95 

7.82 

2.5 

12.6 

.5 

27.2 

46 

54.4 

94 

77.8 

.100 

2.54 

.48 

5.56 

.96 

7.86 

2.6 

12.9 

12. 

27.8 

47 

55.0 

95 

78.2 

.105 

2.60 

.49 

5.62 

.97 

7.90 

2.7 

13.2 

.5 

28.4 

48 

55.6 

96 

78.6 

.110 

2.66 

.50 

5.67 

.98 

7.94 

2.8 

13.4 

13. 

28.9 

49 

56.2 

97 

79.0 

.115 

2.72 

.51 

5.73 

.99 

7.98 

2.9 

13.7 

.5 

29.5 

50 

56.7 

98 

79.4 

.120 

2.78 

.52 

5.79 

IFt. 

8.03 

3. 

13.9 

14. 

30.0 

51 

57.3 

99 

79.8 

.125 

2.84 

.53 

5.85 

1.02 

8.10 

3.1 

14.1 

.5 

30.5 

52 

57.8 

100 

80.3 

.130 

2.89 

.54 

5.90 

1.04 

8.18 

3.2 

14.3 

15. 

31.1 

53 

58.4 

125 

89.7 

.135 

2.95 

.55 

5.95 

106 

8.26 

.3.3 

14.5 

.5 

31.6 

54 

59.0 

150 

98.8 

.140 

3.00 

.56 

6.00 

1.08 

8.34 

3.4 

14.8 

16. 

32.1 

55 

59.5 

175 

106 

.145 

3.05 

.57 

6.06 

1.10 

8.41 

.3.5 

15. 

.5 

32.6 

56 

60.0 

200 

114 

.150 

3.11 

.58 

6.11 

1.12 

8.49 

3.6 

15.2 

17. 

33.1 

57 

60.6 

225 

120 

.155 

3.16 

.59 

6.17 

1  14 

8.57 

3.7 

15.4 

.5 

33.6 

58 

61.1 

250 

126 

.160 

3.21 

.60 

6.22 

1.16 

8.64 

3.8 

15.6 

18. 

34.0 

59 

61.6 

275 

133 

.165 

3.26 

.61 

6.28 

1.18 

8.72 

3.9 

15.8 

.5 

34.5 

60 

62.1 

300 

139 

.170 

3.31 

.62 

6.32 

1  20 

8.79 

4. 

16.0 

19. 

35.0 

61 

62.7 

350 

150 

.175 

3.36 

.63 

6.37 

1.22 

8.87 

.2 

16.4 

.5 

35.4. 

62 

63.2 

400 

160 

.180 

3.40 

.64 

6.42 

1.24 

8.94 

.4 

16.8 

20. 

35.9 

63 

63.7 

4.50 

170 

.185 

3.45 

.65 

6.47 

1.26 

9.01 

.6 

17.2 

.5 

36.3 

64 

64.2 

500 

179 

.190 

3.50 

.66 

6.52 

1.28 

9.08 

.8 

17.6 

21. 

36.8 

65 

64.7 

550 

188 

.195 

3.55 

.67 

6.57 

1.30 

9.15 

5. 

17.9 

.5 

37.2 

66 

65.2 

600 

197 

.200 

3.59 

.68 

6.61 

1.32 

9.21 

.2 

18.3 

22. 

37.6 

67 

65.7 

700 

212 

.21 

3.68 

.69 

6.66 

1.34 

9.29 

.4 

18.7 

.5 

38.1 

68 

66.2 

800 

227 

.22 

3.76 

.70 

6.71 

1.36 

9.36 

.6 

19. 

23. 

38.5 

69 

66.7 

900 

241 

.23 

3.85 

.71 

6.76 

1.38 

9.43 

.8 

19.3 

.5 

,38.9 

70 

67.1 

1000 

254 

.24 

3.93 

.72 

6.81 

1.40 

9.49 

6. 

19.7 

24. 

39.3 

71 

67.6 

.25 

4.01 

.73 

6.86 

1.42 

9.57 

.2 

20.0 

.5 

39.7 

72 

68.1 

.26 

4.09 

.74 

6.91 

1.44 

9.63 

.4 

20.3 

25 

40.1 

73 

68.5 

.27 

4.17 

.75 

6.95 

1.46 

9.70 

.6 

20.6 

26 

40.9 

74 

69.0 

.28 

4.25 

.76 

6.99 

1.48 

9.77 

.8 

20.9 

27 

41.7 

75 

69.5 

540 


HYDRAULICS. 


Fig.  6. 


Art.  8.  On  the  flow  of  water 
tbroug-ta  vertical  opeiiiii^STS  fur- 
nisbed  with  short  tubes.    When  water 

flows   from  a  reservoir,  Fig  6,    through    a  vert  partition 

mm  a  a,  the  thickness  a  m  of  which  is  about  m  or  3  times 

the  least  transverse  dimension  of  the  opening,  (whether 

that  dimension  be  its  breadth,  or  its  height;)  or  when,  if 

the  partition  be  very  thin,  as  n  n,  the  water  flows  through 

a  tube,  as  at  t,  the  length  of  which  is  about  2  or  3  times  its 

least  transverse  dimension,  then  the  eflBuent  stream  will 

entirely  flll  the  opening,  or  the  tube,  as  shown  in  Fig  6  ;  or, 

in  technical  language,  will  run  with  a  full  flow;  or  a  full 

bore ;  and  will  disch  more  water  in  a  given  time,  than  if 

the  tube  were  either  materially  longer  or  shorter.     For  if 

longer  than  3  times  the  least  transverse  dimension,  the 

flow  will  be  impeded  by  the  increased  friction  against  the 

sides  of  the  tube  ;  and  if  shorter  than  about  twice  the  least 

transverse  dimension,  the  water  will  not  flow  in  a  full  stream,  but  in  a  contracted  one,  as  shown  by 

Fig  11.  This  will  be  the  case  whether  the  tube  be  circular,  or  rectilinear,  in  its  cross-section. 

To  find  approximately  the  actual  vel.  and  disch  into  the 
air,  throug-h  a  tube,  or  opening-,  either  circular  or  recti- 
linear  in  its  outline,  or  cross-section  ;  and  whose  leng-th  c  i, 
or  c  e,  in  the  direction  of  the  flow,  is  about  2}4  or  :i  times  its 
least  transverse  dimension ;  when  the  surface-level,  s,  Fig  6, 
remains  constantly  at  the  same  heig-ht;  and  which  height 
must  not  be  below  the  upper  edge  of  the  tube,  or  opening. 

RoTLE  1.    Take  out  the  theoretical  vel  from  Table  10,  corresponding  to  the  head  measured  vert 

from  the  center  (or  more  properly,  the  cen  of  grav)  c,  of  the  opening,  to  the  level  water  surf*.  Mult 
it  by  the  coeff'  of  disch  .81.  The  prod  will  be  the  reqd  vel,  in  ft  per  sec.  Mult  this  actual  vel  by  the 
transverse  area  of  the  opening,  in  sq  ft.  If  circular,  knowing  its  diam,  this  area  will  be  found  in 
Table  3.  The  prod  will  be  the  quantity  of  water  dischd,  in  cub  ft  per  sec;  within,  probably,  S 

or  4  per  cent. 

RoxE  2.  Find  the  sq  rt  of  the  head  in  ft.  Mult  this  sq  rt  by  6.5.  The  prod  will  be  the  actual 
vel  in  ft  per  sec. 

Ex.  An  opening  c  •  ;  or  box-shaped  tube  c  (,  Fig  6,  is  3  feet  vride,  by  .25  of  a  ft  high  ;  and  its  length 
in  the  direction  ci  or  c  e  in  which  the  water  flows  is  about  .62  of  a  ft,  or  about  2^^  times  its  least 
transverse  dimension,  or  its  height.  The  head  from  the  cen  of  grav  c,  of  the  opening,  to  the  constant 
surf-level  s,  is  4  feet.     What  will  be  the  vel  of  the  water  ;  and  how  much  will  be  dischd  per  sec  ? 

By  Rule  1.    The  theoretical  vel  (Table  10,  )  corresponding  to  a  head  of  4  ft  is  16  ft  per  sec. 

And  16  X  .81  =  12.96  ft  per  sec,  the  actual  vel  reqd.  Again,  the  transverse  area  of  the  opening,  or  of 
the  tube,  is  3  ft  X  .25  ft  =  .75  sq  ft.     And  .75  X  12.96  =  9.72  cub  ft;  the  quantity  dischd  per  sec. 

By  Rule  2.  The  sq  rt  of  4  is  2.  And  2  X  6.5  =  13  ft  per  sec.  the  reqd  vel,  as  before ;  the  very  slight 
diff'  being  owing  to  the  omission  of  small  decimals  in  the  coeffs. 

Rem.  1.  If  the  short  tube  *  projects  partly  inside  of  the  vert 
partition  ti  n,  the  disch  will  be  diminished  about  }/g  part.     In  that  case,  use  .71 

or  .7  instead  of  the  .81  of  Rule  1 ;  or  5.7  instead  of  the  6.5  of  Rule  2. 

Rem.  2.  When  the  thickness  a  m  of  the  vert  partition  m  m  a  a;  or  the  length  c  e  of  the  tube  t,  Fig 
6,  is  increased  to  about  4  times  the  least  transverse  dimension  of  the  opening  ;  or  of  the  diam,  when 
circular;  then  the  additional  friction  against  its  sides  begins  appreciably  to  lessen  the  vel  and  disch. 
In  that  case,  or  for  still  greater  lengths,  up  to  100  diams,  they  may  be  found  approximately,  by  nsing 
instead  of  the  coefif  of  disch  .81  iu  Rule  1,  the  following  coeffs,  by  which  to  mult  the  theoretical  vels 
»f  Table  10. 

TABL.E  11. 


Length  of 

Length  of 

Pipe 

Coeff. 

Pipe 

Coeff. 

in  Diams. 

in  Diams. 

4 

.80 

40 

.62 

6... 

...  .76 

50... 

...  .60 

10 

.74 

60 

.57 

15... 

...  .71 

70... 

...  .55 

20 

.69 

80 

.52 

25... 

...  .67 

90... 

...  .50 

30 

.65 

100 

.48 

Rem.  3.  When  the  length  of  the  opening  or  tube,  in  the  direction  in  which  the  water  flows,  becomes 
?.es«  than  about  twice  its  least  transverse  dimension,  the  disch  is  diminished  ;  so  that  for  lengths  from 
1^  times,  down  to  openings  in  a  very  thin  plate,  we  may  use  .61,  instead  of  the  .81  of  Rule  1.  For 
such  openings,  however,  see  Arts  9  and  10. 

Rem.  4.  But  on  the  other  hand,  the  disch  through  such  short  openings  and  tubes  as  are  shown  in 
Fig  6,  may  be  increased  to  nearly  the  theoretical  ones  of  Table  10,  by  merely  rounding  off  neatly  the 
edges  of  the  entrance  end  or  mouth,  as  in  Fig  7 ;  which  is  the  shape,  and  half  actual  size  of  one  with 
■which  Weisbach  obtained  .975  of  the  theoretical  vel  and  discharge,  when  the  head  was  10  ft;  and  .958 


HYDRAULICS. 


541 


■with  a  head  of  one  foot ;  so  that  in  similar  cases,  .975,  and  .958  may  be  used  instead  of  the  coeflf  .8\ 
in  Rule  1. 


Figr.a 


As  much  as  .92  to  .94  may  be  obtained  by  widening  the  opening,  m  n,  toward  its  outer  mouth,  o«. 
Fig,  8,  making  the  divergence,  or  angle  o,  about  5° ;  or  by  widening  it  toward  its  inner  mouth,  as  at 
t  c,  Fig  9 ;  but  increasing  the  angle  of  divergence,  at  h,  to  from  11°  to  16°.  In  all  cases,  we  consider 
the  small  end  as  being  the  opening  whose  area  must  be  multiplied  by  the  vel  to  get  the  discharge. 

In  some  experiments  made  with  larg^e  pyramidal  wooden 
trotig'hs  9.5  ft  long,  with  an  inner  mouth  of  3.'2  X  2.4  ft,  and  a  discharging  one 
of  .62  X  -4^  ft;  and  under  a  head  of  93^  feet,  the  discharge  was  .98  of  the  theoretical  one,  due  to  the 
smaller  end.     Therefore,  .98  may  be  used  in  such  cases,  instead  of  the  .81  of  Rule  1. 

Rem.  5.  By  using  an  adjutage  shaped  as  in  Fig  10,  the  discharge  may  be  increased  to  several 
times  that  due  to  the  head  above  the  center  of  gravity  s  of  the  orifice  mn:  because  in  such  cases,  as 
explained  in  Art  1  w,  the  true  head  at  s,  or  the  head  causing  the  rapid  flow  through  the  nar- 

rowest   portion     mn,    may     be    much 
^  greater  than  the  head  above  e. 


See  the  Venturi  Meter,  Art  4  a. 


Fig.  10. 


Art.  9.     On  the  disch  of  water  throag:h   opening^s  in  thin 
vert  partitions,  with  plane  or  flat  faces,  ee,  or  n  n,  Fig  11.*    If  the 

face  e  e,  or  n  n,  instead  of  being  plane,  and  vert,  should  be  curved, 
or  inclining  in  diflF  dir«ctions  toward  the  opening,  then  the  disch 
will  be  altered.  When  water  flows  from  a  reservoir.  Fig  11,  through 
a  vert  plane  plate  or  partition  nn,  which  is  not  thicker  than  about 
the  least  transversedimension  of  the  opening, whether  thatdimension 
be  its  breadth,  or  its  height  o  o ;  t  or  when,  i'  the  partition  e  e  itself 
is  much  thicker,  we  give  the  opening  the  shape  shown  at  b,  (which 
evidently  amounts  to  the  same  thing,)  then  the  eSiuent  stream  will 
not  pass  out  with  a,  full  flow,  as  in  Fig  6,  but  will  assume  the  shape 
shown  in  Fig  11 ;  forming,  just  outside  of  the  opening,  what  is 
called  the  vena  contracta,  or  contracted  vein.  In  order  that  this 
contraction  may  take  place  to  its  fullest  extent,  or  become  complete, 
the  inner  sharp  edges  of  the  opening  must  not  approach  either  the 
surf  of  the  water,  or  the  bottom  or  sides  of  the. reservoir,  nearer 
than  about  l}4  times  the  least  transverse  dimension  of  the  opening. 
The  contracted  vein  occu/s  at  a  dist  of  about  half  the  smallest  di- 
mension of  the  orifice,  from  the  orifice  itself.  In  a  circular  orifice, 
at  about  half  the  diam  dist;  and  ordinarily  its  area  is  about  .62.  or  nearly  %  that  of  the  orifice  itself^ 
At  this  point  the  actual  mean  vel  of  the  stream  is  very  nearly  (about  .97)  the  theoretical  vel  given  by 
Table  10,  and  hence  the  actual  discha  are  but  .62,  or  nearly  %  of  the  theoretical  ones. 

Case  1.    To  find  the  actual  disch  into  air.J  thron^h  either  a 
circular  or  rectilinear  g  opening;  in  a  thin  vert  plane  parti- 


Fig.  13. 


♦We  believe  that  these  rules  for  thin  plate  are  also  sufiiciently  approximate 
for  most  practical  purposes,  if  the  opening  be  in  the  bottom  of  the  reservoir; 
or  in  an  inclined,  instead  of  a  vert  side. 

t  When  the  side  of  a  reservoir,  or  the  edge  of  a  plank,  &c,  over  which  water 
flows,  has  no  greater  thickness  than  this,  the  water  is  said  to  flow  through, 

or  over,  thih  plate,  or  thin  partition. 

t  Should  the  disch  take  place  under  water,  as  in  Fig  12.  hoth  surf-levels  re- 
maining constant,  then  the  head  to  be  used  is  the  vert  diff  ao.of  the  two 
levels.  After  making  the  calculation  with  this  head,  we  should,  according  to 
"Weisbach,  deduct  the  y^  part;  inasmuch  as  he  states  that  the  disch  is  that 
much  less  when  under  water,  than  when  it  takes  place  freely  into  the  air. 
Other  experimenters,  however,  assert  that  it  is  precisely  the  same  in  both  cases. 

§  If  the  shape  of  the  opening  is  oval,  triangular,  or  irregular,  the  head 
must  be  measured  vert  from  its  cen  of  gray. 


542 


HYDRAULICS. 


^ion,  when  tlie  contraction  is  complete  ;  an€l  when  the  snrf- 
level,  .v,  remains  constantly  at  the  same  height;  water  being 
supplied  to  the  reservoir  as  last  »»s  it  runs  out  at  the  open- 
ing.* 

Rule  1.  When  the  head,  measured  vert  from  the  center  (or  rather  from  the  cen  of  grav)  c,  of  the 
opening,  to  the  surf-level  s  of  the  reservoir,  is  not  less  than  1  ft.  nor  more  than  10  ft ;  and  when  the 
least  transverse  dimension  of  the  opening  is  not  less  than  an  inch,  mult  the  theoretical  vel  in  ft  per 
sec  due  to  the  head.  (Table  10,  )  by  the  coefficient  of  disch  Q-z.     The  prod  will  be  the  acMial 

mean  vel  of  the  water  throuch  the  opening.  Mult  this  vel  by  the  area  of  the  opening  in  sq  ft;  tbe 
prod  will  b6  the  disch  in  cub  ft  per  sec,  approximately. 

"When  the  head  is  greater  than  10  ft,  use  .6,  instead  of  .62. 

Rule  2.  Find  the  sq  rt  of  the  head  in  ft.  Mult  this  sq  rt  by  5  j  the  prod  will  be  the  vel  in  ft  pei 
sec ;  which  mult  by  the  area  as  before  for  the  disch. 

Ex.  What  will  be  the  disch  through  an  opening  in  complete  contraction,  whose  dimensions  are  ft 
ins,  or  .5  ft  vert ;  and  4  ft  hor  ;  the  vert  head  above  the  cen  of  grav  of  the  opening  being  constantly 
«feet?  . 

By  Rule  1.    The  theoretical  vel  (Table  10,  )  corresponding  to  6  ft  head,  is  19.7  ft  per  sec.  And 

19.7  X  .62  =  12.214  ft,  the  reqd  vel.  Again,  the  area  of  the  opening  =  .5  X  4  =  2  sq  ft ;  and  12.214  X 
2  r;  24.428  cub  ft  per  sec ;  the  disch. 

By  Rule  2.  The  sq  rt  of  6  =  2.45  :  and  2.45  X  5  =  12-.25  ft  per  sec,  the  reqd  vel ;  and  12.25  X  2  ::^ 
24.5  cub  ft  per  sec,  the  disch. 

Both  Tery  approx  even  if  the  orifice  reaches  to  the  surface  of  the  issuing  water. 

Rem.  1.    The  coef  .62  Is  a  mean  of  results  of  many  old  experimenters 

In  1874  Genl.  T.  G.  Ellis  of  Massachusetts  conducted  an  elaborate  series  (Trans  Am  Soc  C  E,  Feb 
1876)  on  a  large  scale,  the  general  results  of  which,  within  less  than  1  per  ct,  are  given  in  the  follow, 
ing  table.     See  also  Rem  3.     The  sharp  edged  oriflces  were  in  iron  plates  .25  to  .5  inch  thick. 


Orifice. 

Head  above  Center. 

Coef. 

2  ft  sq. 

2.    to    .H.5  ft. 

.60  to  .61 

2  "  long,  1  ft  high 

1.8  to  11.3  " 

.60  to  .61 

2  "  long.  .5  high 

1.4  to  17.0  " 

.61  to  .60 

2  "  diam.     . 

l.S  to    9.6  " 

.59  to  .61 

Rem.  2.    Extreme  care  is  reqd  to  obtain  correct  results;  but  for  manj 

purposes  of  the  engineer  an  error  of  5  to  10  per  ct  Is  unimportant. 

It  will  rarely  happen  that  greater  accuracy  is  required  than  njay  be  obtained  by  the  foregoini 
rules;  but  when  such  does  occur,  aid  may  be  derived  from  the  following    table    dedUCCu 

from  the  experiments  of  Lrcsbros  and  Poncelet.  on  openings  8  ins 

wide,  of  diff  heights,  and  with  difif  heads.  Use  that  coeff  in  the  table  which  applies  to  the  case,  in- 
stead of  the  .62  of  Rule  1.  In  some  of  the  cases  in  this  table,  the  upper  edge  of  the  opening  is 
nearer  the  surf-level  of  the  reservoir  than  l}4  times  its  least  transverse  dimension. 

TABIiE  12.     Coefficients  for  rectang^ular  opening-s  in  thin 
vertical  partitions  in  full  contraction.* 


Head 

Head 

The  breadth  in  all  the  openings  rr  8  inches. 

above  cen. 
of  grav.  of 

above  cen. 
of  grav.  of 

HEIGHT  OF  OPENING. 

opening 

opening 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

in  Feet. 

in  Inches. 

8 

6 

4 

3 

2 

1 

.4 

.033 

.4 
.8 

1 

13^ 

.70 
.69 
.68 
.68 

.0666 



.65 
.64 

.64 

.0833 

.125 

.61 

,1666 

2 



.60 

.62 

.64 

.68 

.2083 

^H 

.59 

.61 

.62 

.64 

.67 

.250  • 

3 

.60 

.61 

.62 

.64 

.67 

.2917 

3>^ 

.57 

.60 

.61 

.62 

.64 

.66 

.3333 

4 

.58 

.60 

.61 

.63 

.64 

.66 

.3750 

4)^ 

.56 

.59 

.60 

.61 

.63 

.64 

.66 

.4167 

5 

.57 

.59 

.61 

.62 

.63 

.64 

.66 

.6666 

8 

.59 

.60 

.61 

.62 

.63 

.64 

.65 

1 

12 

,60 

.60 

.61 

.62 

.63 

.63 

.64 

3 

36 

.60 

.60 

.61 

.62 

.62 

.63 

.63 

5 

60 

.60 

.60 

.61 

.61 

.62 

.62 

.62 

10 

120 

.60 

.60 

.60 

.60 

.60 

.61 

.61 

Rem.  3.  Careful  experiments  on  openingrs  4^>^  ft  wide,  and  1^ 
ins  hi$::h.  under  heads  of  from  6  to  15  ft,  show  that  the  coeff  .62  will  give  results 
correct  within  -Jjr-  part,  for  openings  of  that  size  also,  under  large  heads;  although  the  thickness  oa 
ibe  partition  varied  on  its  diff  sides,  from  12  to  20  ins.  It  must  be  recollected,  however,  that  nothinf, 
more  than  close  approximations  are  to  be  attained  in  such  matters. 

Rem.  4.  It  has  been  asserted  by  some  writers,  that  when  tw^o  or  more 
cowtiguons  opening's  are  discharging;  at  the  same  time  from  the  same  reser- 
voir, they  disch  less  in  proportion  than  when  only  one  of  them  is  open.  Other  experiments,  how* 
ever,  seem  to  show  that  this  is  not  the  case ;  it  is  therefore  probable,  at  least,  that  the  difif,  if  any, 
is  but  trifling. 

*  See  first  footnote  on  preceding  page. 


HYDRAULICS. 


543 


Ca&e  2.  Tlie  cliscliarg-e  throngrh  tbiii  vert  partitions  in  coin« 
plete  contraction,  when  tlie  surface-level,  niy  Figr  13,  descends 
as  the  w^ater  lloivs  out  into  the  air.    In  this  case,  if  -the  reservoir  is 

prismatic,  that  is,  if  its  iior  sections  are  everywhere  equal ;  audit'  uo  water  is  flowing  into  the  reser- 
voir, to  supply  the  place  of  that  which  flows  out,  then,  to  find  the  time  reqd  to  disch  the  reservoir. 

Bulb.  Inasmuch  as  the  time  in  which  such  a  reservoir  entirely  discharges  itself,  is  twice  that  iu 
wkich  the  same  quantity  would  flow  out  under  a  constant  head,  as  in  Case  1,  therefore,  cal- 

culate the  disch  iu  cub  ft  per  sec  by  Rule  1,  Art  9  ;  div  the  number  of  cub  ft  con- 
tained in  the  reservoir,  above  the  level  g  of  the  bottom  of  the  opening,  Fig  13,  by 
this  disch  ;  the  quot  will  be  the  number  of  sec  in  which  a  volume  equal  to  that  in 
the  reservoir,  to  the  depth  g,  wouW  run  out  iu  Case  1,  of  a  constant  head.  And 
twice  this  number  will  be  the  seconds  reqd  to  empty  the  reservoir  iu  Case  2,  of  a 
varying  head. 

Rem.  If  it  should  be  reqd  to  find  the  time  in  which  such  a  prismatic  reservoir 
would  partly  empty  itself,  as,  for  instance,  from  m  to  n,  Fig  13,  first  calculate,  by 
the  above  rule,  the  sees  necessary  to  empty  it  if  it  had  only  been  filled  to  n ;  and 
afterward  calculate  as  if  it  had  been  filled  to  m.  The  difif  between  the  two  times 
will  evidently  be  the  time  reqd  to  empty  It  from  m  to  n.  If  the  opening  is  not  iu 
complete  contraction,  see  Arts  11,  &c. 

If  the  disch  is  into  a  lower  reservoir,  w^hose 
surf-level  remains  constant,  proceed  in  tlie  same  manner; 
only  use  the  diCf  of  level  of  the  two  surfs  as  the  head,  and  afterward  (according 
to  Weisbach)  increase  the  time  -J-^  part. 

Art.  10.  Disch  from  a  reservoir  R,  Fig"  14,  the  surf-level,  «, 
of  which  remains  constantly  at  the  same  height;  through 
an  opening,  o.  in  thin  vert  partition;  and  in  complete  con- 
traction; but  entirely  under  water;  and  into  a  prismatic 
reservoir,  m. 


Fifif.  13. 


R 


e 

Fig.  14. 


Seconds  req  Uired  /  height  a  c  ^  hor  area  of 

to    discharge    a    quantity  =         ^         jn  ft        '^  m  in  sq  ft 


cda,  the  level  c  remaining  = 
constant. 


,^/height  ac  ^  hor  area  of  ^  „ 

Seconds  required  __  in  ft      ^  m  in  sq  ft  ^* 


to  raise  level  in  wi  from  c  to  a  ' 


a  of  opening  , 
o  in  sq  ft 


Seconds  required 

to  raise  level  in  m  from  c  to  = 
any  other  level,  d. 


V    both  in  ft  j    "^  ^°  «q  f^  ^' 

Area  of  opening  ,^    ko  v  s  •« 
o  in  sq  ft         ^  -"^  ^  "•*' 

Rem.  1.  If  it  should  be  reqd  to  find  the  time  of  filling  m,  from 
its  bottom  c,  up  to  d,  we  may  do  so  very  approximately  by  calculating  by 
the  first  rule  in  Art  9,  the  time  reqd  from  e  to  the  center  of  the  opening  o,  as  if  all  that  portion  of 
the  disch  took  place  into  air;  and  afterward,  from  the  center  of  the  opening  to  d,  by  the  rule  just 
given.  This  case  is  similar  to  that  of  filling  a  lock  from  the  canal  reach  above,  in  which  the  surf- 
level  may  be  considered  constant. 

Rem.  2.  If  the  bottom  of  the  opening  o,  should  coincide  with 
the  bottom  of  the  reservoir,  then  the  coeff  will  become  greater  than  .62. 

See  Art  11,  for  obtaining  coeffs  for  imperfect  contraction. 

Rem.  3.  If  the  opening,  instead  of  being  in  complete  con- 
traction, is  of  any  of  the  shapes  Figs  6  to  9,  then  a  reference  to  Art  8  will  show 
what  coefiF  must  be  substituted  for  .62. 

Case,  3.  l>isch  from  one  prismatic  reservoir.  Fig  15, 1¥,  into 
another,  X,  of  any  comparative  sizes  whatever,  through  an 
opening  o,  in  a  plane  thin  vert  partition,  and  in  complete 
contraction;  when  the  water  rises  in  X,  while  it  falls  in  W. 

To  find  the  time  in  which  the  water,  flowing  from  W  into  X,  through 
0,  will  fall  through  the  dist  as,  so  as  to  stand  at  the  same  level  s  c,  in 
both  reservoirs. 

In  this  case,  the  water  reqd  to  fill  X  from  c  to  d,  (d  being  the  bottom 
of  the  opening  o,)  flows  out  into  the  air ;  and  the  time  necessary  for  it 
to  do  so,  must  be  calculated  separately  from  that  reqd  above  d,  which 
flows  into  water. 

Rule.  First  from  e  to  d.  Find  the  hor  area  of  each  reservoir,  in 
sq  ft.  Mult  the  hor  area  of  X,  by  the  vert  depth  de  in  ft,  for  the  cub 
ft  contained  in  that  portion.  Div  these  cub  ft  by  the  hor  area  of  W. 
The  quot  will  be  the  dist  am,  in  feet,  through  which  the  water  in  W 
must  descend,  in  order  to  fill  X  to  d. 


FifiT.  15. 


Seconds  r«- 
quired  to  low- 
er from  a  to  m,  and 
raise  from  eio  d. 


Twice  the        /     /yT. 
hor  area  of  X   I  "V 
W  iu  aq  ft         ^ 


z      ^/head  mn\ 
~^        in  ft     J 


Area  of  opening  ^  «.,  ^  c 
o  iu  sq  ft       A  .oz  X  t 


544 


HYDRAULICS. 


Seeoiids  required 

to  lower  from  m  to  «,  and  raise 
from  d  to  c.    (Very  approx) 


Hor  area  of  ^  twice  the  hor  area  y.  ,^/head  m  n 

Y  in  en  ff      '^  nf  W  in  an  ff.  ^    ^^  «n  ff 


of  W  in  sq  ft 


Area  or        ,/iaoT  area        nor  area\ 

opening  X  (  of  W   +   of  X  )  X  .62  X  8.03 

oin  sq  ft    \in  sq  ft   in  sq  ft/ 


Fig.  16. 


Ex.  Let  the  hor  area  of  "W  be  100  sq  ft :  and  that  of  X,  60  sq  ft.  Let  an  be  20  ft ;  and  mn  16  ft ; 
and  the  area  of  the  opening  o,  3  sq  ft.  In  what  time  will  the  water  descend  from  a  to  s,  and  rise 
from  e  to  c  ? 

Inasmuch  as  the  method  of  finding  the  time  for  filling  from  e  to<f,  by  the  water  falling  from  a  to 
m,  requires  uo  further  exempiitication,  we  will  coutine  ouraelves  to  the  additional  time  necessary  for 
filling  from  d  to  c,  by  the  water  falling  from  m  to  s.     To  find  this,  we  have,  the  sq  rt  of  the  head 

4  X  100  X  60  X  2  48000 

«.n  =  4ft;andthesumofthe2areas  =  100  +  60  =  160.    Hence,  ^^-^^^^^^^^^^   =    ^^  = 

120.1  see;  the  additional  time  reqd,  very  approximately. 

Note  1.  If  the  opening-,  as  d.  Fig  16,  reaehes 
to  tlie  very  bottom  of  tlie  reservoirs,  we  may 

consider  all  the  water  flowing  from  R  into  T,  as  flowing  into  water. 
Therefore,  using  the  head  am,  we  at  once  calculate  the  time  necessary 
for  the  water  in  the  two  reservoirs  to  arrive  at  the  same  level  s  c,  by 
the  last  process  of  the  preceding  rule;  or,  in  other  words,  by  the  pro- 
cess given  iu  the  preceding  example.  But  in  this  case  it  must  be  borne 
in  mind  that  the  opening  o  is  no  longer  in  complete  contraction,  iuas* 
much  as  the  contraction  along  its  lower  edge  is  suppressed. 

The  disch  will  consequently  be  somewhat  increased;  and  a  coeff 
greater  than  .62  becomes  necessary.  The  method  of  finding  this,  is 
given  in  the  following  Case  4.  A  reference  to  Art  8  will  give  the  coeff 
in  case  the  opening  is  shaped  as  Figs  6  to  9. 

Art.  11.  Case  4.  Tlie  discharge  through  openings  in  plane 
thin  vert  partitions;  but  in  incomplete  contraction. 

The  opening  may  be  such  that  contraction  will  take  place 
along  one  portion  of  its  perimeter,  or  at  the  top  of  the  open- 
ing a,  Fig  17  ;  while  it  is  suppressed  on  another  portion  ;  as 
at  the  bottom  and  two  ends  of  the  opening  a ;  where  suppres- 
sion is  caused  by  the  addition  of  short  side  and  bottom  pieces 
c,  c,  c.  Or  it  may  be  caused  by  the  bottom,  or  ends,  or  both, 
coinciding  with  the  bottom  and  sides  of  the  reservoir.  In 
•uch  cases  the  disch  will  be  greater  than  in  those  of  complete 
contraction ;  but  less  than  in  those  of  full  flow  ;  inasmuch  as 
the  opening  now  partakes  somewhat  of  the  character  of  the 
short  tubes  of  Art  8 ;  and  the  coeff  will  rise  from  .62,  or  that 
which  usually  pertains  to  openings  in  full  contraction  ;  and 
will  approach  .8,  or  that  of  full  flow,  in  proportion  to  the  ex- 
tent of  perimeter  along  which  contraction  is  suppressed ;  or 
even  to  .9  or  .98  by  the  use  of  such  openings  as  are  shown  by 
Figs  7,  8,  9. 

To  find  approximately  a  new  coefi"  of  disch;  and  the  disch 
itself,  in  eases  of  incomplete  contraction. 

Rule.  First  find  by  the  foregoing  rules,  what  would  be  the  disch  in  the  particular  case  that  may 
be  under  consideration,  supposing  the  contraction  to  be  complete.  Then  div  that  portion  of  the 
perimeter  of  the  opening  on  which  contraction  is  suppressed,  by  the  entire  perimeter.  Mult  the  quot 
by  the  dec  .152  if  the  opening  is  rectangular,  or  by  .128  if  circular.  To  the  prod  add  unity,  or  1.  Call 
the  sum,  g.  Then  say,  as  unity,  or  1,  is  to  g,  so  is  the  coeff  for  complete  contraction  in  ordinary  cases 
(usually  .62)  to  the  reqd  new  coeff.  Finally,  repeat  the  original  calculation,  only  substituting  this  new 
coeff  in  the  place  of  .62. 

According  to  this  rule,  we  have  the  following  coeff  of  discharge  for  rectangular  openings  within  pro- 
bably 3  or  4  per  cent,  when  contraction  is  not  suppressed  on  more  than  %  of  the  perimeter.  The  theo- 
retical discharge  multiplied  by  the  corresponding  coeff  will  give  the  aptual  discharge.  "When  the  con- 
traction is  carried  farther,  the  coeff  becomes  extremely  irregular,  and  is  probably  indeterminable. 

For  complete  contraction  (ordinarily) 62 

When  contraction  is  suppressed  on  %  the  perimeter 64 

"  H  "  "        67 

"  ?i  "  "        69 

"  "  "  '*  entirely  around  the  orijice 80 

Intermediate  ones  can  be  estimated  nearly  enough,  mentally. 

Rem.  1.  l¥hen,  instead  of  a  short  spout,  as  in  Fig  17,  the 
opening  is  provided  ivith  an  indefinitely  long  hor  trough, 

similarly  attached,  and  open  at  top,  there  will  be  no  practically  appreciable  diminution  of  disch  below 
that  through  the  simple  opening  as  at  a,  Fig  11 ;  provided  the  head  measured  above  the  cen  of  grav 
of  the  opening  be  at  least  as  great  as  2  or  2}i  times  the  height  of  the  opening  itself.  Therefore,  under 
such  circumstances  the  disch  may  be  calculated  by  the  rules  in  Art  9.  But  with  smaller  heads  the 
disch  diminishes  considerably  ;  so  when  the  head  above  the  center  becomes  but  as  great  as  the  height 
of  the  opening,  it  will  be  but  about  4  of  the  calculated  one.  "With  still  smaller  heads,  the  flow 
becomes  less  much  more  rapidly ;  but  has  not  been  reduced  to  any  rule. 

Rem.  2.    If,  instead  of  being  hor,  the  trough  is  INCIiIN£B 


Fig.  17. 


HYDRAULICS. 


545 


SkS  much  as  1  in  10,  the  disch  will  be  increased  very  slightly,  (some  3  or  4  per 
cent)  over  that  calculated  by  the  rules  in  Art  9.  for  the  plain  opening.  These  results  were  obtained 
by  experiments  on  a  vefy  small  scale ;  and  should  be  considered  as  mere  approximations. 

Art.  12.  In  a  case  like  Fi^  18,  wbere  contraction  is  supposed 
to  be  suppressed  at  the  bottom,  and  at  botb  vert  sides  of  the 
opening;  o,  in  consequence  of  their  coinciding 
with  the  bottom  and  sides  of  the  reservoir;  but  where  the 
front  of  the  reservoir,  iustead  of  being  vert,  is  sloped  as  at/; 
and  when  the  water,  after  leaving  the  opening,  flows  away 
over  a  slightly  sloping  apron, .y,  then  the  disch  in  cub  ft  per 
sec  may  be  approximately  found  by  Rule  1,  Case  1,  Art  9, 
only  substituting  .8  in  place  of  .62,  when  /  slopes  back  45^, 
or  1  to  1 ;  or  .74  when  /  slopes  back  6:^°,  or  with  a  base  of  1 
to  a  rise  of  2.  In  such  cases  of  inclined  fronts,  the  height  of 
the  opening  must  be  measured  vert,  or  rather  at  right  angles 
to  the  floor  of  tlie  reservoir ;  and  not  in  a  line  with  the 
slop!  n  15  front. 

Rem.  HVhen  the  front,  /,  of  t  Jie  reservoir  is  vert,  and  a  sloping; 
apron  or  trouj^h,  gr,  is  used,  having  its  upper  edge  level  with  the  bottom 
of  the  opening,  the  disch  is  not  appreciably  diminished  below  that  which  takes  place  freely  into  the 
air,  provided  the  head  above  the  cen  of  grav  of  the  opening  is  not  less  than  from 

18  to  24  ins,  for  an  opening  6  to  9  ius  high. 
12  to  16  '*       "     "        "        4  ins  high. 

9  "     "        "        2  ins  or  less,  kigh. 

Art.  13.  To  find,  approximately,  the  time  reqd  for  the  emp- 
tying' of  a  pond,  or  any  other  reservoir,  as  Fig- 19,  w^hich  is 
not  of  a  prismatic  shape ;  through  an  opening,  n,  near  the 
bottom. 

Role.  First  ascertain  the  exact  shape  and  dimensions 
of  the  reservoir.  If  large,  and  irregular,  it  must  be  care- 
fully survej-ed ;  and  soundings  taken,  and  figured  upon  a 
correct  plan  and  cross-sections.  Next,  consider  the  entire 
body  of  water  to  be  divided  into  a  series  of  thin  hor  strata, 
A,  B,  C,  D  ;  the  top  line  of  the  lower  one  being  at  least  a 
few  ins  above  the  top  of  the  opening  n.  It  is  not  necessary 
that  these  strata  should  be  of  equal  thickness ;  although 
the  thinner  they  are,  the  more  correct  will  the  result  be. 
The  depth  of  the  lower  one,  D,  will  vary  to  some  extent 
with  the  height  of  the  opening ;  those  next  above  it  should 
not  exceed  about  a  foot  in  thickness,  until  a  depth  of  6  or  8  feet  is  reached ;  then  they  may  conve- 
niently, and  with  sufficient  accuracy,  be  increased  to  about  2  ft,  for  6  or  8  ft  more  ;  and  so  on ;  be- 
coming thicker  as  they  approach  the  surf.  By  aid  of  the  drawings,  calculate  the  content  of  each 
stratum  in  cub  ft.  Now,  since  the  strata  are  thin,  we  may,  without  serious  error,  assume  each  of 
them  to  be  prismatic,  as  shown  by  the  dotted  lines ;  and  may  assume  that  the  head  under  which  ea«h 
stratum  (except  the  lowest)  empties  itself  through  n,  is  equal  to  the  vert  height  from  the  center  of 
the  opening  to  the  center  of  the  stratum.  Thus,  mn  will  be  the  head  of  A  ;  wn.  the  head  of  B  ;  xn, 
the  head  of  C  Then,  for  the  stratum  A.  by  Rule  1,  Art  9,  (only  using  mn  as  the  head  instead  of  on,) 
and  instead  of  the  coefif  .62  of  that  rule  (which  can  only  be  used  if  n  is  in  complete  contraction)  using 
.64,  or  whatever  other  coeff  near  the  end  of  Art  11  applies  to  the  case,  calculate  the  disch  in  cub  ft 
per  sec.  Div  the  content  of  the  stratum  .4  by  this  disch,  and  the  quot  will  be  the  number  of  sec  reqd 
for  discharging  A.  Using  the  head  wn,  proceed  in  precisely  the  same  way  with  the  stratum  B ;  and 
using  the  head  xn,  do  the  same  with  C.  Finally,  for  the  lower  stratum  D.  find  by  Rule  1,  Art  9,  (with 
the  same  caution  as  before  respecting  the  proper  coeff,)  in  what  time  it  would  empty  itself  under  a 
constant  head  equal  to  y  n,  measured  from  its  surf  to  the  center  of  the  opening.  Double  this  time  will 
be  that  reqd  to  empty  itself  in  the  case  before  us,  under  its  varying  head.  Finally,  add  together  all 
these  separate  times  ;  and  their  sum  will  be  the  entire  time  reqd  to  empty  the  pond,  or  reservoir,  ap- 
proximately enough  for  practical  purposes. 


■\- 


Ec|19 


35 


546 


HYDRAULICS. 


^  The  Miner's  Incli. 

The  miner's  inch  is  a  unit  of  rate  of  discharge  of  water,  largely  used  in  the 
far  western  mining  States.  It  is  expressed  in'terms  of  a  standard  orifice,  usually 
1  inch  square  and  vertical,  and  a  standard  head.  The  head  varies  from  about  3 
to  9  ins,  but  6  ins  is  a  commonly  accepted  standard.  Under  that  head,  with 
coefficient  =  0.62,  the  discharge,  through  an  oritice  1  inch  square,  would  be:  per 
sec,  0.0244  cu  ft  =  0.183  U.  S.  gals  (of  231  cu  ins)  ;  per  min,  1.466  cu  ft  =  10.96 
gals;  per  day,  2,111  cu  ft  -=  15,790  gals.  Usually  the  oritice  is  of  fixed  depth 
and  adjustable  length  (see  Fig.),  and  the  standard  head  is  obtained  and  main- 
tained by  such  adjustment.  The  relation  between  area  and  shape  of  orifice  and 
the  discharge  per  sq  in  of  area  is  indicated  by  the  following  table:* 

Discharge,  in  cu  ft  per  min,  per  square  inch  of  opening,  under  6  ins  head. 
Length  of  opening,  ins. 
Opening  2  ins  high,  .  .   . 
4  "      '«^      .   .   . 


4 

6 

8 

12 

24 

100 

1.473 
1.450 

1.480 
1.470 

1.484 
1.481 

1.487 
1.491 

1.490 
1.501 

1.494 
1.509 

r^^^N.          .-^---V  _^-^::>r^ 

M         ^ 

3 

» 

/^^^^<-^:;^^====^=f^ 

i 

j  cu.  ft. 

<      r 

M^S^^^i^ 

0.029 
O.OSBJ^ 

^ 

(^persec. 

\^^^ 

'^-^^^i^-^::=^^=^ 

J         0.627  > 

^ 

] 

.8^ 

j  cu.  ft. 
[per  mln 

r^==^ 

^^:rr:^::^^^;:::;--^^  o. 

m^ 

^ 

^ 

CC^-^ 

t^^;;:;^^ ao25^ 

^^ 

\a^ 

^ 

XL 

-'^^                       0.024  J 

^ 

i.( 

^ 

j  cu.  ft. 

0.023^ 

y^ 

2,. 

m 

^ 

(per day 

0.^ 

o.cm^ 

M. 

^      2,2«0 

2,80< 

2,400^ 
0.21^ 

D.22 

- 

\  gallOQ9 
I  per  sec. 

0.( 

19-^^'^ 

1. 

i^^ 

'^      i 

l,100>"^^ 

0.2( 

^ 

j  gallous 
^per  mla 

^ 

-^     2,000^ 

1 

3^ 

^ 

0.018^ 

1.2 

1,800^ 

,900^ 

0. 

J 

c 

18^ 

;.iy^ 

^ 

^ 

1^ 

,00( 

^ 

j  galloas 
/per day 

\.y^ 

1,700^ 

0.11 

^ 

n 

^ 

18,000^ 

"^ 

0.16 

17, 

m^ 

1,600 

^ 

0.15^ 

15^ 

^  16,000^^ 

0.14^ 

0-j 

^\h 

000^ 

^ 

wm^ 

Discharge 

1        1 

, 

^^y^ 

13,000 

_u 

lucu 
2  V 

,  ft.  per  see. 

2  g  xj  hea 

d  ib  ft. 

12,000^ 

144 

3  4  5  6  7  8  $>  ^ 

Head.  In  laches  i 

Diagram  of  discliarg'e  tliroug*!!  an  orifice  1  inch  square. 

The  thickness  of  plank,  in  which  the  orifice  is  cut,  varies  from  1  to  3  ins,  and 
the  lower  edge  of  the  orifice  is  often  chamfered,  flaring  outwardly.  The  bottom 
of  the  orifice  is  sometimes  flush  with  the  bottom  of  the  box,  but  is  usually  raised 
2  or  3  ins  above  it,  as  in  our  Fig. 


*  Condensed  from  circular  of  Pelton  Water  Wheel  Co. 


HYDRAULICS. 


547 


Art.  14  (a).  On  the  discharge  of  water  over  weirs  or  Over^^ 
falls.  The  weir  affords  a  very  convenient  means  for  gauging  the  flow  of  small 
streams,  for  measuring  the  quantity  of  water  supplied  to  water-wheels,  etc. 

[b)  A  measuring  weir  is  always  arranged  with  its  baclc,  or  up-stream  side,  a  &, 
Fig.  20,  vertical,  aud  as  nearly  as  may  be  at  right  angles  to  the  direction  of  flow 
of  the  stream.  The  ends,  a  A,  a  h.  Figs.  21  and  22,  are  vertical,  and  the  crest  a  a 
is  horizontal. 

(c)  £ii€l  eon  tractions.  When  the  weir  a  a  extends  entirely  across  the 
channel  of  approach,  as  in  Fig.  21,  so  that  its  ends  a  h'a  h  coincide  with,  or  form 
portions  of,  the  sides  65  of  the  channel,  contraction  (Art.  9,  p.  541)  takes  place 
only  on  the  top  and  bottom  of  the  sheet  of  water  passing  over  the  weir,  as  at 
m  c  and  at  a,  Fig.  20,  and  is  entirely  "  suppressed  "  at  the  ends,  so  that  the  water 
flows  out  as  shown  in  Fig.  21  a.  Such  a  weir  is  called  a  suppressed  weir, 
or  a  weir  without  end  contraction.    But  when,  as  in  Figs.  22  and  22  a, 


the  ends  ah,  ah  are  at  a  distance  from  the  sides  5  5  of  the  channel  or  reservoir, 
contraction  takes  place  at  the  ends- of  the  weir,  as  shown  at  a  and  a,  as  well  as 
over  the  crest.  Such  contraction  diminishes  the  discharge.  A  weir  of  this  kind 
is  called  a  weir  with  end  contractions. 

Other  things  being  equal,  the  extent  of  the  contraction,  and  its  effect  upon  the 
discharge,  increase  with  the  head  H.  When  the  length  a  a  or  L  of  the  weir  ex- 
ceeds about  10  times  the  head  H,  the  effect  of  the  end  contractions  upon  the  dis- 
charge is  nearly  imperceptible ;  but  as  the  length  diminishes  in  proportion  to 
the  head,  the  effect  of  the  contraction  increases  rapidly.  Mr.  P'rancis  (Art.  14  m) 
found  that  when  L  =  only  4  X  H,  the  discharge  was  reduced  6  per  cent,  by  com- 
plete end  contractions.  In  view  of  the  uncertainty  as  to  the  effect  of  end  con- 
tractions, it  is  better  to  avoid  them  and  to  use  weirs,  like  Fig.  21,  where  the  con- 
traction is  suppressed  ;  but  if  end  contraction  is  permitted  at  all,  it  must  be  made 
complete;^  for  the  coefficients  given  do  not  apply  to  cases  of  incomplete 
contraction,  i.e.,  with  contraction  only  par^/y  suppressed. 

{d)  In  a  w^eir  w^ithont  end  contraction,  care  must  be  taken  that  the 
air  has  free  access  to  the  space  {w,  Fig.  20,  or  22  h)  behind  the 
falling  sheet  of  water.  Otherwise  a  partial  vacuum  forms  there, 
the  sheet  is  drawn  inward  toward  the  weir,  and  the  discharge 
is  greatly  modified.  At  the  same  time,  the  sheet  should  be  pre- 
vented from  expanding  laterally  as  it  leaves  the  crest.  Both  of 
these  objects  may  be  attained  by  prolonging  the  upper  portion 
only  of  both  sides  of  the  channel  a  little  way  down-stream 
beyond  the  crest  and  the  upper  part  of  the  falling  sheet,  as  in 
Fig.  22  h.  Mr.  Francis  found  that  such  projections,  by  confining 
the  sheet  laterally,  diminished  the  discharge  about  0.4  per  cent. 

(e)  Ordinarily  the  crest  is  *'  in  thin  plate"  or  "in  thin  partition"  (see  foot- 
note t,  p.  541),  so  that  the  sheet  passing  over  the  weir  touches  it  only  at  the  very 
corner,  a.  Fig.  20,  A  rounded  corner  increases  the  discharge,  as  does  the  round- 
ing of  the  edges  of  an  orifice  (Art.  8,  p.  541),  and  a  crest  sufficiently  wide  to  de- 
flect the  falling  sheet  diminishes  the  discharge  (see  coeflBcients  for  this  case  in 
Table  15,  p.  554),  but  both  forms  introduce  much  uncertainty,  and  should  there- 
fore be  avoided. 

*  The  contraction  is  said  to  be  "complete"  when  it  is  practically  as  great  as  it 
could  be  made  by  any  further  increase  of  the  distance  a  s,  Figs.  22  and  22  a ;  and  thlg 
is  believed  to  be  attained  when  a  s  is  made  equal  to  the  head  H. 


Fig.22b 


548  HYDRAULICS. 

(/)  The  lengtli  Ia  of  the  crest.  Figs.  21  to  22  a,  should  be  at  least  three 
times  thr!  head  H,  in  order  to  reduce  the  effect  of  friction  of  the  sides  ss  and 
that  of  end  contractions  where  such  exist.  The  heig'ht  jj,  i^'ig.  20,  of  the 
vertical  back  a  b  in  contact  with  the  water  should  be  not  less  than  twice  the 
head  H;  for,  in  order  to  reduce  the  velocity  of  approach  (see  Art.  14  u),  the 
cross-section  of  the  channel  leading  to  the  weir  should  be  large  in  propor- 
tion to  that  of  the  stream  ac.  The  cross-section  of  the  channel  of  approach 
should  be  as  ri^gaiar  as  possible. 

((/)  The  weir  should  be  stoutly  built,  as  vibrations  of  the  structure  may 
seriously  modify  the  discharge. 

(Ji)  Theoretically,  the  head  is  the  vertical  distance  H',  Fig.  24,  from  the 

crest  a  to  a  point  o'  where  the  water  is  perfectty  still,  and  the  surface  therefore 
hori.zoritaL  But  in  fact  the  head  is  usually  measured  from  the  crest  a  to  a  point  o 
a  few  feet  back  from  the  weir,  where  the  water  is  only  comparatively  still,  the 
velocity  of  approach  being  perceptible.  (See  Art.  14,  u.)  The  difference  between 
the  head  H  actually  measured  and  the  head  H'  to  still  water  is  usually  very 
slight.     It  is  greatly  exaggerated  in  the  figure. 

The  correct  measurement  of  the  head  is  a  delicate  matter,  the  dis- 
charge being  increased  or  diminished  about  l^per  cent,  by  1  per  cent,  of  in- 
crease or  diminution  of  the  head.  Waves  or  ripples  and  other  disturbances  of 
the  surface,  and  capillary  attraction,  are  the  chief  sources  of  error. 

(/)  To  avoid  the  latter  difficulty,  the  hook-g-au^e  is  used  for  measuring 
the  height  of  the  water  surface  in  important  cases.  This  consists  of  a  long  grad- 
uated rod,  provided  at  its  foot  with  an  upturned  hook  or  point,  and  sliding 
vertically  (by  means  of  a  screw  motion)  in  a  fixed  support,  to  which  is  attached 
a  vernier  indicating  on  the  scale  the  height  of  the  point.  The  sliding  rod  is 
first  run  down  until  the  point  is  well  below  the  surface,  and  then  gradually 
raised  by  means  of  the  screw  until  the  point  just  reaches  the  surface,  which  is 
indicated  by  the  first  appearance  of  a  "pimple"  in  the  water  surface  imme- 
diately over  the  hook.  Under  favorable  circumstances  a  good  hook-gauge  may 
be  read  within  from  .0002  to  .0005  foot. 

{j)  To  avoid  inaccuracies  due  to  the  distnrbance  of  the  surface  by 

the  current,  by  wind,  etc.,  the  level  is  sometimes  taken  (with  the  hook-gauge  or 
otherwise)  in  a  side  chamber  which  communicates  with  the  main  channel  of 
approach.  The  surface  in  the  chamber  maintains  the  same  level  as  that  in  the 
channel  itself,  but  is  comparatively  free  from  disturbance.  Or  a  bucket  com- 
municating with  the  channel  by  means  of  a  pipe,  can  be  made  to  serve  in  the 
same  way.  Either  may  of  course  be  sheltered  from  the  wind.  Caution. 
Messrs.  Fteley  and  Stearns  found  that  when  the  bucket  or  chamber  communicated 
with  the  water  n,mr  the  bottom  and  close  behivd  the  weir,\he  head  thus  obtained  was 
generally  somewhat  greater  than  that  found  by  measurement  near  the  surface 
and  (i  feet  back  from  the  weir.  But  Mr.  Francis  found  the  difference  scarcely 
perceptible. 

(k)  Great  care  is  necessary  in  adjusting*  the  hook-gauge  for  the 
height  of  the  crest;  for  any  error  in  this  affects  all  the  subsequent  experi- 
ments. The  hook  is  usually  adjusted  to  the  height  of  the  surface  when  the  latter 
just  reaches  the  level  of  the  crest ;  but  this  method  is  rendered  inaccurate  by 
capillary  attraction  at  the  crest.  A  more  accurate  method  is  to  have,  in  addition 
to  the  hook-gauge,  a  stout  jized  hook,  pointing  upward,  the  level  of  which,  rela- 
tively to  that  of  the  crest,  may  be  ascertained  by  means  of  an  engineer's  level, 
holding  the  rod  on  the  crest  and  also  on  the  point  of  the  fixed  hook.  The  water 
surface  is  then  allowed  to  fall  slowly  until  a  "  pimple  "  just  appears  over  the  fixed 
hook.  It  is  then  kept  at  that  level  and  the  hook-gauge  adjusted  accordingly. 
Or  if  the  gauge-hook  is  a  stout  one,  the  levelling  rod  may  be  set  at  once  upon  its 
point  without  having  recourse  to  a  fixed  hook.  It  is  better  to  adjust  the  hook- 
gauge  so  as  to  read  zero  for  the  crest  level,  which  is  thus  made  the  datum ;  for 
the  reading  of  the  hook-gauge  for  the  water  surface  then  gives  the  head  H  at 
once,  and  without  subtracting  the  height  of  the  crest. 


HYDRAULICS. 


549 


(1)  Formalse  for  weir  cliscbargre. 

Let 
Q,  ^  the  actual  discharge  over  the  weir,  in  cubic  feet  per  second  ;  * 
Q,'  -=  the  theoretical  discharge  over  the  weir,  in  cubic  feet  per  second', 
H  =H'  t  =  the  vertical  distance  or  head  a  vi,  Fig.  24,  p.  556,  in  feet,*  measured 
from  the  crest  a  to  the  korizonlai  surface  o'  of  still  water  up-stream  from 
the  weir ; 
li   =  the  length  a  a  of  the  weir,  in  feet,*  Figs.  21  to  22  a ;  ^ 

g    =  the  acceleration  of  gravity  =  say  82.2  feet*  per  second  per  second*,       ^ 
,     .  „  ,.    ,  actual  discharge  Q 

c     =  coefhcient  of  discharge  =  -r r^ — \ — t-- — ^ =  ^,  J 

^        theoretical  discharge       Q' 

2 


',    =--o}/2g=my2g=  say  5.36  c  =  say  8.025  m. 

Then,  tor  the  tlieoretical  dischargee,  we  have 
2  ^  „  ,^_,-, 


a'=  3  LHv'25'H;^ 

and  for  the  actual  <Ii«$cliarg-e, 

iit  -cQ' 

=  I  c  L  H  y2jH , 


=  mLH  V2gK     

=  X  LH  v'ir=xL/BF=x  LhI 


(1) 


See  foot-notes 


For  the  value  of  the  coefficient  (c,  m,  or  a;  J)  we  have  recourse  to 
experiment,  measuring  the  actual  discharge  and  comparing  it  with  the  theoret- 
ical one,  as  in  the  following  articles.  • 

*  The  formulae  apply  equally  to  any  system  of  measures,  as  the  English,  the  metric, 
etc.  It  is  requisite  merely  that  the  units,  of  length,  of  time,  etc.,  used,  be  the 
same  throughout.     In  metric  measure,  g  =  9.81  meters  per  second  per  second. 

f  For  the  present  we  suppose  the  head  to  be  measured  to  still  water,  so  that  H  =  H'. 
When  this  is  not  the  case,  see  "  Velocity  of  Approach,"  Art.  14  (u),  etc. 

X  It  will  be  noticed  that  the  formulae  (2),  (3)  and  (4),  with  their  corresponding  coef- 
ficients, c,  m,  and  x,  are  really  identical,  dittering  only  in  form.  The  last  is  the  most 
convenient  in  practice,  but  all  are  met  with  in  works  on  bydrauhcs. 

§  When  water  issues,  under  a  head  H,  from  a  horizontal  orifice  in  the  bottom  of  a 
vessel,  the  theoretical  velocity  (Art.  7,  is  =  ]TgK;  and  this  may  be  regarded 

as  true  also  for  vertical  orifices  in  the  sides  of  vessels,  provided  the  head  H  to  the 
center  of  gravity  of  the  orifice  is  at  least  two  or  three  times  the  vertical  dimension 
of  the  orifice;  for  in  both  cases  the  theoretical  velocities  through  the  several  parts 
of  the  orifice  may  be  taken  as  equal.  But  when  a  vertical  orifice  is  nearer  to  the  sur- 
face, or  when  it  reaches  to  the  surface  as  in  the  case  of  a  weir,  we  must  take  into  con- 
sideration the  differences  in  the  velocities  with  which  the  water  issues  from  points  at 
different  depths. 

Theoretically,  the  particles  pass  the  oblique  plane  ao\  Fig.  23,  in  horizontal  lines, 
with  velocities  (=  y  2  gh  =  8.025  \/h)       ^ 

proportional  to  the  square  roots  of   Q  '^V  ft 

their  several  vertical  depths  h  (not 
Indicated  in  fig.)  below  still  water 
surface  at  o'.  Therefore  if  from  a  m 
we  imagine  horizontal  lines  a  a\ 
d  d',  V  v\  c  c\  etc.,  etc.,  to  be  drawn, 
representing  all  these  velocities  to 
any  scale,  then  the  outer  ends  a', 
d',  v\  c',  etc.,  etc.,  of  these  lines 
will  form,  with  a  m  and  a  a\  a 
parabolic  segment  a  m  c' «',  the  area 
of  which  is : 

2  2  '  2  

area  —  -  area  of  rectangle  a  n  (see  Parabola,  p,  192)  =~amY,aa'—      H  |/2  g  H; 

and  this  area  in  square  feet,  multiplied  by  the  thickness  of  the  escaping  sheet  of 


550 


HYDRAULICS. 


(m)  Mr.  James  B.  Francis*  experimented  at"  the  lower  locks,"  Lowell, 
Mass.,  in  1852,  with  weirs  10  feet  long,  5  feet  and  2  feet  high,  under  heads  from  7 
to  19  inches.    To  apply  his  results,  the  following  conditions  must  exist: 

The  head  H,  Fig.  20,  must  be  between  6  and  24  inches.  The  height  jo  of  the 
vertical  back  of  the  weir  above  the  bottom  b  of  the  channel  must  be  at  least 
twice,  the  head  H.  The  crest  a  must  be  "  iu  thin  partition  "  (foot-note  t  p.  541), 
and  its  length  L,  Figs.  21  to  22  a,  must  be  at  least  3  times  the  head  H.  The  ends. 
cuk,  a  h  must  be  vertical,  and,  when  there  is  end  contraction,  "  in  thin  partition." 
wVVhen  there  is  end  contraction,  Mr.  Francis  first  deducts  from  the  actual  length 
L  of  the  weir  one  tenth  f  of  the  head  H  for  each  end  where  contraction  occurs. 
Thus,  if  7i  =  the  number  of  end  contractions  (two  in  Fig.  22), 


InFlg.22,Q  =  a;  (^^-f^)  ^  ^^- 1  =  ^  (^~|^)  ^^ 


(5) 


But  within  the  limits  specified  above,  the  formula  is  very  approximate  wUhout 
correction  for  end  contraction,  provided  the  length  L  of  the  weir  is  at  least  10 
times  the  head  H ;  and  within  6  per  cent,  of  the  truth  when  L  is  =  4  H.  When 
there  is  no  end  contraction,  of  course  no  such  correction  is  required,  and  the 

3 

formula  remains  Q  =  x  L  H  v^ITJ  =  x  L  H    • 
Mr.  Francis  ^ives  x  =  3.33  for/ee^,  §  or 

number  of 


Bischarg^e  =  3.33  X  (length  - 


X 


head  H 


)XH*  t 


end  contractions  '^  10 
the  mean  of  his  88  experiments  being  3.3318.  The  least  value  of  x  obtained  by 
him  was  3.3002,  or  1  in  112  less  than  3.33;  and  the  greatest  was  3.3617,  or  1  in  105 
more  than  3.33.  Hence,  with  x  =  3.33,  the  formula  will  give  the  discharge  for 
each  of  his  experiments  within  1  per  cent.  In  67  out  of  the  88  experiments 
X  ranged  between  3.32  and  3.35,  and  in  53  between  3.32  and  3.34.  When  x  is  3.33, 
m  is  ^  0.415,  and  c  is  =  0.622. 

The  height  of  the  surface  was  measured  six  feet  back  from  the  weir  by  two 
hook-gauges,  one  on  each  side  of  the  channel ;  and  the  mean  of  their  readings 
was  used  in  calculating  the  coefficient  x. 


water,  or  length  L  of  weir,  in  feet,  gives  the  theoretical  discharge  in  cubic  feet  per 
second.    Or 

2 

Q'  =  L  X  area  a  m  c'  a'  =  L  X  — H  y2  g  H. 

Hence,  area  a  m  c'  a'  in  sq.  ft.  represents  the  theoretical  disch.  in  cub.  ft.  per  sec.  over 
1  ft.  length  of  weir,  under  head  H.   The  theoretical  mean  vel.  through  the  section  a  o'  is 
V  =  „'  =  iheorettcaLdisohargeQ',  ^  |  ^^  ^  ^  L  H  =  ^  >'2TH7 

area  am  3  ^    ^  Z 


T^ig.23 


or  two  thirds  of  the  theoretical  hori= 
'  zontal  velocity  a  a'  of  the  particles 
passing  immediately  over  the  weir. 
As  in  the  case  of  orifices  (Art.  9, 
p.  541),  the  actual  vel.  at  the  %mallek 
section  of  the  sheet  after  passing  the 
weir  (corresponding  to  the  "vena 
contracta ")  is  probably  very  nearly 
equal  to  this  theoretical  velocity. 

*  "Lowell  Hydraulic  Experi- 
ments," Van  Nostrand,  New  York, 
1883. 

t  In  Messrs.  Fteley  and-  Stearns'  experiments  this  figure  was  not  constant  at  0.10, 
but  varied  between  0.061  and  0.124,  generally  increasing  as  the  head  decreased. 

%  We  here  suppose  the  head  to  be  measured  to  the  surface  of  still  water,  so  that  H 
and  H'  (see  Art.  14  h,  p.  548)  are  the  same.    See  Velocity  of  Approach,  Art.  14  (m). 

^  Since  1  meter  =  3.2808  ft.,  the  value  of  a:  for  metric  measure  corresponding  to  Mr. 
Francis'  3.33,  is  =  3.33  -r-  v/3.2808  =  3.33  s-  1.8113  =  1.83& 


^2a-B:—^, 


HYDRAULICS. 


551 


Table  13.*  I>iscliarg-e  in  cubic  feet  per  second  for  eacli  foot 
in  leng-th  of  weir  in  thin  plate  and  without  end  contraction,  by  the  Francis 

formula:  Disehar;^e,  Q  --  3.33  L  fit  =  3.33  L  H  yK. 

Very  approximate  also  when  there  is  end  con'iraction,  provided  that  L  is  at 
least  =  10  H ;  and  but  about  6  per  cent*  in  excess  of  the  truth  if  L  =  4  H.  Mr. 
Francis  limits  the  formula  to  heads  H  from  0.5  foot  to  2.0  feet,  but  no  serious 
error  will  result  from  using  the  table  for  any  of  the  heads  given.  For  weirs 
of  other  leng-ths  than  1  foot,  multiply  the  tabular  discharge  by  the  actual 
length  in  feet.    .01  foot  =  0.12  inch  =  scant  %  inch. 


Head,  H, 

Cub.  ft. 

Head,  H, 

Cub  ft. 

Head,  H, 

Cub.  ft. 

Head,  H, 

Cub.  ft. 

Head,  H. 

Cub.  ft. 

in  ft. 

per  sec. 
0.003 

in  ft. 
.51 

per  sec. 
1.213 

in  ft. 
1.01 

per  sec. 

in  ft. 

per  sec. 

in  ft. 

per  sec. 

.01 

3.380 

1.51 

6.179 

2.01 

9.489 

.02 

0.009 

.52 

1.249 

1.02 

3.430 

1.52 

6.240 

2.02 

9.560 

.03 

0.017 

.53 

1.285 

1.03 

3.481 

1.53 

6.302 

2.03 

9.631 

.04 

0.027 

.54 

1.321 

1.04 

3.532 

1.54 

6.364 

2.04 

9.703 

.05 

0.037 

.55 

1.358 

1.05 

3.583 

1.55 

6.426 

2.05 

9.774 

.06 

0.049 

.56 

1.395 

1.06 

3.634 

1.56 

6.488 

2.06 

9.846 

.07 

0.062 

.57 

1.433 

1.07 

3.686 

1.57 

6.551 

2.07 

9.917 

.08 

0.075 

.58 

1.471 

1.08 

3.737 

1.58 

6.613 

2.08 

9.989 

.09 

0.090 

.59 

1.509 

1.09 

3.790 

1.59 

6.676 

2.09 

10.062 

.10 

0.105 

.60 

1.548 

1.10 

3.842 

1.60 

6.739 

2.10 

10.134 

.11 

0.121 

.61 

1.586 

1.11 

3.894 

1.61 

6.803 

2.11 

10.206 

.12. 

0.138 

.62 

1.626 

1.12 

3.947 

1.62 

6.866 

2.12 

10.279 

.13 

0.156 

.63 

1.665 

1.13 

4.000 

1.63 

6.930 

2.13 

10.352 

.14 

0.174 

.64 

1.705 

1.14 

4.053 

1.64 

6.994 

2.14 

10.425 

.15 

0.193 

.65 

1.745 

1.15 

4.107 

1.65 

7.058 

2.15 

10.49S 

.16 

0.213 

.66 

1.786 

1.16 

4.160 

1.66 

7.122 

2.16 

10.571 

.17 

0.233 

.67 

1.826 

1.17 

4.214 

1.67 

7.187 

2.17 

10.645 

.18 

0.254 

.68 

1.867 

1.18 

4.268 

1.68 

7.251 

2.18 

10.718 

.19 

0.276 

.69 

1.909 

1.19 

4..323 

1.69 

7.316 

2.19 

10.792 

.20 

0.298 

.70 

1.950 

1.20 

4.377 

1.70 

7.381 

2.20 

10.866 

.21 

0.320 

.71 

1.992 

1.21 

4.432 

1.71 

7.446 

2.21 

10.940 

.22 

0.344 

.72 

2.034 

1.22 

4.487 

1.72 

7.512 

2.22 

11.015 

.23 

0.367 

.73 

2.077 

1.23 

4.543 

1.73 

7.577 

2.23 

11.089 

.24 

0.392 

.74 

2.120 

1.24 

4.598 

1.74 

7.643 

2.24 

11.164 

.25 

0.416 

.75 

2.163 

1.25 

4.654 

1.75 

7.709 

2.25 

11.239 

.26 

0.441 

.76 

2.206 

1.26 

4.710 

1.76 

7.775 

2.26 

11.314 

.27 

0.467 

.77 

2.250 

1.27 

4.766 

1.77 

7  842 

2.27 

11.389 

.28 

0.493 

.78 

2.294 

1.28 

4.822 

1.78 

7.908 

2.28 

11.464 

.29 

0.520 

.79 

2.338 

1.29 

4.879 

1.79 

7.975 

2.29 

11.540 

.30 

0.547 

.80 

2.383 

1.30 

4.936 

1.80 

8.042 

2.30 

11.615 

.31 

0.575 

.81 

2.428 

1.31 

4.993 

1.81 

8.109 

2.31 

11.691 

.32 

0.603 

.82 

2.473 

1.32 

5.050 

1.82 

8.176 

■   2.32 

11.767 

.33 

0.631 

.83 

2.518 

1.33 

5.108 

1.83 

8.244 

2.33 

11.843 

.34 

0.660 

.84 

2.564 

1.34 

5.165 

1.84 

8.311 

2.34 

11.920 

.35 

0.690 

.85 

2.610 

1.35 

5.223 

1.85 

8.379 

2.35 

11.996 

.36 

0.719 

.86 

2.656 

1.36 

6.281 

1.86 

8.447 

2.36 

12.073 

.37 

0.749 

.87 

2.702 

1.37 

5.340 

1.87 

8.515 

2.37 

12.150 

,38 

0.780 

.88 

2.749 

1.38 

5.398 

1.88 

8.584 

2.38 

12.227 

.39 

0.811 

.89 

2.796 

1.39 

5.457 

1.89 

8.652 

2.39 

12.304 

.40 

0.842 

.90 

2.843  . 

1.40 

5.516 

1.90 

8.721 

2.40 

12.381 

.41 

0.874 

.91 

2.891 

1.41 

5.575 

1.91 

8.790 

2.41 

12.459 

.42 

0.906 

.92 

2.939 

1.42 

5.635 

1.92 

8.859 

2.42 

12.536 

.43 

0.939 

.93 

2.987 

1.43 

5.694 

1.93 

8.929 

2.43 

12.614 

.44 

0.972 

.94 

3.035 

1.44 

5.754 

1.94 

8.998 

2.44 

12.692 

.45 

1.005 

.95 

3.083 

1.45 

5.814 

1.95 

9.068 

2.45 

12.770 

.46 

1.089 

.96 

3.132 

1.46 

5.875 

1.96 

9.138 

2.46 

12.848 

.47 

1.073 

.97 

3.181 

1.47 

5.935 

1.97 

9.208 

2.47 

12.927 

.48 

1.107 

.98 

3.231 

1.48 

5.996 

1.98 

9.278 

2.48 

13.005 

.49 

1.142 

.99 

3.280 

1.49 

6.057 

1.99 

9.348 

2.49 

13.084 

.50 

1.177 

1.00 

3.330 

1.50 

.  6.118 

2.00 

9.419 

2.50 

13.163 

*  Table  13  is  an  extension  of  the  "  original "  table  published  in  our  first  edition, 
1872.  Most  of  the  values  now  given  are  taken,  by  permission,  from  a  table  published 
by  Messrs,  A.  W.  Huuking  and  Frank  S  Hart,  of  Lowell,  Mass.,  in  May,  1884. 


552 


HYDRAULICS. 


(n)  Messrs.  A.  Fteley  and  F.  P.  ^Stearns  *  experimented  at  Boston, 
Mass.,  in  1877-79,  upon  weirs  5  feet  and  19  feet  long,  3  feet  2  inches  and  6  feet  63^ 
inches  high,  and  under  heads  from  U.8  inch  to  19  inches.  For  weirs  in  thin  par- 
tition and  without  end  contraction,  with  a  rectangular  and  uniform  channel  of 
approach  and  under  heads  greater  than  0.07  foot  or  0.84  inch  (other  conditions 
as  specified  in  (6)  and  (tZ;),  their  formula  is: 


^t 


(6). 


Bischarge,  Q  =  3.31  L  H2  +  0.007  L 

=  0.4125  L  H  1/27H  +  0.007  L 
In  their  experiments,  the  heads  were  measured  six  feet  back  from  the  weir. 
The  total  variation  in  the  values  of  the  coeflBcients  obtained  was  about  2^^  per 
cent.    Compare  foot-note  §  below. 

(o)  M.  Baziut-  experimented  at  Dijon,  France,  in  1886-88,  with  weirs  from 
t^  about  13^  to  6]4  feet  long,  from  about  9  inches  to  3  feet  9  inches 

§  high,  and  under  heads  from  2}4  to  21  inches.    The  top  of  the 

weir  is  shown  in  Fig,  23  a.  The  weirs  were  placed  at  diflferent 
points  in  a  rectangular  and  regular  canal  TOO  feet  long, 
smoothly  lined  with  cement.    Compare  foot-note  ^  below. 

While  Mr.  Francis  and  Messrs.  Fteley  and  Stearns  provide 
for  the  effect  of  velocity  of  approach  (see  Art.  14  10  and  v}  hj 
modifying  the  measured  head  H,  M.  Bazin  includes  it  in  th^ 
cojfficient  m  in  the  formula  Q  =  m  L  H  ]/2  g  H.  After  compar- 
ing his  experiments  with  those  of  Messrs.  Fteley  and  Stearns, 
^(Art.  14  n  and  v),  he  gives,  for  m,  in  cases  where  there  is  no 
'velocity  of  approach,  the  values  M  in  the  last  column  of  Table 
14,  or,  very  approximately, 
o  /^'o  velocity  of  approach, \  _,      ^  -iac   ,  ^-^^^ 

(      H  and  H' identical      )  M  =  0.405  +  ^jp. 

When  velocity  of  approach  is  to  be  taken  into  account: 


Fig.  23  a. 

Measurements 
in  meters. 


:M 


[— (wf^n- 


(7); 


where  H  is  the  head  actually  measured  to  running  watei^ 
and  p  is  the  height  a  &  of  the  weir,  Fig.  20.    H  and  p  must  of 
course  both  be  measured  in  the  same  unit,  as  both  in  meters,  or  both  in  feet,  etc. 

M.  Bazin  believes  that  except  in  the  case  of  very  low  weirs  (which  should  be 
avoided)  the  values  of  m  given  by  formula  (7)  and  in  Table  14  calculated  from 
it,  will  be  found  within  1  per  cent,  of  the  truth  for  weirs  in  thin  partition  ancl 
without  end  contraction,  if  the  conditions  of  his  experiments  are  exactly  repro- 
duced, and  provided  especially  that  the  sheet  of  water  is  not  allowed  to  expand 
laterally  after  passing  the  crest  (Art.  14  (d))  and  that  the  air  has  free  access  to 
the  space  lo,  Fig.  20,  behind  the  falling  sheet  of  water. 

For  heads  between  4  inches  and  1  foot,  M.  Bazin  gives,  as  sufficiently  ap- 
proximate, 

when  there  is  no  velocity  of  approach,  M  =  0.425,g 

and,  to  allow  for  velocity  of  approach,  m  =  0.425  -f  0.21  (  ^t-^ —  )• 

\±1  -\-p/ 


*  Transactions,  American  Society  of  Civil  Engineers,  Jan.,  Feb.  and  March,  1883. 

+  See  correction  for  Telocity  of  Approach,  Art.  14-  (ii). 

X  Experiences  nouvelles  sur  I'ecoulement  en  deversoir.  Extrait  des  Anuales  des 
Fonts  et  Chaussees,  Oct.,  1888.  Paris,  Vve  Ch.  Dunod,  1888.  Translation  by  A. 
Marichal  and  John  C.  Trautwine,  Jr.,  presented  to  Engineers'  Club  of  Philadelphia, 
in  1889,  for  publication  in  its  Proceedings. 

g  This  would  make  x  =  3.41  (since  x  =  m  y2'g  =  8.025  m) ;  whereas  Mr.  Francis 
gives  X  =  3.33,  which  agrees  very  well  with  Messrs.  Fteley  and  Stearns,  within  the 
limits  of  Art.  14  (m).  Yet  M.  Bazin  measured  the  head  16  feet  back  from  the 
weir,  while  the  other  experimenters  measured  it  only  6  feet  back,  and  the  slight 
increase  of  head  thus  obtained  by  M.  Bazin  would  of  itself  have  made  his  coefficient 
loiver  than  theirs.  Its  excess  may  be  largel}'  due  to  the  character  of  the  channel  of 
approach,  which  in  his  case  was  from  50  to  700  feet  long,  rectangular  and  regular  in 
cross-section,  and  smoothly  coated  with  cement.  In  the  other  experiments  it  wa» 
much  less  regular. 


HYDRAULICS. 


553 


Table  14.    Tallies  of  m,  in  the  formula 

Q  =  m  L  H  }/2  glL  ...  or 

in^!ib!^ftTe?1ec.  =  '^  X  ^^"S^^'  ^"  ^*-  X  ^^^^'^ .^^^^  ^^  ^t*  X  i/2yH7Tt; 
by  M.  Bazin's  formula  (7) :  m  =  Mf      1  +  0.55  (  =j r    .    For  M,  see  last 

column  of  table.    Or,  very  approximately,  M  =  0\405  +  ~.^  . 

Note.  The  coefficient  to,  for  any  given  case,  remains  the  same  for  English, 
metric  or  other  measure ;  provided  the  head,  the  length  and  g  are  all  measured 
in  the  same  unit,  and  the  discharge  in  the  cube  of  that  unit ;  for  m  is  simply 

2  2  actual  discharge 

3  "~  3  theoretical  discharge  * 

It  will  be  noticed  that  below  the  heavy  lines  the  head  H  is  greater  than 
»^  height  j3,  and  thus  exceeds  the  limit  laid  down  in  (f )  and  (m). 


Head  H,  Fig.  24,  p.  556. 


approximate 

feet. 

inches. 

.05 
.06 
.07 
.08 
.09 

.164 
.197 
.230 
.262 
.295 

1.97 
2,36 
?.76 
3.15 
3.54 

.10 

.328 

3.94 

.12 
.14 

.394 
.459 

4.72 
5.51 

.16 
.18 

.525 
.591 

6.30 
7.09 

.20 

.656 

7.87 

.22 

.722 

8.66 

.26 
.28 

.853 
.919 

10.24 
11.02 

.30 

.984 

11.81 

.32 
.34 
.36 
.38 

1.050 
1.116 
1.181 
1.247 

12.60 
13.39 
14.17 
14.96 

.40 

1.312 

15.75 

.42 
44 
.46 
.48 

1.378 
1.444 
1.509 
1.575 

16.54 
17.32 
18.21 
18.90 

.50 

1.640 

19.69 

.52 
.54 
.56 
.58 
.60 

1.706 
1.772 
1.837 
1.903 
1.969 

20.47 
21.26 
22.05 
22.83 
23.62 

Height,  p,  Pig.  20,  of  crest  of  \reir  above  bed  of  up-stream  channel. 


meters  0.20     0.30     0.40     0.50     0.60     0.80      1.00     1.50 

2.00 

" 

o 

^feet       0.656    0,984    1,312    1.640    l.%9    2.624    3.280    4.920 

6.560 

.i 

n. 

finches  7,87    11.81    15.75    19.69    23.62    31.50    39,38    59,07 

78,76 

in 

.458 
.456 
.455 
.458 
.457 

m 

.453 
.450 
.448 
.447 
.447 

ni 

.451 
,447 
.445 
.443 
.442 

m 

.450 
.445 
.443 
.441 
.440 

ni 

.449 
,445 

.442 
.440 

.438 

m 

.449 
.444 

.441 
.438 
.436 

m 

.449 
.443 
.440 
.438 
.436 

m 

.448 
.443 
.440 
.437 
.435 

in 

.448 
.443 
.439 
,437 
.434 

.4481 
.4427 
.4391 
.4363 
.4340 

.459 

.447 

.442 

.439 

.437 

.435 

.434 

.433 

.433 

.4322 

'.462  1 
.4661 

.448 
.450 

.442 
.443 

.438 
.438 

.436 
.435 

.4.S3 
.432 

.432 
.430 

.430 

.428 

.430 

.428 

.4291 
.4267 

.471* 
.475 

.453 
.456 

.444 
.445 

.438 
.439 

.435 
.435 

.431 
.431 

.429 

.428 

.427 
.426 

.426 
.425 

.4246 
.4229 

.480 

.459 

.447 

.440 

.436 

.431 

.428 

.425 

.423 

.4215 

.484 
.488 

.462 
.465 

HI 

.442 
.444 

.437 

.438 

.431 
.432 

.428 
.428 

.424 
.424 

.423 
.422 

.4203 
.4194 

.492 
.496 

.468 
.472 

.455' 

.457 

,446 

,448 

.440 
.441 

.432 
.433 

.429 
.429 

.424 
.424 

.422 
.422 

.4187 
.4181 

.500 

.475 
.478 
.481 
.4^3 
.486 

.460 
.462 
.464 
.467 
.499 

.450 
.452 
.454 
.466 
.458 

.443 

.434 
,436 
.437 
.438 
.439 

.430 
.430 
.431 
.432 
.432 

.424 
.424 
.424 
.424 
.424 

,421' 
.421 
.421 
.421 
.421 

.4174 

^ 

.444 
.446 

,448 
.449 

.4168 
.4162 
.4156 
.4150 

zz 

.489 
.491 
.494 
,496 

.472 
.474 
.476 
.478 
.480 

.482 
,483 
.485 
.487 
,489 
.490 

.459 
.461 
.463 
.465 
.467 

.468 
.470 
.472 
.473 
,475 
.476 

.451 
.452 
,454 
.456 
.457 

.459 
.460 
.461 
.463 
.464 
.466 

.440 

.433 
,434 
.435 
.435 
.436 

,437 

.424 
.425 
.425 
.425 
425 

.426 
.426 
.426 
.427 
.427 
.427 

.421 
,421 
.421 
.421 
,421 

.421 
.421 
.421 
.421 
,421 
.421 

.4144 

^ 

.441 
,442 
.443 

.444 

.445 
.446 

,447 
.448 
.449 
.451 

.4139 
.4134 
.4128 
.4122 

.4118 

,438 
.439 
,440 
.441 

,4112 

4107 

4101 

::::: 

.4096 
.4092 

Owing  to  the  wide  range  of  the  head  H  and  of  the  height  p  in  these  experi- 
ments, we  find  in  them  a  urider  diverg-ence  in  the  values  of  the  coefficient 
than  resulted  from  the  earlier  investigations.  Thus,  the  smallest  value  of  to 
above  the  heavy  lines  is  0.4092,  or  about  one  nineteenth  less  than  the  mean, 
0.4325;  and  the  greatest  is  0.459,  or  about  one  sixteenth  more  than  0.4325. 

*  In  these  experiments,  the  head  H  was  measured  at  a  point  5  meters  (16,4  ft )  back 
from  the  weir.    The  correction  for  velocity  of  approach  is  contained  in  the  coefficient  m. 

t  M  is  the  value  of  m  when  there  is  no  velocity  of  approach  ;  i.  e.,  where  the  cross- 
•lectlon  of  the  channel  of  approach  is  indefinitely  great  compared  with  that  of  the 
stream  of  water  passing  over  the  weir. 


654 


HYDRAULICS. 


(2>)  From  a  comparison  of  a  number  of  experimental  data,  the  author 
deduced  the  following 
Table  15.    Approximate  values  of  the  coefficient  m  in  the 

formula : 

^  =  mLHi/2yH, 
for  weirs  of  several  different  shapes  and  thicknesses.    (Original.) 


3  ft.  thick ; 

Head,  H. 

Sharp  Edge.* 

2  Inches 
thick. 

smooth ;  slop- 
ing outward 
and  downward, 

3  ft.  thick ; 

smooth  ;  an4 

level. 

Feet. 

Inches. 

from  1  in  12  to 

1  in  18. 

in 

m 

m 

m 

.0833 

1 

.41 

.37 

.32 

.27 

.1666 

2 

.40 

.38 

.34 

.30 

.25 

3 

.40 

.39 

.34 

.31 

.8333 

4 

.40 

.41 

.35 

.31 

.4166 

5 

.40 

.41 

.35 

.32 

.5 

6 

.39 

.41 

.35 

.33 

.5833 

7 

.39 

.41 

.35 

.32 

.6666 

8 

.39 

.41 

.34 

.31 

.8333 

10 

.38 

.40 

.34 

.31 

1. 

12 

.38 

.40 

.33 

.31 

2. 

24 

.37 

.39 

.32 

.30 

3. 

36 

.37 

.39 

.32 

.30 

(q)  To  find  the  head  H,  approximately ;  having  the  discharge  Q. 
According  to  formulae  (3)  and  (4),  Art.  14  (/). 


Hence 


Q  =  m  LH  V/27H  =  a; LH /H  =  m  L  v^|/H3,  =a;LyH3, 
^  =Vm«L«2^="  ^|i^^^  .....    (8) 


Head,  H, 

approximately 


_^      /square  of  discharge  of  stream,  in  cub,  ft.  per  sec. 
^    \  m2  X  length^  X  64.4 


|sq.  of  discharge^ 
*^  'V  ^2~xlength2"' 
The  coefficient  m,  or  x  itself  varies  somewhat  with  the  head ;  but  the  formula 
may  be  usefully  employed  as  an  approximation   by  taking,  for  sharp-crested 
weirs,  m  =  0.415  (m^  =  0.172)  or  a;  =  3.33  (x^  =  n).    For  other  shapes,  see  Table 
15,  above. 
{r)  Submerg-ed  -weirs.  Fig.  23  6,  are  those  in  which  the  surface  of  the 
dotrn-stream  water  at  h,  after  the  construction 
of  the  weir,  is  higher  than  the  crest  a. 

In  a  weir  discharging  freely  into  the  air,  as 
in  Fig.  20,  Mr.  Francis  found  that  with  a  head 
of  1  foot  the  discharge  was  diminished  only 
about  one  thousandth  part  by  placing  a  solid 
horizontal  floor  about  6  inches  below  and  in 
front  of  the  crest  of  the  weir  for  the  water  to 
fall  upon.  Also,  when  the  head  was  10  inches,  and  the  water  fell  freely  through 
the  air  into  water  of  considerable  depth  (as  in  Fig.  20),  the  quantity  discharged 
was  the  same  whether  the  surface  of  the  down-stream  water  was  about  3  inches 
or  about  13  inches  below  the  crest  a. 

In  experiments  by  Mr.  Francis  and  by  Messrs.  Fteley  and  Stearns,  with  air 
freely  admitted  underneath  the  falling  sheet  of  water  just  below  the  crest  a,  th« 
discharge  was  not  appreciablv  affected  by  a  submergence  of  h  =  from  0.017  H  to 
0.023  H.  When  air  was  only  partially  admitted,  the  discharge  was  affected  {in- 
creased) by  less  than  one  per  cent,  while  h  remained  less  than  0.15  H. 


Fij§:.23b 


*  These  values  are  lower  than  those  g.iven  in  Art.  14  (w)  and  (h),  and  much  lowei 
than  those  in  (o). 


HYDRAULICS. 


555 


Bnbnat's  formula  for  submergred  weirs.    Let 

Bl  and  /*  =  the  heads  measured  vertically  from  the  crest  a  of  the  weir  to  the 
surface  of  still  water  *  up-stream  and  down-stream  from  the  weir, 
respectively. 

ci  =  H— A  =  their  difference  =  the  difference  iri  level  between  the  up-stream 
and  down-stream  surfaces  of  still-water  ;* 


:  coefficient  of  discharge  =  ; 


actual  discharge 


Then 


theoretical  discharge* 
Q  =  cl(a-|-|  d)  l/273';t (9)or^ 

Actual  discliarse  _  ^  x    ^^?g^^  ?f  X  8.025  v/rfmlt.X  f  A  in  ft.  -f  |  (f  in  ft.) . 
in  cub.  ft.  per  sec.  weir  in  ft.  ^  \  3  / 

(«)  MessrSi  Fteley  and  iStearns  %  experimented  at  Boston  in  1877  with 
submerg;ed  iveirs  under  up-stream  heads  H  from  about  4  to  10  inches ;  and 
Mr.  Francis  I  at  Lowell  in  1883  under  heads  from  about  1  foot  to  2  feet  4 
inches. 

From  these  experiments  we  deduce  the  following 

Table  16,  of  approximate  values  of  the  coefficient  c  in  the  formula  for 
discharge  over  submerg^ed  weirs. 

Deduced  from  experiments  by  Fteley  and  Stearns  and  by  J.  B.  Francis.  In 
Mr.  Francis'  experiments,  the  value  of  c  for  a  given  value  of  A  -j-  H  generally 
increased  as  H  increased. 


Fteley  and  Stearns. 

J.  B.  Francis. 

(H  =  0.325  to  0.815  feet.) 

(H  =  1  to  2.32  feet.) 

^-*-H 

c 

c 

.05 

.623  to  .632 

.10 

.625  to  .635 

.620  to  .630 

.20 

.618  to  .628 

.610  to  .625 

.30 

.600  to  .610 

.598  to  .615 

.40 

.590  to  -600 

.586  to  .610 

.50 

.585  to  .595 

.5^5  to  .607 

.60 

.583  to  .593 

.585  to   .607 

.70 

•  .580  to  .590 

.585  to  .607 

.80 

.681  to  .591 

.685  to  .607 

.90 

.590  to  .600 

.95 

.610  to  .615 

*  For  velocity  of  approach,  see  Art.  14  (m)  etc. 

t  In  deducing  this  formula,  the  water  that  passes  over  the  weir  between  c  and  6  is 
assumed  to  flow  as  over  a  weir  with  its  crest  at  b,  and  with  free  discharge  into  the  air, 
as  over  the  crest  a  in  Fig.  20 ;  and  for  this  portion,  by  formula  (2)  in  Art.  14  (i),  the 
discharge  would  be : 

Qj  =  cL|d]/27d; 

while  the  water  that  passes  through  the  lower  portion  between  h  and  a  is  regarded  a^i 
flowing  through  a  submerged  vertical  orijice  whose  height  is  6  a  =  ^,  under  a  hsed 
=  d.  For  this  lower  portion,  therefore,  the  discharge  would  be : 
Q^  =  c  L  7i  yWgH. 
It  is  assumed  that  the  coefficient  of  discharge  c  is  the  same  for  the  upper  section 
c  6  as  for  the  lower  one  a  b.  Hence,'adding  these  two  discharges  together,  we  obtain, 
for  the  entire  disharge : 

1  Transactions,  American  Society  of  Civil  Engineers,  March,  1883,  p.  101,  etc. 
g  Transactions,  American  Society  of  Civil  Engineers,  Sept.,  1884,  p.  295,  etc. 


556 


HYDRAULICS. 


(t)  Mr.  Clemens  Herscliel,*  comparing  these  experiments  with  some 
earlier  ones  by  Mr.  Francis,  gives  the  following : 

Having  ascertained  the  depths  H  and  h  of  the  crest  below  the  still-water  levels 
up-stream  and  down-stream  respectively,  divide  h  by  H.  Find  the  quotient,  as 
nearly  as  may  be,  in  the  column  headed  A  -4-  H  in  Table  17.  Take  out  the  cor- 
responding coeflScient  a,  and  multiply  it  by  the  up-stream  head  H.f 

The  product  a  H  is  the  head  which  would  cause  the  given  weir  to  discharge 
the  same  quantity  freely  into  the  air,  as  in  Fig.  20.  Find  the  discharge  into  air 
over  the  given  weir  with  the -bead  a  H ;  and  this  discharge  will  be  approximately 
the  same  as  that  of  the  actual  submerged  weir  under  the  up-stream  head  H  and 
against  the  down-stream  head  h ;  or  (H  being  the  actual  up-stream  head  on  the 
submerged  weir)  the  discharge  is 

Q  =  m  L  a  H  \/TgaR  =  x  L  a  H  ^aBT. (10). 


TABI.I:  17. 


A-H 

a 

^-H 

a 

.72 

.74 
.76 

.78 
.80 

a 

.10 
.20 
.25 
.80 
.35 
.40 

1.000  to  1.010 
0.975  to  0.995 
0.960  to  0.984 
0.945  to  0.973 
0.928  to  0.960 
0.912  to  0.946 

.45 
.50 
.55 
.60 
.65 
.70 

0.894  to  0.930 
0.874  to  0.910 
0.853  to  0.889 
0.829  to  0.863 
0.803  to  0.833 
0.775  to  0.799 

0.762  to  0.784 
0.747  to  0.769 
0.732  to  0.752 
0.713  to  0.733 
0.693  to  0.713 

(u)  Telocity  of  approach. 

si  "^ 


See  Fig.  24.  It  is  generally  impracticable 
to  measure  the  head  H'  to  perfectly  still 
water  o'  up-stream.  The  head  is  usually 
measured  at  a  point  o,  from  2  or  3  to  6  or 
8  feet  or  more  up-stream  from  the  weir, 
according  to  the  size  of  the  latter.    At 

f^X^^'P'-^'        '^~>-.:,     'v         [      such  points  the  velocity  is  generally  ap- 
— ^^Jj-    CJ'^v.^  ■     Sv^.    \  preciable,  and  the  surface  therefore  a 
-'js.     i       v     '^^     \        little  lower  than  at  o'.    Hence  a  formula 
I  .^s^^    _> ^^.^u^   ^  ^    using  the  smaller  head  H  so  measured, 

''  insteadofH',  and  coefficients  based  upon 
H',  will  give  too  small  a  discharge.  Mr. 
Francis  found  that  a  current  of  1  foot 
per  second,  or  nearly  0.7  mile  per  hour, 
at  the  point  o  to  which  the  head  was 
measured,  increased  the  discharge  but  about  2  per  cent,  when  the  head  was 
13  inches;  and  a  current  of  6  inches  per  second  increased  the  discharge  about  1 
per  cent,  when  the  head  was  8  inches. 

If,  however,  the  velocity  of  approach  is  such  as  to  require  consideration,  pro- 
ceed as  follows:  For  the  approximate  mean  velocity  of  approach,  we  have: 


Fig.24: 


a' 


approxi  mate  discharge 


3.33  L  H^ 


'  area  of  entire  cross  section  of  stream  at  o 


and,  for  the  head  due  to  this  velocity,  ^  =  s- 
Then,  for  all  practical  purposes,  we  may  say :  H'  =  H  -f-  A ;  or 


Q  =  m  L  (H  +  A)  ]/2  ^(H  +  h)  =  x  L  (H  -f-  A)2 (11) 

although,  strictly  speaking,  the  difference  of  level  between  o'  and  o  is  really 
(as  shown  in  Fig.  24)  somewhat  greater  than  h^  or  than  v^  -ir2g,  because  some 
head  is  lost  in  friction  between  o'  and  o. 


*  Transactions,  American  Society  of  Civil  Engineers,  May,  1885,  pp.  189,  etc. 

t  Mr.  Herschers  table,  from  which  ours  is  condensed,  gives  a  for  every  0.01  foot  of 
A  -^  H ;  but  the  values  of  a  intermediate  of  those  we  have  selected  may  be  taken  from 
our  table  almost  exactlv  by  simple  proportion.  The  range  in  the  coefficient  a  in  the 
table  for  each  value  ck  h  ^Yi  is  that  indicated  by  the  experiments,  which  varied 
similarly.  We  are  not  instructed  how  to  select  between  these  extremes;  but  in 
most  cases  their  mean  value  is  probably  nearest  right. 


HYDRAULICS. 


557 


(v)  Messrs.  Fteley  and  Stearns  make  H'  =  H  +  1.5  //  for  suppressed 
weirs,  and  H'  =  H  -f-  2.05  h  for  weirs  with  complete  end  contractions,  as  averages  ; 
and  ^Ir.  Hamilton  Smith,  Jr.,*  after  comparing  their  experiments  with  others 
by  Lesbros,  Castel  and  Mr.  Francis,  gives  H.'  =  K  ^ll/^h,  and  H'=  H  +  ].4/i,for 
the  two  cases  respectively. 

i'w)  On  the  other  haud,  Mr.  Francis'  formula,  as  modified  for  velocity  of 
approach, 

Q  =  x  L  t  (YlETW—  yW)  =  m  L  t  ]/2^~(v/(H  +  h)^  —  y/W)  t    .   .    .    (12), 
makes  the  effect  of  H'  less  than  that  of  H  +  A. 

(x)  Messrs.  A.  1*\  Hunking-  and  Frank  S.  Hart,  Civil  and  Hy- 
draulic Engineers,  have  substituted  for  the  expression  (]/(H  -f  h)^  —  yii^)  in 
formula  (12),  the  equivalent  one  K  v  H^,  in  which  K  is  a  coefficient  deduced  from 
the  former  expression,  and  therefore  depending  upon  the  relation  between  H 
and  h,  or,  ultimately,  upon  that  between  the  cross-section  a  s  Fig.  24  at  the  weir 
and  the  entire  cross-section  of  the  stream  at  o. 

Having  found  the  area  of  cross-section  at  o,  divide  it  by  (  L  —  n    ^  ] ,  which 

is  the  length  of  the  weir  corrected  for  contraction.    See  Art.  14  (m).    Call  the 
quotient  D.§    Divide  the  measured  head  H  by  D.     Find  this  last  quotient  in 

the  column  ^_    of  the  table.     Multiply  the  approximate    discharge,  Q  =  3.33 
/l  —  71  ^  I  H2,by  the  corresponding  coefficient  K;  or 

Actual  Discharge  Q  =  3.H3  K  (h  —  n~)  hI.    .  . 
Table  18.    Coefficient  K  in  formula  (13), 


(13) 


H 

H 

H 

H 

H 

K 

K 

K 

K 

K 

D 

D 

D 

D 

D 

.01 

1.0000 

.09 

1.0020 

.17 

1.0072 

.24 

1.0143 

.31 

1.0239 

.02 

1.0001 

.10 

1.0025 

.18 

1.0081 

.25 

1.0155 

.32 

1.0254 

.03 

1.0002 

.11 

1.0030 

.19 

1.0090 

.26 

1.0168 

.33 

1.0271 

.04 

1.0004 

.12 

1.0036 

.20 

1.0100 

.27 

1.0181 

.34 

1.0287 

.05 

1.0006 

.13 

1.0042 

.21 

1.0110 

.28 

1.0195 

.35 

1.0305 

.06 

1.0009 

.14 

1.0049 

.22 

1.0121 

.29 

1.0209 

.36 

1.0322 

.07 

1.0012 

.15 

1.0056 

.23 

1.0132 

.30 

1.0i£4 

.37 

1.0»41 

.08 

1.0016 

.16 

1.0064 

*  *'  Hydraulics,"  John  Wiley  &  Sons,  New  York,  1886. 

f  If  there  are  end  contractions,  L  here  becomes  f  L  —  n  r^  I  •     See  Art.  14  (m). 

X  This  formula  is  deduced  as  follows :  Let  the  area  of  the  parabolic  segment  a  s  a'. 
Fig.  24,  represent  the  theoretical  discharge  over  a  weir  one  foot  long  (as  explained  in 
foot-note  ^  p.  549)  under  the  measured  head  H  =  as,  as  though  there  were  no  current 
at  o.  Let  ms  =  h  =  v^  -i-2  g.  The  theoretical  velocities  of  the  particles  passing  the 
oblique  plane  o  a  under  their  actual  heads,  will  now  be  represented  by  horizontal  lines 
8s",  a  a",  etc.,  etc.,  drawn  from  every  point  in  s  a  to  the  outer  curve  .s"  a"  ;  the  line  s  s" 
representing  V  =  velocity  of  approach  =  \/2  gh,  and  a  a"  representing  y/2  g  (H~+~fe). 
Then,  area  s  s"  a"  a 

2  2 

=  area  ma"  a  —  area  m  s"  s  =  -  -  area  of  rectangle  an  —  — -  area  of  rectangle  s  k 


=  -|-(H-f  ?i)  y2g  (ii  +  /i)_-|^  y2Th,  =   -|y2g  (l/(H  +  ;.)3  -Yh- 
and  the  actual  discharge  is 
Q  =  c  X  length  of  weir  X  area  s  s"  a"  a  =  c  L  —  y2j  (  V(H  -f  h)^  —  }/P  j 

=  m  L  Y2J  (yiW+liY^  —  fhA  =  a:  L  (y'iRT'hJ''  —  j/p)- 
§  In  a  weir  without  end  contraction,  D  =  H  4-  2>. 


)' 


558 


HYDRAULICS. 


K  is  very  approximately  =  1  -f  —  (  -—  j  .    Hence 

=  [3.33  +  0.83(«y](L-nf)Ht. 


.    (14) 


See  Journal  of  the  Franklin  Institute,  Philadelphia,  August,  1884,  from  which 
we  condense  the  above  table. 
(y)  M.  Bazin,  see  Art.  14  (o),  provides  for  the  velocity  of  approach  by  modi- 

(H  \* 
—  J  ; 

while  by  Messrs.  Hunking  and  Hart's  method  (based  upon  Mr.  Francis'  experi- 

/  H  \  2 
ments)  m  becomes  ^^^^  0.415  +  0.10  I  y  1  • 

Art.  15.  Inclined  -weirs.  If  the  up-stream  face  of  the  weir,  instead 
of  being  vertical,  as  in  Fig.  25,  is  inclined  up-stream,  as  in  Fig.  25  a,  or  down- 
stream, as  in  Fig.  25  &,  the  character  and  amount  of  the  discharge  are  modified. 
With  an  up-stream  inclination  (Fig.  25a)  the  lower  side  of  the  sheet  of  water 
passing  over  the  weir  leaps  higher,  and  tends  more  and  more  up-stream  as  the 


Fig.  25  a. 
Inclined  np'Stream 


Fig.  25  6. 
Inclined  down-stream. 


Inclination  is  increased.  With  a  down-stream  inclination  (Fig.  25  b),  on  the  con- 
trary, as  the  inclination  increases  the  upward  leap  of  the  sheet  decreases,  its 
profile  becomes  more  and  more  flattened,  and  the  curve  of  the  upper  surface, 
due  to^the  fall,  extends  farther  up-stream  from  the  crest  of  the  weir. 

An  up-stream  inclination  (Fig.  25  a)  decreases,  and  a  down-stream  one  (Fig. 
25  b)  increases,  the  discharge,  as  is  indicated  by  the  following  coefficients  ob- 
tained by  M.  H.  Bazin  :* 

For  the  <li*«cliarg-e  over  an  inclined  iveir.  having  ascertained  the 
discharge  over  a  vertical  weir  of  the  same  height  and  head  and  under  similar 
fconditions  in  other  respects,  multiply  the  discharge  over  the  vertical  weir  by 
the  following  approximate  coefficients :     . 


Inclination. 

Angle 

Coef- 

Horizontal. 

Vertical. 

with  hor. 

with  vert. 

ficient. 

1 

45° 

45° 

0.93 

Weirs    inclined   up- J 
stream,  Fig.  25  a....  j 

1 
i 

56°  19' 
71°  34' 

33°  41' 
18°  26' 

0.94 
0.96 

Vertical  weirs,  Fig  25... 

0 

90° 

0° 

1.00 

i 

71°  34' 

18°  26' 

1.04 

Weirs  inclined  down-  J 
stream,  Fig.  25  6 ' 

i 

1 

56°  19' 

45° 

33°  41' 
45° 

1.07 
1.10 

2 

26°  34' 

63°  26' 

1.12 

*  "Experiences  Nouvelles  sur  I'^coulement  en  Deversoir,"  2e  Article;  "  Annales 
des  Fonts  et  Chaussees,"  January,  1890,  translated  in  Proceedings,  Engineers'  Club  of 
Philadelphia,  vol.  ix.,  1892. 


HYDRAULICS.  559 

The  discharge  will  be  increased  also  if  the  inner  corner  or  edge  of  the  crest 
be  rounded  off,  instead  of  being  left  sharp ;  or  if  the  sides  of  the  reservoir  con- 
verge more  or  less  as  they  approach  the  weir,  so  as  to  form  wings  for  guiding 
the  water  more  directly  to  it ;  or  if  a  b,  Fig  20,  be  less  than  twice  a  m.  Indeed, 
so  many  modifying  circumstances  exist  to  embarrass  experiments  on  this  and 
similar  subjects  that  some  of  those  which  have  been  made  with  great  care  are 
rendered  inapplicable  as  other  than  tolerable  approximations,  in  consequence 
of  the  neglect  to  take  into  consideration  some  local  peculiarity  which  was  not 
at  the  time  regarded  as  exerting  an  appreciable  etiect._  Unless,  therefore,  cir- 
cumstances admit  of  our  combining  all  the  conditions  mentioned  in  Art.  14  (d), 
(/)  and  (m),  pp.  547,  548  and  550,  thereby  securing  *;er// approximate  results,  we 
must  either  resort  to  an  actual  measurement  of  the  discharge  in*  a  vessel  of 
known  capacity  ;  or  else  be  content  with  rules  which  may  lead  to  errors  of  5, 
10,  or  more  per  cent,  in  proportion  as  we  deviate  from  these  conditions.  Fre- 
quently even  10  per  cent,  of  error  may  be  of  little  real  importance. 

Remark  1.  When  the  water,  after  passing  over  a  weir,  Fig.  26,  instead  of  fall- 


Flg,  26. 

ing  freely  into  the  air,  is  carried  away  by  a  slightly  inclined  apron  or  trough,  T, 
the  floor  of  which  coincides  with  the  crest  a,  of  the  weir,  then  the  discharge  is 
not  appreciably  diminished  thereby  when  the  head  a  m,  is  15  inches  or  more. 
But  if  the  head  a  m  is  but  1  foot,  then  the  calculated  discharge  must  be  reduced 
about  one-tenth;  if  6  inches,  two-tenths;  if  2^  inches,  three-tenths;  and  if  1 
inch,  five-tenths,  or  one-half,  as  approximations. 

Remark  2.  Professor  Thomson,  of  Dublin,  proposed  the  use  of  triangular 
notches,  or  weirs,  for  measuring  the  discharge ;  inasmuch  as  then  the  periphery 
always  bears  the  same  ratio  to  the  area  of  the  stream  flowing  over  it ;  which  is 
not  the  case  with  any  other  form.    Experimenting  with  a  right-angled  triangular 

,-90°.-.. 


Fig.  26  A. 

notch  in  thin  sheet-iron.  Fig.  26  A,  with  heads  of  from  2  to  7  inches,  measured 
Vertically  from  the  bottom  of  the  notch  to  the  level  surface  of  the  quiet  tvaier,  he 
found  discharge  in  cubic  feet  per  second  =  .0051  X  v'fifth  power  of  head  in  inches. 
=  2.54  X  ]/fifth  power  ofHRead  in  feet.*  Or,  in  general,  if  m.  =  coefficient  of 
contraction  (Art.  9,  p.  541)  T  =  tangent  of  half  the  angle  of  the  notch  =  width 
of  water-surface  -r-  the  depth  in  the  notch,  g  —  the  acceleration  of  gravity  =  say 
32.2  feet  per  second,  and  h  =  the  head,  measured  as  above ;  then 

Discharge  =  ^^~  T  V2gM *  =  4.28  m  T  y'h^* 

Remark  3.  In  constructing  the  irrigating  canal,  Canale  Villoresi,  near  Milan, 
in  1881-4,  the  Italian  engineer,  Cesare  Cippoletti,t  adopted  a  trapezoidal 

notcli,  with  its  bottom  edge  horizontal  and  its  ends  sloping  at  -—; — : — r,  in 

*'  ®       1  horizontal 

order  to  avoid  the  necessity  of  either  suppressing  or  allowing  for  end  contrac- 
tions. (See  Art.  14  c,  p.  547,  and  m,  p.  550.)  The  contraction  was  found  to  affect 
only  the  triangular  spaces  over  the  sloping  ends  of  the  weir,  and  the  effective 
length  of  the  weir  thus  remained  constant  (and  equal  to  the  length  of  the  bot- 
tom edge)  for  all  heads.  In  using  these  weirs  the  contraction  is  complete  along 
the  bottom  as  well  as  at  the  ends. 

*  For  such  roots  see  p.  68. 

t  See  his  work,  Canale  Villoresi ;  Modulo  per  la  Dispensa  della  Acqua,  etc.,  Milan, 
1886;  published  by  Societa  Italiana  per  Condotte  d'Acqua.  Results  summarized  by 
L.  G.  Carpenter,  in  Bulletin  No.  13,  Agricultural  Experiment  Station,  Fort  Collins, 
Colorado,  October,  1890. 


560  HYDEAULICS. 

ON  THE  I  I.OW  OF  WATER  IN  OPEX  CHAHrHTEIiS. 

Art.  16.    The  mean  velocity  of  flow  is  an  imaginary  uniform  one, 
wliieh,  if  given  to  the  water  at  every  point  in  the  cross  section,  would  give  th6 
.  same  discharge  that  tiie  actual  un uniform  one  does.    Or 

:,  volume  of  discharge 

mean  velocity  = 7^ -7^ 

area  01  cross  section 

In  channels  of  uniform  cross  section,  tlie  maximum  velocity  is  found 
about  midway  between  the  two  banks,  and  generally  at  some  dist  below  the  sur- 
face. This  dist  varies  in  diff  streams;  but,  as  an  average,  it  seems  to  be  about 
one  third  of  tlie  total  depth.  Where  the  total  depth  is  great  in  proportion  to 
the  width,  (say  ^  the  width  or  more),  the  max  vel  has  been  found  as  deep  as 
midway  between  surf  and  bottom;  while  in  small  shallow  streams  it  appears  to 
approach  the  surf  to  within  from  .1  to  .2  of  the  total  depth.  Many  experiments 
upon  shallow  streams  have  indeed  indicated  that  the  max  vel  was  at  the  surf. 

Tlie  ratio  between  tlie  velocities  in  different  parts  of  tbe 
cross  section  varies  greatly  in  diff  streams;  so  that  but  little  dependence 
can  be  placed  upon  rules  for  obtaining  one  from  the  other.  With  the  same  surf 
vel,  wide  and  deep  streams  have  greater  mean  and  bottom  vels  than  small  shal- 
low ones.  In  order  to  approximate  roughly  to  the  mean  vel  when 
the  greatest  surf  vel  is  given,  it  is  frequently  assumed  that  the  former  is  =  | 
(or  .8)  of  the  latter.  But  Mr.  Francis  found,  in  his  experiments  at  Lowell,  that 
surface  floats  of  wax,  2  ins  diam,  floating  down  the  center  of  a  rectangular  flume 
10  ft  wide,  and  8  ft  deep,  actually  moved  about  6  per  cent  slower  than  a  tin  tube 

2  ins  diam,  reaching  from  a  few  ins  above  the  surf,  down  to  within  1^  ins  of  the 
bottom  of  the  flume;  and  loaded  at  bottom  with  lead,  to  insure  its  maintaining 
a  nearly  vert  position.  While  the  wax  .m;/ float  moved  at  the  rate  of  3.73  ft  per 
sec,  the  rate  of  the  tube  (which  was  evidently  very  nearly  the  same  as  that  of 
the  center  veri  thread  of  water)  was  3.98  ft  per  sec.  Also,  that  in  the  same  flume, 
with  vels  of  the  center  tube  varying  from  1.55  to  4  ft  per  sec,  the  vel  of  the  tube 
was  less  than  that  of  the  mean  vel  of  the  entire  cross  section  of  water  in  the 
flume,  about  ..j.96  to  1,  for  the  lesser  vel;  and  .93  to  1  for  the  greater  vel. 
While,  in  another  rectangular  flume  20  ft  wide  and  8  ft  deep,  with  vels  varying 
from  1.16  to  1.84  ft  per  sec,  that  of  the  tubes  was  greater  than  that  of  the  entire 
mass  of  water,  about  as  1.04  to  1.  In  a  flume  29  ft  wide,  by  8.1  ft  deep,  with  vels 
of  about  3  ft  per  sec,  it  was  as  1  to  .9 ;  and  in  a  flume  36j^  ft  wide,  by  8.4  ft  deep, 
with  vels  of  about  3^  ft  per  sec,  as  1  to  .97. 

Charles  Ellet,  Jr,  C  E,  found  in  the  Mississippi  "  at  diff"  points 
Qn  the  river,  in  depths  varying  from  54  to  100  ft;  and  in  currents  varying  from 

3  to  7  miles  an  hojir  that  the  speed  of  a  float  supporting  a  line  50  ft  long,  is  al- 
most always  grea  er  than  that  of  the  surf  float  alone."  The  same  results  were 
obtained  with  lines  25  and  75  ft  long;  the  excess  of  the  speed  of  the  line  floats 
being  about  2  per  cent  over  that  of  tlie  simple  floats:  and  Mr.  Ellet  concludes, 
therefore,  that  the  mean  vel  of  the  entire  cross  section  of  the  Mississippi,  instead 
of  being  less,  is  absolutely  greater  by  about  2  per  cent,  than  the  mean  surfyeX. 
He,  however,  employed  .8  of  the  greatest  surf  vel  as  representing  approximately, 
in  his  opinion,  the  mean  vel  of  the  entire  cross  section  of  water.  In  shallow 
streams,  he  always  found  the  surf  float  to  travel  more  rapidly  than  a  line  float. 

European  trials  of  the  mean  vel  of  separate  single  verticals,  in  tolerably  deep 
rivers,  have  resulted  in  from  .85  to  .96  of  the  surf  vel  at  each  vertical.  The  mean 
of  all  may  be  taken  at  .9. 

Bottom  velocity.  In  streams  of  nearly  uniform  slope  and  cross  section, 
there  is  a  great  reduction  of  vel  near  the  bottom.  As  a  very  rough  approxima- 
tion, the  deepest  measurable  vel,  in  streams  of  uniform  slope  etc,  appears  to  be 
from  J  to  f  of  the  mean  vel. 

Art.  17.  To  measure  the  surface  velocity,  select  a  place  where 
the  stream  is  for  some  dist  (the  longer  the  better)  of  tolerably  uniform  cross 
section;  and  free  from  counter-currents,  slack  water,  eddies,  rapids,  etc.  Ob- 
serve, by  a  seconds-watch, or  pendulum,  how  long  a  time  afloat  (such  as  a  small 
block  of  wood)  placed  in  the  swiftest  part  of  the  current,  occupies  in  passing 
through. some  previously  measured  dist.  From  50  feet  for  slow  streams,  to  150  ft 
for  rapid  ones,  will  answer  very  well.  This  dist  in  ft,  or  ins,  div  by  the  entire 
number  of  seconds  reqd  by  the  float  to  traverse  it,  will  give  the  greatest  surf  vel 
in  ft  or  ins  per  sec. 

The  surf  vel  should  be  measd  in  perfectly  calm  weather, 
so  that  the  float  may  not  be  disturbed  by  wind;  and,  for  the  same  reason,  the 
float  should  not  project  much  above  the  water.    The  measurement  should  he 


HYDRAULICS.  561 

repeated  several  times  to  insure  accuracy.  In  very  small  streams,  the  banks 
and  bed  may  be  trimmed  for  a  short  dist,  so  as  to  present  a  uniform  channel- 
way.  The  float  should  be  placed  in  the  water  a  little  dist  above  the  point  for 
commencing  the  observation ;  so  that  it  may  acquire  the  full  vel  of  the  water, 
before  reaching  that  point. 

Art.    18.     To    gauge   a   streami 

|R  T  l>y  means  of  its  velocity.    Select 

I  a.place  where  the  cross-section  remains,  for 

"•-^^^^ jl.    U      I   ^       J^        .^  a  short  distance,  tolerably  uniform,  and 

free  from  counter-currents,  eddies,  still 
water,  or  other  irregularities.  Prepare  a 
careful  cross-section,  as  Fig.  27.  By  means 
of  poles,  or  buoys,  n,  n,  divide  the  stream  into  sections,  a,  6,  c,  i&c.  Plant  two  range- 
poles,  R,  R,  at  the  upper  end,  and  two  others  at  the  lower  end,  of  the  distance 
through  which  the  floats  are  to  pass ;  for  observing  by  a  seconds  watch,  or  a  pendu- 
lum, the  time  which  they  occupy  in  the  passage.  Then  measure  the  mean  velocity  of 
each  section  a.  h,  c,  &c.,  separately,  and  directly,  by  means  of  long  floats,  as  F  L, 
reaching  to  near  the  bottom :  and  projecting  a  little  above  the  surface.  The  floats  may 
be  tin  tubes,  or  wooden  rods;  weighted  in  either  case,  at  the  lower  end,  until  they 
will  float  nearly  vertical.  They  must  be  of  different  lengths,  to  suit  the  depths  of 
the  different  sections.  For  this  purpose  the  float  may  be  made  in  pieces,  with  screw- 
joints.  The  area  of  each  separate  ^ection  of  the  stream  in  square  feet,  being  multi- 
plied by  the  observed  mean  velocity  of  its  water  in  feet  per  second,  will  give  the 
discharge  of  that  section  in  cubic  feet  per  second.  And  the  discharges  of  all  the 
separate  sections  thus  obtained,  when  added  together,  will  give  the  total  discharge 
of  the  stream.  And  this  total  didcharge,  divided  by  the  entire  area  of  cross-section 
of  the  stream  in  square  feet,  gives  the  mean  velocity  of  all  the  water  of  the  stream, 
in  feet  per  second. 

Rem.  If  the  eliaiiiiel  is  in  common  earth,  especially  if  sandy, 
the  loss  bysoakage  into  the  soil,  and  by  evaporation,  will  frequently  abstract  so 
much  water  that  the  disch  will  gradually  beLorue  less  and  less,  the  farther  down 
stream  it  is  measured.  Long  canal  feeders  thus  generally  deliver  into  the  canal 
but  a  small  proportion  of  the  water  that  enters  their  upper  ends. 

The  double  float  is  used  for  ascertaining  vels  at  diff"  depths.  It  consists 
of  a  float  resting  upon  the  surface  of  the  water,  and  of  a  heavier  body,  or  "  lower 
float",  which  is  suspended  from  the  upper  float  by  means  of  a  cord.  The  depth 
)f  the  lower  float  of  course  depends  upon  the  length  of  the  suspending  cord 
^which  may  be  increased  or  diminished  at  pleasure  until  the  lower  float  is  be- 
lieved to  be  at  that  depth  for  which  the  vel  is  wanted),  and  upon  its  straight- 
ness,  which  is  more  or  less  afiected  by  the  current.  Owing  to  this  latter  circum- 
stance, it  is  difficult  to  know  whether  the  lower  float  is  really  at  the  proper 
depth.  Moreover  it  is  uncertain  to  what  extent  the  two  floats  and  the  string 
interfere  with  one  another's  motions.     In  deep  water  the  string  may  oppose  a 

greater  area  to  the  current  than  the  lower  float  itself  does.    It  thus  becomes 
oubtful  to  what  extent  the  vel  of  the  upper  float  can  be  relied  upon  as  indicat- 
ing that  of  the  water  at  the  depth  of  the  lower  one. 

Art.  19.    Castelli's  quadrant,  or  hydrometric  pendulum, 

consisted  of  a  metallic  ball  suspended  by  a  thread  from  the  center  of  a  graduated 
are.  The  instrument  was  placed  in  the  current,  with  the  arc  parallel  to  the 
direction  of  flow ;  and  the  vel  was  then  calculated  from  the  angle  formed  be- 
tween the  thread  and  a  vert  line. 

Oauthey's  pressure  plate  was  a  sheet  of  metal  suspended  by  one  of  its 
ands,  about  which  it  was  left  free  to  swing.  The  plate  was  immersed  in  the 
stream,  with  its  face  at  right  angles  to  the  current.  The  vel  was  estimated  by 
means  of  the  weight  required  to  make  the  plate  hang  vert  in  opposition  to  the 
force  of  the  current. 

Pitot's  tube  was  originally  a  simple  glass  tube.  Fig.  27  A,  open 
at  both  ends  and  bent  in  the  shape  of  the  letter  L.  One  leg  of  the 
L  was  held  horizontal  under  water,  with  its  open  end  facing  the 
current ;  and  the  velocity  v  at  the  point  o  where  it  was  placed  was 

measured  by  the  vertical  height  h  i  theoretically  =  —  j  to  which 

the  water  rose  in  the  other  leg  above  the  surface  of  the  stream. 

3e 


562 


HYDRAULICS. 


As  developed  by  M.  l>arcy  and  by  Prof.  S.  "W.  Robinson,*  and 

rudely  indicated  in  Fig.  27  B,  Pilot's  tube  consists  essen- 
tialh^  of  two  horizontal  glass  or  metal  tubes  a  and  b,  of 
very  small  bore,  placed  side  by  side  in  the  current  and 
pointed  up-stream.  Tube  a  receives  the  current  in  its 
open  up-stream  end,  while  b  is  closed  at  its  up-stream 
end,  and  has  small  lateral  openings  only.  The  other  end 
of  each  tube  communicates,  by  means  of  small  metal  or 
rubber  piping,  with  one  leg  of  an  inverted  U-shaped  glass 
gauge  fixed  in  a  boat  or  on  shore.  For  convenience,  the 
iwo  flexible  pipes  may  be  joined  together  into  one  double 
pipe.  -By  sucking  through  a  stop-cock  T  at  the  top,  water 
is  drawn  up  to  any  convenient  height  in  the  two  legs  of 
the  gauge.  When  there  is  no  current,  the  two  columns  of 
course  stand  at  the  same  height ;  but  in  a  current,  the  dif- 
ference h  in  their  heights  is  such  that  v  ==  \"1  g  A,  no  cor- 
rective coefficient  being  required.  The  instrument  is  re- 
markably simple  and  accurate,  and  can  be  used  in  very 
narrow  and  shallow  streams  of  water  or  of  gas.  It  meas- 
ures velocities  as  low  as  4  inches  per  second. 
In  practice,  a  and  b  are  fixed  together  in  one  piece,  and  placed,  when  in  use, 
In  a  metal  frame  which  slides  vertically,  either  upon  a  wire  passing  through  it 
and  provided  with  a  plummet  which  rests  upon  the  bottom  and  keeps  the  wire 
stretched,  or  (in  streams  shallower  than  about  20  feet)  upon  a  vertical  wooden 
rod,  the  lower  end  of  which  holds  in  the  bed  of  the  stream.  In  the  former  case, 
the  frame  is  provided  with  a  long  vane  for  keeping  the  instrument  h«aded  up- 
stream. In  either  case,  means  are  provided  for  showing  the  depth  to  which  the 
instrument  is  submerged. 

By  making  the  gauge  scale  adjustable  vertically,  and  placing  it  (at  each  change 
of  depth  of  instrument)  with  its  zero  opposite  the  top  of  the  lower  column,  we 
obviate  the  necessity  of  observing  the  height  of  both  columns  at  each  reading ; 
for  the  reading  of  the  upper  column  alone  then  gives  the  head  h  at  once. 

Art.  20.  The  wheel  meter  consists  of  a  wheel  which  is  turned  by  the 
current,  and  which  communicates  its  motion,  by  means  of  its  axle  and  gearing, 
to  indices  which  record  the  number  of  revolutions.  The  instrument  may  be 
clamped  to  any  part  of  a  long  pole  reaching  to  the  bottom  of  a  stream,  and  thus 
maj  be  used  at  any  depth.  The  observer,  by  means  of  a  wire,  rod  or  string 
reaching  down  to  the  instrument,  throws  the  registering  apparatus  first  into, 
and  then  out  of,  gear  with  the  wheel  (applying  a  brake  to  the  former  at  the 
instant  it  is  thrown  out  of  gear),  and  carefully  noting  the  times  when  he  does 
so.  The  instrument  is  then  raised,  the  number  of  revolutions  in  the  measured 
time  is  read  ofl^  from  the  indices,  and  from  it  the  velocity  is  calculated.  But  the 
meter  is  often  made  self-reg'istering' ;  the  wheel,  at  each  revolution,  auto- 
matically breaking  and  re-establishing  a  galvanic  current  generated  by  a  bat- 
tery. The  wire  carrying  this  current  is  thus  made  to  operate  Morse  telegraphic 
registering  apparatus  placed  in  a  boat  or  on  shore. 

A  number  of  meters,  so  arranged,  can  be  attached  at  different  points  on  the 
same  pole  at  the  same  time,  and  thus  simultaneous  observations  of 
velocities  at  different  depths  may  be  made  and  registered. 

Meters  are  usually  so  arranged  as  to  swing  freely  about  the  long  vertical  pole 
to  which  they  are  clamped,  and  are  provided  each  with  a  vane  or  tail  similar  to 
that  of  a  windmill,  for  keeping  the  wheel  in  the  proper  position  as  regards  the 
current.  The  wheels  are  generally  made  like  those  of  a  windmill;  i.  «.,  with 
blades  set  at  such  an  angle  as  to  present  a  sloping  surface  to  the  current ;  and 
-with  the  axis  of  the  wheel  parallel  to  the  direction  of  flow.  The  axis  runs  in 
agate  bearings.  When  desired,  the  rim  of  the  wheel  is  furnished  with  an  air- 
chamber,  which  just  counterbalances  the  weight  of  the  wheel,  and  thus  removes 
journal  friction  due  to  it.  Meters  provided  with  electrical  registering 
apparatus  sometimes  have  the  gearing  and  indices,  etc.,  enclosed  in  a  glass  case, 
to  prevent  them  from  becoming  clogged  by  weeds,  sediment,  etc. 

A  wheel  meter  is  rated  by  moving  it  at  a  known  velocity  through  still 
water,  and  noting  the  eflfect  produced.  In  this  way  a  coeflftcient  is  obtained  for 
each  meter,  which,  when  multiplied  by  the  number  of  revolutions  recorded  in 
any  given  case,  gives  the  velocity  for  that  case. 


*See  Van  Nostrand's  Magazine.  March,  1878,  and  August,  1886. 


HYDRAULICS.  663 

Art.  21.  Kutter's  formula  for  the  mean  vel  of  water  flowing  in  open 
channels  of  uniform  cross  section  and  slope  throughout. 

Cautioo.  The  use  of  all  such  formulae  is  liable  to  error  arising  from  the 
difl&culty  of  ascertaining  the  exact  condition  of  the  stream  as  regards  roughness 
of  bed,  surface  slope,*  etc. 

Bem.  1.  Care  must  be  taken  tbat  the  bottoni  Tel  is  not  so 
g^reat  as  to  ^vear  anvay  the  soil.  If  there  is  any  such  danger  artificial 
means  must  be  applied  to  protect  the  channel-way;  or  it  may  be  advisable  to 
reduce  the  rate  of  fall,  and  increase  the  cross  section  of  the  channel ;  so  as  to 
secure  the  same  disch,  but  with  less  vel.  A  liberal  increase  sliould  also  be  madf 
in  the  dimensions  of  such  channels,  to  compensate  for  obstructions  to  the  flow, 
arisiug  from  the  growth  of  aquatic  plants,  or  deposits  of  mud  from  rain- 
washes,  etc ;  or  even  from  very  strong  winds  blowing  against  the  current. 

Rem.  2.  l¥ater  running  in  a  channel  with  a  horizontal  bed, 
or  bottoni,  cannot  have  a  uniform  vel,  or  depth,  throug-h- 
out  its  course;  because  the  action  of  gravity  due  to  the  inclined  plane  of  » 
sloping  bottom,  is  wanting  in  this  case;  and  the  water  can  flow  only  bv  forming 
its  surface  into  an  inclined  plane;  which  evidently  involves  a  diminution  of 
depth  at  every  successive  dist  from  the  reservoir. 


Theory  of  floiv.  It  is  generally  held  that  the  resistances  to  the  flow  ot 
water  in  a  pipe  or  channel  are  directly  proportional  to  the  area  of  the  bed  sur. 
face  with  which  the  water  comes  in  contact  (i  e,  to  the  product  of  the  "  wetted 
perimeter"  as  a &co  Figs  28,  29,  30  mult  by  the  length  of  the  channel,  or  of  the 
portion  of  it  under  consideration) ;  and  to  the  square  of  the  vel  of  the  flowing 
water;  and,  inasmuch  as  the  resistance  at  any  given  point  in  the  cross  section 
appears  to  be  inversely  as  the  dist  of  that  point  from  the  bottom  or  sides,  we 
conclude  that  the  total  resistances  are  inversely  as  the  area  of  the  cross  section  ; 
because  the  greater  that  area,  the  greater  would  be  the  mean  dist  of  all  the  par. 
tioles  from  the  bottom  and  sides.     The  resistance  is  independent  of  the  pre^mre. 

In  short,  the  resistances  are  assumed  to  be  in  proportion  to 

vei2  X  wet  perimeter  X  length  v^pl 

area  of  cross  section  a 

and  the  head  h''  in  feet  or  in  metres  etc,  required  to  overcome  those  resist, 
ances,  is 

resistance      a  coefacient,    vel2  X  wet  perimeter  X  length  ,,.       ^v'''pt 

"*''*"  C  area  of  wet  cross  section  a 

from  which  we  have 


''  =  ^cpl'  ^'  ^^i-V^^'\ 


area  of  wet       resistce 
-       ah"         ,  ll  a  A"  ,  /  1     vv'\/  cross  section         head 

«.=  _.    and    v  =  ^---   or    vel  -^-^  X  \  ^^^  perimeter  ><Ti5ith 


*  "In  measuring  the  slope  of  a  large  river,  the  ordinary  errors  of  the  most  careful  leveling  are  a. 
large  proportion  of  the  whole  fall ;  the  variation  of  level  in  the  cross  section  of  the  surface  is  often  as 
great  as  the  slope  for  ten  miles  or  more  ;  the  exact  point  where  the  level  should  be  taken  is  often 
uncertain  :  the  rise  and  fall  of  the  water  makes  it  extremely  diCBcult  to  decide  when  the  levels  should 
be  taken  at  the  upper  and  lower  points  ;  waves  of  translation  may  affect  the  inclination  to  a  great 
}.nd  uncertain  degree,  and  may  even  make  the  surface  slope  the  reverse  wav."  Genl  T.  G.  Ellis, 
Trans  Am  Soc  Civ  Engrs-  Aug  1877. 


564  HYDRAULICS. 

_  area  of  wet  cross  section  a        ,    .v    «i.   j       i-        j.     » 

But r or        —       is  the  "  hydraulic  radius  "  of 

wet  perimeter  p 

*'  mean  depth  "  or  "  mean  radius,"  R,  of  the  cross  section ; 
and        ^sistance  head  .     ^j.        Jl        jg  ^^^  inclination  or  sl(Jpe,  S,   (fre- 

length  I 

quently  denoted  by  "I")  of  the  hydraulic  grade  line,  or  the  sine  of  the 

angle  wso  Fig  l\.  In  open  channels,  it  is  =  the  fall  of  the  surface  per 

unit  of  length. 

We  therefore  have  velocity  =  A/— tt  X  l/mean  radius  X  slox)e 

or,  by  using  a  coeff  (c)  =  '\—?ri 

velocity  =  coefficient  c  X  l/mean  radius  X  slope 

or       v  =  c  \/WS 

The  earlier  hydraulicians  gave  (each  according  to  the  results  of  his  investiga- 
tions) yixed  values  for  tlie  coeff  c,  (generally  about  95  to  100  for  channels 
in  earth  or  gravel,  as  in  our  early  editions),  making  it,  in  other  words,  a  cori' 
slant,  and  independent  of  the  shape,  size,  slope  and  roughness  of  the  channel. 
But  more  recent  investigators  have  shown  that  the  coefficient  c  is  affected  by 
differences  in  any  of  these  particulars. 

According  to  the  formula  of  Ganguillet  and  Kutter  (generally  called,  for  con- 
venience, "Kutter's  formula-'*)  the  value  of  c  is: 


For  Englisli  measure.  For  metric  measure. 

C     = 


41.6  +  ^+i:^                         23  +  :?^  +  i 
slope n  slope        n 


V^mean  rad  in  feet  |/mean  rad  in  metres 

Tables  g-ivin^  values  of  c  for  diff  grades,  mean  radii  and  degrees  of 
leughness,  and  for  English  and  metric  measures,  are  given  on  pp  566,  etc. 

Here  n  is  a  "  coefficient  of  rou&rliness"  of  sides  of  channel  as  given 
below.  These  values  of  n  were  obtained  from  experience,  by  averaging  a  large 
number  of  experiments  made  under  very  different  circumstances.  They  there- 
fore embrace  all  the  disturbing  effects  arising  from  obstructions  existing  upon 
the  bottom  and  sides  of  the  channel  in  the  cases  experimented  upon.  In  small 
artificial  channels  of  uniform  cross  section  and  slope,  these  obstructions  may  be 
said  to  consist  entirely  of  the  comparatively  minute  roughnesses  of  the  matericU 
of  which  the  bed  of  the  channel  consists.  But  in  rivers  and  earth  canals,  even 
where  the  general  direction,  slope  and  cross  section  are  tolerably  uniform,  (as 
they  were  in  the  cases  upon  which  our  list  is  based),  there  are  still  many  con- 
siderable irregularities  in  the  sides  and  bottom  ;  and  these  exert  a  much  greater 
retarding  effect  upon  the  mean  vel  than  the  mere  roughness  of  the  material  of 
the  banks.  We  therefore  find  larger  values  given  for  n  in  such  cases  than  for 
small  regular  artificial  channels,  although  the  material  of  the  sides  etc  was  in 
many  cases  smooth  mud;  and  we  must  not  apply  to  such  comparatively  irregu* 
lar  channels  the  small  values  of  w  obtained  by  experiments  with  small  and  care- 
fully made  straight  flumes  of  uniform  section  and  slope,  even  if  we  suppose  the 
bottom  and  sides  of  the  former  to  be  made  as  smooth  as  those  of  the  latter. 

No  general  formula  is  applicable  to  cases  of  decided  bends  in  the  course 
of  a  natural  stream,  or  of  marked  irregularities  in  tbe  cross  sec- 
tion. Such  cases  would  require  still  higher  coefficients  to  than  those  here  given 
for  rivers  and  canals  ;  but  they  would  have  to  be  ascertained  by  experiment  for 
each  case,  and  would  be  useless  for  other  cases.  For  such  streams  we  must  there- 
fore depend  upon  actual  measurements  of  the  velocity,  either  direct  or  by  means 
of  the  disch. 

*  See  "  Flow  of  Water,"  translated  from  Ganguillet  and  Kutter,  by  Rudolph 
Hering  and  John  C.  Trautwine,  Jr.,  New  York,  John  Wiley  &  Sons,  1889.  $4.00. 


HYDRAULICS.  565 

There  is  much  room  for  the  exercise  of  judgment  iu  the  selection  of  the 
proper  coefficient  n  for  any  given  case,  even  where  the  condition  of  the 
channel  is  well  known.  It  may  frequently  be  necessary  to  use  values  of  n  inter- 
mediate between  those  given  :  for  careless  brickwork  may  be  rougher  than  well 
finished  rubble;  side  slopes  in  "very  firm  gravel  "  may  have  very  diff  degrees 
of  roughness ;  etc  etc.  The  engineer  should  make  lists  of  values'of  n  from  his 
own  experience,  fully  noting  the  peculiarities  of  each  case,  and  calculating  n 
from  the  tables. 

A  given  diflf  in  the  deg  n  of  roughness  exerts  a  much  greater  effect  upon  the 
coefficient  c,  and  thus  upon  the  velocity,  in  small  channels  than  in  larger  ones. 
It  is  therefore  especially  necessary  in  small  channels  that  care  be  exercised  in 
finding  (by  experiment  if  necessary)  the  proper  value  of  n;  and,  where  a  large 
disch  is  desired,  the  sides  of  small  channels  should  be  made  particularly  smootho 

Table  of  n,  or  coefficient  of  rou^linefss. 

In  any  given  case  the  value  of  n  is  the  same  whether  the  mean 
radius  is  ^iven  in  Eng^lish,  metric  or  any  other  measure. 

Artificial  channels  of  uniform  cross  section. 

Sides  and  bottom  of  channel  lined  with  n  = 

well  planed  timber 009 

neat  cement    (applies  also  to  glazed  pipes  and  very  smooth  iron  pipes).  .010 
plaster  of  1  measure  of  sand  to  3  of  cement;     (or  smooth  iron  pipes).  .011 

unplaned  timber  (applies  also  to  ordinary  iron  pipes) 012 

ashlar  or  brickwork 013 

rubble ^ 017 

Channels  subject  to  irreg^ularity  of  cross  section. 

Canals  in  very  firm  gravel 020 

Canals  and  rivers  of  tolerably  uniform  cross  section,  slope  and  direction, 
in  moderately  good  order  and  regimen,  and  free  from  stones  and 

weeds .' 025 

having  stones  and  weeds  occasionally 030 

in  bad  order  and  regimen,  overgrown  with  vegetation,  and  strewn 

with  stones  and  detritus ~ 035 

Art.  22.   The  following  tables  give  Talues  of  the  coefficient 

e  as  obtained  by  Kutter's  formula  for  diff  slopes  (S)  mean  radii  (E)  and  degreei 
of  roughness  (n).* 

Caution.  Different  values  of  c  must  be  used  with  English  and  with  metric 
measures.    We  give  tables  for  both  measures. 

1st.  Having  the  slope  S,  the  mean  rad  R  and  the  deg  n  of  roughness ;  to 
find  the  coefl*  c.  Turn  to  the  division  of  the  table  corresponding  to  the 
given  slope  S.  In  the  first  column  find  the  given  mean  rad,  R.  In  the  same 
line  with  this  R,  and  under  the  given  n,  is  the  proper  value  of  c* 

2d.    Having  the  slope  S,  the  mean  rad  R  and  either  c  or  the  actual  or  reqd 
vel  v;  to  find  the  actual,  or  the  g-reatest  permissible,  deg  n  of 
roughness  of  channel.    If  the  vel  is  given,  and  not  c,  first  find 
velocity 

e  ==     Turn  to  the  division  of  the  table  corresponding  to 

V^ilope  X  mean  radius 
the  given  S,  and  in  the  first  col  find  the  given  R.    In  the  same  line  find  the 
value  given,  or  just  obtained,  for  c;  over  which  will  be  found  the  reqd  n.* 

3d.  Having  the  slope  S,  the  deg  n  of  roughness,  and  the  actual  or  required 
vel  v\  to  find  the  actual  or  necessary  mean  rad,  R.  Assume  a 
mean  rad;  and  from  the  division  of  the  table  corresponding  to  the  given  S  take 
out  the  value  of  c  corresponding  to  the  given  n  and  the  assumed  R.    Then  say 

t/  =  cso  found  X  i/ assumed  mean  radius  X  slope 

*It  is  ofteu  necessary  to  interpolate  values  of  S,  R,  n  and  c  intermediate  of 
those  in  the  tables ;  this  may  be  done  mentally  by  simple  proportion. 


566 


HYDRAULICS. 


If  this  v'  is  the  same  as  the  given  vel,  or  near  enough  to  it,  take  the  assumed  B 
as  the  proper  one.  Otherwise,  repeat  the  whole  process,  assuming  a  new  R, 
greater  than  the  former  one  iiv'  is  Less  than  the  given  vel,  and  vice  versa.* 

4tli.  Having  the  dimensions  of  the  wetted  portion  (abco  Figs  28,  29,  30,)  of 
the  channel,  the  degw  of  roughness,  and  the  actual  or  reqd  vel;  to  find  tb« 
actual  or  necessary  slope,  S : 

„.      •  ,  ,  _  area  of  wet  cross  section 

Find  the  mean  rad,  R  =  ^ n r ? : : — r-~ 

length,  abco,  of  wet  perimeter 

Assume  one  of  the  four  slopes  of  the  tables  to  be  the  proper  one.  From  the 
correspond! Hi?  division  of  the  table  take  out  the  value  of  c  corresponding  to  the 
given  R  and  n. 

If  R  is  3.28  feet,  or  1  metre,  the  value  of  c  thus  found  is  the  proper  one  (be* 
cause  then  c,  for  any  given  n,  remains  the  same  for  all  slopes) ;  and  the  slope,  S| 
may  be  found  at  onfce,  thus; 


Slope, 


given  velocity     \2 
\c  X  l/mean  radius/ 
But  if  R  is  greater  or  less  than  3.28  feet,  or  1  metre,  say 


■H 


v'  =  c  thus  found  X  l/mean  radius  X  assumed  slope 

If  this  i/  is  near  enough  to  the  given  vel,  take  the  assumed  S  as  the  proper  one. 
Otherwise,  assume  a  new  S,  greater  than  the  former  one  if  i/  is  less  than  the  given 
vel,  and  vice  versa ;  and  repeat  the  whole  process.* 

*  It  is  often  necessary  to  interpolate  values  of  S,  B,  M  and  o  intermediate  of  those  in  the  tablM. 
This  may  be  done  mentally  by  simple  proportion. 


Table  of  coefficient  c,  for  mean  radii  in  j 


a 

Mean 

CoeflBicients  n  of  roughness 

Mean 

£ 

radR 

radR 

I 

5 

feet 

.009 

.010 

.011 

.012 

.013 

.015 

.017 

.020 

.025 

.030 

.035 

.040 

feet 

c 

c 

c 

c 

c 

c 

c 

c 

c 

c 

c 

c 

60 

.1 

65 

57 

50 

44 

40 

33 

28 

23 

17 

14 

12 

10 

.1 

§d 

.2 

87 

75 

67 

59 

53 

45 

38 

31 

24 

19 

16 

14 

.2 

A 

111 

97 

87 

78 

70 

59 

51 

42 

32 

26 

22 

19 

.4 

.6 

127 

112 

100 

90 

81 

69 

60 

49 

38 

31 

26 

22 

.6 

.8 

138 

122 

109 

99 

90 

77 

66 

55 

43 

35 

30 

25 

.8 

0  *^ 

1 

148 

131 

118 

106 

97 

83 

72 

60 

47 

38 

32 

28 

1 

tl* 

1.5 

166 

148 

133 

121 

111 

95 

83 

69 

55 

45 

38 

33 

1.5 

2 

179 

160 

144 

131 

121 

104 

91 

77 

61 

50 

43 

37 

2 

e| 

3 

197 

177 

160 

147 

135 

117 

103 

88 

70 

59 

50 

44 

3 

«-i 

3.28 

201 

181 

164 

151 

139 

121 

106 

91 

72 

60 

52 

46 

3.28 

§11 

Co 

4 

209 

188 

172 

158 

146 

127 

113 

96 

78 

65 

56 

49 

4 

6 

226 

206 

188 

174 

161 

142 

126 

108 

88 

74 

64 

57 

6 

c§ 

8 

238 

216 

199 

184 

171 

151 

135 

117 

96 

82 

71 

63 

8 

1^ 

10 

246 

225 

207 

192 

179 

159 

142 

124 

102 

87 

76 

68 

10 

12 

253 

231 

214 

198 

186 

165 

149 

129 

107 

92 

81 

72 

12 

m 

16 

263 

242 

223 

208 

195 

174 

157 

138 

115 

100 

88 

79 

16 

« 

20 

271 

249 

231 

215 

202 

181 

164 

144 

121 

106 

94 

84 

20 

& 

30 

283 

261 

243 

228 

215 

193 

176 

157 

133 

117 

104 

95 

30 

0 

pit 

50 

297 

274 

257 

241 

228 

207 

190 

170 

147 

130 

117 

107 

50 

« 

75 

306 

284 

267 

251 

238 

217 

200 

180 

157 

140 

127 

117 

75 

,  100 

312 

290 

273 

257 

244  223  1  207  1 187  1 163 

147 

134  124 

100 

HYDRAULICS. 


567 


T^ble  of  coefficient  c. 

for 

mean  radii  in /ee/.— Continued. 

Mean 

Coefficients  n  of  roughness 

Mean 

rad  R 

radR 

a 

feet 

.009 

.010 

.011 

.012  i.013 

.015 

.017 

.020 

.025 

.030 

.035 

.040 

feet 

1 

c 

c 

c 

C 

r 
c 

c 

c 

c 

c 

c 

c 

c 

a 

.1 

78 

67 

59 

52 

47 

39 

33 

26 

20 

16 

13 

11 

.1 

&  . 

.15 

91 

79 

69 

62 

56 

46 

39 

31 

23 

19 

16 

13 

.15 

c^ 

.2 

100 

87 

77 

68 

62 

51 

44 

35 

26 

21 

18 

15 

.2 

.3 

114 

99 

88 

79 

71 

59 

50 

41 

31 

25 

21 

18 

.3 

A 

124 

109 

97 

88 

79 

66 

67 

46 

35 

28 

24 

20 

.4 

•s- 

.6 

139 

122 

109 

98 

90 

76 

65 

53 

41 

33 

28 

24 

.6 

.8 

150 

133 

119 

107 

98 

83 

71 

59 

46 

37 

31 

27 

.8 

SI 

1 

158 

140 

126 

114 

104 

89 

77 

64 

49 

40 

34 

29 

1 

s*^  ■ 

1.5 

173 

154 

139 

126 

116 

99 

87 

72 

57 

47 

40 

34 

1.5 

P-Tt< 

2 

184 

164 

148 

135 

124 

107 

94 

79 

62 

51 

44 

38 

2 

§^ 

3 

198 

178 

161 

148 

136 

118 

104 

88 

71 

59 

50 

44 

3 

o  II 

3.28 

201 

181 

164 

151 

139 

121 

106 

91 

72 

60 

52 

46 

3.25 

4 

207 

187 

170 

156 

145 

126 

111 

95 

77 

64 

56 

49 

4 

6 

220 

199 

182 

168 

156 

137 

122 

105 

85 

72 

63 

56 

6 

n'i 

8 

228 

206 

189 

175 

163 

144 

129 

111 

91 

78 

68 

61 

8 

10 

234 

212 

195 

181 

169 

149 

134 

116 

96 

82 

72 

64 

10 

12 

238 

217 

200 

185 

173 

153 

138 

120 

99 

86 

75 

68 

12 

0 

16 

245 

223 

206 

191 

180 

160 

144 

126 

106 

91 

81 

73 

16 

20 

250 

228 

211 

1^6 

184 

165 

149 

131 

110 

96 

85 

77 

20 

s 

30 

257 

236 

219 

204 

192 

172 

157 

139 

118 

103 

92 

84 

3a 

50 

266 

245 

228 

213 

201 

181 

165 

148 

127 

112 

lOi 

93 

50 

75 

272 

250 

233 

218 

207 

187 

171 

153 

133 

119 

108 

99 

75 

100 

275  1  254 

237 

222 

210 

190 

175 

158 

137 

123 

112 

104 

100 

a 

.1 

90 

78 

68 

60 

54 

"  44 

37 

30 

22 

17 

14 

12 

.1 

"«aJ 

.2 

112 

98 

86 

76 

69 

57 

48 

39 

29 

23 

19 

16 

.2 

O^ 

.3 

125 

109 

97 

87 

78 

65 

56 

45 

34 

27 

22 

19 

.3 

^a 

.4 

136 

119 

106 

95 

86 

72 

62 

50 

38 

31 

25 

22 

.4 

•o^ 

.6 

149 

131 

118 

105 

96 

81 

70 

.57 

44 

3p 

30 

25 

.6 

^^ 

.8 

1.58 

140 

126 

114 

103 

88 

76 

63 

48 

39 

33 

28 

.8 

a  *^ 

1 

166 

147 

132 

120 

109 

93 

81 

67 

52 

42 

35 

31 

1 

SI 

1.5 

178 

159 

144 

130 

120 

103 

89 

75 

59 

48 

41 

35 

1.5 

1- 

2 

187 

168 

151 

138 

127 

109 

96 

81 

64 

53 

45 

39 

2 

3 

198 

178 

162 

149 

137 

119 

104 

89 

71 

59 

51 

45 

3. 

s  „  ■ 

3.28 

201 

181 

164 

151 

139 

121 

106 

91 

72 

60 

52 

46 

3.28 

©II 

4 

206 

186 

169 

155 

143 

125 

111 

94 

76 

64 

55 

49 

4 

©g- 

lll 

6 

215 

195 

i78 

164 

152 

134 

119 

102 

84 

71 

61 

54 

6 

8 

221 

201 

184 

170 

158 

139 

124 

107 

88 

75 

66 

59 

8 

10 

226 

205 

188 

174 

162 

143 

128 

111 

92 

78 

69 

62 

10 

^a 

15 

233 

212 

195 

181 

169 

150 

135 

118 

98 

85 

75 

68 

15 

E 

20 

237 

216 

200 

185 

173 

154 

139 

122 

102 

89 

79 

71 

20 

30 

243 

222 

206 

191 

179 

160 

145 

128 

108 

95 

84 

77 

30 

50 

249 

227 

211 

197 

185 

166 

151 

134 

114 

100 

91 

83 

50 

as 

100 

255 

234 

218 

204 

191 

172 

158 

140 

121 

108 

98 

91 

100 

^ 

r     -1 

99 

85 

74 

65 

59 

48 

41 

32 

24 

18 

15 

12 

.1 

-ad 

.2 

121 

105 

93 

83 

74 

61 

52 

42 

31 

25 

21 

17 

.2 

c::: 

.3 

133 

116 

103 

92 

83 

69 

59 

48 

36 

29 

24 

20 

.3 

-£  a 

.4 

143 

125 

112 

100 

91 

76 

65 

53 

40 

32 

27 

23 

.4 

og 

.6 

155 

138 

122 

111 

100 

85 

73 

60 

46 

37 

31 

26 

.6 

•t^  A 

.8 

164 

145 

131 

118 

107 

91 

79 

65 

50 

41 

34 

29 

.8 

§1 

1 

170 

151 

136 

123 

113 

96 

83 

69 

54 

44 

37 

32 

1 

*,«2 

1.5 

181 

162 

146 

133 

122 

105 

91 

77 

60 

49 

42 

36 

1.5 

^g 

2 

188 

170 

154 

140 

129 

111 

97 

82 

64 

54 

46 

40 

2 

«l«  ^ 

•  3 

200 

179 

163 

149 

137 

119 

105 

89 

72 

69 

51 

45 

3 

©-^ 

4 

205 

185 

168 

155 

143 

125 

111 

94 

76 

63 

55 

48 

4 

§jL 

6 

213 

193 

176 

150 

132 

117 

100 

82 

69 

60 

53 

6 

8 

218 

198 

181 

167 

155 

137 

122 

105 

87 

73 

64 

57 

8 

10 

222 

201 

185 

170 

158 

140 

125 

108 

89 

76 

67 

60 

10 

aB§ 

15 

228 

207 

190 

176 

164 

145 

131 

113 

95 

82 

72 

65 

15 

».2 

20 

231 

210 

194 

180 

168 

149 

134 

117 

98 

85 

76 

68 

20 

S"-^ 

30 

235 

215 

198 

184 

172 

154 

139 

122 

103 

89 

80 

73 

30 

in 

50 

240 

220 

203 

189 

177 

158 

143 

126 

108 

94 

85 

78 

50 

3 

100 

245 

224 

208 

194 

182 

163 

148 

131 

113 

99 

90 

83 

100 

568 


HYDRAULICS. 


Table  of  coefficient  c. 

for 

mean  radii  in  /ee^.— Continued. 

Mean 

Coefficients  n  of  rougliness. 

Mean 

S 

rad  R 

radB 

fi 

feet 

.009 

.010 

.011 

.012 

.013 

.015 

.017 

.020 

.025 

.030 

.035 

.040 

feet 

fa 

c 

C 

C 

C 

C 

C 

C 

C 

C 

C 

c 

c 

<=£ 

.1 

104 

89 

78 

69 

62 

50 

43 

34 

25 

19 

16 

13 

.1 

:S  a 

.15 

116 

101 

90 

80 

71 

69 

50 

40 

29 

23 

19 

16 

.15 

3  (D 

.2 

126 

110 

97 

87 

78 

65 

54 

44 

32 

25 

21 

18 

.2 

^.2 

.3 

138 

120 

107 

96 

87 

73 

62 

50 

37 

30 

24 

21 

.3 

S.S 

.4 

148 

129 

115 

104 

94 

79 

68 

55 

42 

33 

27 

23 

.4 

iH'-i  ■ 

.6 

157 

140 

126 

113 

103 

87 

75 

62 

47 

38 

31 

27 

.6 

©^ 

.8 

166 

148 

133 

121 

no 

93 

81 

67 

51 

42 

35 

30 

.8 

u 

1 

172 

154 

138 

125 

115 

98 

85 

70 

55 

45 

37 

32 

1 

1.5 

183 

164 

148 

135 

124 

106 

93 

78 

61 

50 

42 

37 

1.5 

II  § 

2 

190 

170 

154 

141 

130 

112 

98 

83 

65 

54 

45 

40 

2 

1^ 

3 

199 

179 

162 

149 

138 

119 

105 

89 

71 

59 

51 

45 

3 

4 

204 

184 

168 

154 

142 

124 

110 

94 

76 

63 

55 

48 

4 

6 

211 

191 

175 

161 

149 

130 

116 

99 

81 

69 

60 

53 

6 

10 

219 

199 

183 

168 

157 

138 

123 

107 

88 

75 

66 

59 

10 

s 

20 

227 

207 

190 

176 

164 

146 

131 

115 

96 

83 

73 

66 

20 

50 

235 

215 

198 

184 

173 

154 

139 

123 

104 

91 

82 

75 

50 

100 

239 

219 

203 

189 

177 

158 

143 

127 

108 

96 

87 

80 

100 

!^ 

.1 

110 

94 

83 

73 

65 

54 

45 

36 

27 

21 

17 

14 

.1 

.2 

129 

113 

99 

89 

81 

66 

57 

45 

34 

27 

22 

18 

.2 

•5^ 

.3 

141 

124 

109 

98 

89 

74 

63 

51 

39 

30 

25 

21 

.3 

==d 

.4 

150 

131 

117 

105 

96 

80 

69 

56 

43 

34 

28 

24 

.4 

^<» 

.6 

161 

142 

127 

115 

104 

88 

76 

63 

48 

39 

32 

27 

.6 

^S 

.8 

169 

150 

134 

122 

111 

94 

82 

68 

52 

42 

35 

30 

.8 

S|- 

1 

175 

155 

139 

127 

116 

99 

86 

71 

56 

45 

38 

33 

1 

1.5 

184 

165 

149 

136 

124 

108 

93 

78 

62 

50 

43 

37 

1.5 

©o  ^ 
©o 

2 

191 

171 

155 

142 

130 

112 

98 

83 

66 

54 

46 

40 

2 

3 

199 

179 

163 

149 

138 

119 

105 

89 

71 

59 

51 

45 

3 

II  .s 

4 

204 

184 

168 

154 

142. 

124 

110 

93 

75 

63 

54 

48 

4 

5e^ 

6 

211 

190 

174 

160 

149 

130 

116 

99 

81 

68 

59 

52 

6 

c^  II 

10 

218 

197 

181 

167 

155 

136 

122 

105 

87 

74 

65 

58 

10 

9kd 

20 

225 

205 

188 

175 

163 

144 

129 

113 

94 

81 

72 

65 

20 

5^ 

50 

232 

212 

196 

182 

170 

151 

137 

120 

101 

89 

79 

72 

50 

Zg 

100 

236 

216 

200 

186 

174 

155 

141 

124 

105 

94 

85 

77 

100 

fl" 

.1 

110 

95 

"83" 

"74~ 

66 

54 

46 

36 

27 

21 

17 

14 

.1 

"S>ii 

.15 

122 

105 

93 

83 

75 

62 

52 

42 

31 

24 

20 

17 

.15 

S- 

.2 

130 

114 

100 

90 

81 

67 

57 

46 

34 

27 

22 

19 

.2 

r2^  - 

.3 

143 

125 

111 

100 

90 

76 

64 

52 

39 

31 

25 

22 

.3 

"o  0^ 

.4 

151 

133 

119 

107 

98 

82 

70 

57 

44 

35 

29 

24 

.4 

^  & 

.6 

162 

143 

129 

116 

106 

90 

77 

64 

49 

39 

33 

28 

.6 

'S'§ 

.8 

170 

151 

135 

123 

112 

95 

82 

68 

53 

43 

35 

31 

.8 

p^ 

1 

175 

156 

141  i  128 

117 

99 

87 

72 

56 

45 

88 

33 

1 

sS 

1.5 

185 

165 

149  136 

125 

107 

94 

79 

62 

51 

43 

37 

1.5' 

^s  ■ 

2 

191 

171 

155 

142 

130 

112 

99 

83 

66 

55 

46 

40 

2 

©  11 

3 

199 

179 

162 

149 

138 

119 

105 

89 

71 

59 

51 

45 

3 

3.28 

201 

181 

164 

151 

139 

121 

106 

91 

72 

60 

52 

46 

3.28 

II  o 

4 

204 

184 

167 

154 

142 

123  1  109 

93 

76 

63 

55 

48 

4 

6 

210 

190 

173 

160 

148 

129 

115 

99 

81 

68 

59 

52 

6 

10 

217 

196 

180 

166 

154 

136 

121 

105 

86 

74 

65 

58 

10 

ft^ 

20 

225 

204 

187 

173 

161 

143 

128 

112 

93 

80 

71 

64 

20 

2" 

50 

231 

210 

194 

181 

168 

150 

135 

119 

100 

87 

78 

71 

50 

S 

100 

235 

214 

197  1  184  1  172 

153 

139 

122  1  104  1  91 

82  1  75 

100 

For  slopes  steeper  than  .01  per  unit  of  length,  =  1  in  100  =  52.8  feet 
per  mile,  c  remains  practically  the  same  as  at  that  slope.  But  the  velocitp 
(being  =  c  X  l/mean  radius  X  slope)  of  course  continues  to  increase  as  thfl 
8k)pe  becomes  steeper. 


HYDRAULICS. 


569 


Table  of  coefficient  c,  for  mean  radii  in  metres. 


Mean 

Coefficients  n  of  roughness. 

Mean 

5 

radR 

radR 

s 

metres 

.009 
c 

.010 
C 

.011 

.012 

013 

015 

017 

.020 

.025 

.030 

.035 

.040 

metres 

«^. 

c 

c 

c 

c 

c 

c 

c 

c 

c 

c 

o 

.025 

34 

29 

25 

22 

20 

17 

14 

11 

9 

7 

6 

5 

.025 

*s 

-.05 

44 

38 

33 

30 

27 

22 

19 

16 

12 

9 

8 

7 

.05 

^o 

.1 

58 

50 

44 

40 

36 

30 

26 

21 

16 

13 

11 

9 

.1 

»-o 

.2 

72 

63 

56 

51 

46 

39 

34 

28 

21 

18 

15 

13 

.2 

^i 

^ 

82 

72 

64 

58 

53 

45 

39 

33 

25 

21 

17 

15 

.3 

tta  ■ 

.4 

89 

79 

71 

64 

59 

60 

44 

37 

29 

23 

20 

17 

.4 

.6 

.99 

88 

80 

72 

67 

57 

50 

42 

33 

28 

23 

20 

.6 

1. 

111 

100 

90 

83 

77 

67 

59 

50 

40 

33 

28 

25 

1. 

1.50 

121 

109 

100 

92 

85 

74 

66 

57 

46 

38 

33 

29 

1.50 

e 

2 

127 

115 

106 

98 

91 

80 

71 

61 

50 

42 

37 

32 

2 

1 

3 

136 

124 

114 

106 

99 

87 

78 

68 

56 

48 

42 

37 

3 

4 

142 

130 

120 

111 

104 

93 

83 

73 

61 

52 

46 

41 

4 

0 

6 

149 

137 

127 

119 

111 

100 

90 

80 

67 

58 

51 

46 

6 

10 

158 

145 

135 

127 

120 

108 

98 

88 

75 

66 

69 

53 

10 

i 

15 

164 

151 

141 

133 

126 

114 

104 

94 

81 

72 

64 

59 

15 

20 

167 

155 

145 

137 

130 

118 

108 

98 

85 

75 

68 

62 

20 

.  30 

172 

160  1  150 

142 

135 

123 

113 

103 

90   81 

74 

68 

30 

r   .025 

40 

35 

30 

26 

24 

20 

17 

13 

10 

8 

7 

6 

.025 

O 

.05 

52 

44 

39 

34 

31 

26 

22 

18 

13 

11 

9 

7 

.05 

.1 

65 

57 

50 

44 

40 

34 

29 

24 

18 

14 

12 

10 

.1 

*S  . 

.2 

79 

69 

62 

55 

51 

43 

37 

30 

23 

19 

16 

13 

.2 

.3 

87 

77 

69 

62 

67 

48 

42 

35 

27 

22 

18 

16 

.3 

.4 

93 

83 

74 

67 

62 

53 

46 

38 

30 

25 

21 

18 

.4 

p,c^ 

.6 

102 

90 

82 

74 

69 

69 

52 

43 

34 

28 

24 

21 

.6 

i- 

1. 

111 

100 

90 

83 

77 

67 

59 

50 

40 

33 

28 

25 

1. 

1.5 

118 

107 

97 

90 

83 

73 

65 

55 

45 

38 

33 

28 

1.5 

O  it 

2 

123 

111 

102 

94 

87 

77 

68 

59 

48 

41 

35 

31 

2 

•^ 

3 

129 

117 

108 

100 

93 

83 

74 

64 

53 

45 

40 

35 

3 

yu 

4 

133 

121 

112 

104 

97 

86 

77 

68 

56 

49 

43 

38 

4 

«s 

6 

138 

126 

117  t  109 

102 

91 

82 

72 

61 

53 

47 

42 

6 

t^ 

10 

143  1  131 

122  i  114 

107 

96 

87 

78 

66 

58 

52 

47 

10 

0 

15 

147  1  135 

126 

118 

111 

100 

91 

82 

70 

62 

56 

51 

15 

20 

150 

137 

128 

120 

113 

103 

94 

84 

72 

64 

58 

53 

20 

30 

152 

140 

131 

123 

116 

105 

97 

87 

76 

68 

62 

57 

30 

o 

r   .025 

47 

40 

f  35 

31 

28 

22 

19   15 

11 

9 

7 

6 

.025 

.05 

59 

50 

44 

40 

35 

29 

25 

20 

15 

12 

10 

8 

.05 

'=  o 

.1 

72 

62 

55 

50 

45 

37 

32 

26 

19 

16 

13 

11 

.1 

3  o 

.2 

84 

74 

66 

60 

54 

46 

39 

82 

25 

20 

17 

14 

.2 

«s 

.3 

91 

81 

73 

66 

60 

51 

44 

37 

28 

23 

19 

17 

.3 

*a 

.4 

97 

86 

77 

70 

64 

55 

48 

40 

61 

25 

21 

18 

.4 

©^ 

.6 

104 

92 

83 

76 

70 

60 

53 

45 

35 

29 

25 

21 

.6 

1. 
1.5 

111 
117 

100 
105 

90 
96 

83 

88 

77 
82 

67 
72 

59 
64 

50 

40 

33 
37 

28 
32 

25 

28 

1. 

54 

44 

1.5 

«l 

2 

120 

109 

100 

92 

85 

75 

67 

57 

47 

40 

34 

30 

2 

4 

128 

116 

107 

99 

92 

82 

73 

64 

53 

46 

40 

36 

4 

6 

131 

119 

110 

102 

96 

85 

77 

67 

56 

49 

43 

39 

6 

10 

135 

123 

114 

106 

100 

89 

81 

71 

60 

53 

47 

43 

10 

15 

137 

126 

116 

109 

102 

92 

83 

74 

63 

55 

50 

46 

15 

s 

30 

141 

129 

120 

112 

106 

95 

87 

78 

67 

69 

64 

60 

30 

=«§ 

f   .025 

52 

45 

40 

35 

31 

25 

21 

17 

12 

9 

8 

6 

.025 

as 
g.2 

.050 

63 

55 

48 

43 

39 

32 

27 

21 

16 

12 

10 

8 

.050 

.1 

75 

66 

69 

53 

48 

40 

34 

27 

21 

16 

13 

11 

.1 

.2 

87 

77 

69 

62 

57 

48 

41 

34 

26 

21 

17 

15 

.2 

2^ 

.4 

99 

88 

80 

72 

66 

57 

49 

41 

32 

26 

22 

19 

.4 

•I 

.6 

104 

93 

84 

77 

71 

61 

53 

45 

36 

29 

25 

22 

.6 

II  .fl" 

1 

111 

100 

90 

83 

77 

67 

59 

60 

40 

33 

28 

25 

1 

4^^ 

2 

118 

107 

98 

90 

84 

74 

65 

56 

46 

39 

34 

30 

2 

ft^ 

4 

124 

113 

104 

97 

90 

79 

71 

62 

51 

44 

39 

35 

4 

©i^ 

10 

130 

119 

110 

102 

96 

85 

77 

67 

57 

60 

45 

40 

10 

S'o 

30 

135 

124 

114 

107 

100 

90 

82 

73 

62 

55 

50 

46 

30 

670 


HYDRAULICS. 


Table  of  coefficient 

c,  for  mean  radii  in  wip^re*.— Continued. 

5 

Mean 

Coefacients  n  of  roughness. 

Mean 

feo 

radR 

radB 

.2 

meters 

.009 

.010 

.011 

.012 

.013 

.015 

.017 

.020 

.025 

.030 

.035 

.040 

metres 

© 

c 

C 

c 

c 

c 

c 

c 

c 

c 

c 

c 

c 

{3 

.025 

55 

47 

41 

37 

33 

27 

22 

17 

13 

10 

8 

7 

.025 

.050 

66 

68 

61 

45 

40 

33 

28 

23 

17 

13 

11 

9 

.050 

.1 

78 

68 

61 

55 

50 

42 

35 

28 

21 

17 

14 

12 

.1 

fi- 

.2 

90 

80 

70 

64 

69 

49 

42 

35 

27 

22 

18 

15 

.2 

.3 

95 

85 

76 

70 

63 

64 

47 

39 

30 

24 

21 

17 

.3 

©TH 

.4 

99 

89 

80 

73 

67 

67 

60 

42 

32 

27 

22 

20 

.4 

.6 

105 

94 

85 

78 

72 

62 

54 

45 

36 

30 

25 

22 

.6 

1 

111 

100 

90 

83 

77 

67 

69 

60 

40 

33 

28 

25 

1 

II 

2 

117 

106 

97 

89 

83 

73 

65 

56 

46 

38 

34 

30 

2 

i 

4 

123 

111 

102 

95 

88 

78 

70 

61 

50 

43 

38 

34 

4 

1 

6 

125 

114 

105 

97 

91 

81 

72 

63 

63 

46 

40 

36 

6 

10 

128 

117 

108 

100 

93 

83 

75 

66 

65 

48 

43 

39 

10 

s 

30 

132 

121 

112 

104 

98 

87 

79 

70 

60 

52 

48 

43  I  30 

^ 

.025 

67 

60 

43 

38 

34 

28 

23 

18 

13 

11 

9 

7 

.025 

9d 

.  .050 

69 

59 

52 

47 

42 

34 

29 

23 

17 

13 

11 

9 

.060 

^§ 

.1 

80 

70 

63 

66 

60 

42 

36 

30 

22 

17 

14 

12 

.1 

o,^ 

.2 

90 

80 

72 

65 

60 

50 

43 

35 

27 

22 

18 

16 

.2 

.3 

96 

86 

77 

70 

64 

54 

47 

39 

30 

25 

21 

18 

.3 

.4 

100 

89 

81 

74 

67 

58 

60 

42 

33 

27 

23 

19 

.4 

.6 

104 

94 

85 

78 

72 

62 

64 

46 

36 

30 

25 

22 

.6 

•pSf 

1 

111 

100 

90 

83 

77 

67 

59 

50 

40 

33 

28 

25 

1 

n  i* 

2 

116 

106 

97 

90 

83 

72 

64 

55 

45 

38 

33 

29 

2 

*g 

4 

121 

111 

102 

94 

87 

77 

69 

60 

50 

42 

37 

33 

4 

ft^ 

6 

124 

113 

104 

97 

90 

80 

71 

62 

52 

45 

40 

36 

6 

5o 

10 

127 

115 

106 

99 

92 

82 

73 

64 

54 

47 

42 

38 

10 

3 

30 

130 

119 

110 

102 

96 

86 

77 

68 

58 

51 

46 

42 

30 

.025 

59 

50 

44 

39 

35 

28 

24 

19 

14 

10 

9 

7 

.025^ 

^  . 

.05 

69 

60 

53 

48 

43 

35 

29 

24 

18 

14 

11 

9 

.05 

S| 

.1 

81 

71 

63 

57 

51 

43 

36 

30 

22 

18 

15 

12 

.1 

fc.  ^ 

.2 

91 

81 

72 

65 

60 

50 

44 

36 

27 

22 

18 

16 

.2 

a-= 

.3 

97 

86 

77 

71 

65 

55 

48 

40 

31 

25 

21 

18 

.3 

.4 

101 

90 

81 

74 

68 

58 

50 

42 

33 

27 

23 

20 

.4 

o  II 

.6 

106 

95 

86 

78 

72 

62 

54 

46 

36 

30 

25 

22 

.6 

11  a 

1. 

111 

100 

90 

83 

77 

67 

59 

50 

40 

33 

28 

25 

1. 

1.5 

115 

104 

94 

87 

80 

70 

62 

53 

43 

36 

31 

27 

1.5 

2 

117 

105 

96 

89 

83 

72 

64 

55 

45 

38 

33 

29 

2 

4 

121 

110 

101 

93 

87 

76 

68 

59 

49 

42 

37 

33 

4 

S^ 

10 

126 

114 

105 

98 

91 

81 

73 

64 

53 

46 

41 

37 

10 

30 

129 

118 

108 

101 

95 

84 

77 

67 

57 

50 

45 

41  30 

For  slopes  steeper  tlian  .01  per  unit  of  length,  =  1  in  100,  the  eo' 
efficient  c  remains  practically  the  same  as  at  that  slope.  The  velocity,  however, 
being  =  cX  l/mean  radius  X slope,  continues  to  increase  as  the  slope  becomes 
steeper. 

To  construct  a  diagram,  fig  30  A,  from  which  the  values  g^iven 
by  Kutter's  formula  may  be  taken  by  inspection. 

Draw  xz  hor,  and  say  from  2"to  4  ft  long;  and  oy  vert  at  any  point  o  within 
say  the  middle  third  o\  xz.  On  oy  lay  off,  as  shown  on  the  left,  the  values  of  c 
for  which  the  diagram  will  probably  be  used.  If  a  scale  of  .05  inch,  or  .002 
metre,  per  unit  of  c  be  used,  and  be  made  to  include  c  =  250  for  English  meas- 
ure, or  150  for  metric  measure,  oy  will  be  about  1  ft  long.  For  the  sake  of 
clearness  we  show  only  the  larger  divisions  in  this  and  in  what  follows. 

On  oz  lay  off,  as  shown  on  its  upper  side,  the  square  roots  of  all  the  values  of 
the  mean  rad  R  for  which  the  diagram  is  to  be  used.  One  inch  per  ft,  or  .06 
metre  per  metre,  of  sq  rt,  is  a  convenient  scale.  Ma7'k  the  dividing  points  with 
the  respective  values  of  the  mean  radii  themselves. 

Having  decided  upon  the  flattest  slope  to  be  embraced  in  the  diagram,  say 


w  =  41.6  + 


.0028 


flattest  slope  per  unit  of  length 


for  English  measure. 


HYDRAULICS. 


571 


iFig.SOA. 


50 


13 


w  —  23  +  - 


for  metric  meaaure. 


flattest  slope  per  unit  of  length 
For  each  value  of  n  to  be  embraced  in  ttie  diagram,  say 

y  —  «,-=  -J —  for  English  measure  ;  or  p  —  w  = for  metric  measure. 

'  n  n 

To  each  value  of  y  —  w,  add  w,  thus  obtaining  values  of  y.    We  take  .000026 
per  unit  of  leugth  as  the  flattest  slope,*  and  .01,  .02,  .03  and  .04  for  n.f    Henc9 

.0028 


(using  English  measure)  w  =  41.6  +  .qqqqo^ 
1.811      1.811      1.811      1.811 


y  —  tc  — - 


.01  *       .02 


.04 


=-  41.6  +  112  =  153.6. 
;    =      181.1,  90.5,  60.4,  45.8  respectively, 


*  This  is  about  as  flat  as  is  likely  to  occur  in  practice. 
t  In  most  cases  many  intermediate  ones  would  be  used. 


572 


HYDRAULICS. 


and         y  —  181.1  +  153.6,        90.5  +  153.6,        60.4  +  153.6      and      45.3  -f  153.6; 

or  334.7,  244.1,  214.0  and  198.9  respectively.  Lay  off  these  values  of  y  on  oy  in 
pencil,  as  at  y,  y',  y",  and  y'",  using  the  scale  already  laid  otf  for  c  on  oy. 

From  each  point,  y^y'  etc,  draw  a  hor  pencil  line  yt.  y' t'  etc,  and  mark  on  it, 
in  pencil,  the  value  of  n  used  in  determining  its  height  oy  etc. 

Next  say  x  =  w?  X  greatest  value  of  n.  Make  o  a;  =  x  by  the  scale  of  sq  rts  of  R 
on  0  z.  In  our  case  o  i  =  153.6  X  .04  =  6.144  by  the  scale  of  sq  rts  of  R,  or  =  6.144* 
«=  37.75  by  the  scale  of  R. 

Divide  ox  into  as  many  equal  spaces  (4  in  our  case)  as  .01  is  contained  in 
greatest  n.    Mark  the  dividing  points  with  the  values  of  w,  as  in  our  Fig. 

From  each  dividing  mark  on  ox  erect  a  perpendicular,  {xt'"  etc)  in  pencil,  to 
cut  that  hor  line  {y'"  f'  etc)  which  corresponds  to  the  same  value  ol  n.  The 
interseciions  are  points  in  a  hyperbola.  Join  them  by  straight  lines  t'"  i'\  t"  t\ 
t'  t  etc. 

From  r  iu  oz  (corresponding  to  a  mean  rad  of  3.28  ft,  or  1  metre)  draw  radial 
lines,  rt,  rt\  rt"  etc.  Mark  them  "w  =  .01 ",  "71  =  .02"  etc,  the  same  as  their 
corresponding  lines  yt,  y' t'  etc. 

For  each  slope  (S)  to  be  used  in  the  diagram  (except  the  flattest,  for  which 
this  has  already  been  done)  say 

(0028  \ 
41.6  +  -j —  )  X  greatestn,    for  English  measure. 

z\  x"  etc  ==  (  23  4-  '—, — '—  1  X  greatest  n,    for  metric  measure. 
'  \  slope  / 

Thus,  our  slopes  are  =  .000025,  .00005,  .0001  and  .01  per  unit  of  length.    Hence, 

;^"_(41.6+^)X. 04  =1.675. 

Lay  off  each  value  of  x',  x"  etc  from  oy  on  a  separate  hor  pencil  line  0' x'  etc, 
using  the  scale  of  sq  rts  of  R  as  on  oz. 

Mark  each  line  0' x'  etc  in  pencil  with  the  slope  used  in  fixing  its  length. 

Divide  each  disto'x'etc  into  the  same  number  of  equal  parts  as  ox.  From 
the  dividing  points  (which,  like  thoseof  ox,  represent  the  values  of  w)  erect  perps 
to  cut  the  radial  lines  rt'",  rt"  etc,  each  perp  cutting  that  radial  line  which  cor- 
responds to  the  value  of  n  represented  by  the  point  at  the  foot  of  the  perp.  The 
intersections  corresponding  to  each  line  0'  x'  etc  form  a  hyperbolic  curve.  Mark 
each  curve  with  the  slope  of  its  corresponding  line,  ox,  0' x'  etc. 

The  drawing  is  now  in  the  shape  proposed  by  Mess  Ganguillet  and  Kutter,  and 
is  ready  for  use  in  finding  either  c,  »,  R  or  S  when  the  other  three  are  given. 
Thus : 

Ist.  Having  R,  S  and  n,  to  find  c.  For  example  let  R  =  20  ft,  S  =  .00005, 
n  =  .03.  From  the  intersection  d  of  slope  curve  .00005  and  radial  line  n  =  .03, 
draw*  (i-20  to  the  point  (20)  in  oz  corresponding  to  the  given  R.  At  e,  where 
d-20  cuts  oy,  is  the  reqd  c,  =  96  in  this  case. 

2d.  Having  R,  S  and  c,  to  find  n.  For  example  let  R  =  20  ft,  S  =  .00005, 
c  =  96.  Through  the  points  R  =  20  in  oz,  and  c  =  96  in  oy,  draw*  d-20  to  cut 
curve  .00005.  n  (=  .03)  is  found  by  m«ans  of  the  radial  lines  nearest  to  the  in- 
tersection, d. 

3d.  Having  S,  w  and  c,  to  find  R.  For  example  let  S  =  .00005,  n  =  .03, 
c  =  96.  Find  curve  .00005  and  radial  line  n  =  .03.  From  tlieir  intersection  d 
draw  d-'ik  through  the  point  eshowingc  =  96.  Its  intersection  with  oz  shows 
the  reqd  R,  20  in  this  case. 

*  Instead  of  drawing  these  lines,  we  may  use  a  fine  black  thread  with  a  loop  at  one  end.  Drive  a 
needle  either  into  one  of  the  points  R  or  into  one  of  the  intersections,  d  etc.  Slip  the  loop  over  the 
needle.  The  other  end  of  the  thread  is  held  between  the  fingers,  and  the  thread  is  made  to  cut  the 
other  points  as  reqd.  The  diagram  should  lieperfectly  flat,  and  the  string  be  drawn  tight  at  each  ob- 
serration,  in  order  that  friction  between  string  and  paper  may  not  prevent  the  string  from  forming* 
■traight  line.  Or  the  free  end  of  the  string  may  rest  on  a  pamphlet  or  other  object  about  }^  inch  thick, 
to  keep  the  string  clear  of  the  diagram.  Special  care  must  then  be  taken  to  have  the  eye  perp  over 
the  point  observed. 


HYDRAULICS.  673 

4tli.    Having  R,  c  and  n,  to  find  S.    For  example  let  R  =  20  ft,  c  =  96, 

n  '-=  .03.  Through  R  =  20  and  c  =  96  draw  d-20.  S  (.00005)  is  found  by  means 
of  the  curves  nearest  to  the  point  d  of  intersection  of  d-20  with  radial  line 
n  =  .03. 

The  following  addition  to  Kutter's  diagram,  proposed  by  Mr  Rudolph  Hering, 
Civil  and  Sanitary  Engineer,  Philadelphia,*  enables  us  to  read  tlie  veloc- 
ity from  the  diag-ram. 

Find  the  sq  rt  of  the  reciprocal  of  each  slope  to  be  embraced  in  the  diagram 

=  -x  -. .^    -  , 77  .    Lay  off  these  sq  rts  on  the  right  of  oy,  using 

\  slope  per  unit  of  length  "^  ^  &  ^.  » 

the  scale  of  c  already  laid  off  on  its  left.    In  our  fig  we  have  so  proportioned  the 

two  scales  that  —  -  ==  -^ — .    Mark  the  dividing  points  with  the  slopei 

l/recip  of  S         ^ 
per  unit  of  length. 
On  oz  lay  off  the  vels  to  be  embraced  in  the  diagram,  using  the  scale  of  sq  rts 

of  R  already  laid  off  on  oz,  and  making  — ^  =  —  ^ 

1/R    .  l/recip  of  S 

1st.  Having  R,  S  andw;  to  find  v.  For  example  let  R  =  20  ft,  S  -=  .00005, 
%  =  .03.  From  R  =  20  draw  d-20  to  the  intersection  d  of  curve  .00005  with  radial 
line  w  ==  .03.  d-20  cuts  oy  at  e,  where  c  =  96.  With  a  parallel  ruler  join  R 
— =  20  with  S  =  .00005  on  o  y.  Draw  a  parallel  line  through  c  =  96.  It  cuts  o 2  at 
m,  giving  the  reqd  vel,  3.03  ft  per  sec. 

2d.  Having  R,  S  and  v;  to  find  n.  For  example  let  R  =  20  ft,  S  =  .00005, 
t;  — =  3.03  ft  per  sec.  With  a  parallel  ruler  join  R  =  20  and  slope  .00005  on  oy. 
Draw  a  parallel  line  through  v  =  3.03.  It  cuts  oy  at  e,  where  c  =  96.  Through 
R  =  20  and  c  =  96, draw  d-20  to  cut  curve  .00005.  The  point  dof  intersection, 
being  on  radial  line  n  =  .03,  shows  .03  to  be  the  proper  value  of  n. 

Any  line  drawn  to  the  curves  from  R  =  3.28  ft  or  1  metre,  is  one  of  the  radial 
lines  used  in  making  the  diagram.  It  therefore  necessarily  cuts  all  the  slope 
surves  at  points  showing  the  same  value  of  n. 

3d.    Having  S,  n  and  v;  to  find  R.    For  example,  let  S  =  .00005,  n  =  .03, 

V  =  3.03  ft  per  sec.  Assume  a  value  of  R,  say  10  ft.  Find  curve  .00005  and  radial 
line  n  =  .03.  Join  their  intersection  d  with  R  =  10  ft.  The  connecting  line  cuts 
oy  at  c  =  82.  With  a  parallel  ruler  join  c  ==  82  with  v  =  3.03.  Draw  a  parallel 
line  through  slope  =  .00005  on  oy.  It  cuts  o z  at  R  =  27.3,  showing  that  a  new 
trial  is  necessary,  and  with  an  assumed  R  greater  than  10  ft. 

If  R  thus  found  is  the  same  as  the  assumed  one,  the  latter  is  correct.    If  they 
are  nearly  equal,  their  mean  may  be  taken. 
4tli.    Having  R,  n  and  v  ;  to  find  S.     For  example,  let  R  =  20  ft,  w  =  .08, 

V  =  3.03  ft  per  sec.  Assume  a  slope  (say  .0001).  Find  its  curve,  and  radial  line 
n  =  .03.  Join  their  intersection  with  R  =  20,  and  note  the  value  (89)  of  c  where 
the  connecting  line  cwts  oy.  With  a  parallel  ruler  join  c  =  89  with  v  =  3.03. 
Draw  a  parallel  line  through  R  =  20.  It  cuts  oy  zi  slope  .000058,  showing  that 
a  new  trial  is  necessary,  and  with  an  assumed  S  flatter  than  .0001.  If  R  is  3.28 
ft,  or  1  metre,  the  diagram  gives  the  corrects  at  the  first  trial,  no  matter  what 
S  was  assumed  at  starting.  With  any  other  R,  if  the  diagram  gives  the  Kime  S 
as  that  assumed,  the  latter  is  correct.  If  the  two  differ  but  slightly,  we  may  take 
their  mean. 

•  Trantftctiona  of  the  Auerican  Society  of  Civil  Engineers,  January  ll7t. 


574 


VELOCITIES    IN    SEWERS. 


Table  of  vels  in  Circular  Brick  Sewers  when  running  full,  by 
Kutter's  formula,  but  taking  n  at  .015  instead  of  his  .013,  in  consideration 

of  the  rough  character  of  sewer  brickwork  generally. 

When  riinniii$ir  only  balf  full  the  vel  will  be  the  same  as  when  full, 
but  this  is  not  the  case  at  any  other  depth  whether  greater  or  less.  At  greater 
ones  it  increases  until  the  depth  equals  very  nearly  .9  of  the  diam,  when  it  U 
about  10  per  cent  greater  than  when  either  full  or  half  full.  From  depth  of  .9  of 
the  diam  the  vel  decreases  whether  the  depth  becomes  greater  or  less.  At  depth 
of  .25  diam  the  vel  is  about  .78  of  that  when  full ;  and  then  diminishes  muoll 
more  rapidly  for  less  depths.    All  this  applies  also  to  pipes. 

The  vel  for  any  fall  or  diam  intermediate  of  those  in  the  table  can  be  found  by 
simple  proportion.  Original. 


FaU 

in  ft 

Diameters  in  feet. 

m  ft 

per 
mile. 

2 

3 

4 

6 

8 

12 

16 

20 

per 
100  ft 

Yelocitiea  in  feet  per  second. 

.1 

.19 

.27 

.35 

.50 

.64 

.89 

1.10 

1.34 

.0018 

.2 

.30 

.42 

.53 

.74 

.93 

1.26 

1.56 

1.84 

,0031 

.4 

.46 

.65 

.80 

1.08 

1.39 

1.81 

2.20 

2.60 

.007« 

.6 

.59 

.81 

1.00 

1.35 

1.70 

2.22 

2.70 

3.18 

.0114 

.8 

.69 

.95 

1.17 

1.57 

1.94 

2.56 

3.08 

3.60 

.0161 

1.0 

.79 

1.07 

1.32 

1.77 

2.16 

2.84 

3.43 

3.96 

.0189 

1.25 

.89 

1.21 

1.49 

1.98 

2.42 

3.17 

3.8 

4.5 

.0287 

1.50 

.98 

1.33 

1.64 

2.18 

2.64 

3.5 

4.2 

4.9 

.0284 

1.75 

1.06 

1.44 

1.78 

2.34 

2.85 

3.8 

4.5 

5.3 

.0881 

2.0 

1.15 

1.55 

1.91 

2.53 

3.1 

4.0 

4.8 

5.6 

.0379 

2.5 

1.32 

1.78 

2.18 

2.85 

3.5 

4.5 

5.4 

6.3 

.0473 

3.0 

1.44 

1.94 

2.38 

3.2 

3.8 

5.0 

6.0 

6.9 

.0568 

3.5 

1.58 

2.10 

2.58 

3.4 

4.1 

5.3 

6.5 

7.4 

.0662 

4. 

1.68 

2.2 

2.7 

3.6 

4.4  ■ 

5.7 

6.9 

7.9 

.0768 

5. 

1.90 

2.5 

3.1 

4.1 

4.9 

6.3 

7.6 

8.7 

.0947 

6. 

2.06 

2.7 

3.3 

4.4 

5.4 

6.9 

8.3 

9.6 

.1136 

7. 

2.2 

3.0 

3.6 

4.8 

5.8 

7.5 

9.0 

10.4 

.1326 

8. 

2.4 

3.2 

3.8 

5.1 

6.2 

8.0 

9.7 

11.1 

.1514 

9. 

2.5 

3.4 

4.1 

5.4 

6.6 

8.5 

10.3 

11.8 

.1703 

10. 

2.7 

3.5 

4.3 

5.7 

6.9 

9.0 

10.8 

12.5 

.1894 

12. 

2.9 

3.9 

4.8 

6.3 

7.6 

9.9 

11.9 

13.6 

.2273 

15. 

3.3 

4.4 

5.4 

7.1 

8.5 

11.0 

13.3 

15.3 

.2841 

18. 

3.6 

4.8 

5.9 

7.7 

9.3 

12.1 

14.5 

16.7 

.3409 

21. 

3.9 

5.1 

6.3 

8.4 

10.0. 

13.0 

15.7 

17.9 

.3975 

24. 

4.2 

5.5 

6.8 

8.9 

10.8 

13.9 

16.8 

19.2 

.4546 

27. 

4.5 

5.9 

7.2 

9.5 

11.4 

14.8 

17.9 

20.4 

.5109 

30. 

4.7 

6.2 

7.5 

9.9 

12.0 

15.6 

18.8 

21.5 

.5682 

35. 

5.0 

6.7 

8.2 

10.8 

13.0 

16.8 

20.4 

23.2 

.6629 

40. 

5.4 

7.1 

8.7 

11.5 

13.9 

18.0 

21.7 

24.8 

.7576 

45. 

5.6 

7.5 

9.2 

12.2 

14.8 

19.1 

23.0 

26.3 

.8523 

50. 

5.9 

8.0 

9.7 

12.8 

15.5 

20.1 

24.2 

27.7 

.9470 

60. 

6.5 

8.7 

10.7 

14.1 

17.0 

22.1 

26.5 

30.3 

1.136 

70. 

7.0 

9.4 

11.5 

15.2 

18.4 

23.9 

28.5 

32.8 

1.326 

80. 

7.4 

10.1 

12.3 

16.2 

19.7 

25.5 

31.0 

35.0 

1.515 

90. 

7.9 

10.7 

13.1 

17.2 

20.9 

27.0 

32.3 

37.1 

1.705 

100. 

8.4 

11.3 

13.8 

18.2 

22.0 

28.5 

34.1 

39.1 

1.894 

A  vel  of  10  ft  per  sec  =  600  ft  per  minute  =  36000  ft,  or  6.818  miles  per 
hour.  About  5  ft  per  sec  is  as  great  as  can  be  adopted  in  practice  to  prevent  the 
lower  parts  of  the  sewers  from  wearing  away  too  rapidly  by  the  debris  carried 
along  by  the  water. 


HYDRAULICSc 


575 


Art.  23.    The  rate  at  ivliich  rain  ^i^ater  reaches  a  sewer  or 

culvert,    etc.      Burkli-Zieg-ler    Formula.     See    "European    Sewerage 
Systems,"  by  Rudolph  Hering,  C.  E.,  iu  Trans.  Am.  Soc.  C.  E.,  Nov.  1881. 

^l^^"/h'?/  -'^  coef  Av.  cub.  ft.  of  rainfajl 

ac?e"i  each-  =  .  according    X  per  second  per  acre. 


to  judgment  during  heaviest  fall. 


1         4  /Av.  si 
\  No.of 


slope  of  ground 
feet  per  1000  ft. 


lug  sewer         ""•' — ^^ —  *= »  xNo.of  acres  drained 

Coefficient,  for  paved  streets,  0.75  ;  for  ordinary  cases,  0.625 ;  for  suburbs 
with  gardens,  lawns,  and  macadamized  streets,  0.31. 

Note  that  1  iuch  of  rainfall  per  hour  may  be  taken  as  eqiiivalent  to  1  cubic  foot 
per  second  per  acre.    See  Conversion  Tables,  pp.  235,  etc. 

Example.  If  an  area  of  3100  acres  (nearly  5  square  miles),  with  an  average 
slope  of  5  feet  per  1000  feet,  receives  a  maximum  rainfall  of  3  inches  per  hour, 
then,  assuming  a  coefficient  of  0.5,  the  rate  at  which  the  water  would  reach  the 
mouth  of  a  sewer  at  the  lower  end  of  the  3100  acres  would  be 


0.5  X  3  X    ^-^yq-^  =  0.5  X  3  X  0.203  =  0.305  cubic  feet  per  second  per  acre; 

or  0.305  X  3100  =  945.5  cubic  feet  per  second,  total. 

Let  the  grade  of  the  intended  sewer  be  say  4  feet  per  mile;  and,  to  avoid 
excessive  wear  of  its  brickwork  by  debris  swept  along  by  the  water,  let  its 
velocity  be  limited  to  6.3  feet  per  second,  which  may  be  permitted  on  occasions 
as  rare  as  rains  of  3  inches  per  hour,  althougli,  for  tolerably  constant  flow,  where 
liable  to  debris,  it  should  not  exceed  about  5  feet  per  second. 

Find,  in  table  opposite,  the  diameter,  14  ft.,  corresponding,  as  nearly  as  may 
be,  to  a  velocity  of  6.3,  and  to  a  grade  of  4  feet  per  mile.  The  area  is  154 
square  feet.  Hence,  154  X  6.3  =  970  cubic  feet  per  second  ==  capacity  of  sewer. 
To  allow  for  deposits  in  the  sewer,  make  the  diameter  say  14.5  or  15  feet. 

Table  of  least  Telocities  and  g-rades  for  drain-pipes  and 
sevi^ers  in  cities,  in  order  that  they  may  under  ordinary  circumstances  keep 
themselves  clean,  or  free  from  deposits.     (Wicksteed.) 


Grade. 

Grade. 

Diam. 

Vel.  in  ft. 

Grade, 

Feet  per 

Diam. 

Vel.  in  ft. 

Grade, 

Feet  per 

In  Inches. 

per  Min. 

lin 

Mile. 

in  Inches. 

per  Min. 

lin 

Mile. 

4 

240 

36 

146.7 

18 

180 

294 

18.0 

6 

220 

65 

81.2 

21 

180 

3J3 

15.4 

7 

220 

76 

69.5 

24 

180 

392 

13.5 

8 

220      . 

87 

60.7 

30 

180 

490 

10.8 

9 

220 

98 

53.9 

36 

180 

588 

9.0 

10 

210 

119 

44.4 

42 

180 

686 

7.7 

11 

200 

145 

36.7 

48 

180 

784 

6.8 

12 

190 

175 

30.2 

54 

180 

882 

6.0 

15 

180 

244 

21.6 

60 

ISO 

980 

5.4 

Weight  i)er  foot  run  of  g-1aze<l  terra  cotta  pipes  for  drains,  etc.; 
prices  per  foot  run  adopted  by  the  United  Sewer  Fipe  Makers  of  the  United 
States,  March,  1887.     For  discounts,  see  Price  List. 


Drain  pipe,  with  socket  joint 

Sewer  pipe,  wi 

th  sleeve  joint 

Bore 

Wt 

Price 

Bore 

Wt 

Price 

Bore 

Wt 

Price 

Bore 

Wt 

Prict 

ins 

lbs 

$ 

ins 

lbs 

S 

ins 

lbs 

$ 

ins 

lbs 

$ 

2 

4 

0.14 

6 

18 

0.30 

15 

45 

1.25 

30 

150 

5.50 

3 

7 

0.16 

8 

22 

0.45 

18 

65 

1.70 

36 

195 

7.00 

4 

10 

0.20 

10 

30 

0.65 

21 

89 

2.50 

42 

203 

8.50 

5 

12 

0.25 

12 

33 

0.85 

24 

100 

3.25 

48 

230 

10.50 

The  joints  are  filled  with  cement  mortar;  or,  when  used  for  drainage  only, 
with  clay.  Drain  pipes  (3  to  12  ins  bore)  are  about  |  inch  thick.  A  bend  or 
branch  costs  about  as  much  as  from  3  to  5  feet  of  pipe.  The  48-inch  pipes  are 
about  2  ins  thick. 

Art.  24.  When  the  area  of  cross  section  of  channel  is  re- 
duced at  any  point,  as  by  a  dam  (Fig  33,  p.  576)  or  by  narrowing  it,  either 
at  its  sides  (Fig  32)  or  by  placing  in  it  a  pier  etc.  Fig  34;  a  portion  at  least  of 
vhe  force  of  grav  (which  would  otherwise  be  giving  vel  to  the  water  up-stream 
from  the  point  where  the  obstruction  takes  place),  CRusea  pressvre  against  the 
dam  etc.    This  pres  maintains  the  up-stream  water  at  a  higher  level  than  it 


576 


HYDRAULICS. 


would  otherwise  have.  Said  water  is  then  practically  in  a  reservoir;  i  e,  it  hai 
less  vel  and  greater  pres  than  before.  If  the  reservoir  has  no  outlet,  there  is  no 
yel ;  andaZZ  of  the  head,  or  force  of  grav,  acting  on  the  water  is  expended  in ^re*. 

But  if  there  is  an  outlet,  as  over  the  dam,  or  between  the  piers  etc,  a  portion 
CO,  Figs  31,  33,  34,  of  this  pres  or  head,  is  expended  in  giving  vel  (or  an  accelera- 
tion of  vel)  to  the  water  escaping  by  that  outlet;  after  whicli  only  so  much 
head  (in  the  shape  of  surface  slope)  is  needed  as  will  overcome  the  resistances 
of  the  channel  down-stream  from  the  obstruction,  and  so  maitUain  unijorm  the 
vel  given  to  the  water  by  the  head  c  o. 

Where  a  large  canal,  such  as  those  intended  for  navigation,  is  fed  from  a  reser- 
Toir,  the  fall  coin  feet  is  approximately 

=  mean  velocityS  in  canal,  in  feet  per  second,  X  .017 ; 

and  in  smaller  canals,  such  as  mill  courses, 

=  mean  velocity2  in  canal,  in  feet  per  second,  X  -02. 

The  abruptness  of  the  fall  may  be  diminished  by  rounding  off  or  sloping  the 
edges  of  the  piers,  or  the  corners  at  the  sides  of  the  channel  (Fig  32)  or  the 
approach  to  the  dam 

Fig  33  is  a  cross  section  of  Clegg's  dam,  across  Cape  Fear  River,  N.  C.  It 
is  from  measurements  made  by  EUwood  Morris,  C  E;  by  whom  they  were  com- 
municated to  the  writer.  The  dam  is  of  wooden  cribwork ;  and  its  level  crest, 
S  f t  5  ins  wide,  is  covered  with  plank  ;  along  which  the  water  glides  in  a  smooth 
sheet,  6  ins  deep,  (at  the  time  of  measurement).    At  the  upper  end  of  thfai 

sheet,  and  in  a  dist  of  about  2  ft,  a  head  co  of  9  ins  forms  itself,  as  in  the  fig. 


Fij)  33 


Bo  3k 


Canal 


HYDRAULICS. 


577 


Art.  25.  Sconr.  In  a  channel  of  uniform  and  constant  slope  and  cross 
section,  the  vel  of  the  particles  of  water  immediately  adjoining  the  bottom  and 
sides  is  very  slight;  and  but  little  scouring  takes  place.  But  when  irregular- 
ities in  the  slope  or  cross  section  occur,  as  in  the  lat^t  article,  the  scour  is  greatly 
increased  in  their  inimediate  neighborhood. 

The  erection  of  one  or  two  piers  in  a  quite  large  st'ream,  will  frequently  pro- 
duce an  almost  incredible  amount  of  scour,  if  the, bottom  is  at  all  of  a  yielding 
nature.  The  greatest  scour  of  course  lakes  place  during  freshets ;  and  near  the 
obstrrctiou. 

Scouring;  action  is  supposed  to  be  as  square  of  vel. 

According  to  Smeaton,  a  vel  of  8  miles  an  hour  will  not  derange  quarry  rubble  stones,  not  exeted* 
ing  half  a  cub  ft.  deposited  around  piers,  <fec  ;  except  by  washing  the  soil  from  under  them. 
1  inch  per  sec,  =  5  f t  per  min,  —  .056818  of  a  mile,  or  300  ft  per  hour. 
1  foot  per  sec,  =  60  ft  per  min,  =  .681816  of  a  mile,  or  3600  ft  per  hour. 

To  reduce  inclies  per  sec,  to  feet  per  minute,  multiply  by  5. 

"  ••  <'  "        «<  «<        .i  ..      hour,  "  "    300. 

^  "      .         "  "  "        "        to  miles  per  hour,  divide  by  17.6. 

One  mile  per  hour  :r  88  ft  per  min  =  1.4667  ft,  or  17.6  ins  per  sec. 

The  two  following'  tables  are  (with  many  corrections)  from  Nicholson's 
Architecture  ;  and  must  be  looked  upon  merely  a.s  probable  approximations.  They  suppose  the  piers, 
4c,  to  be  properly  rounded  or  pointed  at  their  upstream  ends,  so  as  to  give  as  free  a  passage  as  pos- 
■ible  to  the  water.  He  says  that  if  iiiey  are  square-ended,  the  head  will  be  increased  about  50  per  ct. 
The  subject  is  an  extremely  intricate  one,  and  admits  of  no  precise  solution.  If  the  increased  vel 
■cours  away  the  bottom  until  the  area  of  water-way  becomes  as  great  as  it  originally  was,  the  head 
disappears ;  and  the  vel  also  becomes  reduced  to  its  original  rate.    This  is  common  in  soft  bottoms. 


TABLE 


Of  beads  produced  by  obstructions  to  streams. 


Original  Vel. 
of  Stream.* 


Per  Sec. 

Ins. 

Ft. 

3 

hi 

6 

Vi 

12 

1 

24 

2 

36 

3 

48 

4 

60 

b 

72 

6 

120 

10 

.170 
.341 
.681 
1.36 
2.04 
2.72 
3.41 
4.09 
6.81 


Kind  of  Bottom 
which  begins  to 
wear  away  under 
Bottom  Vel.  equal 
to  those  in  the 
first  three  cols. 


Ooze,  and  Mud. 

Clay , 

Sand 

Gravel 

Small  Shingle... 
Large        " 

Soft  Shistus 

Stratified  Rocks, 
Hard  Rocks 


Proportion  of  Area  of  original  Water-way, 
occxipied  by  the  Obstrnctions. 

h\i^\\\\\\\\\-\\%\\ 


Head  of  Water  prodnced  at  the  Obstrnctions  j  in 
Peet. 


,0003 

.0004 

.0004 

.0006 

.001 

.0014 

.0033 

.0067 

.0011 

.0014 

.0017 

.0023 

.00  i 

.0058 

.0133 

,0267 

.0045 

.0056 

.0069 

.0091 

.015 

.0231 

.0532 

,1069 

.0182 

.0225 

.0276 

.0364 

.060 

.0924 

.2128 

,4276 

\)m> 

.0507 

.0621 

.0819 

.135 

-2079 

.4788 

.9621 

0728 

.0902 

.1104 

.14.56 

.240 

.3696 

.8412 

1.710 

1137 

.1410 

.1725 

.2275 

.375 

.5775 

1.320 

2.672 

1638 

.2030 

.2484 

.3276 

.540 

.8316 

1.915 

3,848 

4550 

.5640 

.6901 

.9100 

1.50 

2.310 

5.280 

10.69 

2.326 
4.144 
6  475 
9.304 
25.9 


TABTiE  Increased  velocities   produced   at  and    by 

rounded,  or  pointed  obstructions.    If  square,  these  vels  must,  accord- 
ing to  Nicholson,  be  increased  }/^  part. 


Original  Vel. 
of  Stream.* 

Proportion  of  Area  of  Water-way,  occnpied  by  the  Obstrnctions. 

tV  1  tV  1    i    1    4    1    i    1    i    1    4    1    f    1    f 

Per 

Per  Sec. 

Hour. 

Ins, 

Pt. 

Miles. 

Velocity  prodnced  at  the  Obstrnction  in  Feet  per  Second. 

3 

1,^ 

.170 

.28 

,29 

.30 

.32 

.35 

.394 

.52 

.7 

1.05 

6 

1^ 

.341 

.56 

.58 

,60 

.64 

.70 

.788 

1.05 

1.4 

2,1 

12 

1 

.681 

1.13 

1.16 

1,20 

1.26 

1.40 

1.58 

2.1 

2.8 

4.2 

24 

2 

l..'i6 

2.27 

2.33 

2,40 

2.52 

2.80 

3.16 

4.2 

5.6 

8.4 

36 

8 

2.04 

3.39 

3.48 

3,60 

3.78 

4.20 

4.74 

6.3 

8.4 

12.6 

48 

4 

2.72 

4.64 

4.66 

4,80 

5.04 

5.60 

6.32 

8.4 

11.2 

16.8 

60 

5 

3.41 

5.60 

5.80 

6,00 

6.40 

7.00 

7.88 

10.5 

14.0 

21.0 

72 

6 

4.09 

6.78 

6.96 

7.20 

7.56 

8.40 

9.48. 

12.6 

16.8 

25.2 

120 

10 

6.81 

11,3 

11.6 

12.0 

12.6 

14.0 

15.8 

21.0 

28.0 

42.0 

*  A  very  vague  expression.    Does  it  refer  to  the  greatest  surface  vel  at  mid-ohannel ;  er  to  the  mean 
vel  of  the  entire  cross- section? 

37 


578 


HYDRAULICS. 


Art.  26.  The  resistance  of  water  a$;aiiist  a  flat  surface  moT- 
ing'  throns'h  it  at  risj;-lit  aiig'les,  is  nearly  as  the  squares  of  the  Tel;  and, 
according  to  Hutton,  its  amount  in  fibs  per  sq  ft  approx  =  Square  of  vel  in  ft  per 
sec.    Or  like  tlie  pres  of  a  riinning-  stream  against  a  perp  fixed  flat 

Barfaee,  it  is  =  wt  of  a  col  of  water  whose  base  =:  pressed  surf,  and  whose  ht=:head  due  to  the  vel 

The  resist  of  a  sphere  ia  to  that  of  its  great  circle  about  as  1  to  2.9. 

When  the  moTing  surf.  Instead  of  being  at  right  angles  to  the  direction  in  which  it  moves,  forms 
another  angle  with  it,  the  resistance  becomes  less  in  about  the  following  proportions.  Therefore, 
Ivhen  the  surf  ia  inclined,  first  calculate  the  resistance  as  if  at  right  angles :  and  then  mult  bj  the 
following  decimals  ©pposite  the  angle  of  inclination  : 


90°.. 

..1.00   . 

60°... 

.  .88 

*0».. 

.  .58 

20°.. 

.  .16 

80  .. 

..  .98 

55  .. 

.  .83 

35  .. 

.  .46 

15  .. 

.  .10 

70  .. 

..  .95 

50  ... 

.  .76 

30  .. 

.  .34 

10  .. 

.  .06 

65  .. 

..  .92 

45  ... 

.  .68 

25  .. 

.  .24 

5  .. 

.  .02 

Ttie  scour,  or  abrading*  power  of  mioving:  water  is  considered  U 

be  as  the  square  of  its  vel. 

Art.  27.    To  calculate  the  horse-power  of  falling*  w^ater,  on 

the  ordinary  assumption  that  a  horse-power  is  equal  to  33000  lbs  lifted  1  foot  vert  per  min.     That  of 

average  horses  is  really  but  about  %  as  much,  or  22000  S)s,  1  foot  high  per  min.     Mult  together  th« 

number  of  cub  ft  of  water  which  fall  per  min  ;  the  vert  height  or  head  in  feet,  through  which  it  falls; 

and  the  number  62.3,  (the  wt  of  a  cub  ft  of  water  in  lbs  :)  and  dir  the  prod  by  SSOOOT  Or,  by  formula, 

cub  ft     ^  vert  ^     lbs 

The  number  of  ^  per  min   ^   height  in  ft  ^   62.3 

horse.powers  ^^ 

Ex.  Over  a  fall  16  ft  in  vert  height,  800  cub  ft  of  water  are  dischd  per  min.  How  many  horse 
powers  does  the  fall  afford  ? 

cub  ft      ft        a* 
„  800  X  16  X  62.3         797440 

Here.   =  •=  24.17  h-now. 

'  33000  33000  *^ 

TFater-WheelB  do  not  realize  all  the  power  inherent  in  the  water,  as  found  by  out 
rule.  Thus,  underjinots  realize  but  from  >i  to  J^ ;  breast-wheels,  }4 ;  overshots,  from  %  to  Ji  ;  tur- 
bines, Ji  to  .85  of  14; ;  according  to  the  skill  of  design,  and  the  perfection  of  workmanship.  Even  when 
the  wheel  revolves  in  a  close-fitting  casing,  or  breast,  elbow  buckets  give  considerably  more  power 
than  plain  radial  or  center-buckets.  Of  the  power  actually  received  by  a  wheel,  part  is  expended  in 
friction,  &c;  while  the  remainder  does  the  useful  or  paying  network  of  raising  water,  grinding 
grain,  sawing.  &c. 

Observations  by  Oenl  Haupt,  in  1866,  gave  the  following  results  for  a 

small  hydraulic  ram.  Head  of  water  to  ram  =  8.812  ft ;  diam  of  drive-pipe  = 
l^  ins;  length  15  ft.  Diam  of  delivery-pipe  —  %  inch;  length  200  ft.  Vert  height  to  which  the 
water  was  raised  by  the  delivery-pipe,  63.4  feet.  Strokes  of  ram  per  min,  170.  Quantity  of  water 
which  worked  the  ram  =:  768  cub  ins,  =:  3.31  galls,  =  27.73  fi)8  per  min.     Quantity  raised  63.4  ft  high 

tba  water         ft  ft-Qw 

per  min,  =  48  cub  ins,  =  1.736  fts.    Hence  the  power  expended  per  min,  waa  27.73  X  8.812  =  244.35. 

lbs  water        ft  ft-tti8 

And  the  useful  effect,  was  1.736  X  63.4  =  110.06.     Hence  the  ratio  which  the  useful  effect  bears  to  the 

power  in  this  instance,  is  '- — ,  or  .45.    The  actual  power  of  the  ram  is,  however,  greater  than  this, 

inasmuch  as  it  has  to  overcome  the  friction  of  the  water  along  the  delivery-pipe.* 

To  find  the  horse-power  of  a  running  stream.    Water-wheels 

•with  simple  float- boards, t  instead  of  buckets,  are  sometimes  driven  by  the  mere  force  of  the  ordinary 
natural  current  of  a  stream,  without  any  appreciable  fall  like  that  in  the  foregoing  case.  In  such 
eases,  we  must  substitute  the  virtual  or  theoretic  head  ;  which  is  that  which  would  impart  to  it  the 
game  vel  which  it  actually  has.  This  virtual  head  may  be  taken  at  ouce  from  i  aide  p.  539.  Thus,  a 
stream  has  a  vel  of  2.386  miles  per  hour;  or  210  ft  per  min  ;  or  3}^  ft  per  sec  ;  and  in  the  column  of 
heads  in  Table  10,  opposite  to  3.5  vel  per  sec,  we  find  the  reqd  head  .190  of  a  ft.  Having  thus  found 
the  head,  we  must  now  find  the  quantity  of  water  which  passes  any  given  area  of  the  stream  in  a 
min.  Thus,  suppose  that  the  tmmerscci  part  of  a  float  when  vert  is  5  ft  long,  and  1  ft  wide  or  deep; 
then  the  area  of  this  part  which  receives  the  force  of  the  current,  is  5  X  1  =  5  square  feet.    Hence, 

area  vel 

5  sq  ft  X  210  =  1050  cub  ft  per  min.  Having  now  the  cub  ft  per  min,  and  the  vert  height  or  head, 
the  number  of  horse-powers  of  the  stream  of  the  given  area,  is  found  by  the  foregoing  rule,  or  formula. 


*  A  committee  of  the  Franklin  Institute,  in  1850.  gave  .71  as  the 

coeffloient  for  a  ram  at  the  Girard  College,  in  which  the  diam  of  drive-pipe  was  2^  ins  ;  its  length, 
160  ft;  fall,  14  ft.  Delivery-pipe,  1  inch  diam  ;  2260  ft  long  :  vert  rise,  or  height  to  which  the  water 
was  raised.  93  ft.  No  details  of  the  experiment  are  given.  Some  large  rams  in  France  give  a  useful 
effect  of  from  .6  to  .65  of  the  whole  power  expended.  It  is  an  excellent  machine  for  many  purposes; 
and  is  sometimes  used  for  filling  railway  tanks  at  water  stations. 

t  Such  wheels,  for  floating-  mills,  in  £urope,  rarely  exceed  15  ft 

fLiam.  Whatever  the  diam,  they  may  have  about  18  to  20  floats.  The  floats  are  from  8  to  16  ft  long ; 
and  about  4  to  -r^  as  deep  as  the  diam  of  the  wheel.  They  should  not  dip  their  entire  depth  into 
the  water,  but  nearly  so.  They  should  not  be  in  the  same  straight  line  with  the  radii ;  but  should 
incline  from  them  30°  up  stream,  to  produce  their  full  effect.  All  these  remarks  apply  to  wheels 
moving  freely  in  a  wide  or  indefinite  channel ;  as  in  the  case  of  a  floating  mill,  built  on  a  scow,  and 
anchored  out  in  a  stream  :  but  not  to  wheels  for  which  the  water  is  dammed  up,  and  acts  with  a  prac- 
tical fall.  No  great  exactness  is  to  be  expected  in  rules  on  this  subject.  The  best  vel  for  the  wheel 
is  abe«t  .4  that  of  the  stream. 


HYDRAULICS.  579 

cub  ft  per  min  ^  vert  ht  in/t  ^   lbs 
•^__      Xo  of   «  1050  ^  .190         ^  6-2.3  12429 

**'"•  H.  Pow.  33000  ■  =  33000  =  -^'^  ^f  ^  ^-  ^^• 

But  in  practice  the  wheels  actually  realize  but  about  0.4  of  this  power  of 
the  stream.  Therefore,  the  actual  power  of  our  wheel  will  be  but  .377  X  .4  ^ 
0.1508  of  a  horse  power;  or  33000 X  .1508  =  4976  ft-ft)s  per  minute.  Making 
a  rough  allowance  for  the  friction  of  the  machine  at  its  journals,  &c,  we 
should  have  about  4400  ft-ft)s  of  iiseful  power  per  minute  ;  that  is,  the  wheel 
would  actually  raise  about  440  fts  10  ft  high ;  or  44  lbs  100  ft  high,  &c,  pei 
min.  The  vel  of  the  stream  must  not  be  measured  at  the  surface :  but  at 
about  }4  of  the  depth  to  which  the  floats  are  to  dip,  or  be  immersed.  This, 
however,  is  necessary  chiefly  in  shallow  streams,  in  which  the  depth  of 
the  float  bears  a  considerable  ratio  to  that  of  the  water. 

This  power  of  a  running-  stream,  (for  any  given  area  of 
transverse  seetionO  increases  as  tSie  enbes  of  the  vels:  for,  as 
we  have  seen,  the  power  in  ft-ft>s  per  min  is  found  by  mult  together  the 
weight  of  water  which  passes  through  the  section  in  a  min,  and  the  virtual 
head  in  ft ;  and  since  tnis  weight  increases  as  tlie  vel,  and  this  head  as  the 
square  of  the  vel,  the  prod  of  the  two  (or  the  power)  must  be  as  the  cube 
of  the  vel.  Therefore,  if  the  vel  in  the  foregoing  case  had  been  10.5  ft  per 
sec,  or  8  times  3.5  ft,  the  power  of  the  wheel  would  have  been  27  times  as 
great,  or  .1508  X  27  =  4.07  horse  powers. 


580 


DREDGING. 


DEEDGING. 


I>REDQiNG  is  generally  done  by  skilled  contractors,  who  own  the  requisite  machinee, 
■C0W3  or  lighters,  &c ;  and  who  make  it  a  specialty.  It  is  necessary  to  specify  whether 
the  dredged  material  is  to  be  measured  in  place  before  it  is  loosened:  or  after  being 
deposited  in  the  scow:  because  it  occupies  more  bulk  after  being  dredged.  It  was 
found,  in  the  extensive  dredgings  for  deepening  the  River  St  Lawrence  through  the 
Lake  of  St  Peter,  that  on  an  average  a  cub  yd  of  tolerably  stiff  mud  in  place,  makes 
1.4  yds  in  the  scow ;  or  1  in  the  scow,  makes  .715  in  place.  Also  stipulate  whether  the 
removal  of  bowlders,  sunken  trees,  &c,  is  to  constitute  an  extra.  These  often  require 
sawing  and  blasting  under  water.  The  cost  per  cub  yd  for  dredging  varies  much 
With  the  depth  of  water :  the  quantity  and  character  of  the  material :  the  dist  to  which 
it  has  to  be  removed;  whether  it  can  be  at  once  discharged  from  the  machine  by 
means  of  projecting  side-shoots  or  slides :  or  must  be  discharged  into  scows,  to  be  re- 
moved to  a  short  dist  by  poling,  or  to  a  greater  dist  by  steam  tugs ;  whether  it  can  be 
dropped  or  dumped  into  deep  water  by  means  of  flap  or  trap  doors  in  the  bottom  of 
the  hoppers  of  the  scows ;  or  must  be  shoveMed  from  the  scows  ivto  xhalloto  water,  (at 
say  4  to  8  cts  per  yd ;)  cr  upon  land,  (at  say  from  6  to  10  or  20  cts  for  the  shovelling 
ftlone,  or  shovelling  and  wheeling,  as  the  case  may  be;)  whether  much  time  must  be 
consumed  in  moving  the  machine  forward  frequently,  as  when  the  excavation  is 
narrew,  and  of  but  little  depth;  as  in  deepening  a  canal,  Ac  ;  whether  many  bowl- 
ders and  sunken  trees  are  to  be  lifted  ;  whether  interruptions  may  occur  from  waves 
in  storms;  whether  fuel  can  be  readily  obtained,  &c,  &c.  These  considerations  may 
make  the  cost  per  cub  yd  in  one  case  from  2  to  4  times  as  great  as  in  another.  The 
actual  cost  of  deepening  a  ship-channel  through  Lake  St  Peter,  to  18  ft,  from  its  orig- 
inal depth  of  11  ft,  for  several  miles  through  moderately  stiff  mud,  was  14  cts  per 
cub  yd  in  place,  or  10  cts  in  the  scows ;  including  removing  the  material  by  steam 
tugs  to  a  dist  of  about  i^  a  mile,  and  dropping  it  into  deep  water.  'Ibis  includes  re- 
pairs  of  plant  of  all  kinds,  but  no  profit.  It  was  a  favorable  case.  When  the  buckets 
work  in  deep  water  they  do  not  become  so  well  filled  as  when  the  water  is  shallower, 
because  they  have  a  more  vertical  movement,  and,  therefore,  do  not  scrape  along  as 
great  a  distance  of  the  bottom.  Hence  one  reason  why  deep  dredging  costs  more 
per  yard;  in  addition  to  having  to  be  lifted  through  a  greater  height.  Perhaps  the 
following  table  is  tolerably  approximate  for  large  works  in  ordinary  mud,  sand,  or 
gravel ;  assuming  the  plant  to  hare  been  paid  for  by  the  company ;  and  that  common 
labor  costs  $1  per  day. 

Table  of  actual  cost  of  dredging-  on  a  large  scale;  inclnd- 
iiig-  dropping-  the  material  into  scows,  along-side;  or  into 
side-shoots,  on  board.  €onanion  labor  S^l  per  day.  Repairs 
of  plant  are  inclwded ;  but  no  protit  to  contractor.    (Uriginal.J 


Depth 
in  Ft. 

Cts  per  Yard, 
in  place. 

Cts  per  Yard, 
in  .scow. 

Depth 
in  Ft. 

Cts  per  Yard, 
in  place. 

Cts  per  Yapd, 
in  scow. 

L»ss  than  10 
10  to  15 
15  to  20 
20  to  25 

8.4 
9.8 
11.2 
U.O 

6 

7 
8 
10 

25  to  30 
30  to  35 
35  to  40 

18.2 
25.2 
35.0 

13 
18 
25 

For  towing  of  the  SCOWS  by  steam  tugs  to  a  distance  of  ^  mile,  and  dropping 
the  mud  into  deep  water,  add  4  cts.  per  yard  in  the  scow  ;  for  ^  mile,  6  cts. ; 
for  %  mile,  8  cts. ;  for  1  mile,  10  cts.  Add  profit  to  contractor.  On  a  small 
scale  work  is  done  to  a  less  advantage  ;  and  a  corresponding  increase  must 
be  made  in  these  prices.  Also,  if  the  contractor  himself  furnishes  the 
dredgers  and  plant,  a  still  further  addition  must  be  made.  It  is  evident 
that  the  subject  admits  of  no  great  precision.  Small  Jobs,  even  in  favora- 
ble material,  but  in  inconvenient  positions,  may  readily  cost  two  or  three 
times  as  much  per  yard  as  the  above :  and  in  very  hard  material,  as  in 
cemented  gravel  and  clay,  four  or  five  times  as  much  for  the  dredging.  The 
cost  of  towing,  however,  will  remain  as  before,  if  wages  are  the  same. 

The  cost  of  dredgers,  tugs,  &c.,  will  vary  of  course  with  their  capabilities, 
strength  of  construction,  style  of  finish,  whether  having  accommodations 
for  the  men  to  live  on  board  or  not,  &c.    When  for  use  in  salt  water,  the 
bottoms  of  both  dredgers  and  scows  should  be  coppered,  to  protect  them  from  , 
sea-worms ;  and  if  occasionallv  exposed  to  high  waves,  both  should   be  J 
extra  strong.    The  most  powerful  machines  on  the  St.  Lawrence  cost  about  i 


DREDGING.  581 

$45,000  each ;  and  removed  in  10  working  hours  on  an  average  about  1800  cubic 
yards  in  place,  or  2520  in  the  scows.  Good  machines,  capable,  under  similar 
circumstances,  of  doing  as  much,  may,  however,  be  built  for  about  ^25,000 
to  $30,000.  To  remove  this  quantity  to  a  distance  of  3^  to  1  mile,  would  require 
two  steam  tugs,  costing  about  ^000  to  $10,000  each ;  and  4  to  6  scows  (some 
to  be  loading  while  others  are  away),  holding  from  30  to  60  cubic  yards  each, 
and  costing  frf)m  $800  to  $1500  eacli  at  the  shop.  Scows  with  two  hoppers 
are  best.  Such  a  dredger  would  require  at  least  8  or  10  men,  including  cap- 
tain, engineer,  fireman,  and  cook ;  each  tug  4  or  5  men  ;  and  each  scow  2 
men.  The  engineer  should  be  a  blacksmith  ;  or  a  blacksmith  should  be 
added.  In  certain  cases  a  physician,  clerk,  assistant  engineer,  &c.,  may  be 
needed. 

Dredgers  are  often  built  on  the  principle  of  the  Yankee  Excavator,  with  but 
a  single  bucket  or  dipper,  of  from  1  to  2  cubic  yards  capacity.  Hull  about  25 
by  60  feet.  Draft  3  feet.  Cylinder  about  7  or  8  inches  diameter ;  15-  to  18-incli 
stroke ;  ordinary  working'  pressure  50  to  80  lbs.  per  square  inch,  according 
to  hardness  of  material.  Cost  $8000  to  $12,000.  Will  raise  as  an  average 
day's  work  (10  hours)  from  200  to  TjOO  yards  in  place,  or  280  to  700  in  the  scow, 
according  to  the  deptb,  nature  of  the  material,  &c.  Require  5  or  7  men  in 
all  aboard,  including  cook.  Burn  3^^  to  1  ton  of  coal  daily.  Tolerably  large 
bowlders  and  sunken  logs  can  be  raised  by  the  dipper. =!= 

When  the  material  is  hard  and  compacted,  the  buckets  of  dredgers  should 
be  armed  with  strong  steel  teeth  projecting  from  their  cutting  edges.  On 
arriving  at  such  material  every  alternate  bucket  is  sometimes  unshipped. 
By  arranging  the  buckets  so  as  to  dredge  a  few  feet  in  advance  of  the  hull, 
low  tongues  of  dry  land  may  be  cut  away  :  the  machine  thus  digging  its 
own  channel.  The  daily  work  in  such  cases  w  ill  not  average  half  as  much 
as  in  wet  soil. 

On  small  operations,  clred^er»  worked  by  two  or  more  horses, 
instead  of  by  steam,  will  answer  very  well  in  soft  material ;  or  even  in 
moderately  hard,  by  reducing  the  size  and  number  of  the  buckets.  A  two- 
horse  machine  will  raise  from  50  to  100  yards  of  ordinary  mud  in  place,  or 
70  to  140  in  the  scow,  per  day,  at  from  12  to  15  feet  depth. 

Soft  material  in  small  quantity,  and  at  moderate  depth,  may  be  removed 
by  the  slow  and  expensive  mode  of  the  toag:-scoop,  or  ba^-spoon. 

This  is  simply  a  bag  B,  made  of  canvas  or  leather, 
and  having  its  mouth  surrounded  by  an  oval  iron 
ring,  the  lower  part  of  which  is  sharpened  to  form 
a  cutting  edge.  It  has  a  fixed  handle  h,  and  a 
swivel  handle  i.  One  man  pushes  the  bag  down 
into  the  mud  by  A,  while  another  pulls  it  along  by 
the. rope  g ;  and  when  filled,  another  raises  it  by  the 
rope  c,  and  empties  it.  If  the  bag  is  large,  a  wind- 
lass may  be  used  for  raising  it.  The  men  may  work 
from  a  scow  or  raft  properly  anchored.  Or  a  long- 
handled  metal  spoon,  shaped  like  a  deeply-dished  hoe,  may  be  used  by 
only  one  man  ;  or  a  larger  spoon  may  be  guided  by  -a  man,  and  dragged 
forward  and  backward  by  a  horse  walking  in  a  circle  on  the  scow,  &c.,  (fee. 

The  w^eig-ht  of  a  cubic  yard  of  wet  dredged  mud,  pure  sand,  or 
gravel,  averages  about  1^^  tons;  say  111  !bs.  per  cubic  foot ;  muddy  gravel, 
full  l]^  tons ;  sav  125  ibs.  per  cubic  foot.  Pure  sand  or  gravel  dredges  easily ; 
also  beds  of  shells.  Wet  dredged  clay  will  slide  down  a  shoot  inclined  at 
from  5  to  1,  to  3  to  1,  according  to  its  freedom  from  sand,  &c. ;  but  wet  sand 
or  gravel  will  not  slide  down  even  3  to  1,  without  a  free  flow  of  water 
to  aid  it ;  otherwise  it  requires  much  pushing. 

*  The  writer  has  seen  cases  in  which  a  circnlar  saw  for  logs  in  deep 
water  would  have  been  a  very  useful  addition  to  a  dredger.  It  should  be 
worked  by  steam ;  and  be  adjustable  to  different  depths.  It  would  cost  but 
about  $500. 


582  FOUNDATIONS. 


FOUNDATIONS. 

A  VOLUME  might  be  occupied  by  this  important  subject  alone.  We  have  space  for 
©lily  a  few  general  hints  ;  leaving  it  to  the  student  to  determine  liow  far  they  may 
be  applicable  in  any  given  case.  In  ordinary  cases,  as  in  culverts,  retaining  walls, 
ice,  if  excavations,  or  wells,  &c,  in  the  vicinity,  have  not  already  proved  that  the  soil 
is  reliable  to  a  considerable  depth,  it  will  usually  be  a  suflBcient  precaution,  after 
having  dug  and  levelled  off  the  foundation  pita  or  trenches  to  a  depth  of  3  to  5  ft,  to 
test  it  by  an  iron  rod,  or  a  pump-auger ;  or  to  sink  holes,  in  a  few  spots,  to  the  depth 
of  4  to  8  ft  farther ;  (depending  upon  the  weight  of  the  intended  structure ;)  to  ascer- 
tain if  the  soil  continues  firm  to  that  distance.  If  it  does,  there  will  rarely  be  any 
risk  in  proceeding  at  once  with  the  masonry;  because  a  stratum  of  firm  soil,  from  4 
to  8  ft  thick,  will  be  safe  for  almost  any  ordinary  structure ;  even  though  it  should 
be  underlaid  by  a  much  softer  stratum.  If,  however,  the  firm  upper  stratum  is  ex- 
posed to  running  water,  as  in  the  case  of  a  bridge-pier  in  a  river,  care  must  be  taken 
to  preserve  it  from  gradually  washing  away;  or  from  becoming  loosened  and  broken 
up  by  violent  freshets;  especially  if  they  bring  down  heavy  masses  of  ice,  trees,  and 
other  floating  matter.  These  are  sometimes  arrested  by  piers,  and  accumulate  so  as 
to  form  dams  extending  to  the  bottom  of  the  stream  ;  thus  creating  an  increase  of 
velocity,  and  of  scouring  action,  that  is  very  dangerous  to  the  stability  both  of  the 
bottom  and  of  the  structure.  When  the  testing  has  to  be  made  to- a  considerable 
depth,  it  may  be  necessary  to  drive  down  a  tube  of  either  wrought  or  cast  iron,  to 
prevent  the  soil  from  falling  into  the  unfinished  hole.  If  necessary,  this  tube  may 
be  in  short  lengths,  connected  by  screw  joints,  for  convenience  of  driving  ;  and  the 
•arth  inside  of  it  may  be  removed  by  a  small  scoop  with  a  long  handle.* 

Boring's  in  common  soils  or  clay  may  be  made  100  feet  deep  in  a  day 
or  two  by  a  common  wood  auger  1}4  inches  diameter,  turned  by  Iavo  to  four 
men  with  3  feet  levers.    This  will  bring  up  samples. 

in  starting  the  masonry,  the  largest  stones  should  of  course  be  placed  at  the  bot- 
tom of  the  pit,  so  as  to  equalize  the  pressure  as  much  as  possibly;  and  care  should 
be  taken  to  bed  them  solidly  in  the  soil,  so  as  to  have  no  rocking  tendency.  The 
next  few  courses  at  least  should  be  of  large  stones,  so  laid  as  to  break  joint  thoroughly 
with  those  below.  The  trenches  should  be  refilled  with  earth  as  soon  as  the  masonry 
will  permit;  so  as  to  exclude  rain,  which  would  injui-e  the  mortar,  and  soften  the 
foundation.  It  is  well  to  ram  or  tread  the  earth  to  some  extent  as  it  is  being  deposited. 

If  the  tests  show  that  the  soil  (not  exposed  to  running  water)  is  too  soft  to  support 
the  masonry,  then  the  pits  should  be  made  considerablj'  wider  and  deeper ;  and  after- 
ward be  filled  to  their  entire  width,  and  to  a  depth  of  from  3  to  6  or  more  ft,  (de- 
pending on  the  weight  to  be  sustained,)  with  rammed  or  rolled  layers  of  sand,  gravel, 
or  stone  broken  to  turnpike  size ;  or  with  concrete  in  which  there  is  a  good  propor- 
tion of  cement.  On  this  deposit  the  masonry  may  be  started.  The  common  practice 
in  such  cases,  of  laying  planks  or  wooden  platforms  in  the  foundations,  for  building 
upon,  is  a  very  bad  one.  For  if  the  planks  are  not  constantly  kept  thoroughly  wet^ 
they  will  decay  in  a  few  years ;  causing  cracks  and  settlements  in  the  masonry. 

Some  portions  of  the  brick  aqueduct  f  for  supplying  Boston  with  water  gave 
a  great  deal  of  trouble  where  its  trenches  passed  through  running  quicksands  and 
other  treacherous  soils.  Concrete  was  tried,  but  the  wet  quicksand  mixed  itself 
with  it,  and  killed  it.  Wooden  cradles,  &c,  also  failed  ;  and  the  diflficulty  was  finally 
overcome  by  simply  depositing  in  the  trenches  about  two  feet  in  depth  of  strong 
gravel.^  Sand  or  gravel,  when  prevented  from  spreading  sideways,  forms  one  of  the 
best  of  foundations.  To  prevent  this  spreading,  the  area  to  be  built  on  may  be  sur- 
rounded by  a  wall ;  or  by  squared  piles  driven  so  close  as  to  touch  each  other ;  or  in 
less  important  cases,  by  short  sheet  piles  only.     But  generally  it  is  suflBcient  simply 

*  Subterranean  caverns  in  limestone  regions  are  a  frequent  source  of  trouble, 
against  which  it  is  difficult  to  adopt  precautions. 

f  The  Cochituate  aqueduct,  built  1846-48  ;  egg-shape,  6  feet  4  inches  X  5  feet, 
with  semicircular  invert. 

X  Smeaton  mentions  a  stone  bridge  built  upon  a  natural  bed  of  gravel  only 
about  two  feet  thick,  overlying  deep  mud  so  soft  that  an  iron  bar  40  feet  long 
sank  to  the  head  by  its  own  weight.  C)ne  of  the  piers,  however,  sank  while  the 
arches  were  being  turned,  and  was  restored  by  Smeaton.  Although  a  wretched 
precedent  for  bridge-building,  this  example  illustrates  the  bearing  power  of  a 
thick  layer  of  well-compacted  gravel. 


FOUNDATIONS.  583 

to  give  the  trenches  a  good  width  ;  and  to  ram  the  sand  or  gravel  (which  are  all 
the  better  if  wet)  in  layers;  taking  care  to  compact  it  well  against  the  sides  of 
the  trench  also.  Under  heavy  loads,  some  settlenaent  will,  of  course,  take  place, 
as  is  the  case  in  all  foundations  except  rock.  If  very  heavy,  adopt  piling,  &c. 
See  Grillage. 

Wtien  an  unreliable  soil  overlies  a  firm  one,  but  at  such  a 
depth  that  the  excavation  of  the  trenches  (Which  then  must  evidently  be  made 
wider,  as  well  as  deeper)  becomes  too  troublesome  and  expensive;  especially 
when  (as  generally  happens  in  that  case)  water  percolates  rapidly  into  the 
trenches  from  the  adjacent  strata,  we  may  resort  to  piles.  When 

making  deep  foundation-pits  in  damp  clay,  we  must  remember  that  this 
material,  being  soft,  has,  to  a  certain  degree,  a  tendencv  to  press  in  every  direc- 
tion, like  .water.  This  causes  it  to  bulge  inward  at  the  sides,  and  upward  at  the 
bottom.  The  excavations  for  tunnels,' or  for  vertical  shafts,  often  close  in  all 
around,  and  become  much  contracted  thereby  be  fore  thev  can  be  lined  ;  there- 
fore they  should  be  dug  larger  than  would  otherwise  be  nec.essary.  Tlie  bottoms 
of  canal  and  railroad  excavations  in  moist  clay  are  frequently  pressed  upward 
by  the  weight  o(  the  sides.  I>ry  clay  rapidly  absorbs  moisture  from  the  air, 
and  swells,  producing  effects  similar  to  the  foregoing.  Its  expansion  is  attended 
by  great  pressure ;  so  that  retaining-walls  backed  with  dry  rammed  clay  will 
be  in  ^danger  of  bulging  if  the  clay  should  become"  wet.  It  is  a  treacherous 
material  to  work  in.    For  concrete  foundations,  see  pp.  946,  &c. 

As  to  the  g-rentest  load  that  may  safely  be  trusted  on  an  earth  founda- 
tion, Kankine  advises  not  to  exceed  1  to  1.5  tons  per  square  foot.  But  experi- 
ence proves  that  on  good  compact  gravel,  sand,  or  loam,  at  a  depth  beyond 
atmospheric  influences,  2  to  3  tons  are  safe,  or  even  4  to  n  tons  if  a  few  inches  of 
settlement  may  be  allowed,  as  is  often  the  case  in  isolated  structures  without 
tremors.  Years  may  elapse  before  this  settlement  ceases  entirely.  Pure  clay, 
especially  if  damp,  is  more  compressible,  and  should  not  be  trusted  with  more 
than  1  to  2.5  tons,  according  to  the  case.  All  earth  foundations  must  yield  some- 
what. Equality  of  pressure  is  a  main  point  to  aim  at.  Tremor  in- 
creases settlements,  and  causes  them  to  continue  for  a  longer  period,  especially 
in  weak  soils.  Great  care  must  be  taken  not  to  overload  in  such  cases,  even  if 
piled.  Foundations  in  silty  soils  will  probably  settle,  in  years,  at  the 
rate  of  from  3  to  12  inches  per  ton  (up  to  2  tons)  per  square  foot  of  quiet  load, 
if  not  on  piles. 

Figure  2  shows  an  easy  mode  of  obtaining  a  foundation  in  certain  cases.  It 
is  the  *'pierre  perdue"  (lost stone)  of  the  French;  in  English,  "ran- 
dom stone,"  or  rip-rap. 

It  is  merely  a  deposit  of  rough  angular  quarry  stone  thrown  into  the  water ; 
the  largest  ones  being  at  the  outside,  to  resist  disturbance  from  freshets,  ice, 
floating  trees,  &c.  A  part  of  the  interior  may  be  of  small  quarry  chips,  with 
some  gravel,  sand,  clay,  &c.  When  the  bottom  is  irregular  rock,  this  process 
saves  the  expense  of'levelling  it  off  to  receive  the  masonry.  For  2  or  3  feet 
below  the  surface  of  the  water,  the  stones  may  generally  be  disposed  by  hand,  so 
as  to  lie  close  and  firmly.  Small  spawls  packed  between  the  larger  ones  will 
make  the  work  smoother,  and  less  liable  to  be  displaced  by  violence.  Cramps 
or  chains  may  at  times  be  useful  for  connecting  several  of  the  large  stones 
together  for  greater  stability.    Rip-rap,  however,  is  apt  to  settle. 

If  ttie  bottom  is  so  yielding  as  to  be  liable  to  wash  away  in 

freshets,  it  may,  in  addition,  be  protected,  as  in  Fig  2,  by  a  covering  of  the  same  kind 

of  stones,  as  at  c :  extend- 
ing all  around  the  struc- 
ture. Or  the  main  pile 
of  stones  may  be  extend- 
ed as  per  dotted  line  at  d; 
BO  that  if  the  bottom 
should  wash  away,  as  pef 
dotted  line  at  o,  the 
stones  d  will  fall  int« 
the  cavity,  and  thus  pre- 
vent further  damage. 
Sheet-piles,  s  s,  may  be 
jlriven  as  an  additional  precaution.  For  greater  security,  the  bed  of  the  river  may 
be  dredged  or  scooped  under  the  entire  space  to  be  covered  by  the  main  deposit  as 
oer  dotted  lines  in  Fig  3,  to  as  great  a  depth  as  any  scouring  would  be  apt  to  reach ; 


584 


FOUNDATIONS. 


this  excavation  also  to  be  filled  with  stone.    Such  foundations  are  evidently  bttl 
adapted  to  quiet  water.    The  masonry  should  rest  on  a  strong  platform. 

Large  depoBltn  of  stone,  as  In  these 
two  figs,  greatly  increase  the  velocity, 
and  the  scoarlng  actJou  of  the  stream 
around  them,  especially  In  freshets ;  un- 
less the  bottom  on  each  side  from  the  de- 
t>OBlt  be  dredged  oat  t«>  such  an  extent 
that  the  original  area  of  water  shall  not 
be  reduced.  If  the  bottom  isi  treacherous, 
ttls  should  be  done  before  depositing  the 
isoveriog  stones  c,  Fig  2.  Judgment  and 
Experience  are  necessary  In  such  matters, 
lis  in  all  others  connected  with  engineer- 
ing. Mere  study  will  not  guard  against 
•Oustant  failures.  Theory  and  practice 
aunt  guide  each  other. 

Fig  .3  is  another  simple  method;  and 
Vfaen  It  does  not  create  too  great  an  ob- 
struction to  the  navigation  of  the  stream,  or  to  the  escape  of  its  waters  in  time  of  high  flresbeta,  Is  a 
very  effective  one.  Here  lh«  piles  are  first  driven  into  the  river  bottom,  for  the  support  of  the  pier; 
then  the  deposit  of  stone  is  thrown  in,  for  the  support  and  protection  of  the  piles;  preventing  them 
from  bending  under  their  loads;  and  shielding  them  from  blows  from  floating  bodies.  The  tops  of 
the  Diles  being  cut  off  to  a  level,  a  strong  platform  of  timber  Is  laid  on  top  of  them,  as  a  base  for  the 
mas'onrv.  The  top  of  the  platform  should  not  be  less  than  about  12  or  18  ins  below  ordinary  low 
water,  to  prevent  decay.  Mitchell's  Iron  screw  pile ;  or  hollow  piles  of  cast  iron,  may  be  used  Instead 
of  wooden  ones. 

Figs  4  represent  a  convenient  method  of  establishing  a  foundation  in  water,  by 
means  of  a  timber  crib,  A  A«  without  a  bottom.  It  should  be  built  of 
squared  timbers,  notch- 
ed together  at  their 
crossings,  as  shown  at 
Fig  5 ;  each  notch  being 
^  of  the  depth  of  the 
Btick.  By  this  means 
each  timber  is  support- 
ed throughout  its  entire 
length  by  the  one  below 
it;  and  resists  pulling 
in  both  directions.  Bolts 
also  are  driven  at  the 
intersections;  at  least 
in  the  sides  of  the  crib, 
to  prevent  one  portion 
trom  being  floated  off 
from  the  other.  The 
crib  is  thus  divided  into 
square  or  rectangular 
cells,  ft-om  2  to  4  or  6  ffe 
on  a  side,  according  to 
the  requirements  of  the 
case.  The  partitions 
between  the  cells  are 
put  together  in  the 
eame  manner  as  ihose 
at  the  sides  of  the  cribs; 
and  consequently,  like 
the  latter,  form  solH 
wooden  wall* 

The  orlb  may  be  ft-araed  afloat,  at  any  conveoteut  spot;  and  when  finished,  may  be  to«ed  to  tfm 
final  place,  where  it  is  carefully  moored  In  position,  and  then  sunk  by  throwing  stone  into  a  few 
tells  provided  with  platforms,  as  at  c  c.  for  that  purpose.  These  platforms  should  be  placed  a  little 
above  the  lower  edge  of  the  cells,  so  as  not  to  prevent  the  crib  from  settling  slightly  Into  the  soil,  and 
thus  coming  to  a  full  bearing  upon  the  bottom.  After  it  has  been  sunk,  all  the  cells  are  filled  with 
rough  stone.  A  stout  top  platform  may  be  added  or  not,  as  the  case  may  be  ;  also,  a  protection,  1 1,  of 
random  stone,  to  prevent  undermining  by  the  current.  If  the  sides  are  exposed  to  abrasion  from 
Ice,  Ac,  they  may  be  covered  in  whole  or  in  part  with  plank,  or  plate  iron  ;  and  the  angles  strength- 
ened by  iron  straps,  &c.  In  deep  water,  a  foundation  may  be  made  partly  of  random  stone,  as  ia 
Figs  2  and  3 ;  and  on  top  of  this  may  be  sunk  a  crib,  with  Its  top  about  2  It  under  low  water,  as  a  bas« 
for  the  masonry.    This  Is  much  safer  than  random  stone  alone. 

On  uneven  rock  bottom  it  may  be  necessary  to  scribe  the  bottom  of  the 

crib  to  fit  the  rock;  or  the  crib  may  first  be  sunk  by  means  of  a  loaded  platform  on  its  top,  or  by 
filling  some  of  its  cells,  until  its  lowest  timbers  are  within  a  short  distance  above  the  bottom.  Being 
there  kept  in  a  horlrontal  position,  small  stonee  may  be  thrown  luto  the  oells,  and  allowed  to  find 
their  way  under  the  timbers  of  the  crib,  thus  forming  a  level  support  for  It.  The  cells  may  then  be 
ttled;  and  rip-rap  deposited  ootside  around  tbecrlb  to  prevent  the  small  stones  from  being  diapiaoed. 


FOUNDATIONS. 


685 


A  crib  with  only  an  ontside  row  of  cells  for  sinking  it  may  fes 

built  i  and  the  ioterior  chamber  may  be  filled  with  concrete  under  water.  The  masonry  may  then 
rest  on  the  concrete  alone.  If  the  crib  rests  upon  a  foundation  of  broken  stone,  the  upper  interstices 
of  this  stone  should  first  be  levelled  off  by  small  stone  or  coarse  gravel  to  receive  the  concrete  of  the 
inner  chamber. 

Or  a  crib  like  Fig*.  4  maj'  be  sunk,  and  piles  be  driven  in  the  cells,  which 
may  afterward  be  filled  with  broken  stone  or  concrete.  The  masonry  may  then  rest  on  the  piles  only, 
which  in  turn  will  be  defended  by  the  crib.  If  the  bottoni'  is  liable  to  scour,  place  sheet-piles  or 
rip-rap  around  the  base  of  the  crib. 

By  all  means  avoid  a  crib  like  e,  Fig*  5>^,  much  higher  at  one  part 
than  at  another,  if  the  superstructure  s  is  to  rest  on  the  timher  of  the  crib  instead  of  ou 
piles,  or  on  concrete  independent  of  the  timber ;  for  the  high  part  of  the  crib  will  compress  more  under 
its  load  than  the  low  part,  anfi  win  thus  cause  the  superstructure  to  lean  or  to  crack. 

A  crib  either  straight  sided  or  circular,  with  only  an  outer  row  of  cells  for  pnd- 
dling;  may  be  used  as  a  cofferdam  (see  cofferdams, p.  586).   The  joints 

between  the  outer  timbers  should  be  well  caulked  ;  and  care  be  taken,  by  means  of  outside  pile-planks, 
gravel,  &c,  to  prevent  water  from  entering  beueath  it. 

The  cast-iron  Bridg-e  across  the  Schnylkill  at  Chestnut  St, 
Phila,  Mr.  Strickland  Kneass,  Engineer,  aftbrds  a  striking  example  of  crib 
foundation.  The  center  pier  stands  on  a  crib,  an  oblong  octagon  in  plan  ;  31  by  87  feet  at  base  ;  24 
by  80  ft  at  top ;  and  (with  its  platform)  29  ft  high.  Its  timbers  are  of  yellow  pine,  hewn  12  ins 
square  :  and  framed  as  at  Fig  5.  The  lower  timbers  were  carefully  cut  or  scribed  to  conform  to  the 
irregularities  of  the  tolerably  level  rock  upon  which  it  rests.  These  were  ascertained  ("after  the  8  ft 
depth  of  gravel  had  been  dredi?ed  off)  in  the  usual  manner  of  mooring  above  the  site  a  large  floating 
wooden  platform,  composed  of  timbers  corresponding  in  position  with  all  those  of  the  lower  course 
of  the  intended  crib,  both  longitudinal  and  transverse.  Soundings  were  then  taken  close  together 
along  all  these  lines  of  timber.  Most  of  the  celH  are  about  3  by  4  ft  on  a  side,  in  the  clear.  A  few 
of  them  had  platforms  at  the  level  of  the  second  course  from  the  bottom,  for  receiving  stone  for  sink, 
ing  the  crib ;  the  others  are  open  to  the  bottom. 

The  crib  was  built  in  the  water  ;  and  was  kept  floating,  during  its  construction,  with  its  unfinished 
top  continually  just  above  water,  by  gradually  loading  it  with  more  stone  as  new  timbers  were  adied. 
The  stone  required  for  this  purpose  alone  was  300  tons.  When  the  crib  was  towed  into  position,  and 
moored,  150  tons  more  were  added  for  sinking  it.  All  the  cells  were  afterward  filled  with  rongh  dry 
stone,  and  coarse  gravel  screenings ;  making  a  total  of  1666  tons.  A  platform  of  12  by  12  iuch  squared 
timber  covered  the  whole ;  its  top  being  2^  ft  below  low  water.  The  pier  alone,  which  stands  on  this 
crib,  weighs  3255  tons;  and  during  its  construction  it  compressed  the  crib  6^  ins.  The  weight  of 
superstructure  resting  on  the  pier,  may  be  roughly  taken  at  1000  tons  more. 

An  ordinary  caisson  is  merely  a  strong"  scoiv,  or  a  box  with- 
out a  lid;  and  with  sides  which  may  at  pleasure  be  readily  detached  from  its  bottom.  It  is  built  on 
land,  and  then  launched.  The  masonry  may  first  be  built  in  it,  either  in  whole  or  in  part,  while 
afloat;  and  the  whole  being  then  towed" into" place,  and  moored,  may  be  sunk  to  the  bottom  of  tl^e 
river,  to  rest  upon  a  foundation  previously  prepared  for  it,  either  by  piling,  if  necessary;  or  by 
merely  levelliag  off  the  natural  surface,  &c.     The  bottom  of  the  caisson  constitutes  a  strong  limber 

platform,  upon  which  the  miisonry  rests;  and 
is  so  arranged,  that  after  it  is  sunk,  the  sides 
may  be  detached  from  it,  and  removed  to  be 
rebbttomed  for  use  at  another  pier,  if  needed. 
This  detaching  may  be  effected  b_v  some  such 
contrivance  as  that  shown  in  Fig  6,  where 
P  P  m;  is  the  bottom  of  the  caisson ,  to  which 
are  firmly  attached  at  intervals  strong  iron 
eyes  t;  which  are  taken  hold  of  by  hooks  d,  a* 
the  lower  end  of  long  bolts  E  »i,*  reaching  to 
the  top  timbers  S  of  the  crib,  where  they  are 
confined  by  screw  nuts  n.  By  loosening  the 
nuts  n,  the  hooks  d  can  be  detached  from  the 
eyes  t;  and  the  sides  can  then  be  removed 
from  the  bottom;  there  being  no  other  connec- 
tion between  the  two.  These  hooks  and  eyes 
are  usually  placed  outside  of  the  caisson:  the 
screw  nuts  n  being  sustained  by  the  projecting 
ends  of  cross  pieces,  as  tt,  Fig  9.  The  im- 
proper position  given  them  in  our  Fig  was 
merely  for  convenience  of  illustrating  the  prin- 
ciple. It  will  sometimes  be  necessary  to  have 
one  side  detachable  from  the  others,  in  order  to  float  the  caisson  away  clear  from  the  finished  pier; 
unless  it  be  floated  away  before  the  masonry  has  been  built  so  high  as  to  render  the  precaution  use- 
less. Fig  6  shows  one  of  many  ways  of  constructing  a  caisson  ;  with  sides  consisting  of  upright 
corner-posts,  I;  cap  pieces  S,  on  top  ;  and  sills  g  at  bottom,  resting  on  the  bottom  platform  P  P  w ; 
intermediate  uprights  T,  framed  into  the  caps  and  sills;  the  whole  being  covered  outside  by  one  or 
two  thicknesses  of  planking  B,  which,  as  well  as  the  platform,  should  be  well  calked,  to  prevent 
leaking.  Tarpaulin  also  may  be  nailed  outside  to  assist  in  this.  The  greatest  trouble  from  leaking 
is  where  the  sides  join  the  platform.  On  top  of  the  platform  is  firmly  spiked  a  timber  o  o,  extending 
all  around  it  just  inside  of  the  inner  lower  edge  of  the  sides  of  the  caisson.  Its  use  is  to  prevent 
the  sides  from  being  forced  inward  by  the  pressure  of  the  water  outside.  The  details  of  construction 
will  of  course  vary  with  the  requirements  of  the  case.  In  deep  caissons,  inside  cross-braces  or  struts 
from  side  to  side, 'as  at  c  c.  Fig  7,  will  be  required  to  prevent  the  sides  from  being  forced  inward  by 
the  pressure  of  the  water,  as  the  vessel  gradually  sinks  while  the  masonry  is  being  built  within  it. 
As  the  masonry  is  carried  up.  the  struts  are  removed;  and  short  ones,  extending  from  the  sides  of 
the  caisson  to  the  masonry,  are  inserted  in  their  place.  When  the  caisson  is  shallow,  only  the  upper 
course  of  braces  will  be  required,  they  also  support  a  platform  for  the  workmen  and  their  materials. 
In  deep  caissons,  in  order  not  to  be  in  the  way  of  the  masons,  the  outer  planking  of  the  sides  may, 
in  part,  be  gradually  built  up  as  the  masonry  progresses  Ft  may  sometimes  be  expedient  to  build 
the  masonry  hollow  at  first,  «"'tb  ♦bin  trp«sv«rse  wails  inside  to  stiffen  it  if  necessary  ;  and  to  com- 


686 


FOUNDATIONS. 


plete  the  interior  after  sinking  the  caisson.  Indeed,  masonry  or  brickwork,  in  cement,  may  thus  uw 
built  hollow  at  first,  resting  on  the  platform ;  the  masonry  itself  forming  the  sides  of  the  caisson. 
Or  the  sides  may  consist  of  a  water-tight  casing  of  iron,  or  wood,  of  the  shape  of  the  intended  pier, 
&c.  This  casing  being  confined  to  the  platform,  becomes,  in  fact,  a  mould,  in  which  the  pier  may  be 
formed,  and  sunk  at  the  same  time  by  filling  it  with  hydraulic  concrete.  For 
concrete  foundations,  see  pp  946  &c. 

On  rock  bottom  the  under  timbers  of  the  platform  may  be  cut  to  suit  the  irregularities 
as  already  itated  under  "  Cribs."  Or  the  bottom  may  be  levelled  up  by  first  depositing  large  stones 
around  the  area  upon  which  the  caisson  is  to  rest ;  and  then  filling  between  these  with  smaller  stones 
andgravai;  testing  the  depth  by  sounding.  Or  a  level  bed  of  cement  concrete  may,  with  care,  be 
deposited  in  the  water.  If  there  are  deep  narrow  crevices  in  the  rock,  through  which  the  concrete 
may  escape,  they  may  be  first  covered  with  tarpaulin.  Diving  bells  may  often^be  used  to  advantage, 
in  all  such  operations.  But  in  the  case  of  very  irregular  rock,  it  will  often  be  better  to  resort  to  cof- 
fer-dams. 

Valves  for  the  admission  of  water  for  sinking  the  caisson  are 
usually  introduced.  If,  after  sinking,  it  should  be  necessary  to  again  raise  the  whole,  it  is  only 
Accessary  to  close  the  valves,  and  pump  out  the  water.  Guide  piles  may  be  driven  and  braced  along- 
side of  the  caisson,  to  insure  its  sinking  vertically,  and  at  the  proper  spot.  Or  it  may  be  lowered  by 
screws  supported  by  strong  temporary  framework. 

Assuming  the  uprights  I,  T,  &c.  Fig  6,  to  be  sufficiently  braced,  as  at  cc.  Fig  7,  the  following  table 
will  show  the  thickness  of  planking  necessary  for  difiFerent  distances  apart  of  the  uprights,  (in  the 
dear,)  to  insure  a  safety  of  six  against  the  pressure  of  the  water  at  different  depths;  and  at  the 
■ame  time  not  to  bend  inward  under  said  pressure,  more  than  xttt  P^^^  o^  the  distance  to  which 
they  stretch  from  upright  to  upright ;  or  at  the  rate  of  }^  inch  in  10  ft  stretch ;  J^  inch  in  5  ft,  4a. 
Such  a  table  may  be  of  use  in  other  matters. 

Table  of  tbickness  of  white  pine  plank  required  not  to  bend 
more  than  -^^-^  part  of  its  clear  horizontal  stretch,  under 
Clilferent  heads  of  water.    (Original.) 


HEADS  IK  PEET. 

Stretch 

in  Ft. 

40 

30      1       «0      1       10 

1       ' 

Thickness  in  Inches. 

3 

SH 

3 

2H 

2H 

m 

4 

W2 

4 

3^ 

2% 

2M 

6 

6% 

6 

5^ 

4M 

3Ji 

8 

9 

8 

7 

5>^ 

^H 

10 

ii}4 

10 

8H 

7 

634 

12 

1334 

I2}i 

10% 

8>^ 

6% 

15 

163^ 

15 

13 

10>^ 

8H 

20 

22i4 

20 

iiH 

14 

11 

Coffer-dams  are  enclosures  from  which  the  water  may  be  pumped  out,  so  aa 
to  allow  the  work  to  be  done  in  the  open  air.  Their  construction  of  course  varies 
greatly.  In  still  shallow  water,  a  mere  well-built  bank  of  clay  and  gravel ;  or  of 
bags  partly  filled  with  those  materials  when  there  is  "much  current,  will  answer 
©very  purpose ;  or  (depending  on  thfi  depth)  a  single  or  double  row  of  sheet-piles ;  or  of 
squared  piles  of  larger  dimensions,  driven  touching  each  other;  their  lower  ends  a 
few  feet  in  the  soil ;  a,nd  their  upper  ones  a  little  above  high  water,  and  protected 
outside  by  heaps  of  gravelly  soil  or  puddle,  (as  at  P  in  Fig  7,)  to  prevent  leaking. 
The  sheet-piles  may  be  of  wood;  or  of  cast  iron,  of  a  strong  form. 

The  sufficiency  of  a  mere  bank  of  irell-packed  earth  in  still 
^ater,  is  shown  by  the  embankments  or  levees,  thrown  up  in  all  countries,  to  pre- 
vent rivers  from  overflowing  adjacent  low  lands.  The  general  average  of  the  levees 
along  700  miles  of  the  Mississippi,  is  about  6  ft  high;  only  3  ft  wide  on  top;  side- 
•lopes  V^  to  1.  In  floods  the  river  rises  to  within  a  foot  or  less  of  their  tops  ;  and 
frequently  bursts  through  them,  doing  immense  damage.  They  are  entirely  too  slight. 

The  method  of  a  single  row  of  12  by  12  inch  squared  piles,  driven  in  contact  with 
each  other,  (close  pilcs^  a,nd.  simply  backed  by  an  outer  deposit  of  impervious  soil, 
is  very  effective;  and  with  the  addition  of  interior  cross-braces  or  struts,  like  cc,  Fig 
7,  to  prevent  crushing  inward  by  the  outside  pressure  of  the  water  and  puddle  when 
pumped  out,  has  been  successfully  employed  in  from  20  to  25  ft  depth  of  water,  in 
which  there  was  not  sufficient  current  to  wash  away  the  puddle.  The  cross-braces 
are  inserted  successively,  as  the  water  is  being  pumped  out ;  beginning,  of  course, 
with  the  upper  ones.  The  ends  of  these  braces  may  abut  on  longitudinal  timbers, 
bolted  to  the  piles  for  the  purpose.  Another  method  is  a  strong:  crib,  com- 
posed of  uprights  framed  into  caps  and  sills ;  and  covered  outside  with  squared 
timbers  or  plank,  laid  touching  each  other,  and  well  calked ;  as  in  the  caisson,  Fig 
6 ;  but  without  a  bottom.  Between  the  opposite  pairs  of  uprights  are  strong  i  nterior 
atruts,  as  c  c,  Fig  7,  reaching  from  side  to  side,  to  prevent  crushing  inward.    Th« 


FOUNDATIONS. 


687 


np'per  series  of  these  usually  supports  a  platform  for  the  workmen,  windlasses,  fte. 
The  crib  iiaving  been  built  on  land,  is  launched,  taken  to  its  final  place,  and  sunk  bj 
piling  stones  on  a  temporary  platform  resting  on  the  cross-struts  ;  the  bottom  of  the 
stream  having  been  previously  levelled  off,  if  necessary,  for  its  reception. 

To  prevent  leaking  under  the  bottom  of  the  crib,  sheet-piles  may  be  driven  around  it,  their  heads 
extending  a  few  feet  above  its  bottom ;  or  a  small  deposit  of  outside  puddle  may  be  placed  around  it, 
as  shown  at  the  stone  deposits  1 1,  Fig  4.  Or  a  broad  flap  of  tarpaulin  may  be  closely  nailed  around 
and  a  little  above  the  lower  edge  of  the  crib ;  so  arranged  that  it  may  be  spread  out  loosely  on  the 
river  bottom,  to  a  width  of  a  few  feet  all  around  the  outside  of  the  crib ;  and  the  puddle  may  be  placed 
upon  it.  Such  a  tarpaulin  is  also  very  useful  in  case  the  river  bottom  is  somewhat  irregular,  and 
cannot  be  levelled  off  without  too  great  expense ;  in  which  case  the  crib  cannot  come  to  a  full  bearing 
apon  it;  and  consequently  the  water  would  leak  or  flow  beneath  freely.  It  is  especially  adapted  to 
uneven  rock  ;  where  sheet-piles  cannot  be  driven.  An  artificial  stratum  of  impervious  soil  may,  how- 
ever, be  deposited  on  bare  roclc ;  in  which  case  the  sinking  of  the  crib,  and  the  subsequent  operations 
•»ill  be  the  same  as  on  a  natural  stratum.  These  expedients  are  evidently  more  or  less  applicable  ia 
other  oases,  where,  to  avoid  repetition,  they  are  not  specially  mentioned. 


Plan  a (  one  end. 

Flgr  7  is  another  crib  coflTer-dam :  in  which  the  sides,  instead  of  being 
planked  longitudinally,  as  in  the  last  instance,  are  sheathed  with  vortical  sheet-piles 
Sy  driven  after  the  crib  is  sunk.  It  is  much  inferior  to  the  last,  owing  to  its  greater 
liability  to  leak.  In  one  of  this  description.  Fig  7,  successfully  used  in  16  ft  water, 
the  dimensions  of  the  crib  were  34  ft  by  80  ft.  Along  each  long  side  were  7  uprights  t,  t, 
19  ft  long,  12  ins  square,  12^  ft  apart.  Into  each  opposite  pair  of  these  were  notched, 
and  held  by  dog-irons,  6  cross-braces  c  c,  of  12  ins  square.  The  distance  between  the 
two  upper  ones  was  3  ft  in  the  clear ;  gradually  diminishing  to  18  ins  between  the 
two  lower  ones,  on  account  of  the  increased  pressure  of  the  water  in  descending.  On 
the  outside  of  the  uprights,  and  opposite  the  ends  of  the  braces,  were  bolted  longi* 


SECTION 


ludinal  timbers  to  support  the  out«ide  pressure  against  the  3-inch  sheet-piling  s». 
Other  longitudinal  pieces  o  o,  confine  the  heads  of  the  sheet-piles  to  the  top  of  the 
crib  after  they  are  driven.  The  feet  of  the  sheet-piles  were  cut  to  an  angle,  as  at  m ; 
to  make  them  draw  close  to  each  other  at  bottom  in  driving. 

The  sheet-piles  will  drive  in  a  far  more  regular  and  satisfactory  manner,  with  the 
arrangement  shown  in  Figs  8.  Hereooaretheuprigbts;  ccarepairsof  longitudinal 


588 


FOUNDATIONS. 


pieces,  notched  and  bolted  to  the  uprights,  near  both  their  tops  and  their  feet ;  and 
at  as  many  intermediate  points  as  may  be  desired.  The  sheet-piles  I,  are  inserted 
between  these ;  and  of  course  are  guided  during  their  descent  much  more  perfectly 
than  in  Pig  7. 

When  the  current  is  too  strong  to  permit  the  use  of  outside  puddle,  P,  Fig  7,  th« 
principle  of  coffer-dam  shown  in  Fig  9,  is  generally  used  ;  in  which  both  sides  of  the 
puddle  are  protected  from  washing  away.  The  space  to  be  enclosed  by  the  dam  is  sur- 
rounded by  two  rows  of  firmly-driven  main  piles  p  p,  on  which  the  strength  chiefly 
depends.  They  may  be  round.  In  deciding  upon  their  number,  it  must  be  remem- 
bered that  they  may  have  to  resist  floating  ice,  or  accidental  blows  from  vessels,  &c. 
With  reference  to  this,  extra  /ewder-piles  may  be  driven.  A  little  below  the  tops  of 
the  main  piles  are  bolted  two  outside  longitudinal  pieces  w  w,  called  wales ;  and  oppo- 
site to  them  two  inner  ones,  as  in  the  fig.  The  outer  ones  serve  to  support  cross- 
timbers  1 1,  which  unite  each  pair  of  opposite  piles,  and  steady  them ;  and  prevent 
their  spreading  apart  by  the  pressure  of  the  puddle  P.  The  inner  ones  act  as  guides 
for  the  sheet-piles  s  s,  while  being  driven ;  after  which  the  heads  of  the  sheet-piles 
are  spiked  to  them.  In  deep  water  these  sheet-piles  must  be  very  stout,  say  12  ins 
square  ;  to  resist  the  pressure  of  the  compacted  puddle. 

A  g'ang'way  m,  is  often  laid  on  top  of  the  cross-pieces  1 1,  for  the  use  of  the 
workmen  in  wheeling  materials,  &c.  The  puddle  P  is  deposited  in  the  water  in  the 
space,  or  boxing,  between  the  sheet-piles.  It  shonld  be  put  in  in  layers,  and  com- 
pacted as  well  as  can  be  done  without  causing  the  sheet-piles  to  bulge,  and  thus  open 
their  joints.  The  bottom  of  the  puddle-ditch  should  be  deepened,  as  in  the  fig,  in 
case  it  consists,  as  it  often  does,  of  loose  porous  material  which  would  allow  water  to 
leak  in  beneath  it  and  the  sheet-piles.  This  leaking  under  the  dam  is  frequently  a 
source  of  much  trouble  and  expense.  Water  will  find  its  way  readily  through  almost 
any  depth  and  distance  of  clean  coarse  gravelly  and  pebbly  bottom,  unmixed  with 
earth.  Sand  is  also  troublesome ;  and  if  a  stratum  of  either  should  present  itself  ex- 
tending to  a  great  depth,  it  will  generally  be  exped'cat  to  resort  to  either  simple 
cribs,  Fig  4;  or  to  caissons;  with  or  without  piles  in  either  case,  according  to  cir- 
cumstances. But  if  such  open  gravel,  or  any  other  permeable  or  shifting  material, 
as  soft  mud,  quicksand,  &c,  is  present  in  a  stratum  but  a  few  feet  in  thickness,  and 
underlaid  by  stiff  clay,  or  other  safe  material,  leaking* may  be  prevented,  or  at  least 
much  reduced,  by  driving  the  sheeting-piles  2  or  3  ft  into  this  last ;  and  by  deepening 
the  puddle-trench  to  the  same  extent.  It  may  sometimes  be  better,  and  more  con- 
venient, to  dredge  away  the  bad  material  entirely  from  all  the  space  to  be  enclosed 
by  the  dam,  and  for  a  short  distance  beyond,  before  commencing  the  construction  of 
the  latter.  If  the  dam.  Fig  9,  is  (as  it  should  be)  well  provided  with  cross-braces. 
like  c  c.  Fig  7,  extending  across  the  enclosed  area,  the  thickness  or  width  o  o  of  the 
puddle,  need  not  be  more  than  4  or  5  feet  for  shallow  depths ;  or  than  5  to  1 0  ft  for  great 
ones ;  because  its  use  is  then  merely  to  prevent  leaking.  But  if  there  are  no  braces, 
it  must  be  made  wider,  so  as  to  resist  upsetting  bodily;  and  then, with  good  puddle, 
o  o  may,  as  a  rule  of  thumb,  be  ^  of  the  vertical  depth  o  I  below  high  water;  except 
when  this  gives  less  than  4  ft ;  in  which  case  make  it  4  ft ;  unless  more  should  be 
required  for  the  use  of  the  workmen,  for  depositing  materials,  kc.  Or  if  the  excavation 
for  the  masonry  is  sunk  deeper  than  the  puddle,  the  dam  must  be  wider;  elae  it  may 
be  upset  into  the  excavated  pit. 


PLAN 


The    oxcavated    soil  may  be 

raised  in  buckets  by  windlasses,  or  by  hand,  in 
Bucoessive  stages.  The  pumps  may  oe  worked 
by  hand,  or  by  steam,  as  the  case  may  require; 
as  also  the  windlasses  generally  needed  for 
lowering  mortar,  stone,  &c.  More  or  less  leak- 
ing may  always  be  anticipated,  notwithstanding 
every  precaution. 

Where  a  coffer-dam  is  exposed  to  a  violent 
current,  and  great  danger  from  ice,  &c,  the  ex- 
pensive mode  shown  in  Figs  10  may  become 
necessary.  The  two  black  rectangles  c  c,  repre- 
sent two"  lines  of  rough  cribs  filled  with  stone, 
and  sunk  in  position ;  one  row  being  enclosed 
by  the  other ;  with  a  space  several  feet  wide  be- 
tween them.  Sheet-piles  p  p  are  then  driven 
around  the  opposite  faces  of  the  two  rows  of 
cribs ;  and  the  puddle  is  deposited  within  the  boxing  thus  provided  for  it,  as  shown  in  the  fig. 

Where  the  current  is  not  strong  enough  to  wash  away  gravel  backing,  we  may.  on  rock  especially, 
enclose  the  space  to  be  built  on,  by  a  single  quadrangle  of  cribs  sunk  by  stone ;  and  after  adopting 
precautions  to  prevent  the  gravel  from  being  pressed  in  beneath  the  cribs,  apply  the  backing.* 

Figs  IQi^  show  the  plan,  outside  view,  and  transverse  section,  to  a  scale  of  20  ft  to 
an  inch,  of  a  coffer-dam  on  rock,  in  8  to  9  ft  water,  used  successfully  on  the  Schuylkill 
Navigation. 

»  A  pure  clean  coarse  gravel  is  entirely,  unfit  for  such  purpose*.  A  considerable  proportion  •! 
MMrth  Is  essential  for  preventing  leaks. 


FOUNDATIONS. 


689 


CoflfeF-dnni  on  I*OCl<«  Uprights  b,  about  1  ft  square,  and  10  ft  apart  from  center  to  centw 
along  the  sides  of  the  dam  ;  and  10  ft  in  the  clear,  transversely  of  the  dam,  support  two  lines  of  hori- 
zontal stringers,  i  i ;  inside  of  which  are  the  two  lines  of  sheeting-piles,  s  s,  enclosing  between  them 
a  width  of  7  ft  of  gravel  puddle.  Two  flat  iron  bars  (t  t,  of  the  transverse  section)  tie  together  eack 
pair  of  uprights  b  b.  These  bars  are  }4  inch  thick,  by  '2}4  insdeep,  and  9  ft  long.  Their  hooked  endi 
fit  iuto  eye-bolts  c,  which  pass  through  the  uprights  b ;  outside  of  which  they  are  fastened  by  keys,  *, 
(■ee  detail  sketch.)  Between  the  keys  and  b,  were  washers.  At  the  corners  of  the  dam  (see  plan) 
were  additional  tie-bars,  as  shown.  A  small  band  of  straw,  as  seen  at  y,  wrapped  around  the  tie- 
bars  just  inside  of  the  sheet-piles  ;  and  kept  in  place  by  the  puddle  ;  effectually  prevented  the  leaking 
which  generally  proves  so  troublesome  in  such  cases.  The  stout  oblique  braces,  o  o,  were  merely- 
spiked  to  the  outside  faces  of  the  uprights  6.  They  are  not  shown  inthe  transverse  section.  This  dam 
was  built  on  shore;  in  sections  30  to  40  ft  long.  These  were  floated  into  place,  and  weighted  down, 
•heet-piled,  and  puddled  with  gravel.  The  dam  had  sluices  by  which  water  was  admitted  when 
necessary  for  preventing  the  outside  head  from  exceeding  9  ft.  The  lengths  of  the  uprights  6  6  were 
irat  found  by  careful  soundings. 


b  Id 


OUTSIDE, 


PLAN 


The  mooring;-  of  lar^e  caissons  or  cribs,  preparatory  to  sinking 
them,  is  sometimes  troublesome,  especially  in  strong  currents.  It  may  be  necea- 
sary  to  drive  clumps  of  piles;  or  to  temporarily  sink  rough  cribs  filled  with  stone, 
^o  which  to  attach  the  long  guide-ropes  by  which  the  manoeuvring  into  position,  &c, 
js  done.  Frequently  dams  are  left  standing  after  the  work  is  done  ;  if  not  in  the  way 
of  navigation,  or  otherwise  objectionable ;  inasmuch  as  the  materials  are  rarely  worth 
the  expense  of  removal.  But  if  removed,  the  piles  should  not  be  drawn  out  of  the 
ground ;  but  be  cut  off  close  to  river  bottom ;  for  if  drawn,  the  water  entering  their 
holes  may  soften  the  soil  under  the  masonry.  It  is  often  expedient  to  drive  two 
rows  of  piles  from  the  dam  to  the  shore,  for  supporting  a  gangway  for  the  workmen; 
or  even  for  horses  and  carts ;  or  for  a  railway  for  the  easy  delivery  of  large  stones,  &c. 

Coffer-dams  may  be  sunk  tbrou^h  a  soft  to  a  firm  soil,  in 
shape  of  a  box  of  cribwork,  either  rectangular  or  circular,  and  without  a  bottom. 
This  being  strongly  put  together,  and  provided  with  proper  temporary  internal 
bracing,  (to  be  gradually  removed  as  the  masonry  is  built  up,)  is  floated  into  place; 
and  after  being  loaded  so  as  to  rest  on  the  soft  bottom,  is  sunk  by  dredging  out  the 
soft  material  from  ins.ide.  Additional  loading  will  sometimes  be  required  for  over- 
coming the  friction  of  the  soil  against  the  outside ;  or  it  may  even  become  necessary 
to  dredge  away  some  of  the  outer  material  also.  On  rock  it  may  at  times  be 
expedient  to  drill  holes  in  deep  water,  for  receiving  the  ends  of  piles,  or  of  iron  rods, 
Ac.  This  may  be  done  by  means  of  long  drill-rods,  working  in  an  iron  tube  or  pip© 
sunk  as  a  guide  to  the  rod;  with  its  lower  end  over  the  spot  to  be  bored.  Or  a  diving- 
bell  may  be  used.  Or  a  cylin<ler  of  staves  4  to  12  inches  thick,  long  enough  to 
reach  above  the  surface,  and  having  abroad  tarpaulin  flap  or  apron  around  its  lower 
•dge,  to  be  covered  with  gravel  to  prerent  leaking ;  may  be  sunk,  and  the  water 
pumped  out,  to  allow  a  workman  to  descend,  and  work  in  the  open  air. 

Piles.  When  driven  in  close  contact. as  in  Fig  11,  for  preventing  leakage;  for 
confining  puddle  in  a  coffer-dam :  or  for  enclosing  a  piece  of  soft  or  sandy  ground,  to 
prevent  its  spreading  when  loaded ;  or  if  the  outside  soil  should  wash  away  from 


590 


FOUNDATIONS. 


Fig  11 


Generally  these  are  thinuer 
t  t 


around  them,  &c,  they  are  called  sbeet-piles. 

than  they  are  wide ; 
but  frequently  they  are 
square;  and  as  large  as 
bearing  piles ;  and  are 
then  called  close 
piles.  To  make  them 
drive  tight  together  at 
foot,  they  are  cut  ob- 
liquely as  at  /.  Occa- 
sionally, when  driven 
down  to  rock  through 
soft  soil,  their  feet  are 
in  addition  cut  to  an 
edge,  as  at  i,  so  as 
to  become  somewhat 
bruised  when  they  reach  the  rock,  and  thus  fit  closer  to  its  surface.  Their  heads 
are  kept  in  line  while  driving,  by  means  of  either  one  or  two  longitudinal  pieces 
a  and  o,  called  wales  or  stringers.  These  wales  are  supported  by  gauge-piles^ 
or  guide-piles,  previously  driven  in  the  required  line  of  the  work,  and  several 
ft  apart,  for  this  purpose.     See  Figs  8. 

A  dog-iron  d,  of  round  iron,  may  also  be  used  for  keeping  the  edges  of  the 
piles  close  at  top  to  those  previously  driven,  both  during 
and  after  the  driving.    Its  sharp  ends,  c  c,  being  driven  r^^ 

into  the  tops  of  the  wales" it;  w;,  (shown  in  plan,)  it  holds  ^        j^ 

the  descending  pile  o  firmly  in  place.     At  n,  d,  p,  Fig      ^ ^  '^ 

11,  are  other  modes  occasionally  used  for  keeping  the  ^  ^     —i   (  W 

piles  in  proper  line.     At  p,  the  letters  .s  s  denote  small      yi'^i^l^iqi|g 

pieces  of  iron  well  screwed  to  the  piles,  a  little  above  w  T  —■  /  w 

their  feet,  to  act  as  guides  ;  very  rarely  used.    At  m       ^ ' 

are  shown  wooden  tongues  1 1,  sometimes  driven  down  Fig- 12 

between   the  piles  after  they    themselves   have  been 

driven ;  to  assist  in  preventing  leaks.  In  some  cases  sheet-piles  are  employed 
without  being  driven.  A  trench  is  first  dug  to  their  full  depth  for  receiving 
them;  and  the  piles  are  simply  placed  in  these,  which  are  then  refilled.  Closer 
joints  can  be  secured  in  this  manner  than  by  driving. 

When  piles  are  intended  to  sustain  loads  on  their  tops,  whether  driven  all  their 
length  into  the  ground,  or  only  partly  so,  as  in  Fig  3,  they  are  called  bearing 
piles.  They  are  generally  round;  from  9  to  18  ins  diam  at  top;  and  should  be 
straight,  but  the  bark  need  not  be  removed.  White  pine,  spruce,  or  even  hem- 
lock, answer  very  well  in  soft  soils  ;  good  yellow  i)ine  for  firmer  ones  ;  and  hard 
oaks,  elm,  beech,  &c,  for  the  more  compact  ones.  They  are  usually  driven  from 
about  2>^  to  4  ft  apart  each  way,  from  center  to  center,  depending  on  the  char- 
acter of  the  soil,  and  the  weight  to  be  sustained.  A  treaci-wlieel  is  more 
economical  than  the  winch  for  raising  the  hammer,  when  this  is  done  by  men. 
Morin  found  that  the  work  performed  by  men  working  8  hours  per  day,  was 
3900  foot-pounds  per  man,  per  minute  by  the  tread-wheel;  and  only  2600  by  a 
winch. 

After  piles  have  been  driven,  and  their  heads  carefully  sawed  off  to 
a  level,  if  not  under  water,  the  spaces  between  them  are  in  important  cases  filled 
up  level  with  their  tops  with  well  rammed  gravel,  stone 
spawls,  or  concrete,  in  order  to  impart  some  sustaining 
power  to  the  soil  between  the  piles.  Two  courses  of 
stout  timbers  (from  8  to  12  ins  square,  according  to  the 
weight  to  be  carried)  are  then  bolted  or  treen ailed  to 
the  tops  of  the  piles  and  to  each  other,  as  shown  in  the 
Fig,  forming  what  is  called  a  grillage.  On  top  of  these  is  bolted  a  floor  or 
platform  of  thick  plank  for  the  support  of  the  masonry  ;  or  the  timbers  of  the 
upper  course  of  the  grillage  may  be  laid  close  together  to  form  the  floor.  The 
space  below  the  floor  should  also,  in  important  cases,  be  well  packed  with  gravel, 
spawls,  or  concrete.  If  under  water,  the  piles  are  sawed  oflf  by  a  diver,  or 
by  a  circular  saw  driven  by  the  engine  of  the  pile-driver,  and  the  grillage  is 
omitted.  Instead  of  it  the  masonry  or  concrete  may  be  built  in  the  open  air  in 
a  caisson,  which  gradually  sinks  as  it  becomes  filled;  or  on  a  strong  platform 
which  is  lowered  upon  the  piles  by  screws  as  the  work  progresses.  Or  a  strong 
caisson  may  first  be  sunk  entirely  under  water,  and  then  be  filled  with  concrete, 
up  to  near  low  water;  the  caisson  being  allowed  to  remain.  Or  the  cnisson  may 
form  a  cofferdam,  to  be  first  sunk,  and  then  pumped  out.    If  the  ground  is  liable 


FOUNDATIONS.  591 

to  wash  away  from  around  the  piles,  as  in  the  case  of  bridge  piers,  &c,  defend  it 
by  sheet-piles,  or  rip- rap,  or  both. 

The  cost  of  a  floating-  steam  pile  driver,  scow  24  ft  by  50  ft,  draft 
18  ins,  with  one  engine  for  driving,  and  one  (to  ssve  time)  for  getting  another 
pile  ready  ;  with  one  ton  hammer,  is  about  ^6000  ;  and  $^500  more  will  add  a  cir- 
cular saw,  &c,  fur  sawing  off  piles  at  any  reqd  depth.  Requires  engineman,  cook, 
and  4  or  5  others.  Will  burn  about  half  a  ton  of  coal  per  day.  Driving  20  feet 
into  gravel,  and  sawing  off,  will  average  from  15  to  20  piles  per  day  of  10  hours. 
In  mud  about  twice  as  many.     On  land  about  half  as  many  as  in  water. 

In  the  g-unpowder  pile  driver  invented  by  the  late  Mr.  Thomas  Shaw, 
of  Philadelphia,  the  hammer  is  worked  by  small  cartridges  of  powdei',  placed  one 
by  one  in  a  receptacle  on  top  of  the  pile ;  and  exploded  by  the  hammer  itself. 
It  can  readily  make  30  to  40  blows  of  5  to  10  ft  per  minute;  and,  since  the 
hammer  does  not  come  into  actual  contact  with  the  piles,  it  does  not  injure  their 
heads  at  all;  thus  dispensing  with  iron  hoops,  &c,  for  preserving  them.  When 
only  a  slight  blow  is  required,  a  smaller  cartridge  is  used.  To  drive  a  pile  20  ft 
into  mud  averages  about  one-third  of  a  pound  of  powder  ;  into  gravel,  4  times  as 
much.  This  machine  does  not  assist  in  raising  the  pile,  and  placing  it  in 
position,  as  is  done  by  ordinary  steam  pile  drivers  ;  the  latter,  however,  average 
but  from  6  to  14  blows  per  minute. 

Piles  have  been  driven  by  exploding  small  charges  of  dynamite 
laid  upon  their  heads,  which  are  protected  by  iron  plates. 

Steam-hammer  pile  drivers,  operating  on  the  principle  of  that  devised 
by  Nasmyth  about  1850,  are  economical  in  driving  to  great  depths  in  difficult- 
soils  where  there  are  say  200  or  more  piles  in  clusters  or  rows,  so  that  the  machine 
can  readily  be  moved  from  pile  to  pile. 

The  steam  cylinder  is  upright,  and  is  confined  between  the  upper  ends  of  two 
vertical  and  parallel  I  or  channel  beams  about  6  to  12  ft  long  and  18  ins  apart, 
the  lower  ends  of  which  confine  between  tbem  a  hollow  conical  *^  bonnet  cast* 
ing","  which  fits  over  the  head  of  the  pile.  This  casting  is  open  at  top,  and  through 
it  the  hammer,  which  is  fastened  to  the  foot  of  the  piston-rod,  strikes  the  head  of 
the  pile.  Each  of  the  vertical  beams  encloses  one  of  the  two  upright  guide-timbers, 
or  "leaders,"  of  the  pile  driver,  between  which  the  driving  apparatus,  above  do. 
scribed,  is  free  to  slide  up  or  down  as  a  whole. 

When  a  pile  has  been  placed  in  position,  ready  for  driving,  the  bonnet  casting  is 
placed  upon  its  head,  thus  bringing  the  weight  of  the  beams,  cylinder,  hammer, and 
casting  upon  the  pile.  This  weight  rests  upon  the  pile  throughout  the  driving,  the 
apparatus  sliding  down  between  the  leaders  as  the  pile  descends. 

The  steam  is  convened  from  the  boiler  to  the  cyl  by  a  flexible  pipe.  When  it  ia 
admitted  to  the  cyl,  the  hammer  is  lifted  about  30  or  40  ins,  and  upon  its  escape  the 
hanamer  falls,  striking  the  head  of  the  pile.  About  60  blows  are  delivered  per  min- 
ute. The  hammer  is  provided  with  a  trip-piece  which  automatically  admits  steam 
to  the  cylinder  after  each  blow,  and  opens  a  valve  for  its  escape  at  the  end  of  the 
up-stroke.  By  altering  the  adjustment  of  this  trip-piece,  the  length  of  stroke  (and 
thus  the  force  of  the  blows)  can  be  increased  or  diminished.  The  admission  and 
escape  of  steam,  to  and  from  the  cyl,  can  also  be  controlled  directly  by  the  attendant. 
The  number  of  blows  per  minute  is  increased  or  diminished  by  regulating  the  sup' 
ply  of  steam. 

In  making  the  up-stroke,  the  steam,  pressing  against  the  lower  cyl  liead,  of  course 
presses  downward  on  the  pile  and  aids  its  descent. 

The  chief  advantage  of  these  machines  lies  in  the  great  rapidity 
with  which  the  blows  follow  one  another,  allowing  no  time  for  the  disturbed  earth, 
sand,  &c,  to  recompact  itself  around  the  sides,  and  under  the  foot,  of  the  pile.  This 
enables  the  machines  to  do  work  which  cannot  be  done  with  ordinary  pile  drivers. 
They  have  driven  Norway  pine  piles  42  ft  into  sand.  They  are  less  liable  than 
others  to  split  and  broom  the  pile,  so  that  these  may  i>e  of  softer  and  cheaper  wood. 
The  bonnet  casting  keeps  the  head  of  the  pile  constantly  in  place,  so  that  the  piles 
do  not  "  dodge  "  or  get  out  of  line.  Their  heads  have,  in  some  cases,  been  set  on  fire 
by  the  rapidly  succeeding  blows. 

These  machines  consume  from  1  to  2  tons  of  coal  in  10  hours,  and 
require  a  crew  of  5  men.  They  work  with  a  boiler  pressure  of  from 
50  to  75  lbs  per  sq  inch. 


592 


FOUNDATIONS. 


Rules  for  the  Susiaintng-  Power  of  Piles.  ' 

They  diSe:  very  much.  No  rule  can  apply  correctly  to  all  conditions.  The  ground  itself  bctweei 
the  piles,  in  most  cases,  supports  a  part  of  the  load ;  although  the  whole  of  it  is  usually  assigned  to 
the  piles.  Again,  in  very  clayey  soils,  there  is  greater  liability  to  sink  somewhat  with  the  lapse  of 
time,  in  consequence  of  the  admission  of  water  between  the  pile  and  the  clay  ;  thus  diminishing  tb» 
friction  between  them.  The  less  firm  the  soil,  the  more  will  the  piles  be  affocted  by  tremors ;  whie^ 
also  tend  in  time  to  cause  siuking.  In  some  cases  this  sinking  will  not  be  that  of  the  piles  settling 
deeper  into  the  earth  around  them  ;  but  that  of  the  entire  compacted  mass  of  piles  and  earth  int» 
which  they  were  driven,  settling  down  into  the  less  dense  mass  below  them.  Piles  are  sometimes 
blamed  for  settlements  which  are  really  due  to  the  crushing  (flatways)  of  the  timbers  which  rest 
immediately  upon  their  heads. 

lu  the  fine  LiOiiflon  bridg'e  across  the  Thames,  each  pile  under  some  of  the 
piers  sustains  the  very  heavy  load  of  80  tons.  They  are  driven  but  20  feet  into  the 
stiff,  blue  London  clay  ;  and  are  placed  nearly  4  ft  apart  from  center  to  center ;  which 
is  too  much  for  such  piers  and  arches.  At  3  ft  apart  scant,  they  would  have  had  but 
45  tons  to  sustain.  They  are  1  ft  in  diam  at  the  middle  of  their  length.  Ugly  set- 
tlements, some  of  them  to  the  extent  of  about  a  ft,  have  occurred  under  these  piers. 
Blaekfriars  bridgpe,  in  the  same  vicinity,  exhibits  the  same  defect.  By  some 
this  is  ascribed  in  both  cases  to  the  gradual  admission  of  vrater  between  the  clay  and 
the  piles,  perhaps  by  capillary  action  of  the  piles  themselves ;  or  perhaps  by  direct 
leaking.  It  may,  however,  be  owing  in  part  to  the  crushing  of  the  platforms  on 
top  of  the  piles ;  or  to  a  bodily  settlement  of  the  entire  mass  of  piled  clay,  into 
the  unpiled  clay  beneath,  under  the  immense  load  that  rests  upon  it.  This  here 
amounts  to  5}^  tons  per  sq  foot  of  area  covered  by  a  pier ;  and  is  probably  too  much 
to  trust  upon  damp  clay,  when  even  the  slightest  sinking  is  prejudicial. 

maj  J.  Sanders,  U.  S.  Engs,  experimented  largely  at  Fort  Delaware  in  river 
mud ;  and  gave  the  following  in  the  Jour.  Franklin  Inst,  Nov  1851.  For  the  safe 
load  for  a  common  wooden  pile,  driven  until  it  sinks  through  only  small  and 
nearly  equal  distances,  under  successive  blows,  divide  the  height  of  the  fall  in  ins, 
by  the  small  sinking  at  each  blow  in  ins.  Mult  the  quot  by  the  weight  of  the 
hammer,  ram,  or  monkey,  in  tons  or  pounds,  as  the  case  may  be.  Divide  the 
prod  by  8.    He  does  not  state  any  specific  coefficient  of  safety. 

Example.  At  the  CiieBtnnt  St  Bridce«  Philada,  the  greatest  weight  on  any  pile  is  18  tons. 
Mr  Kneass  had  the  piles  driven  until  they  sank  ^,  or  .73  of  an  inch  under  each  blow  from  a  1200  ft 
hammer,  falling  20  ft.  Was  he  safe  in  doing  so  7  Here  we  have  the  fall  in  ins  =  20  X  12  =  240.  And 
240  384000 

-rg-  =  320 ;  and  320  X  1200  =  384000  B)s ;  and  — g =  48000  fts,  =  21.4  tons  safe  load  by  Maj  San- 
ders' rule.     The  soil  was  river  mud. 

Our  own  rule  is  as  follows.    Mult  together  the  cube  rt  of  the  fall  in  ft ;  the  wt  of  hammer  in  lbs ; 

and  the  decimal  .023.     Divide  the  prod  by  the  last  sinking  in  ins.  -\-  1.     The  quotient  will  be  the 

extreme  load  that  will  be  just  at  the  point  of  causing  more  sinking.     For  the  safe  load  take  from 

ene  twelfth  to  one  half  of  this,  according  to  circumstances.    Or,  as  a  formula, 

Cube  rt  of  y  Wt  of  hammer  ^   q.,. 

Extreme  load  _  fall  in  feet  ^     in  pounds       ^  '      ' 

in  tons  ""  Last  sinking  in  inches  -f-  1 

Example.  The  same  as  the  foregoing  at  Chestnut  St  Bridge.  Here  the  cube  rt  of  20  ft  fan  is 
S.714  ft.     Hence  we  have 

Extreme  load  =  2.714  X  laoo  X  .023  ^  U^^  ^^^  ^^^^ 
in  tons  .75  + 1  1.75 

Or  say  half  of  this,  or  21.4  tons,  the  load  for  a  safety  of  2.  Major  Sanders*  rule  makes  the  safe 
load  21.4.     The  actual  one  is  18  tons. 

A  safety  of  2  is  not  enough  for  river  mud. 

But  although  Major  Sanders'  rule  and  our  own  agree  very  well  in  this  instance  if  a  safety  of  2  fee 
taken  for  each,  they  differ  widely  in  some  others.  Thus  at  Neullly  Qrldse*  France,  Perronet's 
heaviest  hammer  weighed  2000  fts,  fall  5  ft,  sinkage  .25  of  an  inch  in  the  last  16  blows ;  or  say  .016 
Inch  per  blow.  The  piles  sustain  47  tons  each.  Our  rule  gives  38.8  tons  for  a  safety  of  2 ;  while  San- 
ders' rule  gives  515  tons  safe  load !  If,  as  we  think  probable,  there  was  no  actual  sinking  at  the  last 
blow,  then  our  rule  gives  39.3  tons  for  a  safety  of  2  ;  while  Sanders'  gives  infinity. 

At  the  Hull  Docks,  England,  piles  10  ins  square,  driven  16  ft  into  alluvial  mud.  by  a  1500  E>  ham- 
mer,  falling  '24  ft,  sank  2  ins  per  blow  at  the  end  of  the  driving.  They  sustain  at  least  20  tfons  each, 
or  according  to  some  statements  25  tons.  Oar  rule  gives  33.2  tons  for  the  extreme  load ;  or  16.6  for  f 
safety  of  only  !.  Sanders  gives  for  safety  12.06  tons.  As  before  remarked,  2  is  not  safety  enough  loi,' 
jn.ud.     In  mud,  it  is  not  primarily  the  piles,  but  the  piled  goU  that  settles,  bodily,  for  years. 

At  the  Boyal  Border  Bridge,  Ei>gland,  piles  were  very  firmly  driven  from  30  to  40  ft  in  s&n^ 
and  gravel,  in  some  ca.ses  wet.  Pine  was  first  tried,  but  it  split  and  broomed  so  badly  under  the  hard 
driving,  that  American  elm  was  substituted,  with  success.  They  were  driven  until  they  sank  but  .05 
inch  per  blow,  under  a  1700  ft  monkey,  falling  16  ft.  They  support  70  tons  each.  Our  rule  gives  47 
tons  for  a  safety  of  2  ;  while  Sanders  gives  364  tons  safe  load  t 

Itisthe  writer's  opinion,  however,  that  the  piles  did  not  aciually  sink,  as  was  (and  always  is,  in 
such  cases)  taken  for  granted  by  the  observers  ;  but  that  they  were  merely  compressed  or  partially 
crushed  by  overdriving.  Most  of  the  piles  were  driven  until  they  sank  (?)  only  an  inch  under  150 
blows ;  but  we  doubt  whether  they  were  any  safer,  or  farther  in  the  ground,  than  when  they  had  re- 
ceived only  one  of  them  ;  and  consider  such  extreme  precaution  wor.se  than  useless. 

In  gome  experiments  (1873)  at  Philada,  a  trial  pile  was  driven  15  ft  into  soft  river  mud,  by  a 
1600  ft  hammer  ;  its  last  sinking  being  18  ins  under  a  fall  of  36  ft.  Only  5  hours  after  it  was  driven 
It  was  loaded  with  6.5  tons  ;  which  caused  a  sinking  of  but  a  very  small  fraction  of  an  inch.    0"»-  rule 


FOUNDATIONS.  593 

?ive,'?  6.4  tons  as  the  extreme  load.  Uuder  9  tons  it  sank  .75  of  an  inch  ;  ajd  under  15  tons,  5  ft.  By 
Maj  Sauders'  rule  its  safe  load  would  be  2.14  tons. 

A  U.  S.  CJovt  trial  pile,  about  12  ins  sq,  driven  29  ft  through  layers  of  silt,  sand,  and  clay,  ham- 
mer 910  lbs,  fall  5  ft,  last  sinking  .375  of  an  inch,  bore  26.6  tons ;  but  sank  slowly  under  27.9  tons. 
Our  rule  gives  26  tons  extreme  load. 

Freneh  eiie:lueerg  consider  a  pile  safe  for  a  load  of  25  tons,  when  it  is  driven  to  the  refusal  of 
1344  as,  falling  4  ft ;  our  rule  gives  24.2  tons  for  safety  2.  They  estimate  the  refusal  by  its  not  sink- 
ing more  than  .4  of  an  inch  under  30  blows.  In  many  important  bridges  &c  they  drive  untU  there  is 
ID  "sinking  under  an  800  tt>  hammer,  falling  5  ft.  Our  rule  here  gives  31.5  tons  extreme  load  ;  or  15.7 
'or  safety  2. 

As  to  the  proper  load  for  safety,  we  think  that  not  more  than  one-half  the  extreme  load  given 
ay  our  rule  should  be  taken  for  piles  thoroughly  driven  in^rm  soils;  nor  more  than  one-sixth  when 
in  river  mud  or  marsh  ;  assuming,  as  we  have  hitherto  done,  that  their  feet  do  not  rest  upon  rock. 

If  liable  to  tremors,  take  only  half  these  loads. 

Piles  may  be  made  of  any  required  size  as  regards  either  length  or  crosa  section,  by  bolt- 
ing and  fishing  together  sidewise  and  lengthwise,  a  number  of  squared  timbers. 

Piles  with  blunt  ends.  At  South  Sfeet  Bridge,  Phila,  1200  stout  piles  of  Nova  Scotia  spruce 
with  blunt  ends  were  driven  15  to  35  ft,  partly  in  strong  eravel,  by  a  common  steam  pile  driver,  at  a 
total  cost  (piles  and  driving)  of  S7  to  $8  each.  At  Wilmington  Harbor,  Cal,  Mr.  C.  B.  Sears,  . 
(J.  S.  Army.  (Jour.  Am.  Soc.  C.  E.,  Dec  1876)  found  that  in  firm  compact  wet  sand,  after  the  first  few 
blows  the  piles  would  not  penetrate  more  than  .5  to  1.5  ins  at  a  blow,  no  matter  how  far  the  2400  ft 
hammer  fell.  The  unpointed  ones  of  which  there  were  many  thousands,  drove  quite  as  readilv  to  aver- 
age  depths  of  15  ft  in  this  sand  as  the  pointed  ones,  and  with  much  less  tendency  to  cant,  'as  a  high 
Fall  had  no  farther  effect  than  to  batter  the  heads  he  reduced  it  to  10  ft.  which  drove  an  average  of 
about  .72  inch  to  a  blow.  To  insure  straight  driving,  the  ends  must  be  at  right  angles  to  the  length. 
Instead  of  driving  piles  to  moderate  depths  it  mav  at  times  be  better  to  merel}  plant  them  butt 
lown  in  holes  bored  by  an  auger  like  Pierce's  Well  Borer. 

The  ultimate  friction  of  piles  even  with  the  bark  on,  and  driven  about  3  ft  apart  fiom  cen 
to  cen  probably  never  much  exceeds  about  1  ton  per  sq  ft  even  when  well  driven  into  dense  moist 
sand  or  loamy  gravel ;  nor  more  than  .5  to  .75  of  a  ton  in  common  soils  and  clays;  or  than  .1  to  .2 
af  a  ton  in  silt  or  wet  river  mud  dependine  on  the  depth  and  density. 

The  frletion  of  cast  Iron  cylinders  seems  to  be  about  .3  that  of  piles. 

There  is  a  great  difference  in  the  penetrability  of  different 

lands.  Thus,  in  the  Lary  bridge,  no  special  difficulty  was  found  in  driving  piles  35  tt  into  deep  wet 
land ;  while,  in  other  wet  localities,  piles  of  very  tough  wood,  well  shod  with  iron.  CHnuot  be  driven 
5  ft  into  sand,  without  being  battered  to  pieces.  The  same  difference  has  been  found  in  the  case  of 
icrew-piles.  At  the  Brandywine  light-house  these  could  not  be  forced  more  than  10  ft  into  the  clean 
wet  sand.  Stiff  wet  clay  (and  clean  gravels)  also  differ  very  much  in  this  respect.  Generally  they 
ire  penetrable  to  any  required  depth  with  comparative  ease ;  but  we  have  seen  stout  hemlock  piles 
Mattered  to  pieces  in  driving  6  ft  through  wet  gravel ;  and  Mr.  Rendel  found  that  at  Plymouth  he 
'could  not  by  any  force  drive  screw-piles  more  than  about  5  ft  into  the  clay,  which  is  not  as  stiff  as 
the  London  clay,"  on  which  the  forementioned  new  London  and  Blackfriars  "bridges  were  founded; 
uad  into  which  even  ordinary  wooden  piles  were  driven  20  ft  without  special  difficulty. 

A  mixture  of  mud  with  the  sand  or  gravel  facilitates  driving  very  much  ;  but  before  beginning  an 
ixtensive  system  of  piling,  a  few  experimental  ones  should  be  driven,  to  remove  doubt  as  to  the 
rouble  and  expense  that  may  be  anticipated.   Mere  boring  will  often  be  but  a  poor  substitute  for  this. 

Afl  a  general  rule,  a  heavy  hammer  with  a  low  fall,  drives  more  pleasantly  than  a  light  one  with  a 
ligh  fall.  Where  a  hammer  of  %  ton  (1500  fts)  falling  25  ft,  in  a  very  strong  ground,  shattered  the 
)iles;  one  of  2  tons,  (4500  fts,)  with  7  ft  fall,  drove  them  satisfactorily.  More  blows  can  be  made  in 
he  same  time  with  a  low  fall ;  and  this  gives  less  time  for  the  soil  to  compact  itself  around  the  piles 
)etween  the  blows.  At  times  a  pile  -may  resist  the  hammer  after  sinking  some  distance  j  but  start 
igain  after  a  short  rest;  or  it  may  refuse  a  heavy  hammer,  and  start  under  a  lighter  one.  It  may 
Irive  slowly  at  first,  and  more  rapidly  afterward,  from  causes  that  may  be  difficult  to  discover.  The 
Iriving  of  one  sometimes  causes  adjacent  ones  previously  driven,  to  spring  upward  several  feet.  A 
aile  is  in  the  most  favorable  position  when  its  foot  rests  upon  rock,  after  its  entire  length  ha.s  been 
Iriven  through  a  firm  soil,  which  affords  perfect  protection  against  its  bending  like  an  overloaded 
K»lumn;  and  at  the  same  time  creares  great  friction  against  its  sides;  thus  assisting  much  in  sus- 
laining  the  load,  and  thereby  relieving  the  pressure  upon  the  foot.  A  pile  may  rest  upon  rock,  and 
fet  be  very  weak  ;  for  if  driven  through  very  soft  soil,  all  the  pressure  is  borne  by  the  sharp  point; 
md  the  pile  becomes  merely  a  column  in  a  worse  condition  than  a  pillar  with  one  rounded  end.  Se« 
L  tg  1.  Strength  of  Iron  Pillars.    In  such  soils  the  piles  need  very  little  sharpening;  indeed, 

had  better  be  driven  without  any ;  or  even  butt  end  down. 

The  driving  of  a  pile  in  soft  ground  or  mud  will  generally  cause  an  adjacent  one  previously  driven, 
bo  lean  outwards  unless  means  be  taken  to  prevent  it. 

In  piling  an  area  of  firm  soil  it  is  best  to  begin  at  its  center  and  work  outwards ;  otherwise  the  soil 
may  become  so  consolidated  that  the  central  ones  can  scarcely  be  driven  at  all. 

£la*^tic  reaction  of  the  soil  has  been  known  to  cause  entire  piled  areaa 
to  rise,  together  with  the  piles,  before  they  were  built  upon. 

In  very  firm  soil,  especially  if  stony;  or 
even  in  soft  soil,  if  the  piles  are  pointed,  and 
are  to  be  driven  to  rock;  their  feet  should 
be  protected  by  shoes  of  either  wrought 
iron,  as  at  a,  .<j,and  h,  Figs  13 ;  spiked  to  the 
pile  by  means  of  the  iron  straps  n,  forged 
to  them ;  or  of  cast  iron,  as  at  c,  where  the 
shoe  is  a  solid  inverted  cone,  the  wide  flat 
base  of  which  affords  a  good  bearing  for  the 
flat  bottom  of  the  pile-point.  The  dotted 
line  is  a  stout  wrought-iron  spike,  well  se- 
cured in  the  cone,  which  is  cast  around  it; 
this  holds  the  shoe  to  the  pile.  Eegular 
38 


594  FOUNDATIONS. 


wrirjught-iron  shoes  will  generally  weigh  18  to  30  lbs ;  but  sheet  iron  may  be  used  when  the  soil  is  but 
moderately  compact;  plate  iron  when  more  bo  ;  and  solid  iron  or  steel  points,  from  2  to  4  ins  square 
at  the  butt,  aud  4  to  8  ins  long,  when  rery  compact  and  stony.  Iffole!<»  may  be 
drilled  in  rock  for  receiving  the  points  of  piles,  and  thus  preventing 'them 
from  slipping  ;  by  first  driving  down  a  tube,  as  a  guide  to  the  drill,  after  the  earth  is  cleaned  out  of 
the  tube.  To  preserve  the  heads  to  some  extent  from  splitting  under  the 
blows  of  the  hammer,  they  are  usually  surrounded  by  a  hoop  h,  Fig  d;  from  ,^  to  1  inch  thick ;  and 
114  to  3  ins  wide.  These  are,  however,  sometimes  but  imperfect  aids  ;  for  in  hard  driving  the  hekd 
will  crush,  split,  and  bulge  out  on  all  sides,  frequently  for  many  feet  below  the  hoop :  moreover,  tht 
hoops  often  split  open.  The  heads,  therefore,  often  have  to  be  sawed,  or  pared  off  several  times 
before  the  pile  is  completely  driven  ;  and  allowance  must  be  made  for  this  loss  in  ordering  piles  for 
any  given  work;  especially  in  hard  soil.  Capt  Turnbull,  U  S  top  Eng,  states  that  at  the  Potomac 
aqueduct,  his  pileheads  were  preserved  from  injury  by  the  simple  expedient  of  dishing  them  out  to  a 
depth  of  about  an  inch,  and  covering  them  by  a  loose  plate  of  sheet  iron ;  as  shown  in  section  at  c, 
Figs  13.  A  very  slight  degree  of  brooming  or  crushing  of  the  head,  materially  diminishes  the  force 
of  the  ram.  Piles  may  be  driven  through  small  loose  rubble  without  much  labor.  Shaw's  driver 
does  not  injure  the  heads.     Piles  which  foot  on  sloping  rock  may  slide  when  loaded. 

To  drive  a  pile  head  below  water  a  wooden  punch,  or  follower,  as 

.  at  p,  Figs  13,  may  be  used.  The  foot  of  this  punch  fits  into  the  upper  part  of  a  casting  //,  round  or 
square,  according  to  the  shape  of  the  pile  ;  and  having  a  transverse  partition  o  o.  The  lower  part 
of  the  casting  is  fitted  to  the  head  of  the  pile  t;  and  the  hammer  falls  on  top  of  the  punch.  When 
driving  piles  vertically  in  very  soft  soil,  to  support  retaining-walls,  or  other  structures  exposed  to 
horizontal  or  inclined  forces,  care  must  be  taken  that  these  forces  do  not  push  over  the  piles  them- 
selves ;  for  in  such  soils  piles  are  adapted  to  resist  vertical  forces  only,  unless  they  be  driven  at  an 
inclination  corresponding  to  the  oblique  force. 

A  broken  pile  may  be  drawn  ont,  or  at  least  be  started,  if  not  very 

firmly  driven,  by  attaching  scows  to  it  at  low  water,  depending  on  the  rising  tide  to  loosen  it.  Or  a 
long  timber  may  be  used  as  a  lever,  with  the  head  of  an  adjacent  pile  for  its  fulcrum.  Or  a  crab 
worked  by  the  engine  of  the  pile  driver.  In  very  difficult  cases  the  method  devised  by  Mr  J.  Monroe, 
C  E,  may  be  used.  A  4  inch  gas  pipe  15  ft  long,  shod  with  a  solid  steel  point,  and  having  an  outer 
shoulder  for  sustaining  a  circular  punch,  was  thereby  driven  close  to  and  2  or  3  ft  deeper  than  two 
piles  driven  12  ft,  in  37  ft  water,  and  broken  off  by  ice.  Four  pounds  of  powder  were  then  deposited 
in  the  lower  end  of  the  pipe,  and  exploded,  lifting  the  piles  completely  out  of  place.  It  will  often  b« 
best  to  let  a  broken  pile  remain,  and  to  drive  another  close  to  it.    May  be  drawn  by  hydranlio  press. 

Ice  adheres  to  piles  with  a  force  of  about  30  to  40  lbs  per  sq  inch,  and  in 

rising  water  may  lift  them  out  of  place  if  not  sufficiently  driven. 

Iron  piles  and  cylinders.  Cast  iron  in  various  shapes  has  been  much 
used  in  Europe  for  sheet  piles:  especially  when  intended  to  remain  as  a  facing  for  the  protection  of 
concrete  work,  filled  in  behind  and  against  them.*  Cast-iron  cylinders,  open  at  both  ends,  may  be 
used  as  bearing  piles ;  and  may  be  cleaned  out,  and  filled  with  concrete,  if  required.  The  friction  in 
driving  is  greater  than  in  solid  piles,  inasmuch  as  it  takes  place  along  both  the  inner  and  the  outei 
surfaces.  This  may  be  diminished  by  gradually  extracting  the  inside  soil  as  they  go  down.  Thev 
require  much  care,  and  a  lighter  hammer,  or  less  fall  than  wooden  ones,  to  prevent  breaking  ;  to 
which  end  a  piece  of  wood  should  be  interposed  between  the  hammer  and  the  pile  ;  or  the  ram  may  b« 
of  wood.  But  it  is  better  to  use  them  inthe  shape  of  screw  cylin<lers,  which, 
moreover,  gives  them  the  advantage  of  a  broad  base,  as  in  the  following. 

Brunei's  process.  He  experimented  with  an  open  cast-iron  cylinder,  3  ft 
outer  diam  ;  1}4  ins  thick;  in  lengths  of  10  ft,  connected  together  by  internal  socket  and  joggle  joints, 
sscured  by  pins,  and  run  with  lead.  It  had  a  sharp-edged  hoop  pr  cutter  at  bottom :  and  a  little 
above  this,  one  turn  of  a  screw,  with  a  pitch  of  7  ins,  and  projecting  one  foot  all  around  the  outsid* 
of  the  cylinder.  By  means  of  capstan  bars  and  winches,  he  screwed  this  down  through  stiff  clay  anq 
sand,  58  feet  to  rock,  on  the  bank  of  a  river.  In  descending  this  distance  the  cylinder  made  143 
revolutions  ;  sinking  on  an  average  about  5  ins  at  each.  The  time  occupied  in  actually  screwing  wai 
48J4  hours;  or  about  ly^  ft  per  hour.  There  were,  however,  many  long  intervals  of  rest  for  clean, 
ing  away  the  soil  in  the  inside.  After  resting,  there  was  no  great  difficulty  in  restarting.  The  next 
fig  will  give  an  idea  of  the  arrangement  of  the  screw. 

The  screw-pile  of  Alex.  Mitchell,  Belfast,  consists  usually  of  a  rolled  iroi) 
shaft  A,  Figs  14,  from  3  to  8  ins  diam;  and  having  at  its  foot  a  cast-iron  screw 
S  S  S,  with  a  blade  of  from  18  ins  to  5  ft  diam.  The  screws  used  for  light-houses, 
exposed  to  moderate  seas,  or  heavy  ice-fields,  are  ordinarily  about  3  ft  diam,  hav« 
1^  turns  or  threads,  and  weigh  about  600  lbs.  The  round  rolled  shafts  are  from 
5  to  8  ins  diam.  They  are  screwed  down  from  10  to  20  ft  into  clay,  sand,  or  coral,  by 
about  30  to  40  men,  pushing  with  6  to  8  capstan  bars,  the  ends  of  which  describe  s^ 
circle  of  about  30  to  40  ft  diam.  For  this  purpose  a  platform  on  piles  has  frequently 
to  be  prepared.  In  quiet  water,  this  may  be  supported  on  scows ;  or  a  raft  well 
moored  may  be  used  when  the  driving  is  easj' ;  or  the  deck  of  a  large  scow  with  ^ 
well-hole  in  the  center  for  the  pile  to  pass  thx'ough.  Roughly  made  temporary 
cribs,  filled  with  stone  and  sunk,  might  support  a  platform  in  some  positions.  The 
platform  must  evidently  be  able  to  i-esist  revolving  horizontally  under  the  great 
pushing  force  of  the  men  at  the  capstan  bars ;  and  on  this  account  it  is  difficult 
to  drive  screws  to  a  sufficient  depth,  in  clean  compact  sand,  by  means  of  a  floating 
platform.    The  feet  of  the  piles  must  be  firmly  secured  to  the  screws,  to  prevent 

*  Cast  iron,  intended  to  resist  sea-vrater,  should  be  close-grained, 

hard,  white  metal.  In  such,  the  small  quantity  of  contained  carbon  is  chemically  combined  with  the 
metal ;  but  in  the  darker  or  mottled  irons  it  is  mechanically  combined,  and  such  iron  soon  becomes 
soft,  (somewhat  like  plumbago,)  when  exposed  to  sea-water.  Hard  white  iron  has  been  proved  to 
resist  for  at  least  40  years  without  any  deterioration  ;  whether  constantly  under  water,  or  alternately 
wet  and  dry.  Copper  and  bronze  are  but  slightly  and  superficially  affected  by  sea-water ;  but  destruc- 
tive galvanic  action  takes  place  if  diff  metals  are  in  contact. 


FOUNDATIONS. 


595 


Figs  14 


their  being  lifted  out  of  them  bj'  the  upward  force  of  waves  against  the  super* 
structure.  At  y  p,  Figs  14,  is  shown  a  mode  of  splicing  or  uniting  the  different 
lengths  or  sections  of  a  pile.  The  point  of  junction  is  at  v;  rr  is  a  stout  iron  ring 
forged  on  to  the  lower  pile  p,  y^ 

about  a  foot  or  18  ins  below  its 
top  V.  A  strong  cylindrical  cast- 
ing n  n,  enclosing  the  ends  of 
the  sections,  rests  on  this  ring, 
and  is  pinned  through  the  piles, 
as  at  tt.  On  this  casting  are 
also  cast  projections  ccc,  for  at- 
taching rods gg,  and  beams  i,  &c, 
necessary  for  bracing  the  struc- 
ture from  pile  to  pile.  The  time 
actually  required  for  driving  a 
•crew  is  from  2  to  10  hours,  in 
favorable  circumstances. 

At  the  Brandywine  lighthouse,  on 
aaand-bank  of  very  pure  sand,  cov- 
ered 6  or  8  ft  at  low  water,  and  from 
11  to  13  ft  at  high,  they  could  not  be 
forced  down,  from  a  fixed  platform, 
for  more  than  10  ft.  At  other  places  20  ft  in  sand  is  reached  without  much  trouble,  where  the  «and 
contains  a  good  deal  of  mud,  but  its  bearlnjs  power  is  then  less.  This  (ultimate)  ranges  between 
about  1  and  6  tons  per  sq  ft  according  to  purity,  depth,  compactness,  &c,  of  the  sand.  In  important 
cases  the  bearing  power  should  be  tested. 

Mitchell's  piles  have  been  screwed  about  40  feet«into  a  mixture  of  clay  and  sand,  with  screws 
4  ft  diam.  They  pass  through  small  broken  stone  and  coral  rock  without  much  difficulty  ;  and  will 
push  aside  bowlders  of  moderate  size.  Ordinarily,  clay  or  sand  will  present  no  great  obstruction; 
but  occasionally  either  of  them  will  do  so.  Perfectly  pure  clean  sand,  as  a  general  rule,  gives  most 
diflSculty.  At  the  Brandywine  shoal  the  driving  was  aided  by  a  spur  and  pinion  placed  as  low  as  the 
water  permitted  ;  and  the  levers  were  worked  by  30  men.  The  danger  #f  twisting  off  the  shaft  is 
the  limit  for  screwing  them.  They  are  much  used  for  the  anchoring  of  chains  for  mooring  buoys,  &c. 
On  land,  small  screws,  with  short  hollow  shafts,  make  good  durable  supports  for  depot  pillars,  cranes, 
wooden  telegraph  poles,  station  signals  in  marine  surveying,  &c,  <fec.  They  can  readily'  be  unscrewed 
for  removal.  Horses  or  oxen  may  be  used  in  driving  large  screws.  The  Brandywine  light-house 
■tands  on  9  screw-piles,  which  are  surrounded  by  30  others  of  5  ins  diam,  as  fenders.  They  have  to 
resist  not  only  moderate  seas,  but  immense  fields  of  floating  ice,  miles  in  extent.  An  u'nflnishe4 
structure  was  destroyed  by  ice,  which  at  times  injures  the  bracing  of  the  standing  one. 

Test  boi'lns-s  s^tioiilil  be  maile  to  ensure  that  the  screws  do  not  stop  just 
above  a  very  weak  stratum  which  may  endanger  their  bearing  power.       So  with  any  piles. 

By  means  of  a  jet  of  water  forcibly  impelled  through  a  tube  by  a  force 
pump,  the  most  obstinate  sands  (but  not  stiflf  clay  or  cemented  gravel)  will  be  loosened,  and  the  sink- 
ing of  screw-piles,  or  wooden  ones,  or  even  the  largest  cylindeis,  be  greatly  facilitated.  In  a  govern- 
ment pier  at  Cape  Heiilopeii  in  very  compact  sand,  in  which  6  out  of  7 
screws  previously  broke  before  Reaching  10  ft,  the  use  of  the  jet  was  found  to  remove  more  than 
three-fourths  of  the  resistance.*  The  pilep  to  be  sunk  ha^ng  first  been  placed  in  position  as  in  Plfc 
15,  the  lower  open  ends  1 1  of  a,  bent  iron  tube  t  s  t  of  one  and  a 
quarter  ins  bore  were  stood  upon  the  unper  face  of  the  screw  disk,  and 
there  held  firmly  by  3  or  4  men  while  the  pile  was  being  screwed  down 
by  the  capstan  c,  which  was  worked  by  a  leading  rope  r.  From  the 
bend  a  of  the  pipe,  a  hose  h,  2  ins  diam.  led  to  the  force  pump,  the 
cylinder  of  which  was  5  ins  bore,  and  9  ins  stroke,  and  worked  about 
80  full  strokes  per  minute,  by  a  mule  walking  on  a  tread  wheel  on  a 
floating  platform/.  There  was  now  no  trouble  in  screwing  the  piles  to 
any  required  depth.  Previous  trials  by  playing  the  jet  beneath  the  disk 
gave  unsatisfactory  results. 

In  mobile  Bay  several  thousands  of  wooden  piles, 
from  18  to  48  ins  diam,  were  sunk  from  10  to  20  ft  into  obstinate  sand, 
at  the  average  sinking  rate  of  about  1  ft  per  second,  entirely  by  means 
of  jets.  The  jet  was  propelled  by  a  city  steam  fire  engine,  on  a  steam- 
boat, through  its  own  hose,  with  a  one  and  a  quarter  inch  nozzle. 
During  the  descent  the  nozzle  n  n  was  held  loosely  in  its  place  near 
the  foot  of  the  pile,  by  two  staples  a  s  and  by  a  string  t  reaching  to  the 
surface.  The  piles  were  suspended  by  their"  heads  from  shears,  by  the 
tackle  of  which  their  descent  was  regulated.  The  sand  settled  firmly 
around  the  piles  in  a  few  minutes  after  they  were  suok.t 

At  Tensas  River,  Alabama,  for  iron  cylinders  6  ft  diam 
(enclosing  piles,  in  deep  light  shifting  sand,  the  jet  was  forced  by  a  small 

rotary  pump  of  200  to  300  revolutions  per  minute,  through  a  canvas  hose  3  ins  diam, 
into  a  central  conical  cast  iron  vessel  10  ins  diam,  from  which  radiated  12  gas  pipes  1 
inch  diam,  and  about  30  ins  long.  At  the  outer  end  of  each  of  these  radii  was  an 
•Ibow  to  which  was  attached  a  long  vertical  pipe  reaching  down  into  the  cylinder, 
and  made  in  10  ft  lengths  with  screw  ends  for  prolonging  them  as  the  cylinder  went 
down.  This  apparatus  was  raised  and  lowered  by  a  light  block  and  line;  and  by  it 
alone  each  cylinder  was  sunk  about  16  ft  into  the  light  sand  in  a  few  hours.} 


M^/5  hn. 


*  Report  Sec  of  War  1872.  t  John  W.  Glenn,  C  E,  Van  Nostrand,  June  1874. 

1  Gabriel  Jordan,  C  E ;  Trans  Am  Soc  C  K,  Feb  1874. 


596  FOUNDATIONS. 


At  the  lievan  Tiadnct,  Mr  James  Brnnlee,  England,  in  a  light 

•andy  marl  of  great  depth,  sunk  hollow  cast  iron  cylinders  of  10  ins  outer  diam,  to  a  depth  of  20  ft, 
by  means  of  a  jet  pipe  2  ins  diam  passing  down  inside  of  the  cylinder,  and  through  a  hole  in  its  base, 
■which  was  a  cast  iron  disk  30  ins  diam,  and  1  inch  thick,  strengthened  by  outside  flanges.  The  con- 
necting flanges  of  the  cylinder  sections  are  outside,  thus  impeding  the  descent,  as  did  also  the  broad 
bottom  disk  ;  still  3  or  4  hours  usually  sufficed  for  the  sinking  of  each,  to  20  ft  depth.  Actual  trial 
showed  that  their  safe  sustaining  power  was  about  5  tons  per  sq  ft  of  bottom  disk. 

At  liOck  Ken  viadnct  each  pier  consists  of  two  cylinders,  open  at  both 
ends;  of  cast  iron,  8  ft  in  diam ;  l]/g  ins  thick ;  in  lengths  of  6  ft,  weighing  4  tons 
each ;  and  bolted  together  by  inside  flanges,  with  iron  cement  between  them.  The 
cylinders  stand  8  ft  apart  in  the  clear ;  and  are  in  36  ft  water.  "  A  strong  staging 
was  erected;  and  4  guide-piles  driven  for  each  cylinder.  The  several  lengths  being 
previously  bolted  together,  these  were  lowered  into  their  places.  Each  cylinder  sank 
bj'  its  own  weight  one  or  two  ft  through  the  top  mud,  and  then  settled  upon  the  sand 
and  gravel  which  form  the  substratum  for  a  great  depth.  Into  this  last  they  were 
sunk  about  8  or  9  ft  farther,  by  excavating  the  inside  earth  under  water,  by  means 
of  an  inverted  conical  screw-pan,  or  dredger,  of  l^  inch  plate  iron.  This  was 
2  ft  greatest  diam,  and  1  ft  deep ;  and  to  its  bottom  was  attached  a  screw  about  1  ft 
long,  for  assisting  in  screwing  it  down  into  the  soil.  Its  sides  had  openings  for  the 
entrance  of  the  soil ;  and  leather  flaps,  opening  inward,  to  prevent  its  escape.  From 
opposite  sides  of  the  pan,  3  rods  of  %  inch  diam  projected  upward  4  feet,  and  were 
there  forged  together,  and  connected  by  an  eye-and-bolt  joint  to  a  long  rod  or  shaft, 
at  the  upper  end  of  which  was  a  four-armed  cross-handle,  by  which  the  pan  wa« 
screwed  down  by  4  men  on  the  staging." 

""When  a  pan  was  full,  a  slide  which  passed  over  the  joint  at  the  bottom  was  lifted;  and  the  pan 
was  raised  by  a  tackle.  This  pan  raised  about  1  cub  ft  at  a  time.  A  smaller  one  of  only  1  ft  diam, 
and  1  ft  deep,  raising  about  34  cub  ft,  was  used  when  the  material  was  very  hard.  By  this  means 
the  cylinders  were  sunk  at  the  rate  of  from  2  to  18  ins  per  day.  The  slow  rate  of  2  ins  was  caused 
by  stones,  some  of  them  of  50  fts.  These  were  first  loosened  by  a  screw-pick,  which  was  a  bar  of 
iron  3  ft  long,  with  circular  arms  12  ins  long  projecting  from  the  sides.  After  being  loosened  by  this, 
the  stones  were  raised  by  the  pan.  The  expense  of  all  this  apparatus  was  very  trifling;  aud  the  ex- 
cavation was  done  easily  aud  cheaply.  After  the  excavation  was  finished,  and  the  cylinder  sunk, 
before  pumping  out  the  water,  concrete  (gravel  2,  hydraulic  cement  1  measure)  was  filled  in  to  the 
depth  of  12  feet,  by  means  of  a  large  pan  with  a  movable  bottom ;  and  about  12  days  were  left  it  to 
harden.  The  water  was  then  pumped  out,  and  the  masonry  built  in  open  air.  In  some  of  the  cylin- 
ders, however,  the  water  rose  so  fast,  notwithstanding  the  12  ft  of  concrete,  that  the  pumps  could  not 
keep  them  clear ;  and  6  ft  more  of  concrete  had  to  be  added  in  those.  Finally  random-stone,  or  rough 
dry  rubble,  was  thrown  in  around  the  outsides  of  the  cylinders,  to  preserve  them  from  blows  and 
DQdermining."  *    The  masonry  extends  20  ft  above  the  cylinders,  and  above  water. 

Tlie  vacuum  and  plenum  processes.  We  can  barely  allude  to 
the  general  principles  of  these  two  modes  of  sinking  large  hollow  iron  cylinders.  In 
the  vacuum  process  of  Dr.  Lawrence  Holker  Potts,  of  London,  the  cylinder 
c.  Fig  16,  while  being  sunk,  is  closed  air-tight  at  top,  by  a 
trap-door,  opening  upward.  A  flexible  pipe  p,  of  India- 
rubber,  Jong  enough  to  adapt  itself  to  the  sinking  of  the 
cylinder,  and  provided  with  a  stopcock  s,  leads  from  the 
cylinder  to  a  vessel  v ;  which  may  be  placed  on  a  raft,  or  a 
scow,  or  on  land,  as  may  suit  circumstances.  The  cylinder 
being  first  stood  up  in  position,  as  in  the  fig,  the  water  is 
JPj^qj[g  pumped  out,  and  the  interior  soil  removed  if  the  cylinder 

•*  has   sunk  some  distance  by  its  own  weight.    The  cock 

s  is  then  closed,  and  the  air  is  drawn  out  from  the  vessel  v 
ky  an  air-pump.  The  cock  is  then  opened,  and  most  of  the  air  in  the  cylinder  rushep 
Into  the  void  veysel  v ;  thus  leaving  the  cylinder  comparatively  empty,  and  therefore 
less  capable  of  resisting  the  downward  pressure  of  the  external  air  upon  its  top 
This  pressure,  as  is  well  known,  amounts  to  nearly  15  Bbs  on  every  sq  inch ;  or  nearly 
1  ton  per  sq  ft  of  area  of  the  top.  Consequently  the  cylinder  is  forced  downward  in 
the  bed  of  the  river,  by  this  amount  of  pressure,  in  addition  to  its  own  weight.  At 
the  same  time,  the  pressure  of  the  air  upon  the  surface  of  the  water  is  transmitted 
through  the  water  to  the  soil  around  the  open  foot  of  the  cylinder ;  so  that  if  this 
soil  be  soft  or  semi-fluid,  it  will  be  pressed  up  into  the  nearly  void  cylinder,  in  which 
is  no  downward  pressure  to  resist  it.  The  descent  varies  from  a  few  inches,  to  4  or  5 
ft  each  time.  The  process  is  then  repeated,  by  admitting  air  again  into  the  cylin- 
der, opening  the  trap-door,  removing  the  water  and  soil,  as  before,  &c.  Additional 
lengths  of  cylinder  may  be  bolted  on,  by  means  of  interior  flanges. 

It  is  a<lapted  only  to  soft  soils,  and  to  wet  sandy  ones ;  but  is  not  sufficient- 
ly powerful  in  very  compact  ones  ;  nor  does  it  answer  where  obatructioas  from  bowlders,  logs,  &c,  occur; 

•Hollow  Iron  Piles  either  cast  or  wrought  with  solid  pointed  feet,  to  be  driven  by  the  hammer 
falling  iJiside  of  them  and  striking  against  the  top  of  the  solid  foot,  are  a  recent  device  of  great  use  io 
many  cases.  The^  are  made  in  sections  of  which  enough  can  be  gradually  united  to  reach  any 
required  depth.  They  avoid  the  danger  of  bending  which  attends  striking  the  top.  The  iron  feet  are 
•welled  outwardly  a  little  to  diminish  earth-friction  against  the  pile  above  them. 


FOUNDATIONS. 


597 


,  in  its  floor,  communi- 


the  removal  of  which  requires  men  to  enter  the  cylinder  to  its  foot ;  which  they  cannot  do  in  the  rarefied 
air.  The  pipe  j?  should  be  ol"  sutticieut  diani  to  allow  the  air  to  leave  the  cylinder  rapidly,  so  that  the 
outer  pressure  may  act  upon  the  top  as  suddenly  as  possible. 

At  the  (ioodwiuSauds  light-liouse,  England,  hollow  cylinders  2}4  ft  in  diam,  were  sunk  34  ft  into 
sand  by  this  process,  iu  about  6  hours  ;  where  a  steel  bar  could  be  driven  only  8  ft  by  a  sledgeham- 
mer.  Others,  I'i  ins  iu  diaiii,  have  been  sunk  16  ft  into  sand  within  less  than  an  hour.  In  this  last 
instance  the  air-pump  had  two  barrels,  4J^  ins  diam,  16  inch  stroke,  worked  by  4  men.  The  pipep 
was  of  lead,  and  only  }4  inch  diam. 

The  plenum  process^,  invented  by  Mr  Tng:er, 
of  France,  consists  in  forcing  air  into  the  cylinder 
C  C,  Fig  17,  to  such  an  extent  as  to  force  out  the 
water,  compelling  it  to  escape  beneath  the  open  foot, 
into  the  surrounding  water.  The  interior  of  the  cylin- 
der being  thus  left  dry  to  the  bottom,  men  pass  down  it 
to  loosen  and  remove  the  soil  at  and  below  its  base.  When 
this- is  done,  they  leave ;  the  compressed  air  is  allowed  to 
escape  ;  and  the  cylinder,  being  no  longer  sustained  by 
the  upward  pressure  of  the  compressed  air  beneath  its 
top,  sinks  into  the  cavity,  or  the  loosened  material  at  its 
foot.  Fig  17  shows  the  simple  arrangement  by  which 
workmen  are  enabled  to  enter  or  leave  the  cylinder, 
without  allowing  the  compressed  air  to  escape;  as  well 
as  the  general  principle  of  the  entire  process. 

L  L  is  a  separate  small  chamber,  tke  alr«Ioek«  which  is 
removed  when  a  new  length  of  pipe  is  to  be  added ;  and  afterward 
replaced  and  firmly  bolted  on.  This  chamber  has  a  small  air-tight 
door  d,  by  which  it  can  be  entered  from  without ;  and  another,  o, 
opening  into  the  cylinder.  The  flaps,  t,  h,  of  both  doors,  open  in- 
ward, or  toward  the  cylinder.     This  chamber  also  has  two  stopcocks, 

eating  with  the  cylinder ;  and  one  e.  above,  communicating  with  the  open  air.  At  s  is  a  bent  tube, 
also  with  acock,  which  passes  air-tight  through  the  side  and  the  bottom  of  the  air-lock.  Through 
it  the  compressed  air  is  forced  into  the  cylinder,  by  an  air  force  pump  or  condenser:  and  through  it 
the  same  air  is  allowed  to  escape  at  a  later  period.  A  siphon  is  shown  at  n  nn.  A  drum  w  is  used 
for  hoisting  the  excavated  material  from  the  bottom,  to  the  air-Jock  ;  its  axle  i  i  passes  air-tight  through 
stufling-boxes  in  the  sides  of  the  lock  ;  the  hoisting  being  done  by  men  outside.  This  is  the  general 
arrangement  employed  by  Mr  W.  J.  McAlpine,  CE,  of  New  York,  at  Harlem  bridge;  and  from  his 
description  of  it,  ours  has  been  condensed.  The  cylinders  were  there  6  ft  diam,  1%  ins  thick,  and  in 
lengths  of  9  ft,  bolted  together  through  inside  flanges  /,  as  the  sinking  went  on.  The  air-lock  is  6  ft 
diam,  by  nearly  6  ft  high  :  with  sides  of  boiler  iron  ;  and  top  and  bottom  of  cast  iron. 

Now  suppose  the  cylinder  C  C  to  be  let  down,  and  steadied  in  position,  as  in  the  fig;  and  the  air- 
lock L  L  to  be  adjusted  on  top  of  it.  The  next  process  is  to  force  in  air  through  the  curved  tube  a  ; 
the  flap  t  of  the  lower  door  o,  and  the  cock  a,  being  previously  closed.  As  the  compressed  air  accu- 
mulates in  the  cylinder,  it  forces  out  the  water;  which  escapes  partly  beneath  the  bottom  of  the  cyl- 
inder, and  partly  by  rising  through  the  siphon  nn,  and  flowing  out  at  g.  The  door  o  being  already 
closed,  and  that  at  d  open,  the  air  in  the  air-lock  is  in  the  same  condition  as  that  outside ;  so  that 
workmen  can  enter  it  readily.  Having  done  so,  they  close  the  door  d,  and  the  cock  c;  and  open  the 
cock  a,  through  which  condensed  air-  from  the  cylinder  rushes  upward,  soon  filling  the  air-lock. 
"When  this  is  done,  the  flap  t  !■  opened,  and  the  men  descend  throifgh  the  door  oby  a  ladder,  or  by  a 
bucket  lowered  by  the  drum  w,  to  the  bottom.  Here  they  loosen  and  excavate  the  material  as  deep 
as  they  can  ;  and,  filling  it  into  a  bucket  or  bag,  they  signal  to  those  outside,  who  raise  it  to  the  air-  • 
lock.  When  done,  they  ascend  to  the  air-lock,  close' the  door  o.  and  the  cock  a;  and  open  the  cock  e, 
through  which  the  condensed  air  in  the  lock  soon  escapes,  leaving  the  internal  air  the  same  as  that 
outoide.  The  door  d  is  then  opened,  the  buckets  of  earth  are  removed,  and  the  men  go  out.  Finally 
the  cock  at«  is  opened,  the  condensed  nir  in  the  cylinder  escapes  through  it  to  the  outside  air,  and 
the  cylinder  sinks  by  its  own  weight  into  the  cavity  and  loosened  soil  prepared  for  it  at  its  base,  and 
which  is  new  forced  up  into  the  cylinder  by  the  rush  of  the  returning  water.  The  process  ia  them 
repeated.  The  sinking  will  often  vary  from  0  to  10  or  more  feet  at  one  operation.  Until  depths  of 
40  or  50  ft,  most  men  can  endure  the  pressure  of  the  condensed  air  ;  but  as  the  depth  increases  thia 
becomes  more  difficult,  and  positively  dangerous  to  life.  Cast-iron  cylinders  15  ft  diam  ;  and  great 
caissons,  Fig  is,  have  been  thus  sunk;  but  at  times  at  great  expense  and  trouble. 

The  cylinder  should  be  g'uided  in  its  descent  by  a  strong  frame,  which 
may  be  supported  by  piles.  Otherwise  it  will  be  apt  to  tilt,  and  thus  give  great  trouble  to  settle  it 
upon  its  exact  place.     Have  been  sunk  in  deep  water  by  divers  undermining  inside. 

The  plenum  process  as  applied  at  the  South  St  bridge,  Philada, 
by  Mr.  John  W.  Murphy,  contracting  engineer,  differs  materially  from  that  described  above  ;  and 
moreover  deserves  notice  on  account  of  the  great  simplicity  and  efficacy  of  his  plant.  This  consisted 
partly  of  two  canal  boats,  decked,  each  100  ft  long,  by  173^  ft  wide,  and  8  ft  depth  of  hold.  They 
were  anchored  parallel  to  each  other,  15  ft  apart.  Supported  by  the  boats,  and  over  the  space  between 
them,  was  a  strong  four-legged  shears  about  50  ft  high;  at  the  top  of  which  was  attached  tackle  for 
handling  the  cast  iron  cylinders.  In  the  hold  of  one  of  the  boats  was  a  Bnrlei|S'h 
Compressor  having  two  pistons  of  10  ins  diam,  and  9  ins  stroke;  together  with 
its  boiler.  On  the  deck  of  the  same  boat  stood  a  vertical  air-tank  or  reg:ulator, 

22  ft  long,  by  2  ft  diam,  made  of  quarter  inch  boiler  iron.  This  served  to  maintain  a  supply  of  com 
pressed  air  in  the  submerged  cyliuder  in  case  of  an  accidental  stopping  of  the  compressor;  which 
otherwise  would  probably  be  fatal  to  the  laborers  in  the  cylinder.  The  condensed  air  flowed  from 
this  air-tank  to  the  air-lock  of  the  cylinder  through  a  hose  4  ins  diam,  made  of  gum  elastic  and  can- 
vas, and  so  long,  and  so  placed,  as  to  extend  itself  as  the  cylinder  went  down,  thus  maintaining  the 
communication  at  all  times.  Entirely  across  both  boats,  and  across  the  interval  between  them,  ex- 
tended two  heavy  wooden  clamps,  each  3  ft  wide  by  18  ins  high ;  each  composed 


598 


FOUNDATIONS, 


of  three  pieces  of  12  X  18  inch  timber  strongly  bolted  together.  At  the  centers  of  these  clamps  the 
two  inner  vertical  sides  which  faced  each  other  were  hollowed  out  to  the  depth  of  a  foot  by  concavi- 
ties corresponding  to  the  curve  of  the  cylinders.  The  distance  apart  of  the  clamps  was  regulated  by 
f^o  strong  iron  rods,  having  screws  and  nuts  at  their  ends  for  that  purpose.  Thus  when  a  section 
of  a  cylinder  was  hoisted  by  means  of  the  shears  into  its  position  over  the  space  between  the  two 
boats,  the  two  concavities  of  the  clamps  were  brought  into  contact  with  it.  and  the  nuts  being  then 
screwed  up,  the  cylinder  was  firmly  held  in  place  by  the  clamps.  The  shears  could  then  be  used  to 
raise  another  section  of  the  cylinder  to  its  place  upon  the  first  one,  that  the  two  might  be  bolted  to- 
gether. By  repeating  this  process  the  height  of  the  cylinder  would  soon  become  too  great  to  allow 
the  shears  to  place  another  section  upon  it ;  in  which  case  the  nuts  of  the  screws  were  slightly 
loosened,  and  the  cylinder  was  allowed  to  slip  down  slowly  into  the  water  until  its  top  was  but  a 
little  above  the  surface.  The  screws  were  then  again  tightened,  and  the  cylinder  again  held  fast 
until  other  sections  were  added  and  bolted  to  it.  When  there  was  danger  that  the  upward  pressure 
of  the  condensed  air  might  lift  a  cylinder,  the  clamps  were  raised  by  the  shears  clear  of  the  boats  ; 
then  tightened  to  the  cylinder,  and  a  platform  of  planks  laid  upon  them,  and  loaded  with  stone. 

The  air-lock  was  so  arranged  as  not  to  require  to  be  removed  when  a  new  sec- 
tion was  to  be  bolted  on.  This  was  eflfected  as  follows.  Sections  of  the  cylinder  were  bolted  together 
in  the  manner  just  described,  until  its  foot  rested  on  the  bottom,  with  its  top  a  few  feet  above,  high 
water.  A  heavy  cast  iron  diaplirag'fii  1^  inches  thick,  to  form  the  floor  of  the 
air-lock,  was  then  placed  on  top.  Then  was  added  another  10  ft  high  section  of  the  cylinder,  to  form 
the  chamber  of  the  air-lock.  These  were  bolted  together;  and  then  another  diaphragm  was  added 
at  top  to  form  the  roof  of  the  air-lock.  These  diaphragms  were  furnished  with  openings,  and  with 
doors  and  valves  corresponding  with  those  shown  in  Pig  17.  and  remained  permanently  in  the 

cylinders  when  the  work  was  finished.  If  the  depth  of  soil  to  be  passed  through  before  reaching 
rock  is  so  great  as  to  require  other  sections  of  cylinder  to  be  bolted  on  above  the  top  of  the  air-lock, 
this  may  be  done  to  any  extent,  inasmuch  as  it  is  immaterial  whether  the  air-lock  is  under  water  or 

not.    To  keep  the  cylinder  both  air-  and  water-tig-ht  the  faces  -of 

the  flanges  before  being  bolted  together  were  smeared  with  a  mixture  of  red  and  white  lead  and  cot- 
ton fiber. 

At  the  South  St  bridgje  the  cylinders  were  4,  6,  and  8  ft  diam  ;  in  lengths 
or  sections  10  ft  long.  They  were  all  l}4  inch  thick.  Inside  flanges  2^  ins  wide,  Vyi  thick,  with  bolt- 
holes  \}4  inch  diam,  by  5  ins  apart  from  center  to  center.  The  bottom  edge  has  no  flange.  A  10  ft 
section  of  an  8  ft  cylinder  weighs  14600  lbs ;  of  a  6  ft  one,  10800 ;  of  a  4  ft  one,  6800.  An  8  ft  dia- 
phragm, 2800  fts  ;  6  ft,  1600;  4  ft,  783.  The  rock  under  the  soil  was  quite  uneven  in  places  :  but  was 
levelled  off  as  the  cylinders  went  down.  These  were  then  bolted  to  it  by  cast  iron  brackets. 
The  ivork  i¥ent  on,  day  and  nig'ht,  summer  and  winter:  with  no  inter- 
ruption from  the  tides,  floods,  or  floating  ice;  and  the  thirteen  eolumns  were  tank,  filled  with  con- 
crete, and  completed  in  11  months  ;  much  of  which  was  consumed  in  levelling  ofiF  the  rock,  and  bolt- 
ing the  brackets.     The  want  of  guides  caused  much  tilting,  trouble,  and  delay. 

Rise  and  fall  of  tide  about  7  ft.  Greatest  depth  of  soil,  gravel,  &c,  passed  through,  30  ft ;  least.  6 
ft.  Depth  of  water  about  25  ft.  The  work  was  under  charge  of  John  Anderson,  a  very  skillful  and 
energetic  superintendent  of  such  matters.  The  entire  neat  cost  of  the  cylin- 
ders in  place,  and  filled  with  hydraulic  concrete,  was  approximately  $92  per  foot  of  total  length 
for  the  8  ft  ones  ;  $64  for  the  6  ft ;  and  $40  for  the  4  ft  diams.  There  were  three  gangs  of  workmen  ; 
and  each  gang  worked  4  hours  at  a  time.  See  a  full  and  very  instructive  description  with  engrav- 
ings, by  D.  M.  Staufifer.  Superintending  Bnglnemr  for  the  city,  in  the  Journal  tt  the  Franklin  Inst, 
Nov.,  1872.  From  i'.  ihe  above  few  items  are  taken.  Mr.  Anderson's  firm  (Anderson  &  Barr,  Tribune 
Building,  N.  Y.)  have  since  successfully  sunk  a  number  of  such  cylinders,  including  (1884-0)  four  of 
wrought-iron,  8  ft  diam,  66  ft  long,  at  an  angle  of  45^  with  the  hor.  intended  as  struts  to  prevent  tht 
movement  of  one  of  the  abut  piers  of  Chestnut  St  bridge,  Phila. 

Cast  iron  cylinders  have  cracked  through, around  their  entire  circum- 
ference, in  many  parts  of  the  U.  S.  in  very  cold  weather;  owing  to  the  diflF  of  contraction  of  the 
Iron,  and  of  the  concrete  filling.    Ignorant  ase  of  them  may  be  attended  by  great  danger. 


The  shaded  part  of  Fig  18  shows  a  transverse  section  of  the  caisson  of  yellow- 
pine  timber  and  cement,  for  the  Brooklyn  tower  of  East  lliver  (N  Y) 

suspension  bridge,  of  1600  ft  clear  span.  It  is  168  ft  long  at  bottom,  and  102  ft  wide. 
A  longitudinal  section  resembles  the  transverse  one,  except  in  being  longer,  and  in 
showing  more  shafts  J.  Of  these  there  are  6,  arranged  in  pairs,  for  expedition  and  as 
a  precaution  against  accident.  Namely,  two  water-shafts  J,  each  7  ft  by  6}/^  ft  across, 
for  removing  by  buckets  and  hoisting  apparatus,  the  material  excavated  beneatli  the 

caisson ;  together  with  such 
water  as  may  accumulate  at 
o  o ;  two  air-shafts  of  21  ins 
diam,  through  which  air  is 
forced  from  above,  to  expel 
the  water  from  the  chamber 
C  S  S  D  below  the  caisson,  so 
as  to  allow  the  laborers  to 
work  there  at  undermining ; 
the  expelled  water  escaping 
under  the  foot  C  D  of  the  cais- 
son, into  the  river  ;  and  two 
supply  shafts  of  42  ins  diam, 
for  admitting  laborers^  tools, 
ftft.  The  sereral  shafts  of  course  have  air-chambers  on  top,  on  the  same  principle  m 
Pig;  17,  to  prevent  the  escape  of  the  compressed  air  in  s  s. 


mqis 


FOUNDATIONS.  699 

The  shafts  are  of  }4  inch  boiler  iron.  The  foot  C  D.  nine  timbers  high,  is  continuous,  extending 
entirely  around  the  caisson;  its  bottom  is  shod  with  cast  iron  ;  its  four.corners  are  strengthened  by 
wooden  knees  20  ft  long. 

From  the  bottom,  up  to  the  line  N,  N,  14  ft,  the  caisson  is  built  of  horizontal  layers  of  timbers  one 
foot  square;  the  layers  crossing  each  other  at  right  angles ;  and  the  timbers  of  each  layer  touching 
each  other  well  forced  and  bolted  together;  and  all  the  joints  filled  with  pitch.  To  aid  in  preventing 
leakage,  the  nuts  and  heads  of  the  screws  have  India-rubber  washers ;  also  all  outside  seams,  as  well 
as  all  the  seams  of  the  layer  of  timbers  N,  N,  are  thoroughly  calked;  and  a  layer  of  tin,  enclosed 
between  two  layers  of  telt,  is  placed  outside  of  each  outer  joint ;  and  over  the  entire  top  of  the  layer 
next  below  N,  N.  . 

When  the  caisson  was  built  up  to  N,  N,  on  land,  it  was  launched,  floated  into  position,  and  anchored ; 
after  which  were  added  for  sinking  it,  fifteen  courses  of  timbers  one  ft  square ;  and  laid  one  ft  apart 
In  the  clear ;  with  the  intervals  filled  with  concrete.  The  top  course  A  B  is  of  solid  timber,  to  serve 
as  a  floor  for  supporting  machinery,  &c.  It  was  sunk  some  feet  below  the  very  bottom  of  the 
river,  in  order  to  avoid  the  teredo. 

Cribs  are  sunk  outside  of  the  caisson,  to  form  temporary  wharves  for  boats  carrying  away  excavated 
material ;  and  for  vessels  bringing  stone,  &c. 

When  the  caisson  was  sunk,  and  the  water  forced  out  from  the  chamber  or  space  CSS  D. workmen 
began  to  excavate  uniformly  the  enclosed  area  of  river  bottom,  so  as  to  allow  the  caisson  to  descsnd 
slowly  until  it  reached  a  firm  substratum.  The  space  C  S  S  D,  as  well  as  the  shafts,  was  then  filled  up 
•olid  with  concrete  masonry.  A  coCfer-dam  was  built  on  top  of  the  caisson;  and  in  it  the  regular 
masonry  of  the  tower  was  started.  The  total  height  of  this  tower  including  the  caisson,  is  about  300 
ft.     For  full  details  see  report,  1873,  of  W.  A.  Roebling  the  chief  engineer. 

Hollow  cylinders,  or  otber  forms  of  brickwork  or  ma- 
sonry, with  a  strong  curb  or  open  ring  of  timber  or  iron  beueutb  tiieux,  may  be 
gradually  sunk  by  undermining  and  excavating  from  the  inside;  and  form  very  stable  loundatious. 
Under  water  this  may  be  done  by  properly  shaped  scoops,  with  or  without  the  aid  of  the  diving-bell, 
according  to  the  depth,  &c.  On  land  it  will  often  be  the  most  economical  and  satisfactory  mode, 
especially  in  firm  soils.  The  descent  may  be  assisted  by  loading  them,  if,  as  sometimes  happens,  the 
friction  of  their  sides  against  the  earth  outside  prevents  their  sinking  by  their  own  weight.  A  brick 
cylinder,  46  ft  outer  diam,  walls  3  ft  thick,  has  been  sunk  40  ft  in  dry  sand  and  gravel,  without  any 
d'ifiBculty.  It  was  built  18  ft  high,  (on  a  wooden  curb  21  ins  thick,)  and  weighed  300  tons  before  the 
sinking  was  begun.  The  interior  earth  was  excavated  slowly,  so  that  the  sinking  was  about  1  ft  per 
day  ;  the  walls  being  built  up  as  it  sank.        Tunnel  shafts  are  at  times  so  sunk. 

On  the  Rhine  for  a  coal  shaft,  a  brick  cylinder  2b}/^  feet  diam  was  first  thus 
sunk  by  its  own  weight  76  ft  through  sand  and  gravel ;  then  an  interior  one,  15  ft  diam,  was  sunk  in 
the  same  way  to  the  depth  of  256  ft  below  the  surface ;  of  which  depth  all  the  180  ft  below  the  first 
cylinder  was  a  running  quicksand.  At  256  ft  friction  rendered  the  cylinder  immovable.  The  quick 
sand  was  removed  by  boring ;  no  pumping  was  done ;  but  the  water  was  permitted  to  keep  the  cyl  full. 

The  entire  foundation  for  a  large  pier  of  masonry  has  been  sunk  in  this  manner,  in  a  single  mass ; 
a  sufiBcient  number  of  vertical  openings  being  left  in  it  for  the  workmen  to  descend,  or  for  tools  to  be 
inserted  for  undermining.  This  is  generally  a  very  slow  and  tedious  operation,  especially  under 
water.  It  may  often  be  expedited  by  diving-bells  or  by  diving-dresses.  It  will  generally  be  better  to 
make  the  mass  wider  at  bottom  than  above  it,  so  as  to  diminish  friction  against  the  outside  earth. 
On  land,  water  may  at  times  be  used  for  softening  the  bottom  earth.  By  keeping  the  interior  of  such 
hollow  masonry  dry,  it  may  even  be  built  downward  from  the  surface ;  by  undermining  only  a  por 
tion  of  its  circumference  at  a  time,  filling  said  portion  with  masonry,  and  then  removing  and  fiUiuji 
the  other  portion  ;  and  so  on  in  successive  stages  of  2  or  3  ft  downward  at  a  time.  This  mode  may  be 
adopted  also  when  friction  has  stopped  the  sinking  of  a  mass  by  its  own  weight  when  undermined.     . 

The  sand  pump  as  used  at  the  St  Louis  bridge  will  often  be  of  service  in  rais- 
ing sand  from  cylinders  while  being  sunk  in  water.  With  a  pump  pipe  of  3.5  ins  bore,  and  a  water 
jet  under  a  pressure  of  150  lbs  per  sq  inch,  20  cub  yds  of  sand  per  hour  were  raised  125  feet.  A  jet  of 
air  has  also  been  successfully  used  in  the  same  way,  as  at  the  East  River,  N  Y,  suspension  bridge,  &c. 

Fascines.  On  marshy  or  wet  quicksand  bottoms,  foundations  may  be  laid  by 
first  depositing  large  areas  of  layers  of  fascines,  or  stout  twigs  and  small  branches, 
strongly  tied  together  in  bundles  from  C  to  12  ft  long,  and  from  6  ins  to  2  ft  in  diam. 

The  layers  or  strata  of  bundles  should  cross  each  other.  A  kind  of  floating  raft  or  large  mattress 
is  first  made  of  these,  and  then  sunk  to  the  bottom  by  being  loaded  with  earth,  gravel,  stones,  &c. 
In  this  manner  the  abutments  and  piers  of  the  great  suspension  bridge  at  Kieff.  in  Russia,  with  spans 
of  440  ft,  were  founded  in  1852,  on  a  shifting  quicksand.  There  the  fascine  mattresses  extend  100  fl 
beyond  the  bases  of  the  masonry  which  rests  upon  them. 

Fascines  may  be  used  in  the  same  way  for  sustaining  railway  embankments,  &c,  over  marshy 
ground,  but  they  will  settle  considerably. 

ISand-piles.  We  have  already  alluded  to  the  use  of  sand  well  rammed  in  layers 
into  trenches  or  foundation  pits ;  but  it  may  also  be  used  in  soft  soils,  in  the  shape 
of  'piles.  A  short  stout  wooden  pile  is  first  driven  5  to  10  feet  or  more,  according  to 
the  case.  It  is  then  drawn  out,  and  the  hole  is  filled  with  wet  sand  well  rammed. 
The  pile  is  then  again  driven  in  another  place,  and  the  process  repeated.  The  inter- 
vals may  be  from  1  to  3  ft  in  the  clear.  Platforms  may  be  used  on  these  piles  as  on 
wooden  ones.  If  the  sand  is  not  put  in  wet,  it  will  be  in  danger  of  afterward  sink- 
ing from  rain  or  spring  water.  In  this  case,  as  with  fascines,  it  is  well  to  test  the 
foundation  by  means  of  trial  loads.  Some  settlement  must  inevitably  take  place 
until  all  the  parts  come  to  a  full  bearing ;  but  it  will  be  comparatively  trifling.  The 
same  occurs  in  every  large  work  to  some  extent ;  as  in  a  roof  or  arch  of  great  span, 
whether  of  wood,  iron,  or  masonry  ;  so  also  with  all  tall  piers,  walls,  &c,  &c.  Sandy 
foundations  under  water  should  be  surrounded  by  stout  well-driven  sheet- piling,  to 
prevent  the  enclosed  sand  from  running  out  in  case  the  outer  sand  is  washed  away ' 
and  should  also  be  defended  by  a  deposit  of  random-stone. 


600  ROCK  -  DRILLING. 

On  bad  bottoms  under  water,  small  artificial  islands  of  good  soil  have 
been  deposited ;  and  the  masonry  founded  upon  them.  Canal  locks  and  other 
structures  may  at  times  be  advantageously  founded  in  this  way  in  marshy  soils. 
If  necessary,  a  depth  of  several  feet  of  the  bad  soil  may  be  dredged  out  before  the 
firmer  soil  is  deposited  ;  and  the  latter  may  be  weighted  by  a  trial  load  to  test  its 
stability. 

The  mode  of  laying  a  foundation  under  water,  by  building  the  masonry  upon 
a  timber  platform  above  water,  upheld  by  strong;  screivs,  and  lowered  into 
the  water  as  the  work  is  finished  in  the  open  air,  a  coui'se  or  two  at  a  time,  has 
of  late  been  much  employed  with  entire  success,  in  large  bridge-piers  in  deep 
water.  It  however  is  not  new.  It  was  suggested  more  than  100  years  ago  by  Belidor. 

Piles  are  driven  6  to  10  ft  apart  arouud  the  space  to  be  occupied  by  the  pier; 
having  their  tops  connected  by  heavy  timber  cap-pieces.  These  last  uphold  the 
screws,  which  work  through  them.     The  whole  is  braced  against  lateral  motion. 

A  CLUMP  OF  PILES  WELL  DRIVEN  ;  and  then  enclosed  by  an  iron  cylinder  sunk 
to  a  firm  bearing,  and  filled  witli  concrete,  is  an  excellent  foundation.  The  piles 
may  extend  to  the  top  of  the  cylinder,  and  tlius  be  enclosed  in  the  concrete.  tSuch 
an  arrangement  has  been  patented  by  S,  B.  Cushing,  C.  E.,  Providence,  R.  I.  The 
cylinder  and  concrete  serve  to  protect  the  piles  from  sea-worms,  and  from  decay 
above  low  water ;  and  are  not  intended  to  support  the  load  above  them. 

STONEWOKK. 

Where  work  is  done  on  a  large  scale,  blasting  can  sometimes  be  done  at  from  10 
to  20  percent  less  cost  per  cubic  yard  by  means  of  machine  drills  and 
dynamite,  than  by  liand  drills  and  g^unpowder.  Ordinarily,  how- 
ever, the  cost  is  about  the  same,  and  the  advantage  of  the  newer  methods 
consists  rather  in  economy  of  time,  convenience,  and  having  the  work  more 
entirely  under  control.  In  ordinary  railroad  work  in  average  hard  rock,  and  when 
common  labor  costs  ^1  per  day  of  ten  hours,  the  cost  per  cubic  yard,  for  loosening, 
will  ordinarily  range  between  30  and  60  cts,  including  tools,  drilling,  powder,  &c. 

Holes  for  blasting,  drilled  by  hand,  are  generally  from  2%  to  4  ft 
deep  ;  and  from  1%  to  2  ins  diam.  Churn -drilling-  is  much  more  expeditious 
and  economical  than  that  hj  jumping , mentioned  below.  The  churn-drill  is  merely 
a  round  iron  bar,  usually  about  1%  ins  diam,  and  6  to  8  ft  long ;  with  a  steel  cutting 
edge,  or  bit,  (weighing  about  a  lb,  and  a  little  wider  than  the  diam  of  the  bar,) 
welded  to  its  lower  end.  A  man  lifts  it  a  few  inches  ;  or  rather  catches  it  as  it 
rebounds,  turns  it  partially  around  ;  and  lets  it  fall  again.  By  this  means  he  drills 
from  5  to  15  feet  of  hole,  nearly  2  ins  diam,  in  a  day  of  10  working  hours,  depend- 
ing on  the  character  of  the  rock.  From  7  to  8  ft  of  holes  1%  ins  diam,  is  about  a 
fair  day's  work  in  hard  gneiss,  granite,  or  compact  siliceous  limestone;  5  to  7  ft 
in  tough  compact  hornblende  ;  3  to  5  in  solid  quartz  ;  8  to  9  in  ordinary  marble 
or  limestone;  9  to  10  in  sandstone;  which,  however,  may  vary  within  all  these 
limits.  When  the  hole  is  more  than  about  4  ft  deep,  two  men  are  put  to  the  drill. 
Artesian,  and  oil  wells,  in  rock,  are  bored  on  the  principle  of  the  churn-drill. 

The  jumper,  as  now  used,  is  much  shorter  than  the  churn-drill.  One  man  (the 
holder)  sitting  down,  lifts  it  slightly,  and  turns  it  partly  around^  during  the  inter- 
vals between  the  blows  from  about  8  to  12  lb  hammers,  wielded  by  two  other  labor- 
ers, the  strikers.  It  can  be  used  for  holes  of  smaller  diameters  than  can  be  made 
by  the  churn-drill ;  because  the  holder  can  more  readily  keep  the  cutting  end  at 
the  exact  spot  required  to  be  drilled.  It  is  also  better  in  conglomerate  rock ;  the 
hard  siliceous  pebbles  of  which  deflect  the  churn-drill  from  its  vertical  direction, 
so  that  the  hole  becomes  crooked,  and  the  tool  becomes  bound  in  it.  The  coal 
conglomerates  are  by  no  means  hard  to  drill  with  a  jumper.  The  jumper  was 
formerly  used  for  large  deep  holes  also,  before  the  churn-drill  became  established. 

Either  tool  requires  resharpening  at  about  each  6  to  18  inches  depth  of  hole; 
and  the  wear  of  the  steel  edge  requires  a  new  one  to  be  put  on  every  2  to  4  d*ays. 
With  iron  jumpers,  the  top  also  becomes  battered  away  rapidlv.  As  the  hole 
becomes  deeper,  longer  drills  are  frequently  used  than  at  the  beginning.  The 
smaller  the  diameter  of  the  hole,  the  greater  depth  can  be  drilled  in  a  given  time ; 
and  the  depth  will  be  greater  in  proportion  than  the  decrease  of  diam.  Under 
similar  circumstances  three  laborers  with  a  jumper  will  about  average  as  much 
depth  as  one  with  a  churn-drill. 

The  hand-drill,  in  which  the  same  man  uses  both  the  hammer  and  the  short 
drill,  is  chiefly  used  for  shallow  holes  of  small  diam.  With  it  a  fair  workman 
will  drill  about  as  many  feet  of  hole  from  6  to  12  ins  deep,  and  about  %  inch  diam, 
as  one  with  a  churn-drill  can  do  in  holes  about  3  ft  deep  and  2  ins  diam,  in  the 
same  time.  Only  the  jumper  or  the  hand-drill  can  be  used  for  boring  holes 
which  are  horizontal,  or  much  inclined. 


COST  OF  STONEWORK.  601 

Cost  of  quarrying'  stone.  After  the  preliminary  expenses  of  purchsjring 
the  site  of  a  good  quarry;  cleaning  off  the  surface  earth  and  disintegrated  top  rock; 
and  providing  the  necessary  tools,  trucks,  cranes,  &c;  the  total  neat  expenses  for 
getting  out  the  rough  stone  for  masonry,  per  cub  yard,  ready  for  delivery,  may  be 
roughly  approximated  thus :  Stones  of  puch  sizes  as  two  men  can  readily  lift,  meas- 
ured in  piles,  will  cost  about  as  much  as  from  }/^  to  }4  the  daily  wages  of  a  quarry 
laborer.  Large  stones,  ranging  from  3^  to  1  cub  yd  each,  got  out  by  blasting,  from 
1  to  2  daily  wages  per  cub  yd.  Large  stones,  ranging  from  1  to  1}^  cub  yds  each,  in 
which  most  of  the  work  must  be  done  by  wedges,  in  order  that  the  individual  stones 
shall  come  out  in  tolerably  regular  shape,  and  conform  to  stipulated  dimensions ; 
from  2  to  4  daily  wages  per  cub  yard.  The  smaller  prices  are  low  for  sandstone, 
while  the  higher  ones  are  high  for  granite.  Under  ordinary  circumstances,  about 
1}/^  cub  yds  of  good  sandstone  can  be  quarried  at  the  same  cost  as  1  of  granite ;  or, 
in  other  words,  calling  the  cost  of  granite  1,  that  of  sandstone  will  he%;  so  that 
the  means  of  the  foregoing  limits  may  be  regarded  as  rather  full  prices  for  sandstone; 
rather  scant  ones  for  granite ;  and  about  fair  for  limestone  or  marble. 

Cost  of  dressing"  stone.  In  the  first  place,  a  liberal  allowance  should  be 
made  for  waste.  Even  when  the  stone  wedges  out  handsomely  on  all  sides  from 
the  quarry,  in  large  blocks  of  nearly  the  required  shape  and  size,  from  }^to}/^ot 
vne  rough  block  will  generally  not  more  than  cover  waste  when  well  dressed.  In 
moderate-sized  blodcs,  (say  averaging  about  3^  a  cub  yard  each.J  ai^d  g«t  out  by 
blasting,  from  3^  to  i^  will  not  be  too  much  for  stone  of  medium  character  as  te 
straight  splitting.  About  the  last  allowance  should  also  be  made  lor  well-scabble(f 
rubble.  The  smaller  the  stones,  the  greater  must  be  the  allowance  for  waste  ia 
dressing.  In  large  oper;j,tions,  it  becomes  expedient  to  have  the  stones  dressed,  as 
far  as  possible,  at  the  quarry ;  in  order  to  diminish  the  cost  of  transportation,  which, 
when  the  distance  is  great,  constitutes  an  important  item — especially  when  by  land, 
and  on  common  roads. 

A  stonecutter  win  first  take  out  of  wind;  and  then  fairly  patent-hammer  dress,  about  8 
to  10  sq  ft  of  plain  face  in  hard  granite,  in  a  day  of  8  working  hours ;  or  twice  as  much  of  such  infe- 
rior dressing  as  is  usually  bestowed  on  the  beds  and  joints  ;  and  generally  on  the  faces  also  of  bridge 
masonry,  &c,  when  a  very  fine  finish  is  not  required.  In  good  sandstone,  or  marble,  he  can  do  about 
34  more  than  in  granite.     Of  finest  hammer  finish,  granite,  4  to  5  sq  ft. 

Cost  of  masonry.  Every  item  composing  the  total  cost  is  liable  to  much 
variation ;  therefore,  we  can  merely  give  an  example  to  show  the  general  principle 
upon  which  an  approximate  estimate  may  be  made;  assuming  the  -wftges  or  a 
laborer  to  be  ^2.00  per  day  of  8  working  hours ;  and  $3.50  for  a  mason.  Tlie 
monopoly  of  quarries  affects  prices  very  much.* 

Cost  of  aslilar  facing  masonry.  Average  size  of  the  stones,  say  5  ft 
long  2  ft  wide,  and  1.4  thick ;  or  two  such  stones  to  a  cub  yd.  Then,  supposing  the 
stone  to  be  granite  or  gneiss,  the  cost  per  cub  yd  of  masonry  at  sucli  wages 
will  be,      Getting  out  the  stone  from  the  quarry  by  blasting,  allowing  yi  for  waste  in 

AreMing;  l^cnbyds,  at  S3.00  per  yard $4.00 

Dressing  U  sq  ft  of  face  at  35  cts *•»« 

*♦        52     "        beds  and  joints,  at  18  cts S-ao 

Neat  cost  of  the  dressed  stone  at  the  quarry 18.26 

Hauling,  say  1  mile;  loading  and  unloading 1-20 

Laying',  including  scaffold,  hoisting  machinery,  superintendence,  &c 2.00 

Neat  cost 21.86 

Profit  to  contractor,  say  15  per  ct ^•^° 

Total  cost V-'V/i ^^"^^  „      .V 

Dressing  will  cost  more  if  the  faces  are  to  be  rounded,  or  moulded.  If  the  stones  are  smaller  thn 
we  have  assumed,  there  will  be  more  sq  ft  per  cub  yd  to  be  dressed,  &c.  .     ,    ,.       ,,,    ».     i,  *». 

If  in  the  foregoing  case,  the  stones  be  perfectly  well  dressed  on  all  sides,  including  the  back,  the 
eost  per  cub  vd  would  be  increased  about  $10;  and  if  some  of  the  sides  be  curved,  as  in  arch  stoncR, 
say  $12  or  $14;  and  if  the  blocks  be  carefully  wedged  out  to  given  dimensions,  $16  or  $18;  thu» 
making  the  neat  cost  of  the  dressed  stone  at  the  quarry  say  $28,  $31,  or  ^Ab  per  cub  yd. 

•  The  blocks  of  granite  for  Bunker  Hill  monument  averaging  2  cub  yds  each,  were 
quarried  by  wedging,  and  delivered  at  the  site  of  the  monument,  at  a  neat  actual  cost  of  $5.40 
per  cub  yd  ;  by  the  Monument  Association  ;  from  a  quarry  opened  by  themselves  for  the  purpose.  The 
Association  received  no  profit;  their  services  being  voluntary.  The  average  contract  offers  for  th» 
«ame  were  $24.30 !  The  actual  cost  of  getting  out  the  rough  blocks  at  the  quarry  was  $2.70.  Load- 
ine  uDon  trucks  at  quarrv,  aljout  15  cts.  Transportation  8  miles  by  railway  and  common  road,  $2.55. 
Total   $5  40     In  1825  to  1845  ;  common  unskilled  labor  averaging  $1  per  day. 


602 


COST  OF  STONEWORK. 


The  item  or  laying  will  be  much  increased  if  the  stone  has  to  be  raised  to  great  heights ;  or  if  it  haa 
to  be  much  handled ;  as  when  carried  in  scows,  to  be  deposited  in  water-piers,  Ac.  Almost  ever) 
large  work  presents  certain  modifying  peculiarities,  which  must  be  left  to  the  judgment  of  the  engi- 
neer and  contractor.  The  percentage  of  contractors*  profit  will  usually  be  less  on  large  works  than 
on  small  ones. 

Cost  of  ashlar  facinsr  masonry.  If  the  stone  be  sandstone 

with  good  natural  beds,  the  getting  out  may  be  put  at  $3.00  per  cubic  yard.  Face  dressing  at  26  cts 
per  sq  ft :  say  $3.64  per  cubic  yd.  Beds  and  joints  13  cts  per  sq  ft ;  say  $6.76  per  cub  yd.  The  neat 
cost,  laid,  $17.00. 

And  the  total  cost  of  lari^e  well  scabbled  rang^ed 
sandstone  masonry  in  mortar,  may  be  taken -at  about  $10  per  cub  yd. 

Cost  of  larg>e  scabbled  grranite  rubble,  such  as  is  generally  used  ai 
backing  for  the  foregoing  ashlar  ;  stones  averaging  about  3^  cub  yd  each  : 

Cost  per 

liabor  at  $1  per  day*  cubjdof 

masonry. 
Getting  out  the  stone  from  the  quarry  by  blasting,  allowing  J^  for  waste  in 

scabbling;  li  cub  yds  at  $3.00 $3.43 

Hauling  1  mile,  loading  and  unloading     , 1.20 

Mortar ;  (2  cub  ft,  or  1.6  struck  bushels  quicklime,  either  in  lump  or  ground  ; 

and  10  cub  ft,  or  8  struck  bushels  of  sand,  or  gravel ;  and  mixing) 1.50 

Scabbiing;  laying,  including  scaffold,  hoisting  machinery,  &c 2.50 

Neat  cost 8.63 

Profit  to  contractor,  say  15  per  ct 1.30 

Total  cost .T, 9.93 

Common  rubble  of  small  stones,  the  average  size  being  such  as  two 
men  can  handle,  costs,  to  get  it  out  of  the  quarry,  about  80  cts  per  yard  of  pile ; 
or  to  allow  for  waste,  say  $1.00.  Hauling  1  mile,  $1.00.  It  can  be  roughly  scabbled, 
and  laid,  for  $1.20  more ;  mortar  as  foregoing,  $1.50.  Total  neat  cost,  ^.70 ;  or,  with 
15  per  ct  profit,  $5.40,  at  ike  above  wages  for  labor, 

W^i th  smaller  stones,  such  as  one  man  can  handle,  we  may  say,  stone  70  cts;  hauling  $1  i 
jaying  and  scaflfold,  tools  &c,  $1;  mortar  $1.50.  Making  the  neat  cost  $4.20;  or  with  15  per  ot  profit,  $4.83. 
Neat  scabbled  irregular  range- work  costs  from  $2  to  $3  more  per  yd  than  rubble;  accoTding  to  the  charac- 
ter of  the  stone  &c.  The  laying  of  thin  walls  costs  more  than  that  of  thick  ones,  such  as  abutments  &c.'>- 

The  eo«t  of  plain  8  inch  thick  ashlar  facing^s  for  dwellings  &c  in 

Philada,  in  1888,  ia  about  as  follows  per  square  foot  showing,  put  up,  including  everything.  Sand- 
■tone,  $1.50  10  $2.25.  Pennsylvania  marble,  $2.50.  New  England  marble,  $2.75  to  $3.25.  Granite. 
$2.25  to  $2.75.  If  6  ins  thick,  deduct  one-eighth  part.  First  ClaSS  artificial  StOUC 
could  be  made  and  put  up  at  one-third  the  price.  K'orth  River  blue  Stoue 

flags,  3  ins  thick,  for  footwalks,  p«t  down,  including  gravel  &c,  70  cts  per  sq  foot.  Bclg'ian 
Street  pavement,  with  gravel,  complete,  ^3.50  per  sq  yard  in  Eastern  cities. 

When  dressed  ashlar  facing  is  backed  by  rubble,  the  expense  per  cub  yard  of  the 
entire  mass  will  of  course  vary  according  to  the  proportions  of  the  two.  Thus,  if 
ashlar  at  $12  per  yd,  is  backed  by  an  equal  thickness  of  rubble  at  $5,  the  mean  cost 
will  be  ($12  -f  $5)  ^  2  =  $8.50 ;  or  if  the  rubble  is  twice  as  thick  as  the  ashlar  then 
($12  -f  $5  +  $5)  ->  3  =  $7.33,  &c.  Such  compound  walls  are  weak  and 
apt  to  separate  in  time,  as  also  walls  of  cut  stone  backed  by  concrete,  or  by  brick ; 
from  unequal  settlement  of  the  two  parts. 

At  times  the  contractor  must  be  allowed  extra  in  opening  new  quarries;  in  forminc 
short  roads  to  his  work  ;  in  digging  foundations  ;  or  for  pumping  or  otherwis*  draining  them,  whe» 
springs  are  unexpectedly  met  with  ;  for  the  centers  for  arches,  &c;  unless  these  items  are  expressly 
included  in  the  contract  per  cub  yd. 


RETAINING-WALLS. 


603 


EETAINING-WALLS. 


Art.  1.  A  retaining'-wall  is  one  for  sustaining  the  pressure 
of  earth,  sand,  or  other  filling  or  hacking,  deposited  behind  it  after  it  is  built ;  in 
distinction  to  a  face-ivall,  which  is  a  similar  structure  for  preventing  the  fall  of 
earth  which  is  in  its  undisturbed  natural  position,  but  in  which  a  vert  or  inclined 
face  has  been  excavated.  The  earth  is  then  in  so  consolidated  a  condition  as  to  exert 
little  or  no  lateral  pres,  and  therefore  the  wall  may  generally  be  thinner  than  a 
retaining  one. 

This,  however,  will  depend  upon  the  nature  and 
position  of  the  strata  in  which  the  face  is  cut.  If 
the  strata  are  of  rock,  with  interposed  beds  of  clay, 
earth,  or  sand  ;  and  if  they  dip  or  incline  toward  the 
■wall,  it  may  require  to  be  of  far  greater  thickness 
than  any  ordinary  retaining- wall ;  because  when  the 
thin  seams  of  earth  become  softened  by  infiltrating 
rain,  they  act  as  lubrics,  like  soap,  or  tallow,  to  fa- 
cilitate the  sliding  of  the  rock  strata ;  and  thus  bring 
an  enormous  pres  against  the  wall.  Or  the  rock  may 
be  set  in  motion  by  the  action  of  frost  upon  the  clay 
seams ;  or,  as  sometimes  occurs,  by  the  tremor  pro- 
duced by  passing  trains.  Even  if  "there  be  no  rock, 
still  if  the  strata  of  soil  dip  toward  the  wall,  there 
will  always  be  danger  of  a  similar  result;  and  addi- 
tional precautions  must  be  adopted,  especially  when 
the  strata  reach  to  a  much  greater  height  than  the 

wall.     A  vertical  urall  has  both  c  o 

and  d  a  vert. 

Experience,  rather  than  theo- 
ry, must  be  our  guide  in  the  building  of 
both  kinds  of  wall.  We  recommend  that 
the  hor  thickness  a  b,  Fig  1,  at  the  base  of  a 
vert  or  nearly  vert  retaining-wall  c  d  b  a, 
which  sustains  a  backing  of  either  sand,  gravel,  or  earth,  level  with  its  top  c  d, 
as  in  the  fig,  should  not  be  less  than  the  following,  in  railroad  practice,  when  the 
foundations  are  not  more  than  about  three  feet  deep. 

When  the  backing^  is  deposited  loosely,  as  usual,  as  when, 
dumped  from  carts  y  cars,  <Scc, 

Wall  of  cut-sione,  or  of  first-class  large  ranged  rubble, 

in  mortar a.b B5  of  its  entire  vert  height  d  b. 

"      good  common  scabbled  mortar-rubble,  or  brick.  A  "  "  "      *' 

"      well-scabbled  dry  rubble 5  "  "  "      " 

With  good  masonry,  however,  we  may  take  the  height  d  s  instead  of  d  b,  and  then 
ihe  above  proportions  of  d  s  will  give  a  sufficient  thickness  at  the  ground-line  o  s. 


When  the  backing:  is  somewhat  consolidated  in  hor  layers, 

each  of  these  thicknesses  may  be  reduced,  but  no  rule  can  be  given  for  this. 

Tke  oflOset  o  e,  in  front  of  the  wall,  is  not  included  in  these  thicknesses. 

When,  however,  the  backing  is  a  pure  clean  sand,  or  gravel,  we  should  use  only  the  full  dimen- 
sions ;  inasmuch  as  the  tremor,  caused  by  passing  trains,  would  neutralize  any  supposed  advantage 
from  ramming  materials  so  devoid  of  cohesion.  Such  sand  may  be  rammed  with  much  adv&.ntage 
for  the  purpose  of  compacting  it  in  foundations  ;  but  a  diff  principle  is  involved  in  that  case.  When 
it  is  done  even  with  cohesive  earths,  with  a  view  of  saving  masonry  in  retaining-walls,  it  is  probable 
that  the  expense  will  generally  be  found  quite  equal  to  that  of  the  masonry  saved. 

The  base  aft-  in  Fig  1.  is  ^  of  the  height  bd.  In  the  foregoing  thicknesses  at  base,  the  back  dh 
of  the  wall  is  supposed  to  be  vert;  and  the  face  ca  either  vert,  or  battered  (sloped  or  inclined  back- 
ward) to  an  extent  not  exceeding  about  13^  inches  to  a  foot;  which  limit  it  is  rarely  advisable  to  ex- 
ceed in  practice,  owing  to  the  bad  effect  of  rain,  &c,  upon  the  mortar  when  the  batter  is  great.  The 
base  of  a  vert  wall  need  not  in  fact  be  as  thick  as  one  with  a  battered  face ;  but  when  the  batter  does 
not  exceed  1.5  inches  to  a  foot,  the  diff  is  very  small.    See  Table,  Art  7. 

Rem.  1.  A  mixture  of  sand,  or  earth,  with  a  larg^e  proportion 

OP  ROUND  BOWLDERS  paving  pebbles,  &c,  will  weigh  considerably  more  than  the  materials  ordinarily 
used  for  backing ;  and  will  exert  a  greater  pres  against  the  wall ;  the  thickness  of  which  should  be 
increased,  say  about  otte-eighth  to  one-sixth  part,  when  such  backing  has  to  be  used. 

Rem.  2.  The  wall  will  be  stronger  if  all  the  courses  of  masonry  be  laid 
w^ith  an  inclination  inward,  as  at  oeb;  especially  if  of  dry  masonry, 
or  if  time  cannot  be  allowed  (as  it  always  should  be,  when  practicable)  for  the  mor- 
tar to  set  properly,  before  the  backing  is  deposited  behind  it.    The  object  of  inclin- 


604 


RETAINING-WALLS. 


ing  the  courses,  is  to  place  the  joints  more  nearly  at  right  angles  to  the  direetlon 
/P,  Figs  6,  7,  and  8,  of  the  pres  against  the  back  of  the  wall;  and  thus  diminish 
the  teaidency  of  the  stones  to  slide  on  one  another,  and  cause  the  wall  to  bulge. 

When  the  courses  are  hor,  there  is  nothing  to  pre- 
vent this  sliding,  except  the  friction  of  the  stones,  one  upon  the  other,  when  of  dry 
masonry ;  or  friction  and  the  mortar,  when  the  last  is  used.  But  if,  as  is  frequently 
the  case,  (especially  in  thick  and  hastily  built  walls,)  this  has  not  had  time  to  harden 
properly,  it  will  oppose  but  little  resistance  to  sliding.  But  when  the  courses  are 
inclined,  they  cannot  slide,  without  at  the  same  time  being  lifted  up  the  inclined 
planes  formed  by  themselves.  In  retaining- walls,  as  in  the  abuts  of  important 
arches,  the  engineer  should  place  as  little  dependence  as  possible  upon  mortar;  but 
should  rely  more  upon  the  position  of  the  joints,  for  stability. 

An  objection  to  this  inclining  of  the  joints  in  dry  (without  mortar)  walls,  is  that  rain-water,  falling 
on  the  battered  face,  is  thereby  carried  inward  to  the  earth  backing;  which  thus  becomes  soft,  and 
settles.  This  may  be  in  a  great  measure  obviated  by  laving  the  outer  or  face-courses  hor ;  or  by 
using  mortar  for  a  depth  of  only  about  a  foot  from  the  face.  The  top  of  the  wall  should  be  protected 
by  a  coping  c  d,  Fig  1,  which  had  better  project  a  few  ins  in  front.  After  the  masonry  has  been 
built  up  to  the  surface  of  the  ground,  the  foundation  pit  should  be  filled  up ;  and  it  is  well  to  con- 
solidate the  filling  by  ramming,  especially  in  front  of  the  wall. 

The  back  «{  &  of  the  wall  [should  be  left  rongrh.    In  brickwork  it 

would  be  well  to  let  every  third  or  fourth  course  project  an  inch  or  two.  This  increases  the  friction 
of  the  earth  against  the  back,  and  thus  causes  the  resultant  of  the  forces  acting  behind  the  wall  to 
become  more  nearly  vert ;  and  to  fall  farther  within  the  base,  giving  increased  stability.  It  also  con- 
iuces  to  strength  not  to  make  each  course  of  uniform  height  throughout  the  thickness  of  the  wall; 
but  to  have  some  of  the  stones  (especially  near  the  back^  suflBciently  high  to  reach  up  through  two  or 
»hree  courses.  By  this  means  the  whole  masonry  becomes  more  eflfectually  interlocked  or  bonded 
together  as  one  mass ;  and  therefore  less  liable  to  bulge.  Very  thick  walls  may  consist  of  a  facing 
of  masonry,  and  a  backing  of  concrete. 

Rbm.  3.  It  is  the  pres  itself  of  the  earth  against  the  back,  that  creates  the  friction,  which  in  turn 
modifies  the  action  of  the  pres  ;  as  the  wt  or  pres  of  a  body  upon  an  inclined  plane  produces  friction 
between  the  body  and  the  plane,  suflBcient,  perhaps,  to  prevent  the  body  from  sliding  down  it,  A  re- 
taining-wall  is  overthrovm  by  being  made  to  revolve  around  its  outer  toe  or  edge  e.  Fig  1,  as  a  ful- 
sram,  or  turning-point;  but  in  order  thus  to  revolve,  its  back  must  first  plainly  rise;  and  in  doing 
so  must  rub  against  the  backing,  and  thus  encounter  and  overcome  this  friction.  The 
friction  exists  the  same,  whether  the  wall  stands  firm  or  not ;  as  in  the  case  of  the 
body  on  an  inclined  plane  ;  the  only  difif  is  that  in  one  case  it  prevents  motion ;  and 
in  the  other  only  retards  it. 

Where  deep  freeasing:  occurs  the  back  of  the  wall  should 

be  sloped  forwards  for  3  or  4  ft  below  its  top  as  at  c  o,  which  should  be  quite  smooth 
so  as  to  lessen  the  hold  of  the  frost  and  prevent  displacement. 

Rem.  4.  When  the  wall  is  too  thin,  it  will  generally  fail 
by  bulg-ing-  outward,  at  about  3/3  of  its  height  above  the 
ground,  as  at  a,  in  Fig  2.  A  slight  bulging  in  a  new  wall 
does  not  necessarily  prove  it  to  be  actually  unsafe.  It  is 
generally  due  to  the  newness  of  the  mortar,  and  to  th© 
greater  pres  exerted  by  the  fresh  backing ;  and  will  often 
cease  to  increase  after  a  few  months.  It  need  not  excite 
^^^^  apprehension  if  it  does  not  exceed  \^  inch  for  each  foot  in 

sv.;v;s!j^^^^!»^ivi^^;^jj^^^        thickness  at  a.    See  Remark  3,  Art  7. 

Art.  2.  The  young  engineer  need  not  in  practice  concern  himself  particularly  about  the  precis* 
■p  OBAV  OF  HIS  BACKING,  or  about  the  ANGLE  OP  SLOPE  at  which  it  will  stand  ;  for  the  material  which 
he  deposits  behind  his  wall  one  day,  may  be  dry  and  incoherent,  so  as  to  slope  at  15-^  to  1 ;  the  nect 
day  rain  may  convert  it  into  liquid  mud,  seeking  its  own  level,  like  water;  the  next  it  may  be  ice, 
capable  of  sustaining  a  considerable  load,  as  a  vert  pillar. 

Moreover,  he  cannot  foretell  what  may  be  the  nature  of  his  backing;  for,  as  a  general  rule,  this 
must  consist  of  whatever  the  adjacent  excavation  may  produce  from  time  to  time  ;  sand  to-day,  rock 
to-morrow,  &c.  Retaining-walls  are  therefore  usually  built  before  the  engineer  knows  the  character 
•f  their  backing;  so  that  in  practice,  these  theoretical  considerations  have  comparatively  but  little 
weight.  Theory,  uncontrolled  by  observation  and  common  sense,  will  lead  to  great  errors  in  every 
department  of  engineering ;  but,  on  the  other  hand,  no  amount  of  experience  alone  will  compensate 
for  an  ignorance  of  theory.     The  two  must  go  hand-in-hand. 

Again,  the  settlement  of  the  backing  under  its  o^v^n  urt,  aided 
by  the  tremors  produced  by  heavy  trains  at  high  speed ;  its  expansion  by  frost,  or 
by  the  infiltration  of  rain ;  the  hydrostatic  pressure  arising  from  the  admission  of 
the  latter  through  cracks  produced  in  the  backing  during  long  droughts  ;  as  well  as 
its  lubricating  action  upon  it,  (diminishing  its  friction,  and  giving  it  a  tendency  to 
slide,)  &c,  exert  at  times  quite  as  powerful  an  overturning  tendency  as  the  legitimate 
theoretical  pres  does.  The  action  of  these  agencies  is  gradual.  Careful  observation 
of  retaining-walls  year  after  year,  will  often  show  that  their  battered  faces  are  be- 
coming vertical.  Then  they  will  begin-  to  incline  outward  ;  and  eventually  the  wall 
win  fail.    Theory  omits  loads  that  may  come  on  backing  increasing  its  pres. 


1  Txi% 


BETAINING-WALL8. 


605 


▲nraming  the  theoretical  views  advanced  by  Professor  Moseley  to  be  correct  as 
theories,  the  thicknesses  which  we  have  recommended  in  Art  1,  for  mortar  walls, 
correspond  to  from  7  to  14  time? ;  and  for  dry  walls  about  10  to  20  times,  the  pres 
assigned  by  him ;  and  we  do  not  consider  ours  greater  than  experience  has  shown 
to  be  necessary.  See  Table  3.  Retaining-walls  designed  by  good  engineers,  but  in 
too  close  accordance  with  theory,  (which  assumes  that  a  resistance  equal  to  twic« 
the  theoretical  pres  is  suflBcient,)  have  failed;  and  the  inference  is  fair  that  many  of 
tiiose  which  stand  have  too  small  a  coefficient  of  safety. 

The  fact  is,  (or  at  least  so  it  appears  to  us,)  there  must  be  defects  in  the  theoretical  assumptions  of 
jM>me  of  the  most  prominent  writers  who  give  practical  rules  on  this  subject.  Thus  Poncelet,  who 
certainly  is  at  their  head,  states  that  his  tables,  for  practical  use,  give  thicknesses  of  base  for  sus- 
taining 1  ^^  times  the  theoretical  pres  ;  and  this  he  considers  amply  safe.  Yet,  for  a  vert  wall  of  cut 
granite,  his  base  for  sustaining  dry  sand  level  with  the  top,  as  in  Fig  1,  is  .35  of  the  vert  height; 
and  for  brick,  .45.  But  the  writer  fouud  that  when  not  subject  to  tremor,  a  wooden  model  of  a  vert 
wall,  weighing  but  28  lbs  per  cub  ft,  aud  with  a  base  of  .35  of  its  height,  balanced  perfectly  dry  sand 
■loping  at  114  to  1,  and  weighing  89  fi»s  per  cub  ft. 

Now,    THK    BESISTANCK    OF    SIMILAR    WALLS,    OP    THE    SAME    DIMKN8IOWB, 

▼AKiES  A8  THEIR  SPECIFIC  GRAVITIES ;  and,  since  granite  weighs  about  165 
lbs  per  cub  foot,  or  6  times  as  much  as  our  model,  it  follows,  we  conceive, 
that  a  wall  of  that  material,  with  a  base  of  .35  of  its  height,  must  have 
a  resistance  of  6  times  any  true  theoretical  pres,  instead  of  only  1.8  . 
times ;  and  that  his  brick  wall  must  have  about  b  times  the  mere  bal- 
ancing resistance.  Our  experiments  were  made  in  an  upper  room  of  a 
•trongly  built  dwelling  ;  and  we  found  that  the  tremor  produced  by  pass- 
ing vehicles  in  the  street,  by  the  shutting  of  doors,  and  walking  about 
the  room,  sufficed  to  gradually  produce  leaning  in  walls  of  considerably 
more  than  twice  the  mere  balancing  stability  while  quiet;  and  it  appears 
te  us  that  the  injurious  effects  of  a  heavy  train  would  be  comparatively 
quite  as  great  upon  an  actual  retaining-wall,  supporting  so  incohesive 
a  material  as  dry  sand. 

Since,  therefore,  Poncelet's  wall  is  in  this  Instance  sufficiently  stable 
for  practice,  it  seems  to  us  that  his  theory,  which  neglects  the  effect  of 
tremors,  Ac,  must  be  defective.  He  also  gives  -i  of  the  height  as  a  suf- 
ficiently safe  thickness  for  a  vert  granite  wall  supporting  stiff  earth;  but 
■we  suspect  that  very  few  engineers  would  be  willing  to  trust  to  that  pro- 
portion, when,  as  usual,  the  earth  is  dumped  in  from  carts,  or  cars ;  espe- 
cially during  a  rainy  period.     If  deposited,  and  consolidated  in  layers, 

theory  could  scarcely  assign  any  thickness  for  the  wall ;  for  the  backing  thus  becomes,  as  it  were,  a 
mass  of  unburnt  brick,  exerting  no  hor  thrust;  and  requiring  nothing  but  protection  from  atmospberie 
influence,  to  insure  its  stability  without  any  retaining-wa.\l.  It  is  with  great  diffidence,  and  distrust 
in  our  opinions,  that  we  venture  to  express  doubts  respecting  the  assumptions  of  so  profound  an  in- 
vestigator and  writer  as  Poncelet;  and  we  do  so  only  with  the  hope  that  the  views  of  more  compe- 
tent persons  than  ourselves,  may  be  thereby  elicited.  Our  own  have  no  better  foundation  than  ex- 
periments with  wooden  and  brick  models,  by  ourselves  ;  combined  with  observation  of  actual  walls. 

Art.  3.  After  a  wall  ab  c  o,  Fig  3,  with  a  vert  back,  has  been  proportioned  by 
our  rule  in  Art  1,  it  may  be  converted  into  one  with  an  offsetted 

back,  as  a  t  n  o.  This  will  present  greater  resistance  to  overturning;  and  yet  con- 
tain no  more  material.  Thus,  through  the  center  t  of  the  back,  draw  any  line  i  n; 
from  n  draw  n  s,  vert ;  divide  i  .<?  into  any  even  number  of  equal  parts ;  (in  the  fig 
there  are  4 ;)  and  divide  s  n,  into  me  more  equal  parts ;  (in  the  fig  there  are  5.)  From 
the  points  of  division  draw  hor,  and  vert  lines,  for  forming  the  offsets,  as  in  the  fig. 

In  the  oflfsetted  wall,  the  cen  of  grav  is  thrown  farther  back  from  the  toe  o,  than 
In  the  other,  thus  giving  it  increased  leverage  and  resistance;  but  within  ordinary 
practical  limits,  the  difif  is  very  small ;  and  since  the  triangle  of  supported  earth  is 
greater  than  when  the  back  is  vert,  its  pres  is  also  greater ;  so  that  probably  no  ap- 
preciable advantage  attends  that  consideration.  The  increase  of  thickness 
near  the  base,  diminishes,  however,  the 
leverag-e  v  a.  Fig-  8,  of  the  pres  /P,  of  the 
earth  against  the  back.  The  center  of  pressure  of 
this  pres  is  in  both  cases  at  3^  the  vert  height  meas- 
ured from  the  bottom ;  and  it  is  therefore  plain  that 
the  farther  back  from  the  front  it  is  applied,  the  shorter 
must  V  a  become.  Moreover,  in  the  offsetted  back,  the 
direction  of  the  pres  becomes  more  nearly  vert  than 
when  the  back  is  upright.  It  is  to  these  causes,  rather 
than  to  the  throwing  back  of  the  cent  of  grav,  that 
the  oflfsetted  wall  owes  its  increase  of  stability  ovar 
oae  with  a  vert  back. 

Art.  4.  ^Vhen,  as  in  Fig  4,  the  backing:  is  higrber  than  the 
wall,  and  slopes  away  from  its  inner  edge  d,  at  the  natural  slope  d  «,  of  13^  to  1,  w« 
are  confident  that  the  following  thicknesses  at  base  will  at  least  bo  found  sufficient 


606 


RETAINING-WALLS. 


for  Tert  walls  with  sand.  They  are  deduced  from  the  experiments  just  alluded  to, 
and  are  but  rude  approximations,  with  no  scientific  basis.  We  should  not  have  in- 
serted them,  but  for  the  fact  that  we  know  of  no  others  for  this  case. 

The  first  column  contains  the  vert  height  s  v,  of  the  earth,  as  compared  with  the 
vert  height  of  the  wall ;  which*  latter  is  assumed  to  be  1 ;  so  that  the  table  begins 
with  backing  of  the  same  height  as  the  wall,  as  in  Fig  1.  These  vert  walls  may  be 
changed  to  others,  with  battered  faces,  by  Art  8 ;  or  without  any  such  proceeding, 
their  faces  may  be  battered  to  any  extent  not  exceeding  1}^  inches  to  a  foot,  or  1  in 
8,  without  sensibly  affecting  their  stability,  without  increasing  the  base. 

TABIiE  1.    (Original.) 


.c'S-o  ', 

^^•^ 

J3«g 

m 

«X3  £ 

n^ 

Wall 

Good 

Wall 

Good 

-2  % 

of 

Mortar 

Wall 

*''S  s 

.      of 

Mortar 

Wall 

'^1 

Cut  Stone, 

Rubble, 

of 

"^1 

Cut  Stone 

Rubble, 

of 

5'?- 

in 

or 

good  dry 
Bubble. 

S"?* 

in 

or 

good  dry 

-SP-o* 

Mortar. 

Brick. 

.Sfrj'S 

Mortar. 

Brick. 

Rabble. 

^2^ 

^£^ 

iP 

|h^ 

Thickne 

ss  at  Base,  in 

parts  of 

Thickness  at  Base,  in 

parts  of 

^§•3 

the  height. 

hS-S 

the  height. 

1. 

.35 

.40 

.50 

2. 

.58 

.63 

.73 

1.1 

.42 

.47 

.57 

2.5 

.60 

.65 

.75 

1.2 

.46 

.51 

.61 

3. 

.62 

.67 

.77 

1.3 

.49 

.54 

.64 

4. 

.63 

.68 

.78 

1.4 

.51 

.56 

.66 

6. 

.64 

.69 

.79 

1.5 

.52 

.57 

.67 

9. 

.65 

.70 

.80 

1.6 

.54 

.59 

.69 

14. 

.66 

.71 

.81 

1.7 

.55 

.60 

.70 

25. 

1.8 

.56 

.61 

.71 

or  more 

.68 

.73 

.83 

Art.  5.  But  when  the  slope  n  r.  Fig  5,  of  V/^  to  1,  starts  from  the  outer  edge  n 
of  the  wall,  greater  thickness  is  required.  Poncelet  gives  the  following  for  thi» 
case,  for  dry  sand. 

TABIiE   2. 


^ 

-*3 

.a  43 

2  «• 

■£  *• 

S® 

o«'S 

r 

i-  \% 

Wall 

of 

Cut  Stone 

Wall 
of 

■  o5 
5'? 

Wall 

of 

Cut  Stone 

^^ 

\        3» 

Wall 
of 

^/f^          /' 

i     ^i. 

Mortar. 

Brickwork. 

'1^ 

in 
Mortar. 

Brickwork. 

n-^^ 

g          / 

p 

1    / 

h8*5 

j  .    1 

.35 

.452 

2.4 

.762 

1.02 

^^       ' 

y^    1-1 

.393 

.498 

3.0 

.811 

1.11 

^^^ 

^^     /        ^^ 

:             1.2 

.4.39 

.548 

4.0 

.852 

1.18 

■^^^ 

^^    /     ^^ 

i              1-3 

.485 

.604 

6.0 

.883 

1.25 

swww 

^^^  / 

;           1.4 

.532 

.665 

11.0 

.909 

1  28 

N\m 

^^ /./'^ 

1.5 

.579 

.726 

21.0 

.922 

1.31 

m 

^   1;? 

.617 

.778 

31.0 

.926 

1..32 

.1645 

.824 

Infinite. 

.934 

1.34 

1.8 

.668 

.847 

Fig.  5. 

1.9 
2.0 

.690 
.707 

.90S 
.930 

When  the  earth  reaches  above  the  top  of  the  wall,  as  in  Figs  4  and  5,  the  wall  is  Sarcharg^ed  ? 

and  the  earth  that  is  above  the  top,  is  called  the  surcharok.  When  the  surcharge  is  carefully  deposited 
above  the  wall,  so  as  to  slope  back  at  a  steeper  angle  than  IJr^  to  1,  as  say  at  1  to  1,  theory  does  not 
require  the  wall  to  be  as  thick.  Notwithstanding  Fencelet's  high  position,  the  writer  cannot  imagine 
that  the  base  of  a  brick  wall  need  bs  so  great  as  IJj  times  its  height  for  any  height  of  sand  whatever. 

Art.  6.    On  the  theory  of  retaining-- walls.    Let  he  am.  Fig  6,  be 

such  a  wall,  upholding  backing  or  filling  c  s  m  g ;  the  upper  surf  c  s  of  which  is 
hor,  and  level  with  the  top  b  c  of  the  wall ;  and  let  m  s  represent  the  nat  slope  of  the 
earth  which  composes  the  backing  ;  m  g  being  hor. 

Abundant  experience  on  public  works  shows  that  this  slope,  whether  for  sand,  gravel,  or  earth». 
when  dry,  may  be  practically  taken  at  1 J^  to  1 ;  that  is,  13^  hor,  to  1  of  vert  measurement;  which 
corresponds  to  an  angle  s  m  g  of  33°  41'  with  the  hor;  which  is  also  about  the  angle  at  which  bricks 
and  roughly  dressed  masonry  begin  to  slide  on  each  other.    This  angle,  however,  varies  cousidera* 


RETAINING-WALLS. 


607 


bly ;  being  greatly  influenced  by  the  degree  of  dryness,  or  dampness,  of  the  material ;  so  that  mode- 
rately damp  sand  or  earth  will  stand  at  a  slope  of  1  to  1,  or  at  an  angle  of  45<^.  Whatever  it  may  be, 
it  is  called  the  angle  of  nat  slope  of  the  material  under  consideration.  In  theoretical  calculations 
for  walls,  it  is  safest  to  assume  (as  we  have  done  throughout)  that  the  backing  is  perfectly  dry,  since 


Fi^.7 


its  pres  ia  then  greatest ;  unless  it  be  supposed  to  be  so  wet  as  to  possess  some  degree  of  fluidity.  The 
triangle  cms  of  earth  above  the  nat  slope  m«,  tends  to  slide  down  said  slope,  but  is  prevented  from 
80  doing  by  the  wall. 

It  is  assumed  in  all  oa^es,  that  the  wall  is  secured  from  sliding  along  its  base,Art9,  thatitis 

thick  enough  to  prevent  failure  by  bulging ;  and  that  it  will  fail  only  by  overturning,  by  rotating 
around  its  toe,  a,  as  a  fulcrum.  The  thickness  necessary  to  insure  safety  against  the  last  will  also  be 
■nfficient  to  prevent  bulging.  Now  referring  only  to  Pig  6  with  a  vert  back,  if  the  angle  oms,  con- 
tained between  the  natural  slope  m  s,  and  a  vert  line  m  o,  drawn  from  the  inner  bottom  edge  m  of 
the  wall,  be  divided  by  a  line  m  t,  into  two  equal  angles,  o  m  t,  tm  s,  then  the  angle  o  m  t  is  called 
THE  ANOLE,  and  m  t  the  slope,  of  maximum  pbessoke.  The  triangular  prism  of  earth,  of  which 
omtis  a  section,  or  an  end  view,  is  called  the  prism  op  max  p^es  ;  because,  if  considered  as  a  wedge 
acting  against  the  back  of  the  wall,  it  would  produce  a  greater  pres  upon  it  than  would  the  entire 
triangle  c  m  s  of  earth,  considered  as  a  single  wedge.  For  although  the  last  is  the  heaviest,  yet  it  is 
more  supported  by  the  earth  below  it.  Calculation  shows  that  if  we  consider  the  earth  o  m  «  to  be 
thus  div  into  wedges  by  any  line  m  t,  the  wedge  that  will  press  most  against  the  waU  is  that  formed 
when  m  t  divides  the  angle  o  m  s,  or  the  arc  o  i,  into  two  equal  parts.    But  see  Art  11. 

Since  m  g  is  hor,  and  m  o  vert,  the  two  form  an  angle  of  90° ;  consequently  the  angle  of  max  pres 
is  plainly  found  by  taking  the  angle  smg  ot  nat  slope  from  90°,  and  div  the  rem  by  2.     Thus  a  nat 

slope  of  IM  to  1,  or  33^  41',  taken  from  90^,  leaves  56°  18';  and  ^Jt?l  ==  28°  9',  tlie  Cor- 
responding ang^le  o  m  t  of  max  pres. 

For  ease  of  calculation,  only  one  foot  of  the  length  of  the  wall,  and  of  its  backing,  is  usually  con- 
sidered. The  number  of  cuh  ft  of  wall,  or  of  backing,  is  then  equal  to  that  of  the  square  feet  in 
their  respective  profiles,  or  cross-sections. 

Now,  according  to  Moseley,  if  we  assume  the  particles  of  earth  composing  the 
backing  to  be  perfectly  dry,  and  devoid  of  cohesion,  (or  tendency  to  stick  to  each 
other,)  which  is  very  nearly  the  case  in  pure  sand;  and  if  we  suppose  the  wall  to  b© 
suddenly  removed,  then  the  triangle  of  earth  cmt,  comprised  between  the  slope  m t 
of  max  pres,  and  the  vert  back  c  m  of  the  wall,  Fig  6,  would  slide  down,  under  the  in- 
fluence of  a  force  which  may  be  represented  by  y  P,  acting  in  a  direction  y  P,  at  right 
angles  to  the  face  c  m  of  the  triangle  of  earth  ;  (or  in  other  words,  at  right  angles 
to  the  back  of  the  vert  wall,)  its  center  of  force  being  at  P,  distant  }/^  way  between 
m  and  c,  measured  from  the  bottom  ;  and  its  amount  equal  to  either  of  the  followiDg: 
Pert)  pres  ^^  ^^  '^  triangle  of  earth  c  m  t  X  o  t 
y  P  vert  depth  o  m 


STol. 


-;  or 


No  2. 


Perp  pres 
yP       = 


Wt  of  a  single  cm?)  ^  „  .y.  „  ^ 
ft  of  the  backing  X  ^^  ^^  ^  ^ 


Art.  11. 


In  view  of  the  great  uncertainty  involved  in  the  matter  of  the  actual  pressure  oi 
earth  against  retainiug-walls  in  practice  (see  Art  2.  ),  and  in  order  to  furnish 

a  simple  rule  which,  although  entirely  unsupported  by  theory,  is  still  (in  the  writer's 
opinion)  suflSciently  approximate  for  ordinary  practical  purposes,  we  shall  assume 
that  No  1  of  the  two  foregoing  formulas  applies  near  enough  to  walls  with  in- 
clined backs  c  r/i,  also,  as  Figs  7  and  8,  (precisely  as  they  are  lettered,)  at  least 
until  the  back  of  the  wall  inclines  forward  as  much  as  6  ins 
hor,  to  1  foot  vert,  or  at  an  angle  cmo  of  26°  34'.  What  follow^s  on 
retaining- walls  will  involve  this  incorrect  assumption,  and 
must  be  regrardecl  merely  as  g-iving*  safe  approximation. 

Some  appear  to  assume  this  perp  pres  to  be  the  only  one  acting  against  the  back 
of  the  wall ;  and  hence  arrive  at  erroneous  practical  conclusions.  For  when,  in 
order  to  prevent  this  force  from  causing  the  triangle  of  earth  to  slide,  we  place  a 
retaining-wall  in  front  of  it,  then,  instead  of  motion,  the  force  will  produce  pres  of 
the  earth  against  the  wall,  causing  friction  between  the  pressed  surfaces  of  th« 


608  RETAINING-WALLS. 

earth  and  wall.  That  is,  If  a  wall  were  to  begin  to  overturn  around  ita  toe  a  as  a 
fulcrum,  its  back  c  m  must  of  course  rise,  and  in  so  doing  must  rub  against  the 
earth  filling  in  contact  with  it ;  and  this  rubbing  would  evidently  act  to  impede  the 
or-erturuing.  So  long  ae  the  wall  does  not  move,  the  same  friction  assists  in  pre- 
venting overturning.  To  ascertain  the  amount  and  effect  of  this  friction,  let  z/P,  Fig 
8,  represent  by  scale  the  force  perp  to  the  back  c  m;  and  supposed  to  havb  been  pre- 
viC'isly  calculated  by. the  foregoing  formula  No  1.  Make  the  angle  ^P/  equal  to 
the  angle  of  wall  friction,*  draw  yf  at  right  angles  to 
2/  P,  or  parallel  to  m  c ;  make  P  x  equal  to  yf,  and  com- 
plete the  parallelogram  P  yfx.  Then  will  x  P  represent 
by  the  same  scale,  tlie  amount  of  tbe  friction 
agraiiist  the  back  of  tlie  wall. 


Hence  we  have  acting  at  P,  two  forces  ; 
namely,  the  perp  force  y  P,  and  the  friction  x  P ;  conse- 
quently, by  comp  and  res  of  force,  the  diag  /P  of  the 
parallelogram  P  yfx,  if  measured  by  the  same  scale,  will 
give  us  the  amount  of  their  resultant ;  ivliich  Is  the 
approx  single  theoretical  foree^  both  in 
amount  and  in  direction,  which  the  wall 
has  to  resist,  including  the  wall  friction. 

But  this  force, /P,  is  also  always  equal  to  the  perp 
force  y  P,  mult  by  the  nat  sec  of  the  angle  y  P/  of 
the  wall  friction ;  (or  divided  by  its  nat  cosine)  and  of 
course  may  be  ascertained  thus : 

wt  of  triangle  v  «  f  v  "'^^  **^  "/  angle  y  P  f  w^  °/  v  n  t 

Approx  theoreti-  _  c  m  t         A  o  t  A        of  wall  friction        _      cpit  "^  °* 

cal  pres  f  P  »er«  depth  o  m  cos  j  Pf  Xom 

Or  finally,  if  it  is  assumed,  as  we  do  throughout,  that  the  earth  is  perfectly  dry  (in 
nsmuch  as  its  pressure  is  then  the  greatest)  and  that  the  angles  of  nat  slope,  and 
of  wall  friction  are  then  each  33°  41'  or  1.5  to  1,  then  in  Figs  6,  7  and  8,  if  the  angle 
cm  o  betweea  thij  back  c  m  and  the  vert  o  m  does  not  exceed  about  26°  34'  we  may 
assume 

'^^PP'^'j^resflp  ***''**  =  wt  of  triangle  c  m  t  X  .643 

which  includes  the  action  of  the  friction  of  the  earth  against  the  back  of  the  wall. 

Rem.  1.    When  the  back  of  the  wall  is  offsetted  or  stepped,  as 

in  Fig  3,  instead  of  being  simply  battered,  as  in  Figs  7  and  8,  the  direction  of  the 
pres  of  the  earth  will  be  the  same  as  if  the  back  had  the  batter  i  n. 

Eem.  2.  Now  to  find  both  the  overturning'  tendency  of  the 
earth,  and  the  resistance  of  the  wall  against  being  overturned  around  its  toe  a  as 
a  fulcrum,  first  find  the  cen  of  grav  g  of  the  wall  and  through  it  draw  a 

vert  line  g  ?i.  Prolong /P  towards  v  and  draw  a  r  perp  to  it.  By  any  scale  make 
«o  =  wt  of  wall,  and  *i  =  calculated  pres  /P.  Complete  the  parallelogram  sino, 
and  draw  its  diagonal  sn,  which  will  be  the  resultant  of  the  pres/P  and  of  the  wt 
of  the  wall ;  and  should  for  safety  be  such  that  aj  be  not  less  than  about  one-fifth 
of  a  m,  even  with  best  masonry  and  unyielding  soil. '  Otherwise  the  great  pressure  so 
near  the  toe  a  may  either  fracture  the  wall  or  compress  the  soil  near  that  point 
so  that  the  wall  -^m  lean  forward.  In  walls  built  by  our  rule,  Art  1,  or  by  table, 
p  610,  aj  will  be  more  than  one-fifth  of  a  m.  The  pres  /  P  if  mult  by  its  leverage 
a  V  will  give  th j  moment  of  the  pres  about  a ;  and  the  wt  of  the  wall  mult  by  its 
leverage  e  a  will  give  that  of  the  wall.  The  wall  is  safe  from  overturning  in  pro- 
portion as  its  moment  exceeds  that  of  the  pres.  It  is  assumed  to  be  safe  against 
sliding,  breaking,  or  settling  into  the  coil. 

♦This  angrle  of  w^all  friction  is  that  at  which  a  plane  of  masonry  must 

b«inclin«"^  to  the  horizontal  so  that  dry  sand  or  earth  would  slide  down  it.  It  Js  about  the  same  aa 
the  nat  slope,  or  33°  41',  or  1.5  to  1 ;  and  its  nat  secant  is  1.202,  and  its  nat  cos  .832. 


RETAINING- WALLS. 


609 


Rem.  4.  If  the  eartki  slopes  downward  from  C,  as 
at  A  or  B,  instead  of  being  hor  as  in  Figs  6,  7,  8,  use  the  wt  of  the 
earth  c  m  n  instead  of  c  m  ^,  m  n  being  the  slope  of  max  pressure. 
In  A  the  point  of  application  will  still  be  at  P  (at  one-third  of 
m  c)  as  in  6,  7,  8 ;  but  in  B  it  will  be  a  little  higher  as  explained 
below  for  Fig  9. 

ISiircharg-ed  w^alls  are  those  in  which  the  earth  backing 
extends  above  the  tops  of  the  walls. 

According  to  theory,  when  as  in  Fig  9,  there  is  a  surcharge 
V  c  k  oi  backing,  sloping  away  from  c  at  its  natural  slope  c  v, 
the  max  pres  against  the  wall  is 
attained  when  the  earth  reaches  to 
the  level  of  d,  where  the  slope  m  t  d 
of  max  pres  intersects  the  face  of  the 
nat  slope  cv;  so  that  if  afterward  the 
earth  is  raised  to  v,  or  to  any  greater 
height,  no  additional  pres  is  thereby 
thrown  against  the  back  of  the  wall. 
So  also  if  the  earth  slopes  from  6,  or 
from  between  c  and  6,  except  that 
then  the  slope  m  d  of  max  pres  must 
extend  up  to  meet  this  other  slope. 

Tbe  approximate  amount 

of  the  oblique  pres,  when  the  wall  is 
surcharged,  (as  in  any  of  the  Figs  4, 
5,  9,)  may  be  found  on  the  same  prin- 
ciple as  when  the  earth  is  level  with 
its  top;  namely,  instead  of  the  trian- 
gle cmt  of  earth,  Figs  6,  7,  8,  9,  find 
the  wt  of  all  the  earth  d  s  rn  i,  Fig  4, 
dmtr,  Fig  5,  or  c  d  m,  Fig  9  (if  the 
surcharge  reaches  to  d  or  r,  or  higher), 
between  the  slope  m  d.  Fig  9,  m  t,  Figs  4  and  5,  of  max  pres,  the  back  of  the  wall,  and 
the  front  slope ;  omitting  any  which,  like  den.  Fig  5,  rests  on  the  top  of  the  wall 
(and  thus  adds  to  its  stability)  when  the  slope  starts  in  front  of  c.  Having  found 
this  weight,  then  for  dry  backing  the 


Pressure        \  _ 
approximately  j  ' 


'  Wt  of  the  earth  X  .643, 


including  the  action  of  the  friction  of  the  earth  against  the  back  of  the  wall ;  near 
enough  (in  the  writer's  opinion)  for  practical  purposes  in  so  uncertain  a  matter; 
but  essentially  empirical. 

The  direction  of  the  pressure  thus  found  will  be  the  same  as  when  the 
earth  is  level  with  the  top  be;  namely,  as  in  Figs  6  and  7,  first  draw  a  line,  as  Py, 
perp  to  the  back  C7?i,  whether  vert  or  inclined.  Then  draw  another  line,  as  P/, 
making  the  angle  y  P/=  trie  angle  of  wall  friction,  which  we  all  along  assume  to 
be  33°  41',  or  1.5  to  1.  Then  P/  will  give  the  direction  of  the  pressure.  But  its 
point  of  application  will  not  always  be  at  P  (one-third  of  the  height  of  the  wall 
above  m)  as  heretofore;  for  in  ail  cases  it  will  be  at  that  point  P,  or  at  some 
hig'her  one  as  h,  where  the  back  is  cut  by  a  line  Z  P  or  e  7i,  Fig  9,  drawn  from  the 
cen  of  grav  of  the  sustained  earth  (omitting  any  that  rests  immediately  on  the  top 
b  c),  and  parallel  to  the  slope  m  d  of  max  pres ;  and  such  a  line  will  strike  at  one- 
third  the  height  of  the  wall  only  when  the  sustained  earth  tcm  or  d cm  forms  a 
complete  triang;le,  one  of  whose  angles  is  at  the  inner  top  edge  c  of  the  wall. 
in  ail  other  cases  said  line  for  a  surcharge  will  strike  above  P. 


W 


610 


KETAINING-WALLS. 


Art.  7.  On  page  603,  Fig  1,  we  recommend  that  the  base  o  s  at  the  ground- 
line  of  well  built  vertical  Walls  should  not  be  less  than  .35,  or  A,  or  .5  of  the 
height  d  s  above  said  line,  depending  on  the  kind  of  masonry.  But  a  wall  with  a 
battered  (inclined)  front  or  face  as  found  by  Art  8,  (by  which  the  following 

table  was  prepared),  will  be  as  strong,  and  at  the  same  time  contain  less  masonry 
than  a  vert  wall,  although  the  battered  one  will  have  the  thickest  base  o  s. 

Table  3,  of  thicknesses  at  base  o  s,  Figr  1,  and  at  top  c  d^  of 
walls  with  battered  faces,  so  as  to  be  as  strong  as  vertical 
ones  which  contain  more  masonry. 

For  the  cub  yds  of  masonry  above  o  s  per  foot  run  of  wall,  mult  the 
square  of  the  vert  height  d  s  by  the  number  in  the  column  of  cub  yds.  ihen 
a^  the  foundation  masonry  below  o  s. 

(Original.) 


All  the  M 

mils  helow  have  the 

All  the  walls  below  have  the 

AH  the  walls  below  have  the  same  strength 

a  vert  one 

same   strength   as    a  vert 

as  a  vert  one  whose  base  o  s, 

figl=.35 

base   o  s 

fig  1=.4 

one  whose  base  os,  fig  1  = 

of  its  htds 

of  itsht  ds. 

.5  of  its  ht  d  s. 

Cut  stone 

Mortar  rubble. 

Dry  rubble. 

Batter,  in 
ins  to  a  ft. 

Base,  in 

pts  of 

ht. 

Top,   in 

pts  of 

ht. 

0  yds  per 
ft  run. 

Base,  in 

pts  of 

ht. 

Top,    in 

pts  of 

ht. 

0  yds  per 
ft  run. 

Base,  in 
pts  of 

ht. 

Top,    in 

pts  of 

ht. 

0  yds  per 
ft  run. 

0 

350 

.350 

.01296 

.400 

.400 

.01482 

.600 

.500 

.01852 

^ 

352 

.810 

.01226 

.401 

.359 

.01407 

.501 

.459 

.0177a 

.355 

.270 

.01158 

.403 

.320 

.01339 

.503 

.420 

.0170^ 

V^ 

.359 

.234 

.01098 

.408 

.283 

.01280 

.506 

.381 

.01645 

.364 

.197 

.01089 

.413 

.246 

.01220 

.510 

.343 

.01580 

fi 

.371 

.163 

.00989 

.419 

.210 

.01165 

.516 

.308 

.01526 

.379 

.129 

.00941 

.425 

.175 

.01111 

.522 

.272 

.01470 

fA 

.389 

.096 

.00898 

.435 

.143 

.01070 

.528 

.236 

.01415 

.400 

.066 

.00863 

.445 

.110 

.01028 

.537 

.204 

.01372 

5 

.425 

.007 

.00800 

.468 

.051 

.00961 

.555 

.138 

.01283 

Triangle 

.429 

.000 

.00794 

.490 

.000 

.00907 

.612 

.000 

.01133 

Moselev  and  others  quote  Gadroy,  for  a  dry  sand  sloping  at  21°.  It  would  be  better  to  cease  from 
circulating  such  evident  mistakes.'  Dry  sand  will  stand  at  no  less  angle  for  a  savant  than  for  any 
body  else.  For  practical  purposes,  we  may  say  that  dry  sand,  gravel  and  earths,  slope  at  33^  41'  or 
m  to  1 ;  as  abundant  experience  on  railroad  embkts  proves.  Poncelet  gives  tables  for  walls  to  sup- 
port  dry  earth  sloping  at  1  to  1,  or  45^:  but  as  we  do  not  believe  in  the  existence  of  such  earth,  we 
omit  such  tables.  Sand,  gravel,  and  earths  may  be  moistened  to  diff  degrees,  so  as  to  stand  at  any 
angle  between  hor  and  vert;  and  by  moistening  and  ramming,  the  earths  may  be  converted  into  com- 
pact masses,  exerting  little  or  no  pres  ;  and  may  even  so  continue  after  they  become  dry  ;  being  then, 
in  fact,  a  itind  of  air-dried  brick.  It  is  sometimes  difficult  to  know  whether  earth  or  sand  is  perfectly 
dry  or  not;  and  an  exceedingly  small  degree  of  moisture  will  cause  them  to  stand  at  1  to  1,  in  smaU 
heaps,  such  as  have  probably  been  observed  by  the  authorities  on  the  subject.  The  writer  found  that 
fine  sand  from  the  sea-shore,  and  under  cover,  would  stand  at  l}4  to  1  during  warm  dry  weather,  and 
at  1  to  1  when  the  air  was  damp.  Yet  no  difif  whatever  in  its  degree  of  moisture  was  perceptible  to 
the  feeling.  Its  susceptibility  to  dampness  was  of  course  owing  to  salt.  A  few  handfuls  of  dry  earth 
may  perhaps  be  coquetted  into  standing  at  1  to  1  on  a  table ;  but  so  far  as  our  observation  extends, 
when  it  is  dumped  in  large  quantities  from  carts  and  wheelbarrows,  its  slope  is  about  IJ^  to  1 ;  and 
this  we  consider  the  proper  one  to  be  used  in  practical  calculations,  where  safety  is  the  consideration 
of  paramount  importance. 

The  less  the  nat  slope^  the  g-reater  is  the  pres :  and  since  the 
slope  is  least  when  the  backing  is  perfectly  dry,  (omitting  of  course  its  condition 
when  so  absolutely  wet  as  to  become  partially  fluid.)  we  have,  on  the  score  of  safety, 
confined  our  tables  to  dry  backing.  As  stated  in  Art  1,  we  cannot  recommend  dimen- 
sions less  than  those  there  given,  when  we  consider  the  rough  treatment  to  which 
masonry  is  exposed  on  public  works. 

In  carrying  a  road  along:  dangerous  precipices,  we  should 
rather  be  tempted  at  times  to  make  thicker  vralls.  We  imagine,  for  instance,  that 
the  centrifugal  force  of  a  heavy  train,  whirling  around  a  sharp  curve,  convex  on  the 
dangerous  side,  should  not  be  overlooked  in  designing  walls  for  such  localities.  This 
force  is  hor;  and  is  applied  near  the  top  of  the  wall :  and,  consequently,  its  leverage 
may  be  considered  as  equal  to  the  height;  whereas  the  theoretical  pres  of  the  earth 
is  oblique  ;  and  is  applied  at  }^  of  the  height  from  the  bottom  ;  so  that  its  leverage 
about  the  toe  of  the  wall  is  very  short.  Moreover,  the  simple  weight  of  the  train,  pro- 
duces pres  against  the  wall ;  as  well  as  that  of  the  backing.  All  such  considerations 
are  omitted  by  theorists.    The  dangerous  pres  caused  by  tremors,  &c,  cannot  be 


RETAINING-WALL8. 


611 


assumed  to  be  applied  at  }^  of  the  height  from  the  bottom ;  nor  indeed,  can  it  be 
calculated  at  all. 

Rem.  2.  Wharf  walls  are  an  instance  where  the  thickness  should  be  increased, 
notwithstandinji;  that  the  pres  of  the  water  in  front  helps  to  sustain  them.  The  earth 
behind  such  walls,  is  not  only  liable  to  be  very  heavily  loaded  when  vessels  are  dis- 
charging; but  is  apt  to  become  saturated  with  water,  especially  below  low-water 
level ;  and  thus  to  exert  a  very  great  pres  again.st  the  walls.  Moreover,  the  water 
gets  under  the  wall ;  and  by  its  upward  pressure  virtually  reduces  its  weight,  and 
consequently  its  stability.  The  same  cause  of  course  diminishes  the  friction  of  the 
wall  upon  its  base.  Such  walls  are,  therefore,  very  liable  to  slide,  if  the  foundation 
is  smooth,  and  horizontal ;  and  have  done  so  even  when  the  foundation  had  a  con* 
siderable  inclination  backward,  as  in  Fig  1.     See  Art  9, 

Rem.  3.  A  retaining-wall  is  usually  in  greater  danger  for  a  few  months  after  its  completion,  than 
after  time  has  been  allowed  for  the  mortar  to  harden  perfectly ;  and  for  the  backing  to  settle.  When 
there  are  suspicions  of  the  safety  of  a  new  wall,  it  would  be  well  to  place  strong  temporary  shores 
against  it,  at  about  3^  to  3^  of  its  height  above  ground.  In  some  cases,  permanent  buttresses  of 
masonrv  may  be  built  for  the  purpose.     They  should  be  well  bonded  into  the  wall. 

Rem.  4.  The  pres  of  the  earth  backing  will  be  much  reduced,  if  the  first  few  feet  of  its  height  be 
made  up  in  thin  hor  layers,  to  be  consolidated  by  being  used  by  the  masons  instead  of  scaffolding;  as 
shown  at  ft,  Fig  1.    Frequently  this  can  be  done  without  inconvenience  ;  and  at  very  trifling  cost. 

Art.  8.  To  change  a  vert  retaining-wall,  into  one  with  a 
battered  face,  which  shall  present  an  eqnal  resistance 
ag-ainst    overturning;    although    requiring   less    masonry* 

This  is  sometimes  termed  a  transformation  of  profile.    (Original.) 

Let  aboi,  Pig  10,  be  the  vert  wall.  Mult  its  ba8« 
01,  by  1.225 ;  (1.22475  is  nearer ;)  the  prod  will  be  the 
base  o  e,  of  a  triangular  wall  boe,  possessing  the 
same  stability ;  and  yet  not  requiring  much  more 
than  half  the  masonry  of  the  vert  one.  See  Rem  1. 
This  being  done,  suppose  a  wall  to  be  desired  with  a 
face  batter,  of  say  3  ins  to  a  ft ;  or  1  in  4.  From  the 
point  w,  where  the  face  of  the  triangular  wall  inter- 
sects that  of  the  vert  one,  step  off  vert  any  4  short 
equal  spaces ;  aiyi  from  the  upper  one  m,  step  oflF  one 
space  hor,  to  v.  Through  v  and  n  draw  the  dotted 
line  s  t,  which  evidently  will  batter  1  in  4.  Then  is 
b  s  t  0  approximately  the  reqd  wall ;  but  a  little 
thicker  than  necessary.  To  reduce  it,  from  t  draw 
the  dotted  line  t  b.  Mark  the  point  c,  where  this 
line  intersects  the  face  a  i,  of  the  vert  wall;  and 
through  c  draw  d  I,  parallel  to  s  t.  Then  is  b  dl  o 
the  reqd  wall.  Our  fig  is  drawn  in  an  exaggerated 
manner,  so  as  to  avoid  confusion,in  the  lines.  The 
base  o  e  of  the  triangular  wall,  would  not  in  reality 
be  near  so  great  as  it  is  represented. 

It  will  be  observed  that  as  the  base  increases,  the  quantity  of  masonry  diminishes. 

Rem.  1.  The  battered  wall  will  in  fact  be  safer  than  the  vert 
one.  The  battered  wall  has  the  same  moment  of  stability  as  the  vert  one  ;  and  the 
pres  of  the  earth  against  it  also  remains  unchanged,  but  the  moment  or  tendency  of  the  pies  to  upset 
the  wall  has  become  less.  For  let  a 6  win,  Fig  11,  represent  a  vertical  wall;  and /o  the  amownt and 
direction  of  pres  behind  it.  (For  ease  of  illustration,  we  have  placed  o  above  the  true  cen  of  pres  of  the 
3anh  filling,  which  would  be  at  one-third  of  a  n  above  n.)  V 
iiends  to  overturn  the  wall  around  its  toe  m,  is  the  dist  m  s, 
measured  from  the  toe  or  fulcrum  m,  and  at  right  angles  to 
ihc  direction  fosc  of  the  pres  ;  and  this  leverage  mult  by 
the  force  /  o,  gives  the  overturning  tendency  or  moment  of 
said  force.      See  "  Moments  and  leverage."  Again, 

let  any,  represent  a  triangular  wall  of  the  same  stability 
as  the  other,  as  found  by  our  rule.  Here  we  still  have  the 
same  »mount/o.  and  direction  fo  $  c,  of  pres  force  against 
the  wall;  but  it  now  acts  to  overturn  the  wall  any 
around  the  toe  y ;  and  therefore,  with  the  reduced  leverage 
y  c.  Consequently,  its  overturning  tendency  is  less  than 
before.  Therefore,  in  ordinary  language,  we  may  say  that 
the  wall  is  stronger  than  before,  although  its  moment  of 
stability,  or  standing  tendency,  has  in  itself  undergone  no 
change."  If  the  pres/o  against  the  vert  back  were  hor,  as 
in  the  case  of  water,  then  its  leverage  would  evidently  be 
the  same  in  both  walls;  and  the  proportion  between  the 
overturning  moment  of  the  pres,  and  the  momenta  of 
stability  of  the  two  walls,  would  be  constant. 

Rem.  2.  In  attempting  to  reduce  the  masonry  by  adopt- 
ing a  wall,  o  h  e.  Fig  10.  of  a  triangular  section ;  or  of  one 
nearly  approaching  a  triangle,  special  attention  should  be 
given  to  the  quality  of  the  masonry  near  the  thin  toee; 
which  will  otherwise  be  apt  to  crack,  or  fail  under  the  pres. 


Fig/0 


,  the  leverage  with  which  this  pres 


612  RET  AINING- WALLS. 

Rem  3.  Morkovkb,  when  common  mohtar  is  itsed  without  an  admixture  op  cement,  which  it  never 
should  be,  la  retaiuiug-wallSjWheredurability  is  au  object,  a  great  batter  is  objeo* 
tioaable ;  inasmuch  as  the  rain,  combined  with  frost,  &c,  soon  destroys  the  mor- 
tar.  Insuchcases,  therefore,  the  baiter  should  not  exceed  1  or  13^  ins  to  a  ft;  and 
even  then,  at  least  the  pointing  of  the  joints,  and  a  few  feet  in  height  of  both 
the  upper  and  the  lower  courses  of  masonry,  should  be  done  with  cement,' or 
cement-mortar.  We  have  observed  a  most  marked  ditf  in  the  corrosion  of  the  mor« 
tar,  where,  in  the  same  walls,  with  the  same  exposure,  one  portion  has  been  built 
with  a  vert  face ;  and  another  with  a  batter  of  but  1}4  inch  to  a  foot.  Common 
mortar  will  never  set  properly,  and  continue  firm,  when  it  is  exposed  to  mois- 
ture from  the  earth.  This  is  very  observable  near  the  tops  and  bottoms  of 
abuts,  retaining- walls,  &c;  the  lime-mortar  at  those  parts  will  generally  be 
found  to  be  rendered  entirely  worthless.  A  profile  somewhat  like  Fig  12,  may 
at  times  prove  serviceable,  instead  of  the  triangular.  This  is  the  form  of  the 
Gothic  buttress;  which  probably  had  its  origin  in  the  cause  just  spoken  of. 


Art.  9.    A  retai]iitig--wall   may  slide,  without 

T?'^   1Q        losing^  its  verticality  ;  and,  indeed,  without  any  danger 

X  Jp    IX'       of  being  overturned.    This  is  very  apt  to  occur  if  it  is  built  upon 

*-^  a  hor  wooden  platform ;   or  upon  a  level  surf  of  rock,  or  clay, 

Without  other  means  than  mere  friction  to  prevent  sliding.    This  may  be  obviated 

by  inclining  the  base,  as  in  Fig  1 ;  by  founding  the  wall  at  such  a  depth  as  to  pro- 

Sride  a  proper  resistance  from  the  soil  in  front;  or  in  case  of  a  platform,  by  securing 

one  or  more  lines  of  strong  beams  to  its  upper  surf,  across  the  direction  in  which 

sliding  would  take  place.    On  wet  clay,  friction  mav  be  as  low  as  from  .2  to 

}/3  the  weight  of  tne  wall ;  on  dry  earth,  it  is  about  }^  to  ^ ;  and  on  sand  or  gravel, 

about  %  to  %.    The  friction  of  masonry  on  a  wooden 

IcI  Icl  [c]         platform,  is  about  -6^.  of  the  wt,  if  dry ;  and  %  if  wet. 

r_ I  Counterforts,  shown  in  plan  at  c  c  c,  Fig  13,  consist  in 

an  increase  of  the  thickness  of  the  wall,  at  Us  back,  at  regular  inter- 

'IJ^'      4  O  '^^^^  ®^  ^^^  length.    We  conceive  them  to  be  but  little  better  than  a 

Jl  ICI  J.5  waste  of  masonry.    When  a  wall  of  this  kind  fails,  it  almost  in- 

J    '  variably  separates  from  its  counterforts ;  to  which  it  is  connected 

merely  by  the  adhesion  of  the  mortar ;  and  to  a  slight  extent,  by  the 

bonding  of  the  masonry.    The  table  in  Art  7  shows  that  a  very  small  addition  to  the  base  of  a  wall,  is 

attended  by  a  great  increase  of  its  strength ;  we  therefore  think  that  the  masonry  of  counterforts 

would  be  much  better,  and  more  chea^ply  employed  in  giving  the  wall  an  additional  thickness,  along 

Its  entire  length ;  and  for  the  lower  third  of  its  height.     Counterforts  are  very  generally  used  ia 

retaining-walls  by  European  engineers;  but  rarely,  if  ever,  by  Americans. 

Buttresses  are  like  counterforts,  except  that  they  are  placed  in  front  of  a  wall  instead  of  be- 
hind it ;  and  that  their  profile  is  generally  triangular,  or  nearly  so.  They  gf  eatly  increase  its  strength  ; 
but,  being  unsightly,  are  seldom  used,  except  as  a  remedy  when  a  wall  is  seen  to  be  failing. 

liand-ties,  or  long  rods  of  iron,  have  been  employed  as  a  makeshift  for  upholding  weak  re* 
taining-walls.  Extending  through  the  wall  from  its  face,  the  land  ends  are  connected  with  anchors 
of  masonry,  cast-iron  or  wooden  posts;  the  whole  being  at  some  dist  below  the  surface. 

Retaining*  walls  with  curved  profiles  are  mentioned  here  merely  to  cau- 
tion  the  young  engineer  against  building  them.  Although  sanctioned  by  the  practice  of  some  high 
authorities,  they  really  possess  no  merit  sufficient  to  compensate  for  the  additional  expense  and  trou- 
ble of  their  construction. 

Art.  10.     Among  military  men,  a  retaining-wall  is  called  a  revetment.     When  the 

earth  is  level  with  the  top,  a  scarp  revetment;  when  above  it,  a  counterscarp 

revetment,  or  &  demi-revetment.  When  the  face  of  the  wall  is  battered,  a  sloping;  and  when  the  back 
ts  battered,  a  coxmtersloping  revetment.    The  batter  is  called  the  taluS. 

Art.  11.  The  pres  against  a  wall  Fig  6,  from  sand  etc  level  with  its  top,  is  not 
diminished  by  reducing  the  quantity  of  sand,  until  its  top  width  c  s  becomes  less  than 
that  (c  t)  pertaining  to  the  angle  cmt  of  maximum  pres.  The  pres  then  begins  to  di- 
minish, but  in  practice  the  diminution  is  not  appreciable  until  the  width  is  reduced  to  about 
one  sixth  of  that  (c  s)  pertaining  to  the  angle  cms  of  natural  slope,  or  about  half  of 
e  t.    The  pres  then  begins  to  decrease  rapidly  as  the  width  is  further  reduced. 

Table  4,  of  contents  in  cub  yards  for  each  foot  in  leng-th 
of  retaining;-walls,  with  a  thickness  at  base  equal  to  .4  of  the  vert  height. 
if  the  back  is  vert.    If  the  back  is  stepped  according  to  the  rule  in  Art  3,  the 

proportionate  thickness  at  base  will  of  course  be  increased.  Face  batter,  1}/^  inches 
to  a  foot;  or  J/g^h  of  the  height.  Back  either  vert,  or  stepped  according  to  the  rule 
in  Art  3,  Fig  3.  The  strength  is  very  nearly  equal  to  ihat  of  a  vert  wall  with  a 
base  of  .4  its  height.  Experience  has  proved  that  such  walls, 

when  composed  of  well-scabbled  rnortar  rubble,  are  safe  under  all  ordinary  circum- 
stances for  earth  level  with  the  top.  Steps  or  offsets,  o  e,  at  foot,  Fig  1,  are  not  here 
included. 


STONE   BRIDGES. 


613 


TABIiE  4.    (Original.) 


Ht. 

Cub. 

Ht. 

Cub. 

Ht, 

Cub. 

Ht. 

Cub. 

Ht. 

Cub. 

Ht. 

Cub. 

Ft. 

Yds. 

Ft. 

Yds. 

Ft. 

Yds. 

Ft. 

Yds. 

Ft. 

Yds. 

Ft. 

Yds. 

1 

.013 

lOJ^ 

1.38 

20 

5.00 

29V«f 

L  10.9 

48 

28.8 

74 

G8.5 

^ 

.028 

11 

1.51 

% 

5.25 

30 

11.3 

49 

30.0 

76 

72.2 

2 

.050 

3^ 

1.65 

21 

5.51 

31 

12.0 

50 

31.3 

78 

76.1 

^ 

.078 

■12 

1.80 

H 

5.78 

32 

12.8 

51 

32  5 

80 

80.0 

3 

.113 

^ 

1.95 

22 

6.05 

33 

13.6 

52 

33.8 

82 

84.1 

}^ 

.153 

13 

2.11 

Vi 

6.33 

34 

14.5 

53 

35.1 

84 

88.4 

4 

.200 

H 

2.28 

23 

6.61 

35 

15.3 

54 

36.5 

86 

92.5 

M 

.253 

14 

2.45 

>^ 

6.90 

36 

16.2^ 

55 

37.8 

88 

96.8 

5 

.313 

% 

2.63 

24 

7.20 

37 

17.1 

56 

39.2 

90 

101.3 

^ 

.378 

15 

2.81 

}4 

7.50 

38 

18.1 

57 

40.6 

92 

105.8 

6 

.450 

^ 

3.00 

25 

7.81 

39 

190 

58 

42.1 

94 

110.5 

J^ 

.528 

16 

3.20 

J4 

8.13 

40 

20.0 

59 

43.5 

96 

115.2 

7 

.613 

3.40 

26 

8.45 

41 

21.0 

60 

45.0 

98 

120.1 

3^ 

.703 

17 

3.61 

^A 

8.78 

42 

22.1 

62 

48.1 

100 

125.0 

8 

.800 

34 

3.83 

27 

9.12 

43 

23.1 

64 

51.2 

102 

130.1 

J^ 

.903 

18 

4.05 

H 

9  45 

44 

24.2 

66 

54.5 

104 

135.2 

9 

1.01 

H 

4.28 

28 

9.80 

45 

25.3 

68 

57.8 

106 

140.5 

M 

1.13 

19 

4.51 

>^ 

102 

46 

26.5 

70 

61.3 

10 

1.25 

H 

4.75 

29 

10.5 

47 

27.6 

72 

64.8 

STONE  BRIDGES. 


Art.  1.  In  an  arch  sts,  Fig  1,  the  dist  eo  is  called  its  span;  ia  its  rise;  t  its 
crown  ;  its  lower  boundary  line,  eao,  its  soffit,  or  intrados  ;  the  upper  one, 
Vtr,  its  back,  or  extrados.  The  terms  soflBt  and  back  are  also  applied  to  the 
entire  lower  and  upper  curved  surfaces  of  the  whole  arch.  The  ends  of  an  arch,  or 
the  showing  areas  comprised  between  its  intrados  and  extrados,  are  its  faces ;  thus 
the  area  sis  a  is  a  face.  The  inclined  surfaces  or  joints,  re,  ro,  upon  which  the/ee< 
of  the  arch  rest,  or  from  which  the  arch  springs,  are  the  skei¥backs.  Lines 
level  with  e  and  o,  at  right  angles  to  the  faces  of  the  arch,  and  forming  the  lower 
edges  of  its  feet,  (see  nn.  Fig  2^,)  are  the  springing-  lines,  or  spring's.  The 
blocks  of  which  the  arch  itself  is  composed,  are  the  areb-stones,  or  voussoirs. 
The  center  one,  ta,  is  the  keystone;  and  the  lowest  ones,  ss,  the  spring'ers. 
The  term  archblock  might  be  substituted  for  voussoir,  and  like  it  would  apply  to 
brick  or  other  material,  "as  well  as  to  stone.  The  parts  tr,  tr,  are  the  haunches  ; 
and  the  spaces  tr  I,  trb,  above  these,  are  the  spandrels.  The  material  deposited 
in  these  spaces  is  the  spandrel  filling^ ;  it  is  sometimes  earth,  sometimes  ma- 
sonry ;  or  partly  of  each,  as  in  Fig  1. 

In  large  arches,  it  often  consists  of  several  parallel  spandrel-walls,  tl.  Fig  23^,  running  lengthwise 
of  the  roadway,  or  astraddle  of  the  arch.  They  are  covered  at  top  either  by  small  arches  from  wall  to 
■wall,  or  by  flat  stones,  for  supporting  the  material  of  the  roadway.  They  are  also  at  times  connected 
together  by  vert  cross-walls  at  intervals,  for  steadying  them  laterally,  as  at  tt,  Fig  23^.  The  parts 
gpen,  gpon,  Fig  1,  are  the  abutments  of  the  arch;  en,  on,  the  faces;  gp,  gp,  the  backs;  and 
pn,  pn,  the  bases  of  the  abuts.  The  bases  are  usually  widened  by  feet,  steps,  or  offsets,  d  d,  for  dis- 
tributing the  wt  of  the  bridge  over  a  greater  area  of  foundation  ;  thus  diminishing  the  danger  of  set- 
tlement.   The  distance  t  a  in  any  arch-stone,  is  called  its  depth. 

The  only  arches  in  common 
use  for  bridges,  are  the  circular, 
(often  called  segmental);  and 
the  elliptic. 

Art.  2.  To  find  the 
depth  of  keystone  for 
first-class  cut -stone 
arches,  whether  cir- 
cular or  elliptic."^ 

Find  the  rad  c  o.  Fig  1,  which 
will  touch  the  arch  at  o,  a,  and 
c.  Add  together  this  rad,  and 
half  the  span  o  t.  Take  the  sq 
rt  of  the  sum.  Div  this  sq  rt 
by  4.  To  the  quot  add  5^^  of  a 
ft.    Or  by  formula, 


Fitfl 


*  Inasmuch  as  the  rules  which  we  give  for  arches  and  abuts  are  entirely  original  and  novel,  it  may 
not  be  amiss  to  state  that  they  are  not  altogether  empirical ;  but  are  based  upon  accurate  drawinji 


614 


STONE   BRIDGES. 


Depth  of  key  _      V Rad  -\-  half  span      i  q  9  fg^i 
in  feet       ~  4  -r    -y 

For  second-class  work,  this  depth  may  be  increased  about  3>^th  part;  or 
for  brick  or  fair  rubble,  about  J^rd.    See  table  of  Keystones. 

In  large  arches  it  is  advisable  to  increase  the  depth  of  the  archstones  toward  the 
springs  ;  but  when  the  span  is  as  small  as  about  60  to  80  or  100  feet,  this  is  not  at  all 
necessary  if  the  stone  is  good ;  although  the  arch  will  be  stronger  if  it  is  done.  Iq 
practice  this  increase,  even  in  the  largest  spans,  does  not  exceed  from  }^  to  3^  the 
depth  of  the  key ;  although  theory  would  require  much  more  in  arches  of  great  rise. 

Rem.  To  find  tlie  rad  c  o,  whether  the  arch  be  circular  or  elliptic.  Square 
half  the  span  e  0.  Square  the  whole  rise  i  a.  Add  these  squares  together;  div  the 
sum  by  twice  the  rise  i  a.  Or  it  may  be  found  near  enough  for  this  purpose  by  the 
dividers,  from  a  small  arch  drawn  to  a  scale. 


Amount  of  pressure  sustained  by  arclistones.  In  bridges  of 
the  same  width  of  roadway ;  if  all  the  other  parts  bore  to  each  other  the  same  propor- 
tion as  the  spans,  the  total  pres  would  increase  as  the  squares  of  the  spans,  while  the' 
pressure  per  square  foot  would  increase  as  the  spans.  But  in  practice  the  depth  of  the 
archstones  increases  much  less  rapidly  than  the  span ;  while  the  thickness  of  the 
roadway  material,  and  the  extraneous  load  per  sq  ft,  remain  the  same  for  all  spans. 
Hence  the  total  pressures,  at  key  and  at  spring,  increase  less  rapidly  than  the  squares 
of  the  spans  ;  but  more  rapidly  than  the  simple  spavs ;  as  do  also  the  pressures  per 
square  foot.  Thus  in  two  bridges  of  the  same  width,  but  with  spans  of  100  and  200  ft, 
with  depths  of  archstones  taken  from  our  table  and  uniform  from  key  to 

spring;  supposed  to  be  filled  up  solid  with  masonry  of  160  lbs  per  cub  ft,  to  a  level  of 
about  15  Inches  above  the  crown,  (including  the  stone  paving  of  the  roadway);  with 
au  extraneous  load  of  100  lbs  per  sq  ft;  the  pressures  will  be  approximately  as  fol- 
lows: 


Span  100  ft.                         1 

Span  200  ft. 

AT  KEY. 

AT  SPRING.        II 

AT  KEY.           1 

AT  SPRING. 

For  1  ft  in 
width  of 
its  entire 
depth. 

Per  sq  ft. 

For  1  ft  in 
width  of 
its  entire 
depth. 

Per  sq  ft. 

For  1  ft  in 
width  of 
its  entire 
depth. 

Per  sq  ft. 

For  1  ft  in 
width  of 
its  entire 
depth. 

Per  sq  ft. 

Rise. 

Tons. 

Tons, 

Tons. 

58 

Tons. 

18?i 

Tons. 
126 

Tons, 
29M 

Tons. 
179 

Tons. 
42 

^ 

86% 

12^ 

57 

19 

112 

'-^7^ 

181 

44 

i 

18 

11 

9 

57« 
67>^ 

20 

22>i 

25 

97 
80M 

57^ 

24}^ 

21 

15}^ 

]88 
207 
230 

47^ 
54>^ 
61}^ 

It  will  be  seen  that  with  the  same  span,  the  pres  at  the  key  becomes  less,  while  that 
at  the  spring  becomes  greater,  as  the  rise  increases.  Also  that  when  the  archstones 
are  of  uniform  deptii,  the  pres  at  either  spring  of  a  semicircular  arch  is  about  4  times 
as  great  as  at  the  key  ;  whereas  when  the  rise  is  but  one-sixth  of  the  span,  the  pres  ac 
spring  averages  but  about  one-third  greater  than  at  the  key.  These  proportions  vary 
feomewhat  in  different  spans. 

The  greater  pres  per  sq  ft  at  the  springs  may  be  reduced  by  increasing  the  depth  of 
the  archstones  towards  the  springs.  This  however  is  not  necessary  in  moderate  spans, 
inasmuch  as  good  stone  will  be  safe  even  under  this  greater  pres. 

By  usini^  parallel  spandrel  ivalls,  see  Fig  2^,  or  by  partly  fill- 

ing with  earth  instead  of  masonry,  the  pres  on  the  archstones  may  be  diminished, 
say,  as  a  rough  average,  about  \  part. 

and  calculations  made  by  the  writer,  of  lines  of  pres,  &c,  of  arches  from  1  to  300  ft  span,  and  of  every 
rise,  from  a  semicircle  to  j^^  of  the  span.  Prom  these  drawings  he  endeavored  to  find  proportions 
which,  although  they  might  not  endure  the  test  of  strict  criticism,  would  3till  apply  to  all  the  cases 
•ritli  an  accuracy  sufiBcient  for  ordinary  practical  purposes. 


STONE   BRIDGES. 


615 


Table  1.    Of  some  existing'  arches,  -with  both  their  actual  and  their 

calculated  depths  (by  our  rule)  of  keystone.  Where  two  depths  are  given  in  the  column  of  keys  the 
smallest  is  for  first  class  cut-stone,  and  the  largest  for  good  rubble,  or  brick.  Those  also  which  are 
not  specified  are  of  first-class  cut-stone.  C  stands  for  circular,  E  for  elliptic.  For  2d  class  work,  add 
about  J^th  part;  and  for  brick,  or  fair  rubble,  about  34tb. 


^  ^    i^    K!    &ii    rrtrri  *^ 


^  X«O0C«C 


fc   ■*  Tjt  ■* -*  ■*    ,-1      "^    M -W!  eo    •^    c4  eo  CO  i4   -^ 


^.     "5     t'*?     oq>ONoo         a  Si  Si 

rv,    -*'  O  C^'  X  O     t-^        £J    S  ^'  '*'     ©     M  «'  X  'r. 


***vft-*N05pH     CO         «  COCO     — <     rHCOCOi-l     rH     r-  I>«  i-l     rH     «      M     rt  r- 


I  e^oioio^    S      ■*    e^  ^4  ^    rH    »3j^S   *    t-t-t-    t-   S    «   §   *    tOta    lO    in   ■*    -^  eo  n 


OOKHO    O      K    CKO    O        OOO  O    OOO    B    K    O   O    O    HO    W    U   O    OOK 


II 

*^0  so-g  a    « 


ill 


c 

:  1 

:  1 

!    c 

il 

:  p: 
:  [c 

H 

c 

E 

'c 

1 

c 

t 

C 

J! 

K     £f"^    «. 


.•2  «        So  o^ 


s    3   .2  -S.  ' 


.  S:  p3    ;« 


CB  0 


o>    li 


fl      C3      -  ti  e  c3       J? 


.2    a    c  «  a 


•  fc  S  £   &^  - 

S,   -S    Eh   ^     <     . 
S    5     .  -S     ??S 


•S  .>; 


K    P3    OS    M   U    Sj 


^  i  § 

«!     Hj    Eh 


*  See  Experimental  Arch  at  Souppes,  France. 

t  Suines  bridge.    Some  authorities  give  2  ft  4  ins  as  the  depth  at  k 


616 


STONE   BRIDGES. 


Experimental  Arcb  at  Souppes,  France.     See  Table, 

Span  =  about  18  X  rise. 


Span. 

Eise. 

Radius  of 
intrados. 

Depth  of  arch-stones 

at  spring. 

at  key 

Width. 

on  faces. 

betw  faces. 

Meters 

37.886 
124.30 

2.125 

85  5 

1.10 
3.61 

1.10 

0  80 

3  5 

Feet 

6.97 

280.52 

3.61 

2.624 

11.5 

Arch  of  granite.  The  centers  rested  (for  four  months)  on  sand  in  16  cylinders,  1  ft 
diameter,  1  ft  high,  of  g'^-inch  sheet  iron.  The  unloaded  arch  settled  15  millimeters 
(0.59  inch)  on  striking  the  centers.  The  additional  settlements  under  extraneous 
loads  were  as  follows: 


Extraneous  load. 

Increase  of  settlement. 

Kilograms. 

Pounds. 

Millimeters. 

Inches. 

Distributed 

3670fX) 

4975 

132600 

809000 

11000 

292000 

21 
0.3 
1.2 

0.8 

Center 

0.012 

Distributed 

0.047 

With  the  distributed  load  of  367000  kilog,  a  load  of  4975  kilog,  falling  0.3  m  (11.8 
ins)  on  key,  caused  vibrations  of  2.8  mm  (OJl  inch).  Annates  des  Fonts  et  Chaussees, 
1866  Part  2,  1868  Part  2.      ^ 

The  ureh  on  the  BotrBBOnvAis  Bailwat,  is  probably  the  boldest;*  and  ths  GABnt  Jomr  asch,  by 

Capt,  now  Gen'l  M.  O.  Meigs,  U  S  Army,  the  grandest  stone  one  in  existence.  Pomt-t-Pktdd,  in 
Wales,  is  a  common  road  bridge,  of  very  rude  construction  ;  with  a  dangerously  steep  roadway.  It 
was  built  entirely  of  rubble,  in  mortar,  by  a  common  country  mason,  in  1750  ;  and  is  still  in  perfect 
condilion.  Only  the  outer,  or  showing  arch-stones,  are  2.5  ft  deep ;  and  that  depth  is  made  up  of  two 
stones.  The  inner  arch-stones  are  but  1.5  ft  deep  ;  and  but  from  6  to  9  inches  thick.  The  stone  quar« 
ried  with  tolerably  fair  natural  beds;  and  received  little  or  no  dressing  in  addition.  The  bridge  is  a 
flne  example  of  that  ignorance  which  often  passes  for  boldness.  Pont  Napoleon  carries  a  railroad 
across  the  Seine  at  Paris.  The  arches  are  of  the  uniform  depth  of  4  ft,  from  crown  to  spring.  They 
are  composed  chiefly  of  smaU  rough  quarry  chips,  or  spawls ;  well  washed,  to  free  them  from  dir» 
and  dust;  and  then  thoroughly  bedded  in  good  cement;  and  grouted  with  the  same.  It  is  \n  fact  an 
arch  of  cement-concrete.  The  Pont  de  Alma,  near  it,  and  built  in  the  same  way,  has  elliptic  arches 
of  from  126  to  141  ft  span  ;  with  rises  of  i-  the  span.  Key  4.9  ft.  These  two  bridges,  considering  th« 
want  of  precedent  in  this  kind  of  construction,  on  so  large  a  scale,  must  be  regarded  as  very  bold; 
and  as  reflecting  the  highest  credit  for  practical  science,  upon  their  engineers,  Darcel  aud  Couche. 
Some  trouble  arose  from  the  unequal  contraction  of  the  different  thicknesses  of  cement.  They  shoitf 
what  may  be  readily  accomplished  in  arches  of  moderate  spans,  by  means  of  small  stone,  and  good 
hydraulic  cement  when  large  stone  fit  for  arches  is  not  procurable.  In  Pont  Napoleon  the  depth  of 
arch  is  less  than    our  rule  gives  for  second  class  cut-stone. 

Art.  3.  Tbe  keystones  for  larg^e  elliptic  arches  by  the  best  en- 
gineers, are  generally  made  about  3^3  part  deeper  than  our  rule  requires  ;  or  than  is 
considered  necessary  for  circular  ones  of  the  same  span  and  rise ;  in  order  to  keep  the 
line  of  pres  well  within  the  joints  ;  although  the  elliptic  arch,with  its  spandrel  filling, 

has  slightly  less  wt ;  and  that  wt  ha» 
a  trifle  less  leverage  than  in  a  circular 
one  ;  and  consequently  it  exerts  less 
pres  both  at  the  key,  and  at  the  skew- 
back.  See  London,  Gloucester,  and 
Waterloo  bridges,  in  the  preceding 
table. 

Ebm.  Young  engineers  are  apt  to  affect  shallow  arch-stones ;  but  it  would  be  far 
better  to  adopt  the  opposite  course ;  for  not  only  do  deep  ones  make  a  more  stable 
structure,  but  a  thin  arch  is  as  unsightly  an  object  as  too  slender  a  column.  Accord- 
ing to  our  own  taste,  arch-stones  fully  3/3  deeper  than  our  rule  gives  for  first-class 
cut  stone,  are  greatly  to  be  preferred  when  appearance  is  consulted.  Especially 
when  an  arch  is  of  rough  rubble,  which  costs  about  the  same  whether  it  is  bui/t  up 
as  arch,  or  as  spandrel  filling,  it  is  mere  folly  to  make  the  arches  shallow.  Stability 
and  durability  should  be  the  objects  aimed  at;  and  when  they  can  be  attained  even 
to  excess,  without  increased  cost,  it  is  best  to  do  so. 

*  Built  like  that  at  Souppes. 


STONE   BRIDGES. 


617 


Table  2.  Depths  of  keystones  for  arches  of  first-class  cut  stone, 
by  Art  2.  For  second  class  add  full  one-eighth  part ;  and  for  superior  brick  one* 
fourth  to  one-third  part,  if  the  span  exceeds  about  15  or  20  ft.    Original. 

Hise,  in  parts  of  the  span. 


Span. 
Feet. 

i 

i 

i 
4 

i    ■ 

* 

i 

A 

Key.  Ft. 

*  Key.  Ft. 

Key.  Ft. 

Key.  Ft. 

Key.  Ft. 

Key.  Ft. 

Key.  Ft, 

2 

.55 

.56 

.58 

.60 

.61 

.64 

.68 

4 

-.70 

.72 

.74 

.76 

.79 

.83 

.88 

6 

.81 

.83 

.86 

.89 

.92 

.97 

1.03 

8 

.91 

.93 

.96 

1.00 

1.03 

1.09 

1.16 

10 

.99 

l.Ol 

1.04 

1.07 

1.11 

1.18 

1.26 

15 

1.17 

1.19 

1.22 

1.26 

1.30 

1.40 

1.50 

20 

1.32 

1..35 

1.33 

1.43 

1.48 

1.59 

1.70 

25 

1.45 

1.48 

1.53 

1.58 

1.64 

1.76 

1.88 

30 

1.57 

1.60 

1.65 

1.71 

1.78 

1.91 

2.04 

35 

1.68 

1.70 

1.76 

1.83 

1.90 

2.04 

2.19 

40 

1.78 

1.81 

1.88 

1.95 

2.03 

2.18 

2.33 

50 

1.97 

200 

2.08 

2.16 

2.25 

2.41 

2.58 

60 

2.14 

2.18 

2.26 

2.35 

2.44 

2.62 

2.80 

80 

2.44 

2.49 

2.58 

2.68 

2.78 

2.98 

3.18 

100 

2.70 

2.75 

2.86 

2.97 

3.09 

3.32 

3.55 

120 

2.94 

2.99 

3.10 

3.22 

3.35 

3.61 

3.88 

140 

3.16 

3.21 

3.33 

3.46 

3.60 

3.87 

4.15 

160 

3.36 

3.44 

3.58 

3.72 

3.87 

4.17 

180 

3.56 

3.63 

3.75 

3.90 

4.06 

4.S8 

200 

3.74 

3.81 

3.95 

4.12 

4.29 

220 

3.91 

4.00 

4.13 

4.30 

4.48 

240 

4.07 

4.15 

4.30 

4.48 

260 

4.23 

4.31 

4.47 

4.66 

280 

4.38 

4.46 

4.63 

300 

4.53 

4.62 

4.80 

Art.  4.  To  proportion  the  abuts  for  an  arch  of  stone  or 
brick,  whether  circular  or  elliptic.    (Original.) 

The  writer  ventures  to  oflFer  the  following  rule,  in  the  belief  that  it  will  be  found 
to  combine  the  requirements  of  theory  with  those  of  economy  and  ease  of  applica- 
tion, to  perhaps  as  great  an  extent  as  is  attainable  in  an  endeavor  to  reduce  so  com- 
plicated a  subject,  to  a  simple  and  reliable  working  rule  for  prac- 
tical bridg'e-builflers.  This  is  all  that  he  claims  for  it.  Notwithstanding  its 
sirflplicity.  it  is  the  result  of  much  labor  on  his  part.  It  applies  equally  to  the  smallest 
culvert,  and  to  the  largest  bridge  ;  whatever  may  be  the  proportions  of  span  and  rise ; 
and  to  any  height  of  ahut  whatever.  It  applies  also  to  all  the  usual  methods  of  filling 
above  the  arch ;  whether  with  solid  masonry  to  the  level  vf,  Fig  2,  of  the  top  of  the 
arch;  or  entirely  with  earth ;  or  partly  with  each,  as  represented  in  the  fig;  or  with 
parallel  spandrel-walls  extending  to  the  back  of  the  abut,  as  in  Fig  2}^.  Altiiough 
the  stability  of  an  abut  cannot  remain  precisely  the  same  under  all  these  conditions, 
yet  the  diff  of  thickness  which  would  follow  from  a  strict  investigation  of  each  par- 
ticular case,  is  not  suflBcient  to  warrant  us  in  embarrassing  a  rule  intended  for  popu- 
lar use,  by  a  multitude  of  exceptions  and  modifications  which  would  defeat  the  very 
object  for  which  it  was  designed.  We  shall  not  touch  upon  the  theory  of  arches, 
except  in  the  way  of  incidental  allusion  to  it.  Theories  for  arches,  and  their  abuts, 
omit  all  consideration  of  passing  loads ;  and  consequently  are  entirely  inapplicable 
in  practice  when,  as  is  frequently  the  case,  (especially  in  railroad  bridges  of  moderate 
spans,)  the  load  bears  a  large  ratio  to  the  wt  of  the  arch  itself  Hence  the  theoretical 
line  of  thrust  bas  no  place  in  such  cases.  Our  rule  is  intended  for  common  practice : 
and  we  conceive  that  no  error  of  practical  importance  will  attend  its  application  to 
any  case  whatever ;  whether  the  arch  be  circular  or  elliptic. 

It  gives  a  thickness  of  abut,  which,  without  any  backing 
of  earth  behind  it.  is  safe  in  itself,  and  in  all  cases,  against 
the  pres,  w^hen  the  bridge  is  unloaded.  Moreover,  in  very  large  arches, 
in  which  the  greatest  load  likely  to  come  upon  them  in  practice  is  small  in  comparison 
with  the  wt  of  the  arch  itself,  and  tlie  filling  above  it,  our  abuts  would  also  be  safe 
from  the  loaded  bridge,  without  any  dependence  upon  the  earth  behind  them  ;  but 
as  the  arches  become  less,  and  consequently  the  wt  of  the  load  becomes  greater  in 
proportion  to  that  of  the  arch,  and  of  the  filling  above  it,  we  must  depend  more  and 
more  upon  the  resistance  of  the  earth  behind  the  abuts,  in  order  to  avoid  the  neces- 
sity of  giving  the  latter  an  extravagant  thickness.  It  will  therefore  be  understood 
thrnughnut  that,  excpt  luhen  parallel  spandrel  walls  are  used,  oxir  rides  svppose  that 
after  the  bridge  is  fudshed^  earth  will  he  dfposited  behind  the  abuts,  <tud  to  the  Height 
of  the  roadway^  as  usual. 


618 


BTONE  BRIDGES. 


In  small  bridges  and  large  colverts  of  firat  class  railroads,  subject  to  the  jarrtna 
of  heavy  trains  at  high  speeds,  the  comparative  cheapness  with  which  an  exoeas  of 
strength  can  thus  be  given  to  important  structorea,  has  led,  in  many  cases,  to  tba 
use  of  abutments  from  one-fourth  to  one-half  thicker  than  by  the  following  r«l6w 
If  of  roiig:li  rubble  add  6  Ids  to  insure  full  thickness  in  every  part. 


Thicks  o  n  of  abut  at  spring^ 

in  ft,  when  the  height  o  s     1 

does  not  exceed  1}/^  times  the  | 

base  8  p  J 


Had  in  ft 


rise  in  ft 
+        10        + 


2  ft. 


Mark  the  points  n  and  y  thus  ascertained.  Next,  from  the  center  i,  of  the  spau 
or  chord  eo,  lay  o£f  ih,  equal  to  -^-^  part  of  the  span.  Join  a  h ;  and  through  w,  and 
parallel  to  ah,  draw  the  indefinite  line  gnp  of  the  abut.  Do  the  same  with  the 
other  abut.  Make  y  m  and  ng  each  equal  to  half  the  entire  height  i  t  of  the  arch; 
and  from  g  draw  a  straight  line  ^  a;,  touching  the  back  of  the  arch  as  high  up  as  pos- 
sible; or  still  better,  as  shown  at  tm,  with  a  rad  dt  or  dm,  (to  be  found  by  trial,) 
describe  an  arc  tm.  Theugx  or  tm  will  be  the  top  of  the  masonry  filling  above  the 
arch;*  and  this  should  be  completed  before  striking  the  centers;  before  which, 
also,  the  embkt  should  be  finished,  at  least  up  to  y  n. 


Eg2 


Now  find  by  trial  the  point  s.  Fig  2,  at  which  the  thickness  sp  is  equal  to  two 

*  Except  when  the  rise  is 
but  about  i  op  the  span,  ok 
less;  in  which  case  oarry  the 
masonry  up  solid  to  the  level 
vtf,  of  the  top  of  the  arch.-  Or 
if  the  arch  is  a  large  one,  ex- 
ceeding say  about  60  ft  span; 
and  especially  if  its  rise  is 
greater  than  about  i  of  its 
span,  it  is  better  to  economize 
masonry  by  the  use  of  parallel 
interior  spandrel-walls.  1 1,  Fig 
7}i,  carried  up  to  v  tf.  Fig  2. 
Indeed,  such  interior  walls  may 
often  be  advantageously  intro- 
duced in  much  smaller  arches. 
When  high,  they  are  steadied 
by  occasional  cross-walls,  as  tt. 
Fig  2^.  Their  feet  should  be 
spread  by  offsets,  as  shown  at  o  o  o,  so  as  to  bear  upon  the  whole  surf  of  the  back  of  the  arch  :  thus 
eaualizing  the  pres  upon  it.  On  top  of  the  walls  flagstones  may  be  laid,  or  small  arches  may  be 
turned  from  wall  to  wall,  for  supporting  the  ballast,  &c,  of  the  roadway.  The  spaces  below  are  left 
hollow.  In  Fig  2H  the  dark  part  «;  w;  is  supposed  to  be  a  section  across  an  abutment ;  but  omitting 
the  second  cross-wall,  similar  to  1 1.     In  R  R  bridges,  put  a  spaudrel-wall,  1 1,  under  each  rail. 


STONE   BRIDGES.  619 

thirds  of  the  corresponding  vert  height  os,  and  draw  sp.  Then  will  the  thickness 
on  or  ey  he  that  at  the  springing  line  of  the  given  circular  or  elliptic  arch  of  any 
rise  and  span;  and  the  Hue gp  will  be  the  back  of  the  abut;  provided  its  height  os 
does  not  exceed  li^  times  sp:  or  in  other  words,  provided  sp  is  not  less  than  %  of 
o  s.  In  practice,  o  «  will  rarely  exceed  this  limit ;  and  only  in  arches  of  considerable 
rise.  But  if  it  should,  as  for  instance  at  07,  then  make  the  base  qu  equal  to  sp,  added 
to  one-fourth  of  the  additional  height  sq-,  and  draw  the  back  uw,  parallel  to  g p; 
and  extending  to  the  same  height,  &c,  as  in  Fig  2.  If,  however,  this  addition  of  "%. 
of  s  q  should  in  any  case  give  a  base  q  w,  less  than  one-half  the  total  height  0  g,  (which 
will  very  rarely  happen  in  practice,)  then  make  q  u  equal  to  half  said  total  height ; 
drawing  the  back  parallel  to  gji,  and  extending  it  to  the  same  height  as  before.  The 
additional  thicknesses  thus  found  below  sp,  have  reference  rather  to  the  pres  of  the 
earth  behind  the  abut,  than  to  the  thrust  of  the  arch.  In  a  very  high  abut,  the  inner 
\me  g  p  would  give  a  thickness  too  slight  to  sustain  this  earth  safely. 

When  the  height  0  6,  Fig  2,  of  the  abut  is  less  than  the  thickness  0  w  at  spring,  a 
email  saving  of  masonry  (not  worth  attending  to,  except  in  large  flat  arches)  may  be 
effected  by  reducing  the  thickness  of  the  abut  throughout,  thus :  Make  o  k  equal  to 
on,  and  draw  kl.  Make  oz  equal  to  ^4  of  ^^^  ^^^  draw  Iz.  Then,  for  any  height 
06  of  abut  less  than  on,  draw  6v,  terminating  in  Iz.  This  hv  will  be  suflBcient  base, 
if  the  foundations  are  firm.  The  back  of  the  abut  will  be  drawn  upward  from  v, 
parallel  to  g  v,  and  terminating  at  the  same  height  as  g  or  w. 

Rem.  1.  All  the  abuts  thus  found  will  (with  the  provisions  in  Art  6)  be  safe, 
without  any  dependence  upon  the  wing-walls ;  no  matter  how  high  the  embkt  may 
extend  above  the  top  of  the  arch.  If  the  bridge  is  narrow,  and  the  inner  faces  of 
the  wing-walls  are  consequently  brought  so  near  together  as  to  afford  material  as- 
sistance to  the  abuts,  the  latter  may  be  made  thinner ;  but  to  what  extent,  must 
depend  upon  the  judgment  of  the  engineer. 

We,  however,  caution  the  young  practitioner  to  be  careful  how  he  adopts  dimensions  less  than  those 
given  by  our  rale.  There  are  certain  practical  considerations,  such  as  carelessness  of  workmanship; 
newness  of  the  mortar;  danger  of  undue  strains  when  removing  the  centers;  liability  of  derange- 
ment during  the  process  of  depositing  the  earth  behind  the  abuts,  and  over  the  arch  ;  &c,  which  must 
not  be  overlooked;  although  it  is  impossible  to  reduce  them  to  calculation. 

Whenever  it  can  be  done,  the  centers  should  remain  in  place  until  the  embkt  is  finished;  and  for 
some   time   afterward,  to  allow  the  mortar  to  set  well.     But  for  more  on  this  see  Rem  4. 

Rbm.  2.  A  good  deal  of  liberty  is  sometimes  taken  in  reducing  the  quantity  of  masonry  above  the 
springing  line  of  arches  of  considerable  rise,  and  of  moderate  spans.  When  care  is  taken  to  leave 
the  centers  standing  until  the  earth  filling  is  completed  above  the  arch,  and  behind  its  abuts,  so  thaJ' 
It  may  not  be  deranged  by  accident  during  that  operation  ;  and  when  good  cement  is  used  instead  oi 
common  mortar,  such  experiments  may  be  tried  with  comparative  safety ;  especially  with  culvert 
arches,  in  which  the  depth  of  arch-stone's  is  great  in  proportion  to  the  span.  They  must,  hpwever,  be, 
left  to  the  judgment  of  the  engineer  in  charge  ;  as  no  specific  rules  can  be  laid  down  for  them.  They' 
can  hardly  be  regarded  as  legitimate  practice,  and  we  cannot  recommend  them.  We  have  known 
nearly  semicircular  arches,  of  30  to  40  ft  span,  to  be  thus  built  successfully,  with  scarcely  a  particle 
of  masonry  above  the  springs  to  back  them.  Such  arches,  however,  are  apt  to  fall,  if  at  any  future 
period  the  earth  filling  is  removed,  without  taking  the  precaution  to  first  build  a  center  or  some  other 
support  for  them.  Even  when  the  embkt  can  be  finished  before  the  centers  are  removed,  we  cannot 
recommend  (and  that  only  in  small  spans)  to  do  less  than  to  make  ng.  Fig  2,  equal  to  "%  of  the  total 
height  t  «  of  the  arch  ;  and  from  g  so  found,  to  draw  a  straight  line  touching  the  back  of  the  arch  as 
high  up  as  possible. 

Rem.  3.  We  have  said  nothing  about  battering  the  faces  of  the  abuts, 

because  in  the  crossing  of  streams,  the  batter  either  diminishes  the  water-way ;  or 
requires  a  greater  span  of  arch.  Such  a  batter,  however,  to  the  extent  of  from  i^ 
to  134  ins  to  a  ft,  is  useful,  like  the  offsets,  for  distributing  the  wt  of  the  structure, 
and  its  embkt,  over  a  greater  area  of  foundation  ;  ^specially  when  the  last  is  not 
naturally  very  firm  ;  or  when  the  embkt  extends  to  a  considerable  height  above  the 
arch.  In  our  tables,  Nos  3  and  5,  of  approximate  quantities  of  masonry  in  semi- 
circular bridges  of  from  2  to  50  ft  span,  the  faces  are  supposed  to  be  vert. 

Art.  5.  Abutment-piers.  When  a  bridge  consists  of  several  arches,  sus- 
tained by  piers  of  only  the  usual  thickness,  if  one  arch  should  by  accident  of  flood, 
or  otherwise,  be  destroyed,  the  adjacent  ones  would  overturn  the  piers ;  and  arch 
after  arch  would  then  fall.  To  prevent  this,  it  is  usual  in  important  bridges  to  make 
some  of  the  piers  sufficiently  thick  to  resist  the  pres  of  the  adjacent  arches,  in  case 
of  such  an  accident ;  and  thus  preserve  at  least  a  portion  of  the  bridge  from  ruin. 
Such  are  called  abutment-piers. 

Our  formula  of 4-  • +  2  ft,  for  the  thickness  at  spring;  with  the  back  battering  as  betbre, 

5  '  10  ' 
at  the  rate  of  ^^  ot  the  span  to  the  rise ;  face  vert ;  will  of  itself  (without  any  modification  for  great 
heights)  give  a^perfectly  safe  abut-pier,  for  any  unloaded  bridge;  arid  to  any  height  whatever;  doe 
regard  being  had,  however,  to  the  consideration  alluded  to  in  the  next  Art,  Thus,  for  an  abut-pier 
as  high  as  o  q,  Fig  2;  or  of  any  greater  height ;  it  is  only  necessary  first  to  find  the  thickness  o  n  at 
spring  as  before  ;  and  then  draw  the  battered  back  gnp:  extending  it  down  to  the  base  at  B ;  with- 
out adding  J4  of  the  additional  height  s  q.    This  addition  is  made  in  the  case  of  abuts,  that  they 


620 


STONE   BRIDGES. 


may  be  secure  from  the  pres  of  the  earth  behind  them ;  as  well  as  from  the  pres  of  the  arch ;  a  oon 
sideratiou  which  does  not  apply  to  abut-piers ;  in  which  only  the  pres  of  the  arch  is  to  be  resisted. 

But  although  the  abut-pier  thus  found  by  our  formula,  would  be  abundantly 
safe,  yet  its  shape  a  6  c  o,  Fig  3,  is  inadmissible.  In  practice  it  would  ble 
changed  to  one  somewhat  like  that  shown  by  the  dotted  lines ;  having  an  equal 
degree  of  batter  on  both  faces.  This  of  course  requires  more  masonry,  with 
but  little  increase  of  stability;  but  that  cannot  be  avoided. 

When  an  abut-pier  is  built  in  deep  water,  or  in  a  shallow  stream  sub- 
ject to  high  freshets,  care  must  be  taken  that  water  cannot  find  its  way  under 
the  pier,  and  thus  produce  an  upward  pres,  which  will  either  diminish,  or 
entirely  counteract  its  efficiency  as  an  abut.  See  Remark  2,  Art  4,  of  Hy- 
drostatics. 

Art.  6.   Inclination  of  tbe  courses  of  masonry 
below  the  spring-s  of  an  arch.    Although  our  fore- 
going rule  gives  a  thickness  of  abut  which  cannot  be  overturned, 
or  upset,  by  the  pres  of  the  arch,  yet  if  the  arch  be  of  large  span, 
and  small  rise,  its  great  lior  thrust  may  produce  a  sliding  out- 
ward of  the  masonry  near  the  level  of  the  springs,  if  the  stones 
are  laid  in  hnr  courses ;  especially  if  the  mortar  has  not  set  well. 
This  danger,  it  is  true,  could  be  avoided  by  confining  the  courses  together 
by  iron  bolts  and  cramps ;  or  by  increasing  considerably  the  thickness  of  the 
abuts ;  but  the  expense  of  doing  either  of  these,  leads  to  the  cheaper  expedient 
of  inclining  the  masonry,  as  shown  between  o  and  n,  Fig  4 ;  the  courses  near  o 
being  steeper;  and  gradually  becoming  less  steep  near  n. 
By  this  process  the  arch  is  virtually  prolonged  i  n to  the  body 
of  the  abut,  so  far  that  when  the  inclination  of  the  lowe» 
masonry  ceases,  as  at  n,  the  direction  of  the  theoretical 
line  of  thrust,  or  of  pres  of  the  arch,  (rudely  represented 
by  the  dotted  curved  line  o  n)  is  nearly  at  right  angles  to 
the  joints  of  the  hor  masonry  below  «;  and  consequently, 
said  thrust  is  unable  to  produce  sliding  at  that  point.  Be- 
tween o  and  n,  the  line  of  pres  is  everywhere  so  nearly  at 
right  angles  to  the  variously  inclined  joints,  as  to  preclude 
the  possibility  of  sliding  in  that  interval  also. 

The  abut  being  thus  safe 
throughout  from  both  overturning  and  sliding,  can  fail 
only  from  defective  foundations;  or  from  the  inferiority 
of  the  stone  of  which  it  la  built;  and  which,  if  soft,  may 
be  crushed. 

This  inclination  of  the  masonry  is  as  neces- 
sary in  an  elliptic  arch,  Fig  4^,  as  in  a  circular 


FiiA- 


STONE    BRIDGES.  621 

The  elliptic  form  is  plainly  unfavorable  for  uniting  the  arch-stones  with  the  inclined  masonry  near 
the  springs,  so  as  to  receive  the  thrust  properly  ;  or  about  at  right  angles  to  its  resultant.  In  ordi- 
nary cases  this  difficulty  may  be  overcome  by  making  the  joints  of  only  the  outside  or  showing  arch- 
stones  to  conform  to  the  elliptic  curve;  as  between  e  and  a;  while  the  joints  of  the  inner  or  hidden 
ones,  may  have  the  directions  shown  between  g  and  u,  nearly  at  right  angles  to  the  line  of  thrust.  It 
will  rarely  happen,  however,  that  the  young  engineer  will  have  to  construct  elliptic  arches  of  sufla- 
cient  magnitude  to  require  either  this,  or  any  equivalent  expedient.  For  spans  less  than  50  ft,  with 
rises  not  less  than  about  i  of  the  span,  nothing  of  the  kind  is  actually  necessary,  if  the  mortar  is 
good,  and  has  time  to  harden.! 

In  order  to  incline  the  masonry  of  any  abut  with  sufficient  accuracy,  it  would 
be  necessary  first  to  trace  the  curved  line  of  pres  of  the  given  arch, 

so  as  to  arrange  the  bed  joints  about  at  right  angles 
to  it  at  every  point  of  its  course;  but  we  offer  the  following  process  as  suflBcing  for  all 
ordinary  practical  purposes;  while  its  simplicity  places  it^vithin  the  reach  of  the  com- 
mon mason.  In  actual  bridges  the  direction  of  the  actual  thrust  changes  as  the  load 
Js  passing;  therefore,  in  practice  no  given  degree  of  inclination  of  the  abut  masonry 
can  conform  to  it  precisely  during  the  entire  passage.  Consequently,  any  excess  of 
refinement  in  this  particular,  becomes  simply  ridiculous ;  especially  in  small  spans. 

Rule  for  inclining:  tlie  beds  of  the  masonry  in  the  abnts. 
Add  together  the  rad  cm,  Fig  4;  and  the  span  of  the  arch.  Div  the  sum  by  6.  To 
the  quot  add  3  ft.  Make  o  t,  on  the  rad,  equal  to  the  last  sum.  Then  is  ^  a  central 
point,  toward  which  to  draw  the  directions  of  the  beds,  as  in  the  fig.  Draw  t  s  hor. 
and  from  <  as  a  center,  describe  the  arc  oy,  o  being  the  center  of  the  depth  of  the 
springers.  From  y  lay  off  on  the  arc  the  dist  yn,  equal  to  one-sixth  part  of  ty:  draw 
in  a.  It  will  never  be  necessary  to  incline  the  masonry  below  this  tn  a.  Neither 
need  the  inclination  extend  entirely  to  the  face  m  i  of  the  abut;  but  may  stop  at  e, 
about  half-way  between  i  and  n.  From  e  upward,  the  inclination  may  extend  for- 
ward to  the  line  e  m. 

Cement  should 
be  freely  used,  not  only  in  the  arches  themselves,  and  in  the  masonry  above  them,  as  a 
protection  from  rain-soakage ;  but  in  abuts,  wing-walls,  retaining- walls,  and  all  other 
important  masonry  exposed  to  dampness.  The  entire  backs  of  important  brick  arches 
should  be  covered  with  a  layer  of  good  cement,  about  an  inch  thick.  The  want  of  it 
can  be  seen  throughout  most  of  our  public  works.  The  common  mortar  will  be 
found  to  be  decayed,  and  falling  down  from  the  soffits  of  arches;  and  from  the  joints 
of  masonry  generally,  within  from  3  to  6  ft  of  the  surface  of  the  ground.  The  mois- 
ture rises  by  capillary  attraction,  to  that  dist  above  the  surf  of  the  nat  soil ;  or 
descends  to  it  from  the  artificial  surf  of  embankments,  &c;  therefore,  cement-mortar 
should  be  employed  in  those  portions  at  least.  The  mortar  in  the  faces  of  battered 
Walls,  even  when  the  batter  is  but  1  to  \%  inches  per  foot,  is  far  more  injured  by  rain 
and  exposure,  than  in  vert  ones ;  and  should  therefore  be  of  the  best  quality.  See 
Mortar,  &c. 

We  have,  however,  seen  a  quite  free  percolation  of  surface  water  through  brick 
arches  of  nearly  3  ft  in  depth,  even  when  cement  was  freely  used.  In  aqueduct 
bridges,  we  believe  that  cement  has  not  been  found  to  prevent  leaks,  whether  the 
arches  were  of  brick,  or  even  of  cut-stone.  May  not  this  be  the  effect  of  cracks 
produced  by  settlement  of  the  arch ;  or  by  contraction  and  expansion  under  atmos- 
pheric influence  ?    Cement  at  any  rate  prevents  the  joints  from  crumbling. 

T  The  feel  of  both  elliptic  and  semicircular  arches  are  always  made  hor  ;  but  it  is  plain  from  Fig 
i*-^,  that  this  practice  is  at  variance  with  correct  principles  of  stability  in  the  case  of  the  ellipse.  It 
Is  the  same  in  the  semicircle.  In  ordinary  bridges  of  the  latter  form,  the  vert  pres,  or  weight  resting 
on  each  skewback,  is  (roughly  speaking)  usually  about  from  3J^  to  4  times  the  hor  pres  on  the  same  : 
and  the  total  pres  is  about  4  times  as  great  as  the  pres  on  the  keystone.  Therefore,  theoretically,  the 
Hkewback  should  usually  be  about  4  limes  as  deep  as  the  keystone ;  and  its  bed,  instead  of  being  hor, 
should  be  inclined  at  the  rate  of  about  I  vert  to  4  hor. 


622 


STONE   BRIDGES. 


When  the  arch  is  flat,  this  inclination  may  become  so  steep,  especially  in  the  upper  parts,  that 
struts,  or  shores  of  some  kind,  must  be  used  for  preventing  the  ma- 
sonry from  sliding  down,  until  the  completion  of  the  arch  secures  it 
from  doing  so.  The  hor  courses  between  the  face  m  i,  and  the  line 
o  e,  will  aid  somewhat  in  this  respect. 

This  method  should  be  applied  to  all  very  large  arches  whose 
rise  is  one-third,  or  less,  of  the  span.  As  before  remarked,  it 
is  not  actually  necessary  in  arches  not  exceeding  about  50  ft  span, 
and  not  flatter  than  -J-  of  the  span.  Indeed,  if  the  earth  filling  can 
be  deposited  b«fore  the  centers  are  removed,  these  limits  may  be  con- 
siderably extended  without  danger.  Still,  since  a  certain  degree  of 
inclination  is  attended  with  very  little  trouble  or  expense,  we  would 
recommend  for  even  such  arches,  a  process  somewhat  like  thefollow- 
inc :  From  half  the  span  take  the  rise.  Div  the  rem  by  3.  Make  o  t, 
Fig  5,  equal  to  the  quot.  Draw  t  n,  and  o  m,  hor.  Div  the  angle 
8  0  m  into  two  equal  parts,  by  the  line  o  a.  lucliue  the  masonry  so 
as  to  be  parallel  to  o  a,  as  tar  down  as  t  n.  The  inclined  courses 
may  extend  out  to  the  face  o  t,  or  not,  at  pleasure. 

Rem.  1.    To  find   tlie  leii^j^th  (ah,  Fig  7) 
from  face  to  face  of  a  culvert.     From 

the  height  ht  of  the  embkt,  take  the  above-ground  height  n  a 
of  the  culvert;  the  rem  will  be  the  height  ho  of  the  embkt 
above  the  culvert.  Then  the  reqd  length  a  6  is  plainl}'  equal 
to  the  top  width  id  of  the  embkt,  added  to  the  two  dists  as, 
cb,  which  correspond  to  its  steepness  of  side-slopes.  Thus,  if 
the  side-slope  is,  as  usual,  1}^  tol,  then  as  andc6  will  each  be 
equal  to  IH  times  o  h;  or  the  two  together  will  be  3  times  o  h. 
So  that  if  the  width  id  is  lift,  and  A  o  5  ft,  the  length  a  b  will  be 
14  -f  (5  X  3)  =  14  -f  15  =  29  ft. 

Art.  7.    The  following:  tables,  3.  4,  and  5,  of  quantities,  will 

be  found  useiul  for  expediting  preliminary  estimates ;  for  wliich  purpose  chiefly  they 
are  intended  ;  hence  no  pains  have  been  taken  to  make  them  scrupulously  correct, 
but  rather  a  little  in  excess  of  the  truth.    The  first  column  of  Table  3  contains  the 

total  vert  height  o  c,  Pig  6,  from  the 

a ,a 


Erf  5 


iJtJt 

S     0   c 


.ZL 


ai 


Fig7 


iirf.6 


crown  o  of  a  semicircular  arch,  to 
the  foundation  or  base  g  m  of  its 
abut.  The  other  columns  give  ap- 
proximately the  number  of  cub  ydsi 
contained  in  each  running  foot,  or 
foot  in  length  of  the  culvert  or 
bridge,  measured  from  end  to  end 
(face  to  face)  of  the  arch  proper; 
and  including  only  the  arch  and  its 
abuts,  as  shown  in  Fig  1 ;  or  in  the 
half  section  opragy  in  Fig  6;  in- 
cluding footings  to  the  abuts,  but 
omitting  the  wing-walls  {iv  n),  and 
the  spandrel-walls  («),  Figs  6  and 
2}^.  At  the  foot  of  each  column  is  the  approximate  content  in  cub  yds  of  the  two 
spandrel- walls  by  themselves;  one  over  each  face  of  the  arch. 

These  spandrel- walls  are  calculated  on  the  supposition  that  their  thickness  at  base,  at  their  jano- 
tion  with  the  wing- walls,  where  their  height  is  greatest,  is  equal  to  -^-^  of  their  height  at  that  point: 
except  where  that  proportion  gires  a  less  thickness  at  top  than  23^  ft;  and  that  they  extend  2  ft  (•  a) 
above  the  top  o  of  the  arch.  At  the  top  of  the  arch,  they  are  all  supposed  to  be  2}4  ft  thick  at  top; 
that  being  assumed  to  be  about  the  least  thickness  admissible  in  a  rubble  wall  in  such  a  position. 
Both  the  back  and  the  face  are  supposed  to  be  vert.  The  contents  of  these  spandrel-walls  will  vary 
somewhat,  however,  even  in  the  same  span,  with  the  height  of  the  abut  and  the  arrangement  of  the 
wings.  They,  however,  constitute  so  small  a  proportion  of  the  entire  contents  given  in  Table  5.  that 
-  this  consideration  may  be  neglected  in  preliminary  estimates.  They  are  so  firmly  bonded  into  the 
masonry  of  the  wings  at  their  highest  points,  and  so  strongly  connected  by  mortar  with  the  backing 
of  the  arch  at  their  bases,  that  they  require  no  greater  thickness  however  high  the  emb  may  be. 

Tlie  contents  of  the  four  wins:- walls,  of  which  njw  h.  Fig  6,  is  one, 
will  be  found  in  a  table  (No.  4)  immediately  following  that  for  the  body  of  the  cul- 
vert. We  have  also  added  a  table  (No.  5)  for  complete  semicircular  culverts  of 
various  lengths,  including  their  spandrel  and  wing  walls. 


STONE   BRIDGES. 


623 


Rem.  1.  Although  the  thickness  of  wing-waills  increases  in  all  parts  with  their 
height,  they  are  not  made  to  show  thicker  at  nj  than  at  ti,  Fig  6 ;  but  (as  seen  in  the 
fig)  are  ofifsetted  at  their  back  ^w,  a  little  below  their  slanting  upper  surf  tj,  so  as 
to  give  a  uniform  width  for  the  steps  or  flagstones,  as  the  case  may  be,  with  which 
they  are  covered.  In  the  fig  the  covering  is  supposed  to  be  of  flagstones  ;  but  steps 
are  preferable,  being  less  liable  to  derangement.  To  prevent  the  flagstones  from 
sliding  down  the  inclined  plane  J  i,  tlie  lower  stone  i  should  be  deep  and  large,  and 
laid  with  a  hor  bed.  The  flags  are  sometimes  cramped  together  with  iron,  and  bolted 
down  to  the  wall.    Steps  require  nothing  of  that  kind,  as  seen  at  s,  Fig  11. 

Rem.  2.  The  tables  show  the  inexpediency  of  too  mneh  con- 
tracting: the  width  of  water-way,  with  a  view  to  economy,  by  adopting 
a  small  span  of  arch,  when  a  culvert  of  greater  span  can  Jbe  made,  of  the  same  total 
height. 

For  the  wings  must  be  the  same,  whether  the  span  be  great  or  small,  provided  the  total  height  is 
the  same  in  both  cases  ;  and  since  the  wings  constitute  a  large  proportion  of  the  entire  quantity  of 
masonry,  in  culverts  of  ordinary  length,  the  span  itself,  within  moderate  limits,  has  comparatively 
little  effect  upon  it.  Thus,  the  total  masonry  in  a  semicircular  culvert  of  3  ft  span,  8  ft  total  height, 
and  60  ft  long  between  the  faces  of  the  arch,  is,  by  Table  5,  1513^  cub  yds  ;  while  that  of  a  5  ft  span, 
of  the  same  height  and  length,  is  152.4.  A  semicircular  bridge  of  25  ft  span,  24  ft  total  height,  and 
40  ft  between  the  faces  of  the  arch,  contains  1031  cub  yds ;  while  one  of  35  ft  span,  of  the  same  height 
and  length,  contains  1134  yds ;  so  that  in  this  case  we  may  add  nearly  50  per  cent  to  the  water-way, 
by  increasing  the  masonry  of  the  bridge  but  yVth  part. 

Rem.  3.  Partly  for  the  same  reason,  and  partly  because  the  cnlverts  for  (fc 
double-track  road  are  not  tivice  as  long:  as  those  for  a  sing:le« 
track  one,  the  quantity  of  culvert  masonry  for  the  former  will  not  average  more 
than  about  from  3^  to  3^  part  more  than  that  for  the  latter;  so  that  it  frequently 
becomes  expedient  to  finish  the  culverts  at  once  to  the  full  length  required  for  a 
double  track,  although  the  embkts  may  at  first  be  made  wide  enough  for  only  a 
single  one,  with  the  intention  of  increasing  them  at  a  future  time  for  a  double  one. 

Thus,  the  average  size  of  culverts  for  a  single  track  may  be  roughly  taken  at  6  ft  span,  30  ft  long 
from  face  to  face,  and  10  ft  total  height;  and  such  a  one  contains,  by  Table  5,  140  cub  yds.  For  a 
double  track,  it  would  require  to  be  about  12  feet  longer;  and  we  see  by  Table  3  that  this  will  ad(* 
2.67  X  12  =  32  cub  yds;  making  a  total  of  172  yds  instead  of  140;  thus  adding  rather  less  than  34 
part.  When  the  culverts  are  under  very  high  embkts,  and  consequently  much  longer,  the  additiou 
for  a  double  track  becomes  comparatively  quite  trifling. 

Table  3,  of  approximate  numbers  of  cub  yds  of  masonry 
per  foot  run,  contained  in  the  arches  and  abutments  only,  as 

shown  in  Fig  1  (omitting  wings,  and  the  spandrel-walls  over  the  faces  of  the  arches} 
of  semicircular  culverts  and  bridges,  of  from  2  to  50  ft  span,  and  of  diflTerent  total 
heights,  h  t.  Fig  1,  or  o  c.  Fig  6.  It  will  be  seen  that  in  many  cases,  a  bridge  of  larger 
span  contains  less  masonry  than  one  of  smaller  span,  when  their  total  heights  are  the 
same.  There  is  a  liberal  allowance  for  footings  or  offsets  at  the  bases  of  the  abuta 
TABIiK  3.    (Original.) 


Total 
Height. 

Span 
2  ft. 

Span 
3  ft. 

Span 
4  ft. 

Span 
5  ft. 

Span 
6  ft. 

Span 
8ft. 

Span 
10  ft. 

Span 
12  ft. 

Span 
15  ft. 

Feet. 
2 
3 

Cub.  y. 
.42 
.60 
.79 
.99 
1.28 
1.62 
2.01 
2.45 
2.94 

Cub.  y. 

Cub.  y. 

Cub.  y. 

Cub.  y. 

Cub.  y. 

Cub.   y. 

Cub.  y. 

Cub.   y. 

.63 

.83 
1.04 
1.28- 
1.59 
1.96 
2.38 
2.85 
3.38 
3.98 

.67 
.87 
1.08 
1.28 
1.55 
1.91 
2.31 
2.76 
3.26 
3.82 
4.42 
5.08 

.92 
1.15 
1.37 
1.64 
1.95 
2.29 
2.72 
3.19 
3.72 
4.29 
4.90 
5.57 
6.30 

.97 
1.21 
1.46 
1.72 
1.99 
2.27 
2.67 
3.12 
3.62 
4.17 
4.77 
5.42 
6.12 
6.87 
7.69 

5 
6 
7 
8 
9 
10 
11 
12 
13 

i.58 
1.85 
2.13 
2.42 
2.77 
3.16 
3.57 
4.10 
.4.67 
5.30 
5.97 
6.70 
7.48 
8.32 
9.20 

1.69 
1.97 
2.26 
2.56 
2.87 
3.19 
3.52 
4.02 
4.57 
5.17 
5.82 
6.52 
7.27 
8.07 
8.92 
9.82 
10.8 

2.12 
2.38 
2.65 
2.9? 
3.2S 
3.55 
3.86 
4.41 
5.01 
5.56 
6.2ft 
7.01 
7.71 
8.5S 
9.46 
10.3 
11.3 
12.3 

'■8!62'* 
8.34 
8.6T 
4.01 
4.36 
4.72 
5.09 
5.69 
6.34 
7.04 
7.69 
8.49 
9.34 

10.2 

11.1 
•  12.1 

13.2 

14.2 

14 

15 

16 

17 

18 

19 

20 

21 

22 

23 

24 

25 

26 

^ 

Contents  of  the  two  spandrel-walls,  over  the  two  ends  of  the  arch,  in  cu^,  yds. 
I      2.9      I      3.7      I      4.4      I      5.2      |      5.8      |      7.9      |      9.8      |      12.       1     16. 


624 


STONE   BRIDGES. 


TABIiE  3.    (Continued.) 

Total 

Spaa 
20  ft. 

Span 

Total 

Span 

Total 

Span 

Height. 

25  ft. 

Height. 

35  ft. 

Height. 

50  ft. 

Feet. 

Cub.  y. 

Cub.  y. 

Feet. 

Cub.  y. 

Feet. 

Cub.  y. 

12 
13 

4.60 
4.98 

20 
21 

10.5 
11.0 

27 

28 

18.0 

18!7 

14 

5.37 

'"'e.'io"' 

22 

11.6 

29 

19.4 

15 

5.77 

6.41 

23 

12.2 

30 

20.1 

16 

6.18 

6.76 

24 

12.7 

31 

20.9 

17 

6.60 

7.16 

25 

13.3 

32 

21.6 

18 

7.03 

7.61 

26 

13.8 

33 

22.4 

19 

7.47 

8.10 

27 

14.5 

34 

23.1 

20 

8.12 

8.60 

28 

15.1 

35 

23.9 

21 

8.82 

9.02 

29 

15.7 

36 

24.7 

22 

9.57 

9.72 

30 

16.3 

37 

25.5 

23 

10.4 

10.4 

31 

17.0 

38 

26.3 

24 

11.3 

11.2 

32 

18.1 

39 

27.1 

23 

12.2 

12.1 

33 

19.2 

40 

28.0 

26 

13.1 

13.0 

34 

20.4 

41 

28.8 

27 

14.1 

14.0 

35 

24.7 

42 

30.0 

28 

15.2 

15.0 

36 

23.0 

43 

31.5 

29 

16.3 

16.1 

37. 

24.3 

44 

33.0 

30 

17.4 

17.2 

38 

25.7 

45 

34.6 

31 

18.6 

18.4 

39 

27.2 

46 

36.3 

32 

19.9 

19.6 

40 

28.7 

47 

38.1 

33 

21.2 

20.9 

41 

30.2 

48 

39.8 

34 

22.6 

22.2 

42 

31.8 

49 

41.6 

35 

24.0 

23.6 

43 

33.5 

50 

43.6 

36 

25.4 

25.0 

44 

35.2 

51 

45.5 

37 

26.9 

26.5 

45 

36.9 

52 

47.4 

38 

28  5 

28.0 

46 

38.7 

53 

49.4 

39 

30.1 

29.5 

47 

40.6 

54 

51.6 

40 

31.7 

31.2 

48 

42.5 

55 

53.7 

41 
42 

32.8 
34.5 

49 
50 

44.4 
46.4 

56 

57 

55.9 

58^1 

43 

36.3 

58 

60.4 

44 

38.1 

59 

62.7 

45 

40.0 

60 

65.1 

Contents  of  the  two  spandrel-walls,  over  ths  two  ends  of  the  arch,  in  cub  yds. 

28.  I     42.  II ;      85.        (I I    195. 

Art.  8.  The  following'  table  of  contents  of  wing-walls,  or  wings,  will, 
like  the  preceding  one,  be  useful  in  making  preliminary  estimates.  The  wings 
no.  no^  shown  in  plan  at  Fig  8,  are  supposed  to  form  an  angle  aoc,  of  120°,  with  the 
face,  or  end  o  o  of  the  culvert.  Their  outer  or  small  ends  n  n,  are  ail  assumed  to  be  of 
the  dimensions  shown  on  a  larger  scale  at  E.  Thickness  at  base  at  every  part  equal 
to  -^Q  of  the  height  of  the  wall  at  said  part;  except  when  that  proportion  becomes 
too  small  to  allow  the  width  or  thickness  at  top  to  be  2.5  ft ;  in  which  case  it  is  en- 
larged at  such  parts  sufficiently  for  that  purpose.  See  Remark  2.  This  happens  only 


Fig.  8 

when  the  height  m  m,  Fig.  E,  of  the  wing,  becomes  less  than  9  ft.  Batter  of  face,  1}^ 
fns.  to  a  ft. ;  or  1  in  8.  Back  vert. ;  but  ofl'setted,  if  necessary,  for  a  short  dist.  below 
the  top,  BO  as  to  give  a  uniform  showing  top  thickne.*s  of  2}/^  ft.  The  masonry  is 
supposed  to  be  good  well-scabbled  mortar  rubble.  The  height  given  in  the  first 
column  is  the  greatest  one  ;  or  that  at  o  o  (or  wj,  Fig.  6),  where  the  wing  joins  the 
face  of  the  culvert.  In  the  table  no  allowance  is  made  for  footings  (offsets  or  steps) 
at  the  base  of  the  wings ;  as  these  are  frequently  omitted  in  wings  on  good  founda- 


STONE   BRIDGES. 


625 


tions.    In  taking  out  quantities  from  the  table,  bear  in  mind  that  the  height  of  the 
wings  is  usually  a  little  greater  than  that  of  the  culvert  itself. 

The  plan  shown  at  C  is  the  common  one,  but  that  at  D  is  greatly  preferable  for 
culverts;  for  the  shoulders  at  o  o  in  Fig.  C,  apart  fromtlieir  greater  liability  to  catch 
branches  of  trees,  etc.,  floating  down  stream,  offer  of  themselves  a  much  greater  re- 
sistance to  the  flow  of  the  water  into  the  culvert  than  do  the  mere  corners  at  o  o, 
Fig.  D. 

Table  4,  of  approximate  contents,  in  cub  yds,  of  the  four 
wing-walls  of  a  culvert,  or  bridge.    (Original ) 

The  heights  are  taken  where  greatest;  as  a,tjw,  Fig  6 


Height 

Length 

Cub.  yds. 

Height 

Length 

Cub.  yds. 

of 

.of 

in 

of 

of 

in 

wing. 

one  wing. 

4  wings. 

wing. 

one  wing. 

4  wings. 

Feet. 

Feet. 

Feet. 

Feet. 

6 

1.73 

4.04 

30 

43.3 

818 

7 

3.46 

8.85 

32 

46.8 

997 

8 

5.20 

14.6 

34 

50.3 

1192 

9 

6.93 

21.5 

36 

53.7 

1414 

10 

8.66 

30.2 

38 

57.2 

1661 

11 

10.4 

40.9 

40 

60.7 

1928 

12 

12.1 

53.7 

42 

64.2 

2220 

14 

15.6 

85.2 

44 

67.6 

2552 

16 

19.1 

128 

46 

71.1 

2912 

18 

22.5 

183 

48 

74.6 

3306 

20 

26.0 

247 

50 

78.0 

3741 

22 

29.5 

329 

55 

86.7 

4942 

24 

32.9 

426 

60 

95.3 

6404 

26 

36.4 

541 

65 

104 

8131 

28 

39.8 

672 

70 

113 

10155 

To  reduce  cub  yds  to  perches  of  25  cub  ft,  mult  by  1.080. 
To  reduce  perches  to  cub  yds,  mult  by  .926,  or  div  by  1.08. 


The  contents  for  heights  intermediate  of  those  in  the  table  may  be  found  approximately  by  simple 
proportion. 

Rem.  1.  It  is  not  recommended  to  actually  prolong  all  wings  until  their  dimen- 
sions become  as  small  as  shown  at  E,  in  Fig  8.  In  large  ones  it  will  generally  be 
more  economical  to  inci-ease  their  end  height  mm,  a,  few  feet.  The  contents,  how- 
ever, may  be  readily  found  by  the  table  in  that  case  also.  Thus  suppose  the  height 
of  the  wings  at  one  end  to  be  30  ft,  and  at  the  other  end  8  ft ;  we  have  only  to  sub- 
tract the  tabular  content  for  8  ft  high,  from  that  for  30  ft  high.  Thus,  818  — 14.6  = 
803.4  cub  yds  required  content. 

Rem.  2.  It  might  be  supposed  that  inasmuch  as  the  wings  of  arches  often  have  to 
sustain  the  pressure  from  embankments  reaching  far  above  their  tops,  they  should, 
like  ordinary  retaining-walls,  be  made  much  thicker  in  that  case.  But  the  fact  that 
they  derive  great  additional  stability  from  being  united  at  their  high  ends  to  the 
body  of  the  bridge  or  culvert,  renders  such  increase  unnecessary  when  proportioned 
by  our  rule ;  no  matter  how  far  the  earth  may  extend  above  them ;  as  shown  by 
abundant  experience. 

Relying  upon  this  aid,  we  may  indeed,  when  the  earth  does  not  extend  above  the  top,  reduce  the 
base  at  o  to  one  third  of  the  ht,  as  shown  at  o  t;  and  by  dotted  line  t  s.  ExpA-ience  shows  that  we 
may  also  do  the  same  even  when  the  earth  reaches  to  a  great  height  above  the  top ;  provided  that 
the  wings,  instead  of  being  splayed  or  flared  out,  as  at  o  n,  o  n,  merely  form  straight  prolongations 
of  the  aljutments  of  the  arch,  as 'shown  by  the  dotted  lines  a.t  o  g  w.  In  this  case  the  pressure  of  the 
earth  against  the  wings  is  less  than  when  they  are  splayed.  We  have  known  the  thickness  at  o 
to  be  reduced  in  such  cases  to  less  than  one-third  the  height,  when  the  wings  were  15  ft  high,  and 
the  height  of  the  embankment  above  their  tops  16  feet  in  one  case,  and  36  ft  in  another.  In  another 
instance,  similar  wings  253^  ft  high,  and  with  29  ft  of  embankment  above  their  top,  had  their  bases 
at  0  rather  less  than  ^  of  the  height.  In  all  these  cases,  the  uniform  thickness  at  top  was  2.5  feet; 
backs  vertical.  We  mention  them  because  this  particular  subject  does  not  seem  to  be  reducible  to 
any  practical  rule.  The  last  wall  appears  to  us  to  be  too  thin  ;  especially  if  the  earth  is  not  deposited 
in  layers  ;  and  after  allowing  the  mortar  full  time  to  set.  The  labor,  however,  required  in  compact- 
ing the  earth  carefully  in  layers,  may  cost  more  than  is  thereby  saved  in  the  masonry.  The  young 
practitioner  must  bear  this  in  mind  when  he  wishes  to  economize  masonry  by  such  means ;  and  also 
that  the  thin  wall  may  bulge,  or  fail  entirely,  if  the  earth  backing  is  deposited  while  the  mortar  ia 
Imperfectly  set. 

40 


626 


STONE   BEIDGE8. 


Table  5.  Approximate  eontents  in  cubic  yards,  of  com* 
plete  semicircular  culverts  and  bridges  of  from  2  to  50  feet 
span  ;  including  the  2  spandrel  walls ;  and  the  4  wings  ;  all  proportioned  by  the 
foregoing  directions ;  and  taken  from  the  two  preceding  tables.  The  height  in  the 
second  column,  is  from  the  top  of  the  keystone,  to  the  bottom  of  the  foundation.  The 
wings  are  calculated  as  being  2  ft  higher  than  this,  including  the  thickness  of  the 
coping.  The  wings  are  frequently  carried  only  to  the  height  of  the  top  of  the  arch; 
thus  saving  a  good  deal  of  masonry.  Table  4,  of  wings  alone,  will  serve  to  make  the 
proper  deduction  in  this  case. 

The  several  lengths  are  from  end  to  end,  or  from  face  to  face,  of  the  arch  proper-. 
The  contents  for  intermediate  lengths  may  be  found  exactly ;  and  those  for  inter* 
mediate  heights,  quite  approximately,  by  simple  proportion.  In  this  table,  as  in 
No.  3,  it  will  be  observed  that  when  the  heights  are  the  same  in  both  cases,  a  largei 
span  frequently  contains  less  masonry  than  a  smaller  one.  A  semicircular  culvert 
or  bridge  contains  less  masonry  than  a  flatter  one,  when  the  total  height  is  the  same 
in  both  cases ;  therefore,  the  first  is  the  most  economical  as  regards  cost ;  but  it  does 
not  afford  as  much  area  of  water-way ;  or  width  of  headway. 
(Original.) 


^ 

J3     ' 

fl*  . 

ji  ' 

■^  J 

-3« 

jaS 

J3*5 

h3  .J 

^*i 

JS-J 

XI 

ftS 

Mi, 

^S 

bOi, 

S)S 

^^:m 

U>^ 

^'■^ 

^=^ 

t^'- 

■s,(* 

^^ 

Qf2 

•3 

a 

J3 

|s 

J^ 

1? 

is 

^s 

S| 

g5 

II 

Jl 

so 

Ft. 

Ft. 

Cub.Y. 

Cub.Y. 

Cub.Y. 

Cub.Y. 

Cub.Y. 

Cub.Y. 

Cub.Y. 

Cub.Y. 

Cub.Y. 

Cub.Y. 

Cub.Y. 

Oub.Y. 

5 

27 

32 

42 

52 

72 

92 

112 

132 

152 

172 

192 

212 

9, 

« 

37 

43 

56 

69 

94 

120 

146 

171 

197 

222 

248 

274 

7 

49 

57 

73 

89 

122 

154 

187 

219 

251 

284 

316 

349 

8 

63 

73 

93 

113 

153 

193 

283 

273 

313 

353 

393 

433 

10 

101 

116 

145 

175 

234 

291 

351 

410 

469 

527 

586 

645 

5 

28 

34 

44 

54 

75 

96 

117 

138 

158 

179 

200 

221 

6 

38 

44 

57 

70 

95 

121 

146 

172 

198 

223 

249 

275 

3 

7 

49 

57 

73 

89 

121 

153 

184 

216 

247 

280 

312 

343 

8 

63 

73 

93 

112 

152 

191 

230 

269 

308 

348 

387 

426 

10 

101 

115 

143 

172 

229 

286 

343 

400 

457 

514 

571 

628 

12 

149 

169 

208 

248 

328 

407 

487 

567 

646 

726 

806 

885 

5 

30 

35 

46 

57 

78 

100 

122 

143 

165 

186 

208 

229 

6 

38 

45 

58 

70 

96 

122 

147 

173 

198 

224 

•250 

275 

4 

7 

49 

57 

73 

88 

119 

150 

181 

212 

243 

274 

305 

336 

8 

63 

73 

92 

111 

149 

188 

226 

264 

302 

340 

379 

417 

10 

100 

114 

141 

169 

224 

279 

335 

390 

445 

500 

555 

611 

12 

147 

166 

204 

243 

319 

395 

472 

548 

625 

701 

777 

854 

14 

209 

234 

285 

336 

437 

539 

641 

742 

844 

945 

1047 

1149 

6 

41 

47 

61 

75 

102 

130 

157 

184 

212 

239 

267 

294 

7 

52 

60 

76 

93 

125 

158 

191 

224 

257 

289 

822 

355 

5 

8 

65 

75 

94 

114 

153 

192 

231 

270 

309 

348 

387 

426 

10 

100 

114 

141 

168 

223 

277 

331 

386 

440 

495 

549 

603 

12 

146 

165 

202 

239 

314 

388 

463 

537 

611 

686 

760 

835 

14 

7 

207 

231 

280 

329 

427 

525 

623 

721 

819 

917 
303 

1015 

1113 

53 

62 

79 

96 

131 

165 

200 

234 

268 

337 

372 

8 

66 

76 

96 

116 

156 

196 

236 

276 

316 

356 

396 

436 

6 

10 

100 

113 

140 

167 

220 

274 

327 

380 

434 

487 

541 

594 

12 

146 

164 

200 

236 

308 

381 

453 

526 

598 

670 

743 

815 

14 

206 

219 

277 

325 

420 

516 

611 

706 

802 

897 

993 

1088 

16 

281 

311 

373 

434 

556 

679 

801 

923 

1046 

1168 

1291 

1413 

7 

57 

67 

85 

104 

141 

178 

215 

252 

289 

326 

363 

400 

8 

70 

81 

102 

124 

166 

209 

251 

294 

337 

379 

422 

464 

8 

10 

104 

118 

145 

173 

228 

284 

339 

395 

450 

505 

561 

616 

12 

147 

135 

200 

236 

308 

379 

450 

522 

593 

664 

736 

807 

14 

206 

j30 

276 

323 

416 

510 

603 

696 

790 

883 

977 

1070 

16 

281 

310 

370 

430 

549 

669 

788 

908 

1027 

1146 

1266 

1385 

18 

367 

405 

480 

554 

704 

854 

1003 

1153 

1302 

1452 

1602 

1751 

8 

74 

85 

108 

131 

176 

221 

266 

311 

357 

402 

447 

492 

10 

107 

121 

150 

179 

236 

294 

351 

408 

466 

523 

581 

63? 

10 

12 

148 

166 

201 

236 

306 

377 

447 

518 

588 

658 

729 

799 

14 

207 

229 

275 

321 

412 

504 

595 

686 

778 

869 

961 

1052 

16 

280 

309 

368 

426 

542 

659 

775 

891 

1008 

1124 

1241 

1357 

18 

366 

402 

475 

548 

693 

839 

984 

1129 

1275 

1420 

1565 

1711 

10 

110 

125 

154 

183 

242 

301 

359 

418 

476 

535 

594 

652 

12 

151 

168 

204 

239 

310 

381 

452 

523 

594 

665 

736 

807 

12 

14 

206 

228 

272 

317 

405 

493 

581 

669 

758 

846 

934 

1022 

16 

279 

306 

362 

418 

529 

640 

751 

862 

•974 

1085 

1196 

1307 

18 

364 

399 

469 

540 

680 

820 

960 

noo 

1241 

1381 

1521 

1661 

20 

470 

512 

598 

684 

855 

1026 

1197 

1368 

1540 

1711 

1882 

2053 

STONE   BRIDGES. 


627 


Table  5- 

-  (Continued.)  (Original.) 

J3  ^ 

J3^" 

^**s 

.a*5 

ja^ 

5  -^ 

J3  ■« 

.c- 

J3-J 

§ 

t^B. 

Wife 

■&&. 

^^ 

■Sjfe 

bC^ 

U^^ 

n^ 

Vc^ 

fr^ 

^^ 

o 

Cin 

?.s^ 

s? 

^^ 

y? 

t^ 

^,^ 

"2 

?>5 

?,^ 

f,S 

^8 

X 
Ft. 

^■^ 

1-3 

^ 

h3® 

ij-^ 

ij-* 

h5" 

1-;^ 

Ft. 

Cub.Y. 

Cub.Y. 

eub.  Y. 

Cub.Y. 

Cub.Y. 

Cub.Y. 

Cub.Y. 

Cub.Y. 

Cub.Y. 

Cub.Y. 

Cub.Y. 

Cub.Y. 

12 

162 

182 

222 

262 

342 

422 

502 

583 

663 

743 

823 

903 

U 

215 

239 

286 

333 

427 

522 

616 

711 

805 

899 

994 

1088 

15 

16 

285 

313 

370 

427 

541 

654 

768 

882 

996 

1110 

1223 

1337 

18 

369 

404 

474 

545 

686 

826 

967 

1108 

1249 

1390 

1530 

1671 

20 

473 

515 

600 

685 

855 

1024 

1194 

1364 

1534 

1704 

1873 

2043 

22 
14 

595 

646 

748 

850 
371 

1054 

1258 

1462 

1666 

1870 

2074 

2278 

2482 

237 

264 

317 

478 

586 

693 

801 

908 

1015 

1123 

1230 

16 

304 

335 

397 

458 

582 

706 

829 

953 

1076 

1200 

1324 

1447 

20 

18 

381 

416 

486 

556 

697 

838 

97a 

1119 

1259 

1400 

1541 

1681 

20 

479 

520 

601 

682 

844 

1007 

1169 

1332 

1494 

1656 

1819 

1981 

22 

598 

646 

741 

837 

1028 

1220 

1411 

1603 

1794 

1985 

2177 

2368 

24 

739 

795 

908 

1021 

1247 

1473 

1699 

1925 

2151 

2377 

2603 

2829 

• 

16 

327 

360 

428 

496 

631 

766 

901 

1036 

1172 

1307 

1442 

1577 

18 

403 

441 

617 

694 

746 

898 

1050 

1202 

1355 

1507 

1659 

1811 

^5 

20 

500 

543 

629 

715 

887 

1059 

1231 

1403 

1575 

1747 

1919 

2091 

22 

614 

663 

760 

857 

1051 

1246 

1440 

1635 

1829 

2023 

2218 

2412 

24 

751 

807 

919 

1031 

1255 

1479 

1703 

1927 

2151 

2375 

2599 

2823 

26 

909 

974 

1104 

1234 

1494 

1754 

2014 

2274 

2534 

2794 

3054 

3314 

28 
22 

1085 

1160 

1310 

1460 

1760 

2060 

2360 

2660 

2960 

3260 

3560 

3860 

686 

743 

859 

975 

1207 

1439 

1671 

1903 

2135 

2367 

2599 

2831 

24 

817 

880 

1007 

1134 

1388 

1642 

1896 

2150 

2404 

2658 

2912 

3166 

85 

26 

969 

1033 

1181 

1309 

1585 

1861 

2137 

2413 

2689 

2965 

3241 

3517 

28 

1130 

1205 

1356 

1507 

1809 

2111 

2413 

2715 

3017 

3319 

3621 

3923 

30 

1327 

1408 

1571 

1734 

2060 

2386 

2712 

3038 

3364 

3690 

4016 

4342 

B2 

1549 

1639 

1820 

2001 

2363 

2725 

3087 

3449 

3811 

4173 

4535 

4897 

35 

1946 

2054 

2271 

2488 

2922 

3356 

3790 

4224 

4658 

5092 

5526 

5960 

30 

1494 

1594 

1795 

1996 

2398 

3800 

3202 

3604 

4006 

4408 

4810 

5212 

32 

1711 

1819 

2035 

2251 

2683 

3115 

3547 

3979 

4411 

4843 

5275 

5707 

34 

1956 

2071 

2302 

2533 

2995 

S457 

3919 

4381 

4843 

5305 

5767 

6229 

50 

36 

2228 

2350 

2597 

2844 

3338 

8832 

4;«6 

4820 

5314 

5808 

6302 

6796 

38 

2519 

2650 

2913 

3176 

3702 

4228 

4J54 

5280 

5806 

6332 

6858 

7384 

40 

2835 

2975 

3255 

3535 

4095 

4655 

5215 

5775 

6335 

6895 

7455 

8015 

42 

3197 

3347 

3647 

3947 

4547 

5147 

5747 

6347 

6947 

7547 

8147 

8747 

45 

3818 

3991 

4337 

4683 

5375 

6067 

6759 

7451 

8143 

8835 

9527 

10219 

50 

5063 

5281 

5717 

6153 

7025 

7897 

8769 

9641 

10513 

11385 

12257 

13129 

Art.  9.  Especial  pains  should  be  taken  to  secure  an  unyielding'  fonn« 
elation  «»r  culverts  and  drains  under  hig^b  embfets  §  otherwise 
the  superincumbent  weight,  especially  under  the  middle  of  the  embkt,  may  squeez** 
them  into  the  soil  below,  if  soft  or  marshy;  and  thus  diminish  the  area  of  water- 
way, or  at  least  cause  an  ugly  settlement  at  the  midlength  of  the  culvert.  Also,  in 
soft  ground,  the  embkt  Inay  press  the  side  walls  closer  together,  narrowing  the 
channel.  This  may  be  prevented  by  an  inverted  arch,. or  a  bed  of  masonry,  between 
the  walls.  A  stratum  from  3  to  6  ft  thick,  of  gravel,  sand,  or  stone  broken  to  turn- 
pike size,  will  generally  give  a  sufficient  foundation  for  culverts  in  treacherous 
marshy  ground ;  or  quicksand,  with  but  a  moderate  height  of  embkt.  It  should  ex- 
tend a  few  feet  beyond  the  masonry  in  ev^ry  direction,  and  should  be  rammed ;  the 
sand  or  grayel  being  thoroughly  wet,  if  possible,  to  assist  the  consolidation.  Piling 
will  sometimes  be  necessary.  If  the  masonry  is  built  upon  timber  platforms,  or  a 
smooth  surface  of  rock,  care  must  be  taken  to  prevent  it  from  sliding,  from  the  pres 
of  the  earth  behind  it.  -This  same  pres  may  even  overthrow  the  piles,  if  they  are 
not  properly  secured  against  it. 
ArtlO.  ^Drains.  c, 


Drains  of  the  dimen- 
sions in  Fig  11,  con- 
tain 1  perch,  of  25 
cub  ft ;  or  .926  of  a 
cub  yd,  per  ft  run. 

They  are  frequently 
built  of  dry  scabbled 
rubble,  and  paved  with 
spawls.  When  bhere  is 
much  wash  through 
them,  with  a  consider- 
able slope,  it  is  better  to 
continue  the  foundation 


JI 


J 


3. 


Tx^li 


628 


STONE   BRIDGES. 


solid  clear  across.  This  Is  often  done  without  those  causes,  inasmuch  as  the  additional  masonry  Is  a 
mere  trifle ;  ami  the  excavation  of  a  single  broad  foundation-pit  is  less  troublesome  than  that  of  two 
narrow  ones.  A  deep  flag-stoue/at  the  entrance,  and  others  at  short  dists  of  the  length,  may  be  in- 
troduced in  both  drains  and  culverts,  to  protect  from  undermining. 

These  drains  extend  under  the  entire  width  of  the  embkt,  from  toe  to  toe ;  and  may  terminate  la 
steps,  as  in  the  side  view  at  S.  They  are  of  course  better  when  built  with  mortar,  with  an  admixture 
of  cement  to  prevent  the  water  when  full  from  leaking  into  and  softening  the  embankment. 

Sometimes  two  or  three  such  drains  may  be  placed  parallel  to  each  other,  instead  of  a  culvert. 
When  two  are  so  placed,  they  contain  only  IJ^  times  the  masonry  of  one  ;  still  their  use  will  generally 
involve  no  saving  of  masonry  over  a  culvert.     A  man  can  crawl  through  Fig  11  to  clean  it. 

Art.  11.  The  drainage  of  the  roartways  of  stone  bridges  of  several 
arches,  is  generally  elfected  by  means  of  open  gutters,  which  descend  slightly  fron? 
the  crowns  of  the  arches,  each  way,  until  they  reach  to  near  the  ends  of  the  re- 
spective spans. 

There  they  discharge  Into  vertical  iron  pipes  built  into  the  masonry.  The  upper  ends  of  the 
pipes  should  be  covered  by  gratings.  When  inconvenience  would  result  from  the  water  falling  upoa 
persons  passing  under  the  arches,  these  pipes  may  be  carried  down  tlie  entire  height  of  the  piers; 
but  when  such  is  not  the  case,  they  may  extend  only  to  the  soflit,  or  under  face  of  the  arch ;  allowing 
the  water  to  fall  freely  through  the  air  from  that  height. 

Table  6,  of  approxiimate  contents,  in  cub  yds,  of  a  solid 
pier  of  masonry,  6  ft  by  22  ft  on  top;  and  battering  1  inch  to  a  ft  on  each  of 
Its  4  faces.  The  contents  of  masonry  of  such  forms  must  be  calculated  by  the  prismoidal  formula ; 
and  not  by  taking  the  length  and  breadth  of  the  pier  at  half  its  height  as  an  average  length  and 
breadth,  as  is  sometimes  done.  This  incorrect  method  would  give  only  6492  cub  yds  as  the  content 
of  the  pier  200  ft  high;  instead  7178  yds,  its  true  content.  High  piers  may  for  economy  be  built  hol- 
low, with  or  without  interior  cross- walls  for  strengthening  them,  as  the  case  may  require;  and  tho 
batter  is  generally  reduced  to  ^  inch  or  less  to  a  foot.  Hollow  piers  require  good  well-bedded  ma* 
^^^^y-  (Original.)  t 


Ht. 

Lgth 

Bdth 

Cubic 

lit. 

Lgth 

Bdth 

Cubic 

Ht. 

Lgth 

at 
base. 

Bdth 

at 
base. 

Cubic 

Ft. 

at 
base. 

at 
base. 

yards 

Ft. 

at 

base. 

at 

base. 

yards 

Ft. 

yards. 

6 

23. 

7. 

32.5 

52 

30.67 

14.67 

537 

128 

43.33 

27.33 

2759 

7 

.17 

.17 

38.6 

54 

31. 

15. 

570 

130 

.67 

.67 

2848 

8 

.33 

.33 

44.9 

56 

.33 

.33 

605 

132 

44. 

28. 

2940 

9 

23.5 

7.5 

51.3 

58 

.67 

.67 

641 

134 

.33 

.33 

3032 

10 

.67 

.67 

58. 

60 

32. 

16. 

679 

136 

.67. 

.67 

3126 

11 

.83 

.83 

64.8 

62 

.33 

.33 

717 

138 

45. 

29. 

3222 

12 

24. 

8. 

71.7 

64 

.67 

.67 

757 

140 

.33 

.33 

3320 

13 

.17 

.17 

79. 

66 

33. 

17. 

798 

142 

.67 

.67 

3420 

14 

.33 

.33 

86.4 

68 

.33 

.33 

840 

144 

46. 

30. 

3521 

15 

24.5 

8.5 

94. 

70 

.67 

.67 

884 

146 

.33 

.33 

3623 

16 

.67 

.67 

102 

72 

34. 

18. 

928 

148 

.67 

.67 

3728 

17 

.83 

.83 

110 

74 

.33 

.33 

973 

150 

47. 

31. 

3835 

18 

25. 

9. 

118 

76 

.67 

.67 

1021 

152 

.33 

.33 

3944 

19 

.17 

.17 

127 

78 

35. 

19. 

1070 

154 

.67 

.67 

4050 

20 

.33 

.33 

135 

80 

.33 

.33 

1120 

156 

48. 

32. 

4168 

21 

25.5 

9.5 

144 

82 

*   .67 

.67 

1171 

158 

.33 

.33 

4284 

22 

.67 

.67 

153 

84 

36. 

20. 

1224 

160 

.67 

.67 

4402 

28 

.83 

.83 

163 

•  86 

.33 

.33 

1278 

162 

49. 

33. 

4520 

24 

26. 

10. 

172 

88 

.67 

.67 

1334 

164 

.33 

.33 

4640 

25 

.17 

.17 

182 

90 

37. 

21. 

1392 

166 

.67 

.67 

4763 

26 

.33 

.33 

192 

92 

.33 

.33 

1451 

168 

50. 

34. 

4887 

27 

26.5 

10.5 

202 

94 

.67 

.67 

1510 

170 

.33 

.33 

5014 

28 

.67 

.67 

212 

96 

38. 

22. 

1569 

172 

.67 

'  .67 

5143 

29 

.83 

.83 

223 

98 

.33 

.33 

1631 

174 

51. 

35. 

5275 

30 

27. 

11. 

234 

100 

.67 

.67 

1695 

176 

.33 

.33 

5409 

31 

.17 

.17 

245 

102 

39. 

23. 

1761 

178 

.67 

.67 

5545 

32 

.33 

.33 

256 

104 

.33 

.33 

1829 

ISO 

52. 

36. 

5680 

33 

27.5 

11.5 

268 

106 

.67 

.67 

1899 

182 

.33 

.33 

5820 

34 

.67 

.67 

280 

108 

40. 

24. 

1968 

184 

.67 

.67 

5962 

35 

.83 

.83 

292 

110 

.33 

.33 

2041 

186 

53. 

37. 

6106 

36 

28. 

12. 

304 

112 

.67 

.67 

2115 

188 

.?3 

.33 

6252 

38 

.33 

.33 

329 

114 

41. 

25. 

2191 

190 

.67 

.67 

640C 

40 

.67 

.67 

356 

116 

.33 

.33 

2269 

192 

54. 

38. 

6552 

42 

29. 

13. 

383 

118 

.67 

.67 

2346 

194 

.33 

.33 

6704 

44 

.33 

.33 

411 

120 

42. 

26. 

2424 

196 

.67 

.67 

6859 

46 

.67 

.67 

441 

122 

.33 

.33 

2504 

198 

55. 

39. 

7016 

48 

30. 

14. 

472 

124 

.67 

.67 

2587 

200 

.33 

.33 

7178 

50 

.33 

.33 

504 

126 

43. 

27. 

2672 

202 

.67 

.67 

7339 

BRICK   ARCHES. 


629 


u 


Art.  12.  Brick  Arc  lies.  Since  even  good  brick  fit  for  large  arches  has 
far  less  crushing  strength  than  good  granite  or  limestone,  and  is  inferior  even  to 
good  sandstone,  while  its  weight  does  not  differ  very  materially  from  stone,  it  is 
plain  that  it  cannot  be  used  in  arches  of  as  great  span  as  stone  can.  Some  of 
those  already  built,  and  which  have  stood  for  many  years,  have  a  theoretical  co- 
efficient of  safety  of  but  about  3 ;  whereas  the  authorities  direct  us  not  to  trust  even 
stone  with  more  than  one-twentieth  of  its  crushing  load.  This  last,  however,  ap- 
pears to  the  writer  to  be  one  of  those  hasty  assumptions  which,  when  once  aa- 
mitted  into  professional  books,  are  difficult  to  be  got  rid  of.  It  is  his  opinion  that 
with  good  cement,  and  proper  care  in  striking  the  centers,  one-tenth  of  the  ulti- 
mate strength  is  sufficiently  secure  against  even  the  abnormal  strains  caused  by 
the  settling  at  crown,  and  rising  at  the  haunches  when  the  centers  are  struck.  It 
is  useless  to  attempt  to  fix  limits  of  safety  for  bad  materials  poorly  put  together. 
Rem.  1.  The  common  practice  of  building  brick  arches  in  a  series  of  con- 
centric ring's,  as  at  a  c  e  e,  Fig  12,  with  no  other  bond  between  them  than 

that  afforded  by  the  mortar,  is  censured  by 
authorities,  on  the  ground  that  the  line  of 
pressure  in  passing  from  the  extrados  to 
the  intrados  tends  to  separate  the  rings, 
and  thus  weaken  the  arch  by,  as  it  were, 
splitting  it  longitudinally.  The  reason 
for  using  these  rings,  instead  of  making 
the  radial  joints  continuous  throughout 
the  depth  m  w  of  the  arch,  as  at  b,  is  to 
avoid  the  thick  mortar-joints  at  the  back  of  , 
the  arch,  and  shown  in  the  Fig.  If  the 
center  of  an  arch  built  as  at  b  be  struck 
too  soon,  the  soft  mortar  in  these  thick 
joints  will  be  so  much  compressed  as  to  cause  great  settlement  at  the  crown, 
throwing  the  arch  out  of  shape,  and  creating  such  inequality  of  pressure  as 
might  even  lead  to  its  fall,  especially  if  flat.  As  a  compromise  between  rings 
and  continuous  joints,  they  are  sometimes  employed  together,  so  as  to  get  rid  of 
some  of  the  long  radial  joints  ;  and  at  the  same  time  to  break  at  intervals 
the  continuity  of  the  rings.  Thus  in  Fig  12,  which  is  supposed  to  be  brick-and- 
l-half  deep,  beginning  at  the  abutment  a,  we  may  lay  half-brick  rings  as  far  as 
say  to  e  0  e;  then  cutting-  aivay  the  brick  o  to  the  line  e  e,  we  may  lay  from 
e  e  to  m  n  a  block  of  bricks  with  continuous  radial  joints,  the  same  as  at  b;  and 
then  start  again  with  three  rings;  and  so  on  alternately.  A  still  better,  but 
more  expensive,  mode  would  be  to  fill  e  e,  m  n  with  a  regular  cut-stone  voussoir. 
The  proper  intervals  for  changing  from  rings  to  blocks  will  depend  upon  the 
number  of  the  rings  and  the  depth  c  a  of  the  arch ;  reference  being  also  had  to 
reducing  the  amount  of  brick  cutting  as  much  as  possible. 

These  points  can  be  best  decided  on  from  a  drawing  of  a  portion  of  the  arch 
on  a  scale  of  3  or  4  ins  to  a  foot.  Generally  the  rings  are  made  only  half-brick,  or 
about  4  to  4.5  ins  thick,  as  at  a  c  ;  and  in  Brunei's  Maidenhead  viaduct  of  two  ellip- 
tic brick  arches  of  128  ft  span,  and  24.25  ft  rise ;  the  boldest  brick  arches  yet  at- 
tempted ;  but  which  have  been  estimated  to  have  a  co-efficient  of  safety  of  but 
three  against  crushing  at  the  crown. 

So  many  others  of  from  70  to  100  ft  span  have  been  successfully  built  entirely  in 
rings  of  either  half  or  whole  brick  thick,  as  to  justify  us  in  attaching  but  little  weight 
to  the  above  theoretical  objection,  provided  first  class  cement  be  used,  and  time 
allowed  it  to  become  nearly  or  quite  as  hard  as  the  bricks  themselves,  before 
striking  the  centers.  Under  such  circumstances  we  should  not  object  to  a  series 
of  rings  even  1.5  bricks  thick,  laid  alternately  header  and  stretcher,  as  at  b. 

If  tlie  bricks  were  Tonssoir-shaped,  that  is,  a  little  thicker  at  one 
end  than  the  other,  then  rings  a  whole-brick  thick  could  be  used  without  any  in- 
crease in  thickness  of  mortar-joint  at  the  back  of  each  ring.  Still  with  more 
than  one  ring,  the  radial  joints  would  not  be  continuous,  as  at  b,  but  broken  as  at 
ac.  Such  bricks  however  would  be  more  expensive  to  make ;  and  moreover,  in 
order  fully  to  answer  the  intended  purpose,  they  would  have  to  be  made  of  many 
patterns,  so  as  to  conform  to  the  many  radii  used  in  arches;  and  even  to  the 
radii  of  the  different  rings,  when  the  depth  of  the  arch  required  several  of  them. 


630 


BRICK   ARCHES. 


Rem.  2.  J¥et  the  bricks  before  laying. 

Rem.  3.  When  the  ends  or  faces  of  a  brick  arch  are  to  be  finished  with  cnt* 
stone  TOnssoirs,  these  had  better  not  be  inserted  until  some  time  after  the 
completion  of  the  brickwork,  the  hardening  of  the  mortar,  and  a  partial  easing 
of  the  centers ;  lest  they  be  cracked  or  spawled  by  the  unequal  settlements  of  them* 
selres  and  the  bricks. 


Rem.  Brick  arches,  from  their  great  number  of  joints  are  apt  to  settle 
much  more  than  cut  stone  ones  when  the  centers  are  removed,  and  thereby  to 
derange  the  shape  of  the  arch,  and  at  times,  without  due  care,  even  to  endanger 
its  safety,  especially  if  it  be  large  and  flat.  When  the  span  exceeds  about  80  to  35 
ft,  and  particularly  if  flat,  use  only  brick  of  superior  quality  in  good  cement 
mortar.  With  even  best  materials  and  work  we  advise  the  young  engineer  not 
to  attempt  brick  arches  for  railroad  bridges  of  greater  spans  than  about  the  fol- 
lowing. Considerably  larger  ones  than  some  of  them  have  been  built,  and  have 
stood ;  but  their  coefs  of  safety  are  not  in  all  cases  satiafactory.  In  this  table  the 
rise  is  in  parts  of  the  span. 


R. 

S. 

R. 

S, 

R. 

S. 

R. 

S. 

R. 

s. 

.5 

100 

1^ 

88 

.225 

68 

% 

50 

.134 

35 

.4 

97 

.29 

82 

f 

60 

.155 

45 

^ 

30 

.36 

93 

% 

75 

.183 

55 

^ 

40 

On  the  Filbert  Street  Extension  of  the  Penna  R  R,  in  Phila, 

are  four  brick  arches  of  50  ft  1  inch  span,  and  with  the  very  low  rise  of  7  ft.  They 
are  2  ft  6  ins  thick,  except  on  their  showing  faces,  where  they  are  but  2  ft.  The 
joints  are  in  common  mortar,  and  about  ^  inch  thick.  These  .four  arches,  about 
200  yards  apart,  with  a  large  number  of  others  of  26  ft  span,  form  a  viaduct.  The 
piers  between  the  short  spans  are  4  ft  3  ins  thick.  Those  at  the  ends  of  the  50-ft 
spans,  18  ft  6  ins.  The  springing  lines  of  all  the  arches  are  about  6  to  8  ft  above  the 
ground.  One  of  the  50-ft  arches  settled  3  ins  upon  prematurely  striking  the 
centers;  but  no  further  settlement  has  been  observed,  although  the  viaduct  has, 
since  built  (1880)  had  a  very  heavy  freight  and  passenger  traffic,  at  from  10  to  20 
miles  per  hour.    Boadbed,  about  100  ft  wide,  giving  room  for  9  or  10  tracks. 


CENTERS   FOR   ARCHES. 

CENTEES  FOE  AEOHES. 


631 


^  Art»  1.  A  center  is  a  temporary  wooden  structure  (built  lying  flat,  on  a  full 
size  dravfing,  on  a  fixed  platform,  under  cover  or  not)  for  supporting  an  arch 
while  it  is  being  built.  It  consists  of  a  number  of  trusses  or  frames,/,/,  Fig.  1, 
placed  from  1  to  6  ft  apart  from  cen  to  cen,  and  covered  with  a  flooring  I,  I,  of 
rough  boards  or  planks,  usually  laid  close,  and  called  the  sheeting'  or  lag- 
ging*, immediately  upon  which  the  archstones  are  laid.  In  Fig  3,  the  lag- 
ging is  not  laid  close.  There  is  no  great  economy  in  placing  the  frames  very 
far  apart,  on  account  of  the  greater  required  amount  of  lagging,  the  thickness 
of  which  increases  rapidly.    For  the  thickness  of  lagging  see  Rem  9. 


::^3S 


Figsl. 


The  centers  rest  by  the  ends  of 
their  chords,  c,  upon  wooden 
striking  -wedges  w,  Fig  1, 

supported  by  standards  com- 
posed of  posts  p,  whose  tops  are 
connected  by  cap-pieces  o; 
and  whose  feet  rest  on  string- 
ers s;  the  whole  being  braced 
diagonally  as  shown. 

If  the  ground  is  very  firm,  and 
the  arch  light,  the  standards  may 
rest  on  it,  with  the  interposition 
of  adjusting-blocks,  n,  be- 
low the  stringer,  to  accommodate 
irregularities  or  the  surface  of  the  ground,  as  in  the  Fig.  These  blocks  should 
be  somewhat  double- wedge-shaped,  so  that  by  driving  them  the  standard  may 
be  raised  at  any  point  in  case  it  should  settle  a  little  into  the  ground.  But  for 
fieavy  arches  the  standards  must  rest  on  a  much  firmer  foundation,  such  as  short 
blocks  of  brickwork  sunk  a  few  feet  into  the  ground,  or  some  other  device 
adapted  to  the  case.  Frequently  projecting  offsets  or  footings,  or  at  times  re- 
cesses, are  provided  in  the  masonry  of  the  abutments  and  piers  for  this  express 
purpose ;  and  with  a  view  to  this  it  is  well  to  design  the  center  at  the  same  time 
as  the  arch. 

Up  to  spans  of  50  or  60 
ft  a  single  row  of  posts  (one  under  each  end  of 
each  frame)  will  suffice ;  but  for  much  larger  ones 
two  or  three  rows,  2  or  more  feet  apart  may  be- 
come expedient,  as  in  the  lower  Fig  2. 

Tlie  striking  or  lowering-nredges 
before  alluded  to  are  for  striking  or  lowering  the 
center  after  the  completion  of  the  arch.  They 
consist  of  pairs  of  wedge-shaped  blocks,  w  to,  at  A, 
Figs  2,  of  hard  wood,  from  1  to  2  ft  long,  about  half 
as  wide,  and  a  quarter  or  moie  as  thick,  (sufficient 
to  lower  the  center  from  say  2  to  6  or  more  inches, 
according  to  span  and  other  circumstances,)  rest- 
ing on  the  cap  o,  of  the  standard,  while  the  chord 
c  of  the  frame  rests  on  them.  When  the  end  of  a 
frame  is  supported  by  two  or  more  posts  p,  as  at  B, 
Fig  2,  instead  of  upon  one,  the  striking-wedges  are 
sometimes  made  as  there  shown ;  and  where  B  v 
is  one  long  wedge  at  right  angles  to  the  abutment, 
and  acting  as  four  wedges  which  may  all  be  low- 
ered together  by  blows  against  the  end  B. 

Up  to  spans  of  60  or  80  ft,  all  the  frames  may  rest 


BY~--S^^^S^^^^S^n^ 


GO  but  two  wedges  like  B  V 


632  CENTERS  FOR  ARCHES. 

,  each  80  long  as  to  reach  transversely  across  the  entire  arch.  Then  all  the 
frames  can  be  lowered  at  one  operation,  as  described  near  end  of  Art  9. 

If  we  had  to  consider  only  the  friction  of  dry  wood  against  dry  wood,  the  taper 
of  these  wedges  might  be  as  steep  as  1  vert  to  3  hor,  without  any  danger  of  their 
sliding  upon  each  other  of  their  own  accord;  and  they  would  then  require  very 
moderate  blows  to  start  them,  or  even  to  entirely  separate  them,  when  the  center 
had  finally  to  be  lowered.  But  it  is  of  the  utmost  importance,  especially  in  large 
arches,  that  the  centers  should  be  lowered  very  slowly,  otherwise 
the  momentum  acquired  by  so  heavy  a  body  as  the  arch  in  descending  suddenly 
even  but  2  or  3  ins,  might  possibly  affect  its  'shape,  or  even  its  safety. 

Therefore  the  wedges  should  not  have  a  taper  steeper  than  about  1  in  6  or  8  for 
arches  of  less  than  about  50  ft  span ;  or  than  1  in  8  or  10  for  larger  spans.  VerticaJ 
lines  at  equal  dists  apart  should  be  drawn  on  the  long  sides  of  the  wedges  as  a 
guide  for  lowering  them  all  to  the  same  extent  at  a  time  ;  and  this  should  not  ex- 
ceed in  all  about  half  an  inch  a  day  in  intervals  of  about  an  eighth  of  an  inch,  for 
50  ft  spans ;  or  about  .1  to  .25  of  an  inch  per  day  in  all,  for  spans  over  100  ft. 
Slowness  is  especially  to  be  recommended  in  brick  arches,  not  only  because 
their  greater  number  of  joints  exposes  them  to  greater  derangement  of  shape, 
but  because  even  good  brick  has  much  less  than  the  average  crushing  strength 
of  good  granite,  limestone,  or  sandstone,  and  therefore  is  far  more  liable  than 
they  to  crack,  or  even  to  crush  (as  the  v^riter  has  seen)  when  the  strains  are 
thrown  almost  entirely  upon  their  edges,  as  described  in  Art  3. 

At  Gloucester  Brid&e,  England,  of  first  class  cut  stone,  span  150  ft,  rise 
35  ft,  the  centers  were  entirely  struck  within  the  very  short  space  of  3  hours ;  and 
the  crown  of  the  arch  descended  10  ins  I  At  Orosvenor  Bridge,  England, 
of  first  class  cut  stone,  span  200  ft,  rise  42  ft,  such  care  was  taken  in  easing  the 
centers  that  the  crown  of  the  arch  settled  but  2.5  ins.  This  case  however  was 
marked  by  two  or  three  peculiarities,  all  of  which  contributed  to  this  favorable 
result.  Namely,  the  center  instead  of  being  a  series  of  frames  supported  as  usual 
by  their  ends,  and  of  course  involving  an  appreciable,  although  small,  degree  of 

sagging  or  settlement,  consisted 
essentially  of  vertical  and  in- 
clined posts  or  struts,  see  Fig  3, 
footing  on  four  temporary  piers 
of  masonry,  7  or  8  feet  thick,  built 
in  the  river,  parallel  to  the  abut- 
ments, and  as  long  as  they.  These 
piers  supported  six  frames  (or 
rather  six  series)  about  7  ft  apart 
cen  to  cen,  of  such  struts,  footing 
on  cast  iron  shoes.  Fig  3  shows 
half  of  one  series.  Each  frame 
or  series  consisted  of  four  fan-like 
sets  of  posts,  all  in  the  same  ver- 
tical plane.  The  long  horizontal  pieces  seen  extending  from  side  to  side  of  the 
arch  were  bolted  to  the  struts  to  increase  their  stiffness ;  and  other  pieces  for  the 
same  purpose  united  the  six  series  transversely.  Here  each  strut  sustains  its  own 
share  of  the  weight  of  the  archstones,  and  transfers  it  directly  to  the  unyielding 
foundation  of  the  pier;  whereas  in  the  usual  trussed  centers,  the  entire  load  rests 
upon  the  frames,  and  is  finally  transferred  to  the  comparatively  unstable  support 
of  the  posts  at  their  ends. 

The  topsp  of  the  posts  of  a  series  varied  about  from  5  to  8  ft  apart  cen  to  cen ; 
and  were  connected  by  a  continuous  curved  rib,  rr,  of  two  thicknesses  of  4  inch 
plank,  bent  to  conform  approximately  to  the  curve  of  the  arch.  On  this  rib  were 
placed  pairs  of  striking-wedges  w  like  Fig  2,  about  16  ins  long,  10  to  12  ins  wide,  and 
tapering  1.5  ins,  so  near  together  (varying  about  from  2.5  to  3.5  ft  cen  to  cen)  that 
there  was  a  pair  under  each  joint  of  the  archstones,  a  a.  On  these  wedges,  and  ex- 
tending over  all  six  of  the  frames,  were  the  lagging  pieces  /,'  4.5  ins  thick. 

This  peculiar  arrang>enient  of  the  striking^-wedg:es  and  lag- 
ging has,  in  large  spans,  great  advantages  over  the  usual  one  of  placing  them  only 
at  the  ends  of  the  frames.  In  the  last  the  entire  center  and  the  entire  arch  are 
lowered  together,  without  giving  an  opportunity  to  rectify  any  slight  derange- 
ments of  shape  or  inequality  of  bearing  that  may  have  occurred  in  the  arch  during 
its  construction.  This  center,  designed  by  Mr.  Trubshaw,  admits  of  lowering 
either  the  whole  equally,  or  any  one  part  a  little  more  or  less  than  the  others. 
He  had  much  experience  in  large  arches,  and  stated  that  during  the  striking  he 
found  that  he  had  an  arch  under  belter  control,  or  could  humor  it  better,  by  keep- 
ing the  haunches  a  little  down  and  the  crown  a  little  up,  until  near  the  end  of 
the  operation. 


CENTERS  FOR  ARCHES.  633 

Rem.  1.  Instead  of  piers  of  masonry  for  supporting  the  feet  of  the 
posts,  wooden  cribs  or  piles  may  often  be  used  if  the  arch  is  over  water. 

Tlie  principle  of  supporting  even  trussed  frames  by  struts 
at  points  of  the  chord  as  far  from  the  abutments  as  "circumstances  will  admit  of 
(in  addition  to  those  at  the  very  ends)  should  always  be  applied  when  possible, 
in  order  to  reduce  their  sagging  to  a  minimum.  Steps  or  offsets  in  the 
masonry  of  the  abutments  and  piers  may  be  provided  for  receiving  the  feet 
of  such  struts,  when  they  are  inclined. 

Rem.  2.  Screws  may  be  used  instead  of  wedges  for  lowering  centers.  At 
the  Pont  d'Alma,  Paris,  ellipse  of  141.4  ft  span,  and  28.2  ft  rise,  the  frames  were  sup- 
ported by  wooden  pistons  or  plungers,  the  feet  of  which  rested  on  sand  con- 
fined in  plate-iron  cylinders  1  ft  in  diam  and  height,  and  having  near 
the  bottom  of  each  a  plug  which  could  be  withdrawn  and  replaced  at  pleasure^ 
thus  regulating  the  outflow  of  the  sand  and  the  descent  of  the  center.  This  de^ 
vice  succeeded  perfectly,  and  is  well  worthy  of  adoption  under  arches  exceeding 
about  60  ft  span.  When  much  larger  than  this  the  driving  of  the  wedges  on 
striking  requires  heavy  blows,  and  becomes  a  somewhat  awkwara  operation,  re- 
quiring at  times  a  battering-ram,  even  when  the  wedges  are  lubricated.  In  rail- 
road cuttings  crossed  by  bridges,  the  earth  under  the  arch  has  been 
made  to  serve  as  a  center,  by  dressing  its  surface  to  the  proper  curve,  and  then 
embedding  in  it  curved  timbers  a  few  feet  apart,  and  extending  from  abut  to  abut, 
for  supporting  the  close  plank  lagging. 

Rem.  3.  All  centers  must  yield  or  settle  more  or  less  under  the  wt 
of  the  arch,  especially  when  supported  only  near  their  ends ;  and  since  the  arch 
itself  also  settles  somewhat  not  only  when  the  centers  are  struck,  but  for  some 
time  after,  it  is  advisable  to  make  them  at  first  a  little  higher  than  the  finished 
arch  is  intended  to  be.  This  extra  height,  when  the  supports  are  at  the  ends, 
naay  be  from  2  to  4  ins  per  100  ft  of  span  for  cut  stone  arches  (according  to  time 
of  striking,  character  of  masonry,  workmanship,  etc.),  and  about  twice  as  much  in 
brick  ones. 

Rem.  4.  The  proper  time  for  striking  centers  is  a  disputed 
point  among  engineers,  some  contending  that  it  should  be  done  as  soon  as  the 
arch  is  finished  and  suffi'^iently  backed  up ;  and  others  that  the  mortar  should 
first  be  given  time  to  harden.  It  is  the  writer's  opinion  that  inasmuch  as  in 
cut-stone  arches  the  mortar  joints  should  be  very  thin ;  and  since,  in  such,  the 
mortar  is  at  best  of  very  little  service,  it  is  of  no  importance  when  they  are  struck; 
provided  the  masonry  backing,  and  the  embkt  up  to  y  n  Fig  2,  p  618,  have  been  com 
pleted;  but  that  in  brick  or  rubble,  the  numerous  joints  of  both  of  which  require 
much  mortar,  (which  for  hardness  should  cousist  largely  of  cement,)  3  or  4  months, 
or  longer,  if  possible,  should  be  allowed  it  to  harden  suflBciently  to  prevent  undue 
compression  and  consequent  settlement  when  the  centers  are  struck.  The  con- 
tinuance of  the  centers  need  not  interfere  with  traffic  over  the  bridge. 

Art.  2.  The  pressure  of  -archstones  against  a  center  is  very  trifling  until  after 
the  arch  is  built  up  so  far  on  each  side  that  the  joints  form  angles  of  25°  or  30^ 
with  the  horizontal.  Theoretical  discussions  on  this  pressure  make  no  allowance 
for  accidental  jarrings  in  laying  the  archstones,  or  by  the  accumulation  of  material 
ready  for  use,  laborers  working  on  it,  &c.  Without  going  into  any  detail,  we  merely 
advise  on  the  score  of  safety  not  to  assume  it  at  less  than  about  the  following  pro- 
portions or  ratios  to  the  weight  of  the  entire  arch,  namely,  in  a  semicircular  arch 
.47;  rise  .35  span,  .61;  rise  .25  span,  .79;  rise  .2  span,  .86;  rise  .167  span,  or  less,  1, 
or  equal  to  the  wt  of  the  arch.  This  gives  the  pressure  of  a  semicircular  arch 
upon  its  centers  rather  less  than  half  its  wt.  The  wt  of  the  centers 
themselTes  when  supported  only  near  the  ends  must  be  considered  as  part 
of  the  load  borne  by  them. 

Art.  3.  We  have  seen  that  as  an  arch  aaais  being  gradually  built  upward  on 
both  sides,  after  passing  the  points  «,  e.  Fig  4,  where  its  joints  form  angles  a  s  e,  of 
about  30°  with  the  horizontal  a  a,  the  arch  begins  to  press  more  and  more 
upon  the  centers ;  thereby  tending  to  flatten  them  at  the  haunches,  as  shown  at  h 
in  the  dotted  line ;  and  consequently  to  raise  them  at  the  crown,  as  shown  at  c. 
But  as  the  building  goes  on  still  higher,  the  added  stones  press  much  more  heavily 
upon  the  centers  than  those  below  had  done,  and  thereby  tend  to  a  final  derange- 
ment of  the  centers  just  the  reverse  of  that  caused  by  the  lower  ones;  namely  to 
depress  them  at  the  crown  a,  as  at  o;  and  consequently  to  raise  the  haunches  as 
at  w;  and  this  the  more  because  the  upper  stones  actually  tend  to  lift  or  ease  the 
lower  ones  from  the  lagging.  In  some  cases  where  this  tendency  has  been  in- 
creased by  forcing;  the  keystones  into  place  by  too  hard  driving,  the  lagging 
under  the  haunches  could  be  drawn  out  without  any  trouble  before  the  centers 
were  eased  at  all.    On  striking  the  centers  this  tendency  to  sink  at  crown  and 


634 


CENTERS    FOR   ARCHES. 


rise  at  haunches  is  very  apt  to  exhibit  itself  more  or  less  dangerously  in  the  arch- 
stones  themselves,  as  in  Fig  5,  causing  those  near  the  crown  to  press  very  hard 
together  at  the  extrados,  and  to  separate  from  each  other  at  the  intrados ;'  while 
near  the  haunches  the  reverse  takes  place.  Hence  the  angles  of  the  stones  are 
frequently  split  and  spawled  off  near  c  and  h  by  this  unequal  pressure.    These 


Fig.  4. 


derangements  are  of  course  much  more  likely  to  be  serious  in  high  arches  than 
in  flat  ones,  especially  if  their  spandrels  are  not  sufficiently  built  up  before 
lowering  the  centers. 

In  the  Grosvenor  bridge,  before  alluded  to,  of  200  ft  span,  this  dangerous  excess 
of  pressure  near  c  and  h  was  prevented  by  covering  the  skewback  joint  of  the 
springing  course  at  each  abutment  with  a  wedge  of  lead  1.5  ins  thick  at  the  in- 
trados of  the  arch,  and  running  out  to  nothing  at  the  extrados.  Beside  this  a 
strip  9  ins  wide  of  sheet  lead  was  laid  along  the  intrados  edge  of  every  joint  until 
reaching  that  point  at  which  it  was  judged  that  the  line  of  pressure  would  pass 
from  the  intrados  to  the  extrados ;  after  which  similar  strips  were  laid  along  the 
extrados  edges  of  the  joints,  up  to  the  crown.  Hence  when  the  centers  were 
struck,  this  excess  of  pressure  merely  compressed  the  lead,  and  was  thus  enabled 
to  distribute  itself  more  evenly  over  tne  entire  depth  of  the  joints.  See  Trans 
Ins  Civ  Eng  London,  vol  i.    See  also  top  of  p  921. 

At  tlie  bridg-e  iat  BTeullly,  France  (of  5  ellij)tic  arches  of  120  ft  span, 
and  30  ft  rise),  the  centers  were  so  radically  defective  in  design  that  the  arches 
sank  13.25  ins  at  crown  during  the  time  of  building;  and  10.5  ins  more  during 
and  immediately  after  the  striking ;  or  say  2  ft  in  all.  Their  construction  made 
the  striking  very  tedious  and  hazardous ;  greatly  endangering  the  lives  of  the 
workmen  and  the  existence  of  the  arches.  Sorhe  of  the  joints  at  the  extrados 
at  the  haunches  opened  an  inch  each ;  and  those  at  the  intrados  of  the  crown  .25 
of  an  inch.  By  the  exercise  of  great  care  and  humoring  in  lowering  the  centers, 
these  openings  were  much  reduced. 

Rem.  1.  Cbaiiiferin^  tbe  edges  of  the  arcbstones  diminishes 
the  danger  of  their  spawling  off  from  unequal  pressure ;  as  does  also  the  scrap- 
ing out  of  the  mortar  of  the  Joints  for  an  inch  or  two  in  depth  be- 
fore striking  the  centers. 

Rem.  2.  It  is  evident  that  in  order  to  prevent,  or  at  least  to  diminish  the 
alternate  derangements  of  the  center,  those  of  its  web  members  which  at  first 
acted  as  struts  near  the  haunches,  Fig.  4,  to  prevent  them  from  sinking  as  at 
h,  must  afterwards  act  as  ties  to  prevent  them  fK)m  rising  as  at  n;  while  those 

which  at  first  acted  as  ties  near 
the  crown  a,  to  prevent  it  from 
rising  as  at  c,  must  afterwards 
act  as  struts  to  prevent  it  from 
sinking  as  at  0.  In  other  words, 
the  principle  of  counter- 
bracing'  must  be  attended 
to  as  well  in  a  frame  or  truss 
for  a  center,  as  in  one  for  a 
bridge. 


Art.  4.  From  the  foregoing  it  is  plain  that  a  simple  unbraced  wooden 


CENTERS    FOR   ARCHES. 


635 


arch,  or  curved  rib  is,  on  account  of  its  great  flexibility,  about  as  unfit  a  form 
as  could  be  chosen  for  a  center,  except  for  very  small  spans,  where  a  great  propor- 
tional depth  of  rib  can  be  readily  secured.  Still  the  writer  has  seen  it  used  for  a 
cut-stone  semicircular  arch  of  35  ft  span,  with  archstones  2  ft  deep.  Fig  6  shows 
one  rib  r  r,  and  the  arch,  a  a,  drawn  to  a  scale.  Each  rib  consisted  of  two  thicknesses 
of  2  iuch  plank  in  lengths  of  about  6.5  ft,  treenailed  together  so  as  to  break  joint, 
as  at  B.  Each  piece  of  plank  was  12  ins  deep  at  middle,  and  8  ins  at  each  end ; 
the  top  edge  being  cut  to  suit  the  curve  of  the  arch.  The  treenails  were  1.25  ius 
in  diam;  and  12  of  them  showed  to  each  length.  These  ribs  were  placed  17  ins 
apart  from  cen  to  cen,  and  steadied  together  by  a  bridging  piece  of  inch  board,  13 
ins  long,  at  each  joint  of  the  planks,  or  about  3.25  ft  apart.  Headway  for  traffic 
being  necessary  under  the  arch,  there  were  no  chorda  to  unite  the  opposite  feet 
of  the  ribs.  The  ribs  were  covered  with  close  board  lagging,  which  also  assisted, 
in  steadying  them  together  transversely.  As  the  arch  approached  about  two- 
thirds  of  its  height  on  each  side,  the  ribs  began  to  sink  at  the  haunthes,  as  at  A, 
Fig  4 ;  and  to  rise  at  the  crown,  as  at  c.  This  was  rectified  by  loading  the  crown 
with  stone  to  be  used  in  completing  the  arch ;  which  was  then  finished  without 
further  trouble. 

A  still  more  striking-  example  of  the  useof  a  simple  unbraced  wooden 
rib,  was  in  the  old  "National  Turnrike  bridge  over  VVills  Creek,  at  Cumberland,  Md, 
This  bridge,  of  which  one  arch  with 
its  center  is  shown  in  Fig  7  drawn 
to  a  scale,  consisted  of  two  elliptic 
cut  stone  arches  26.5  ft  wide  across 
roadway,  and  of  60  ft  span,  and  15 
ft  rise.  The  archstones  were  3  ft 
deep  at  crown,  and  4  ft  deep  at 
skewbacks.  Each  frame  of 
tlie  center  was  a  simple  rib  6 
ins  thick,  composed  of  three  thick- 
nesses of  2  inch  oak  plank  in  different  lengths  (about  7  to  15  ft)  to  suit  the  curve, 
and  at  the  same  time  to  preserve  a  width  of  about  16  ins  at  the  middle  of  each 
length,  and  12  ins  at  each  of  its  ends.  The  thicknesses  were  well  treenailed  to- 
gether, breaking  joint  and  showing  from  10  to  16  treenails  to  a  length. 

Here,  as  in  Fig  6,  there  were  no  chords,  owing  to  the  violence  of  the  floods  in 
the  creek.  These  ribs  were  placed  18  ins  from  cen  to  cen,  and  steadied  against 
one  another  by  a  board  bridging-piece  1  ft  long,  at  every  5  ft.  These  were  of 
course  assisted  by  the  lagging. 

When  the  archstones  had  approached  to  within  about  12  ft  of  each  other  near 
the  middle  of  the  span,  the  sinking  at  the  crown,  and  the  rising  at  the  haunches 
had  become  so  alarming  that  pieces  of  12  X  12  oak,  00,  were  hastily  inserted  at 
intervals,  and  well  wedged  against  the  archstones  at  their  ends.  The  arch  was  - 
then  finished  in  sections  between  these  timbers,  which  were  removed  one  by  one 
as  this  was  done. 

Rem.  1.  Such  instances  of  partial  failure  are  very  instructive. 
It  is  indeed  by  such,  rather  than  by  theoretical  deductions,  that  the  proper  dimen- 
sions are  arrived  at  in  a  vast  number  of  cases  pertaining  to  engineering,  ma- 
chinery, &c.*  Thus  we  might  with  entire  confidence  of  no  serious  mishap,  apply 
ribs  of  the  foregoing  dimensions  to  spans  only  half  as  great. 

Rem.  2.  Assuming  the  rib-planks  to  be  12  ins  wide,  it  would,  as  a  matter  of 
detail,  be  better  to  make  them  about  10  ins  wide  at  the  ends  instead  of  the  8  ins 
in  Fig  6  making  top  curve  2  ins.  To  secure  this,  their  lengths,  depending  on  the 
radius  of  the  rib,  must  not  exceed  those  in  tlie  folloi»iring  table: 


Bad 
of  Arch. 

Greatest  Length. 

Bad 
of  Arch. 

Greatest  Length. 

Feet, 
5 
10 
15 
20 
25 

Feet  and  Ins. 

2  "      5 

3  "      4 

4  "       2 

5  "       0 
5       "       9 

Feet. 
30 
35 
40 

45 
50 

Feet  and  Ins. 

6  "      4 

7  "      0 
7      "      6 

7  "    10 

8  "      2 

•  The  young  engineer  should  stake  and  preserve  full  notes  in  detail  of  all  such  as  may  fall  within 
his  notice* 


636 


CENTERS    FOR   ARCHES. 


If  cut  IK  times  as  long  as  this  table,  they  will  be  very  approxipaately  8  ins 
wide  at  ends  ;  or  each  will  on  top  curve  4  ins. 

Art.  5.    In  cases  where  all  possible  lieadTray  is  essential 

during  the  building  of  the  arch,  as  in  the  two  foregoing  ones,  the  writer  would 

suggest  the  expedient  rudely  illustrated  by 
Fig  8;  namely  to  place  the  centers 
above  the  arch,  instead  of  belovir 
it ;  and  after  the  arch  is  completed  in  sec- 
tions, a  a,  instead  of  loivering  the  cen' 
ters,  to  take  them  apart. 


a 


a 


RgS. 


a 


a 


/«  Fig  8  is  a  transverse  section  through  part 

of  the  center,  and  of  the  arch  a  a.  Here 
rc,rc,rc,  are  frames  of  the  center  say  5  or  6 
ft  apart ;  and  of  any  depth  and  construction 
whatever  that  may  be  necessary  to  insure  absolute  safety ;  and  1 1  is  the  lagging- 
Having  built  the  arch  from  abutment  to  abutment  in  a  series  of  sections  a,  a,  a,  ne- 
cessarily separated  say  a  foot  or  more  by  the  deep  frames,  we  may  take  the  centers 
apart,  and  then  lill  in  the  narrow  intermediate  sections  upon  a  lagging  suspended 
by  iron  rods  from  the  already  completed  sections.  Good  concrete  might  be  used 
for  these  narrow  sections.  In  some  cases  it  might  be  well  to  use  deep  plate- 
iron  ribs  of  I  section,  resting  the  lagging  on  the  lower  flange.  Fart  of  the 
web  might  be  left  remaining  embedded  in  the  masonry;  and  the  upper  part  and 
both  flanges  removed  after  the  arch  is  finished. 

Art.  6.  Centers  irith  hor  chords  e  c  Fig  9  are  objectionable  (notwith- 
standing their  strength)  in  large  spans  of  great  rise,  as  on  right  side  of  the  Fig,  on 

account  of  the  excessive  length 
required  for  the  web  members; 
and  hence  it  will  in  such  cases 
usually  be  found  expedient  to 
adopt  something  s^nalogous  to 
what  is  shown  on  the  left  hand 
of  the  Fig.  Here  a  truss/,  shorter 
and  shallower  than  that  on  the 
right  hand,  is  substituted  for  the 
latter.  At  its  ends  provision  m  ast 
be  made  for  supporting  not  only 
itself,  but  the  archstones  below 
it.  As  the  pressure  of  these  lowi. 
er  archstones  is  comparatively- 
small,  this  may  usually  be  eflecte^ 
by  resting  the  end  of  the  frame 
/  upon  another  and  shallower  frame  o  a.  This  may  in  large  spans  be  aided  by 
either  inclined  or  vertical  struts,  either  single  or  braced  together ;  or  as  the  trestles 
on  p  756.  Sometimes  one  shallow  truss  uke  /  is  sustained  upon  another  truss 
throughout  its  entire  length.  The  striking-wedges  for  these  various  supports  may 
be  placed  at  either  their  tops  or  their  feet,  as  may  be  most  convenient. 

Art.  7.  For  flat  arches  of  10  feet  clear  span,  a  mere  board  o  o 
Fig  10,  12  ins  deep,  by  1.5  ins  thick,  with  another  piece  c  of  the  same  thickness 
on  top  of  it,  trimmed  to  the  curve,  and  con-, 
fined  to  0  0  by  nailing  on  two  cleats  of  nar^ 
row  board,  will  answer  every  purpose,  with 
intervals  of  18  ins  from  cen  to  cen.  If  the 
upper  piece  also  is  as  much  as  12  ins  deep  at 
its  center,  the  clear  span  may  be  extended 
to  15  ft. 

For  spans  of  10  to  15  ft,  and  of  any 
rise,  two  thicknesses  of  plank  from  1  to  2  ins 
thick  according  to  span;  8  to  12  ins  wide  at 
middle  of  each  piece,,  in  lengths  as  per  table.  Rem  2,  Art  4,  well  nailed  or 
spiked  together,  according  to  span,  breaking  joint  as  in  Fig  6,  will  answer  for 
distances  of  2  to  3  ft  apart  cen  to  cen.  For  greater  dists  apart  increase  the  thick- 
ness of  the  planks  proportionally. 

If  the  centers  have  to  be  moved  from  place  to  place,  to  serve 
for  other  arches,  then,  to  preserve  them  from  injury  in  handling,  their  feet  should 
be  united  by  nailing  on  one  or  both  sides  of  each  frame  a  chord  piece  of  about  1 


CENTERS   FOR   ARCHES. 


637 


inch  board ;  and  also  a  vertical  piece  or  pieces  of  the  same  size  from  the  center 
of  the  chord  to  the  top  of  the  frame. 

Even  when  they  are  not  to  be  moved,  the  chord  pieces  are  useful 
even  in  so  small  spans,  inasmuch  as  they  render  the  striking  easier,  by  not  allow- 
ing the  feet  of  the  ribs  to  give  trouble  by  spreading  outward  and  pressing  against 
the  abutments. 

For  spans  of  15  to  30  ft,  and  for  any  rise  not  less  than  one  sixth  of  the 
span,  the  following  dimensions,  varying  with  the  span,  may  be  used  for  distances 
apart  of  3  ft  from  cen  to  cen. 
See  Fig  11.  For  the  bow  b, 
two  thicknesses  of  1  to  2  inch 
plank  from  9  to  12  ins  wide 
at  the  middle;  and  from  7  to 
10  ins  at  each  end,  well  spiked 
together  breaking  joint  as  at  B, 
Fig  6.  For  the  chor<l  c,  two 
thicknesses  of  plank  of  same 
size  as  the  bow  at  its  middle  • 
placed  on  outsides  of  bow,  and 
well  spiked  to  its  ends.  A 
vertical  v,  in  one  piece  as 
wide  as  a  bow  plank,  and  twice 

as  thick.  Its  top  is  placed  under  the  bow,  and  is  confined  to  it  by  two  pieces,  o,  o, 
of  bow  plank  twice  as  long  as  the  bow  plank  is  deep,  and  spiked  to  both  v  and  the 
bow.  The  foot  of  v  passes  between  the  two  thicknesses  of  the  chord  c,  and  is 
spiked  to  them.  Tw^o  oblique  tie-stmts,  s,  each  of  two,  pieces  of  bow 
plank,  outside  of  the  bow  and  vertical  v ;  footing  against  each  other ;  and  spiked 
to  bow  and  v.    These  with  v  divide  the  bow  into  4  parts. 

Rem.  1.  The  above  dimensions  are  suitable  to  a  rise  of  one  sixth.  If  the 
rise  is  one  fourth,  the  thickness  only  of  the  planks  may  be  reduced  one  third 
part ;  and  for  a  rise  of  one  third  or  more,  we  may  reduce  to  one  half. 

Rem.  2.  If  in  the  larger  of  these  spans  the  struts  s  should  show  any  incli- 
nation to  bend  sideways,  nail  on  some  pieces  t  from  frame  to  frame.  Also  in  the 
larger  ones  with  rises  exceeding  one  third,  insert  four  double  struts  s,  instead 
of  two ;  thus  dividing  the  bow  into  6  parts,  as  at  left  side  of  Fig.  11.  For  spans  of 
25  to  35  ft,  add  also  two  struts  like  a  a,  of  same  size  as  v. 

Art.  8.  For  spans  jg-reater  than  abont  30  ft,  the  writer  believes 
that  as  a  general  rule  (Hab!^  to  modifications  according  to  the  judgment  of  the 
engineer  in  charge)  the  following  ideas  will  lead  to  safe  practice.  Namely,  to 
adopt  a  bowstring  truss  with  a  simple  Warren  or  triangular  web,  as  at  /  on  the 
left  side  of  Fig  9.  The  bow  to  rest  on  the  chord,  and  each  to  be  of  a  single  thick- 
ness. The  web  members  (especially  in  large  spans)  to  be  also  of  single  tnickness, 
and  placed  below  the  bow,  resting  on  the  chords,  and  well  strapped  to  both,  so  as 
to  act  as  either  ties  or  struts.  In  smaller  spans  the  web  members  may  each  bo  in 
two  thicknesses,  one  bolted  or  treenailed  to  each  side  of  the  bow  and  chord.  Other 
modes  will  suggest  themselves ;  but  we  have  not  space  for  such  details. 

Or  a  web  of  the  Howe,  or  of  the  Pratt  system,  as  on  the  right  side  of  Fig  9  may  be 
used.  But  in  reference  to  both  of  these  it  may  be  remarked  that  the  use  of 
long*  iron  rods  in  centers  of  large  spans  is  highly  objectionable,  owing 
to  the  diflferent  rates  of  expansion  between  iron  and  wood.  Therefore  if  these 
systems  are  used,  all  the  members  should  be  of  wood.  The  lattice  may  be  used. 
Even  when  the  rise  of  the  arch  exceeds  .25  of  the  span,  it  is  better  not  to  let 
that  of  the  centers  exceed  that  limit ;  but  adopt  the  expedient  shown  at 
the  left  side  of  Fig  9,  with  a  rise  of  about  one  sixth  of  the  span. 

Rem.  1.  To  fix  on  the  number  of  w^eb  triang^les  in  a  Warren 
truss  or  frame  for  a  center,  find  the  square  root  of  the  span,  and  to  it  add  one 
tenth  of  the  span.  Divide  their  sum  by  2,  and  call  the  quotient  n.  Divide  the 
span  by  n.  If  this  quotient  is  a  whole  number  use  it;  or  if  the  quotient  is  partly 
decimal,  use  the  whole  number  nearest  to  it,  as  a  distance  in  feet  to  be  stepped  oflf 
along  the  chord ;  thus  dividing  the  chord  into  a  number  of  equal  parts.  All  the 
points  thus  found  on  the  chord,  are  the  places  for  the  feet  of  the  triangles. 
Next,  from  half  way  between  each  two  of  these  points,  draw  vertical  lines  to  the 
bow.  The  points  thus  found  along  the  bow,  are  the  places  of  the  tops  of  the 
triangles.  This  rule  will  be  used  in  connection  with  the,following  Table  of  Areas 
of  Bows,  as  the  two  are  dependent  on  each  other. 

In  large  arches  the  timber  of  the  bow^  should  not  be  wasted  by 
trimming  its  upper  edges  to  the  curve  of  the  arch,  but  should  be  left  straight ;  and 
separate  pieces  so  trimmed,  like  c  in  Fig.  10,  should  be  spiked  on  top  of  them. 


638 


CENTERS   FOR   ARCHES. 


The  transverse  area  of  tbe  bow,  in  square  inches,  may  be  taken  from 
the  folloMnug  table  ;  and  may  in  practice  be  assumed  to  be  uniform  throughout 
its  entire  length  ;  which  in  fact  it  is  quite  approximately.    See  Rem  2. 

TABI.E  FOR  BOWISTRII^O   CENTERS. 

Table  of  areas  in  square  inches  at  the  crown  of  each  Bow,  of  properly 
trussed  Bowstring  frames  for  centers  of  stone  or  brick  arches.  The  frames  to 
be  placed  5  feet  apart  from  cen  to  cen.  With  these  areas,  the  combined  weights 
of  arch,  center  (of  oak),  and  lagging,  will  in  no  case  in  the  table  strain  the  Bow 
at  crown  of  the  greatest  spans  quite  1000  lbs  per  square  inch ;  diminishing  grad- 
•  ually  to  600  or  700  lbs  in  the  smallest  spans,  which  are  more  liable  to  casualties. 

Although  centers  of  moderate  span  are  usually  made  of  white  or  yellow 
pine,  spruQe,  or  hemlock,  all  of  which  are  considerably  lighter  than  oak,  we  have 
for  safety  assumed  them  to  be  of  oak,  in  preparing  our  table. 
For  spans  of  from  10  to  20  feet  use  the  same  sizes  as  for  20  feet. 


Original. 

Rise  in  parts  of  tbe  Span 

.5 

.4 

.35    .3     .25 

.2 

.15 

.1 

Span 

in  feet. 

Areas  of  transverse  section  of  Bow, 

in  square  inches. 

20 

14 

17 

19 

21 

24 

29 

38 

59 

25 

18 

22 

25 

28 

33 

40 

53 

80 

30 

23 

28 

32 

37 

43 

51 

71 

103 

35 

28 

34 

40 

45 

54 

64 

87 

125 

40 

34 

41 

48 

65 

65 

77 

106 

150 

45 

40 

49 

57 

65 

76 

92 

126 

175 

60 

47 

57 

66 

76 

89 

107 

146 

203 

55 

63 

64 

75 

87 

102 

121 

166 

233 

60 

60 

73 

85 

99 

115 

135 

187 

263 

65 

68 

81 

95 

110 

129 

151 

209 

294 

70 

75 

90 

105 

122 

143 

168 

233 

325 

75 

83 

99 

115 

133 

157 

184 

256 

357 

80 

91 

108 

125 

145 

171 

201 

279 

390 

85 

99 

117 

136 

157 

185 

218 

302 

423 

90 

108 

127 

147 

169 

199 

235 

325 

457 

95 

115 

136 

158 

181 

214 

252 

348 

490 

100 

123 

146 

169 

194 

229 

270 

372 

524 

110 

133 

166 

191 

219 

260 

307 

420 

592 

120 

155 

187 

213 

246 

291 

345 

470 

660 

130 

172 

208 

237 

274 

323 

384 

520 

140 

190 

230 

263 

303 

357 

424 

572 

.  150 

209 

252 

289 

333 

393 

466 

160 

229 

276 

315 

365 

430 

509 

170 

250 

299 

343 

399 

469 

180 

272 

323 

373 

435 

511 

190 

294 

347 

403 

472 

200 

318 

372 

435 

509 

Rem.  2.  The  square  root  of  any  of  these  areas  gives  in  inches  the  side  of 
a  square  bow^  of  that  area.  The  distances  apart  of  the  triangles  which  form 
the  web  of  the  frame,  having  first  been  found  by  Rem  1  (for  said  Rem  and  this 
table  are  dependent  on  each  other),  the  above  areas  for  bows  5  ft  apart  from  cen 
to  cen,  suffice  not  only  to  resist  the  pressure  along  the  bow,  but  also,  as  square 
beams,  to  sustain  with  a  safety  in  no  case  less  than  about  5,  the  load  of  arch- 
stones  resting  upon  them  between  the  adjacent  tops  of  two  triangles ;  and  with 
very  trifling  deflections.  It  is  therefore  unnecessary  to  deepen  the  ribs  for  that 
purpose;  although  it  may  be  done  (preserving  the  same  area)  in  case  consider- 
ations of  detail  should  render  it  desirable. 

As  before  suggested,  it  wull  generally  be  best,  in  spans  exceeding  30  or  40  ft,  to 
give  the  bow  a  rise  not  expeeding  about  one  fifth  or  one  sixth  of  the  span  ;  and 
to  support  the  frames  as  at/,  Fig  9. 

Tbe  size  of  tbe  cbord  may  be  the  same  as  that  of  the  bow ;  and  like  it 
uniform  from  end  to  end ;  care  however  being  taken  that  it  be  not  materially 
weakened  by  footing  the  bow  upon  its  ends ;  or  (when  too  long  for  single  tim- 
bers) by  the  splicing  necessajy  to  prevent  its  being  stretched  or  pulled  apart  by 


CENTERS   FOR   ARCHES. 


63& 


the  thrust  of  the  bow.  When,  however,  the  chord  can  be  placed  at,  or  a  little 
below  the  springs  of  the  arch,  all  danger  of  this  kind  may  be  avoided  by  simply 
wedging  its  ends  well  against  the  faces  of  the  abutments. 

As  to  the  size  of  tlie  "web  members,  when  a  bowstring  truss  is 
fully  loaded  on  top  of  the  bow,  (as  is  approximately  the  case  with  a  center 
and  its  archstones,)  the  strains  on  the  web  members  are  quite  insignificant,  and 
arise  chiefly  from  the  weight  of  the  center  itself;  but  ivbile  it  is  bein^  so 
loaded,  they  are  not  only  greater,  but  are  constantly  changing,  not  only  in 
amount,  but  also  in  character — being  at  one  period  compressive,  and  at  another 
tensile. 

Hence  it  would  be  very  tedious  to  calculate  the  dimensions  of  the  web  members. 
Fortunately  the  necessity  for  doing  so  is  in  a  great  measure  obviated  by  the  fact 
that  a  center  being  but  a  temporary  structure,  the  timbercomposing  it  is  not  ulti- 
mately wasted  if  a  greater  quantity  of  it  is  used  than  is  absolutely  required. 
Moreover  facility  of  workmanship  is  secured  by  not  having  to  employ  timbers 
of  many  different  sizes. 

Hence  the  writer  will  venture  to  suggest,  entirely  as  a  rule  of  thumb,  to  g^ive 
eacb  web  member  balf  the  transverse  area  of  the  bow, 
taking  care  to  make  each  of  them  a  tie-strut. 

Rem.  3.  As  to  details  of  joints,  we  refer  to  the  Figs  on  pages  735, 
736 ;  merely  suggesting  here  the  use  of  long  and  wide  iron  shoes  where  timbers 
are  subjected  to  great  pressure  sideways. 

_  Rem.  4.  To  prevent  the  thrust  of  the  bow  when  its  rise  is  small,  from  split- 
ting off  the  ends  of  the  chords,  the  two  may  be  united  by  many  more  bolts  than 
are  employed  in  roof  trusses,  &c,  where  only  one  is  generally  placed  near  each  end 
of  the  chord.  But  they  may  when  required  be  inserted  at  intervals  extending  to 
many  feet  from  the  ends.  They  should  have  strong  large  washers ;  and  may  have 
about  the  same  inclination  as  the  shortest  web  member. 

Another  way  of  securing  the  same  end  in  smaller  spans,  is  by  completely  en- 
casing the  two  sides  of  the  bow  and  chord,  to  a  distance  of  a  few  feet  from  their 
ends,  in  short  pieces  of  board  or  plank  spiked  to  both  of  them,  and  having  about 
the  same  inclination  as  just  suggested  for  bolts. 

Rem.  5.  Build  up  Doth  sides  of  the  arch  at  once,  in  order  to  strain  the  cen- 
ters as  little  as  possible. 

Rem.  6.  When  a  bridge  consists  of  more  than  one  arch,  and  they  are  to  be 
built  one  at  a  time,  there  must  be  at  least  two  centers ;  for  a  center  must  not 
be  struck  until  the  contiguous  arches  on  both  sides  are  finished,  for  fear  of  over- 
turning the  outer  unsupported  pier.  Therefore  if  there  are  but  two  arches,  they 
must  be  built  at  once,  requiring  two  centers. 

Rem.  7.  Always  use  supports  either  vertical  or  inclined  (and  pro- 
vided with  striking- wedges)  under  tne  frames,  and  intermediate  of  the  end  sup- 
ports, when  possible ;  even  if  they  can  extend  out  but  a  few  feet  from  the  abut- 
ments, as  at  the  left  side  of  Fig  9. 

Rem.  8.  The  weig-ht  of  large  centers  and  their  lagging  is  greater 
for  flat  arches  than  for  high  ones  of  the  same  span ;  and  also  approaches  nearer 
to  that  of  the  supported  arch. 

Rem.  9.  Thickness  of  lagging.   The  following  table  gives  thicknesses 
which  will  not  bend  more  than  an  eighth  of  an  inch  under  the  weight  of  any 
probable  archstones  adapted  to  the  respective  spans ;  and  generally  not 
so  much. 
TABIiC:  OF   liAOGINO.— Original. 


Distance  apart 

of  frames, 
in  the  clear. 


Feet. 
6 
5 
4 
3 
2 


10. 


Span  of  center  in  feet. 
20.  50.  100.  150. 


200. 


Thickness  of  close  lagging  not  to  bend  more  than  %  inch, 


Ins. 


Ins. 
5 

3% 

2% 

2 

1^ 


Ins. 

3 

2 
1^ 


With  thicknesses  three  quarters  as  great  as  these,  the  bending  may  reach 
a  full  quarter  inch ;  which  may  be  allowed  in  dists  apart  of  3  or  more  ft. 

Rem.  10.  Centers  are  framed,  or  put  together,  (like  iron  bridges)  on  a 
firm,  level  temporary  floor  or  platform,  ou  which  a  full-size  drawinsr  of  a  frame  is 


640 


CENTERS   FOR   ARCHES. 


L?J  Figaa  L^ 


first  made.    As  each  frame  is  finished,  it  is  removed  to  its  place  on  the  piers  ol 
abuts. 

Art.  9.  The  Wissabickon  Bridg^e  of  the  Reading  R  R,  at  Philadelphia, 
has  five  arches  of  65  ft  span,  23  ft  rise,  28  ft  wide  (archstones  3  ft  deep,  with  dressed 
beds  and  joints,  in  cement  mortar) ;  with  four  cutstone  piers  9.5  ft  thick  at  top,  and 
from  35  to  50  ft  high.  It  contains  about  15400  cub  yds  of  masonry.*  £acli  center 
consisted  of  7  frames  or  trusses  of  hemlock  timber,  of  the  Bowstring  pattern,  with 
lattice  web-members ;  and  as  nearly  as  may  be,  of  the  same  span  and 

rise  as  the  arches.    They  were  placed  4.5  ft  apart  from  center  to  center;  and  were 

supported  near  each  end  /,  Fig  13 
(a  transverse  section  to  scale)  by 
a  hemlock  post  p,  12  ins  square. 
The  boiv  was  of  two  thicknessea 
bb  of  hemlock  plank,  6  ins  apart 
clear,  in  lengths  of  6  ft,  with  their 
upper  edges  cut  to  suit  the  curve 
I  of  the  arch.  Each  piece  was  4  ins 
thick,  by  13.5  ins  deep  at  its  middle, 
and  12  ins  at  its  ends.  These  pieces 
did  not  break  joint;  but  at  each 
joint  were  four  ^  inch  bolts,  with 
nuts  and  washers,  uniting  tliem 
with  chocks  or  filling-in  pieces. 
The  bow,  b  h,  footed  on  top  of  the  ends  of  the  chords  /;  and  the  angle  formed  by 
their  meeting  (seen  only  in  a  side  view)  was  (for  about  2.5  ft  horizontal  and  5.5  ft 
vertical)  filled  up  solid  with  vertical  pieces,  to  afford  a  firmer  base  for  resting  the 
frame  on  n;  beyond  which  it  extends  (in  a  side  view)  about  18  ins. 

The  chords  /were  of  two  thicknesses  of  4  X  1^  hemlock  plank,  6  ins  apart 
clear,  and  most  of  them  in  two  or  three  lengths:  breaking  joint,  and  with  two  ^ 
inch  bolts,  with  nuts  and  washers,  at  each  joint,  for  bolting  them  together,  and  to 
filling-in  pieces.  The  'web  members  of  each  frame  were  26  lattices,  o,  of 
B  X  12  inch  hemlock,  crossing  each  other  about  at  right  angles,  at  intervals  of  about 
3.5  ft  from  center  to  center,  and  passing  between  the  two  thicknesses  bb  of  the  bow, 
and //of  the  chords.  A  few  of  the  lattices  were  in  two  lengths,  and  the  joints  were 
not  at  the  crossings.  The  lattices  were  connected  at  each  crossing  by  two  hard  wood 
treenails  9  ins  long,  and  2  ins  diam ;  and  one  such,  18  ins  long,  passed  through  the 
intersection  of  each  end  of  a  lattice  with  a  bow  or  chord.  The  first  lattice  foots 
about  4  ft  from  the  end  t  f  a  chord.  They  do  not  extend  above  the  top  of  the  bow. 
All  the  spaces  between  the  two  thicknesses  of  bow  or  chord,  where  not  occupied  by 
the  ends  of  lattices,  were  completely  filled  by  chocks,  well  spiked. 

Each  frame  contained  about  360  cub  ft  of  timber;  and  weighed  about  f 
tons.  They  were  very  flexible  laterally  until  in  place,  and  braced  together  by  4 
transverse  horizontal  planks  spiked  to  their  chords ;  and  by  5  others  above  them, 
spiked  to  the  lattices. 

Until  the  keystones  were  placed,  all  the  joints  of  the  frames  continued  tight,  under 
the  pressure  from  the  arch,  and  from  the  unfinished  backing  to  the  height  of  about 
14  ft  above  the  springing  line ;  but  after  the  keystones  were  set,  all  the  joints  of  the 
chords  alone  opened  from  .25  to  .76  of  an  inch ;  and  at  the  same  time  the  lagging  un- 
der the  haunches  of  the  arches  became  slightly  separated  from  the  sofiit  of  themasonry, 
£ach  center  sank  but  a  full  inch  at  the  middle,  under  the  pressure  from 
the  arck  and  14  ft  of  backing. 

The  portion  of  the  bridge  above  the  piers  was  about  two  thirds  completed  before 
the  centers  were  struck. 

There  was  one  "wedge  w,  w,  (82.5  ft  long,  of  12  X  12  inch  oak)  under  each 
end  of  a  center.  It  was  trimmed  to  form  7  smaller  ones  w,  \v,  each  4.5  ft  long,  and 
tapering  7  *ns ;  one  under  each  end  of  each  frame  /.  They  played  between  tapered 
blocks  a,  a,  of  oak,  2  ft  long,  1  ft  wide,  let  1  inch  into  the  cap  c,  or  into  the  piece  n, 
on  which  last  the  frames  /,/,  rested.  The  sliding  surfaces  were  well  lubricated  with 
tallow  when  put  in  place. 

The  wedg-es  were  struck  with  ease,  at  one  end  of  a  center  at  a  time,  by  aa 
oak  log  battering-ram  18  ft  long,  and  nearly  a  ft  in  diam,  suspended  by  ropes,  and 
swung  and  guided  by  4  men.  They  generally  yielded  and  moved  several  inches  at 
the  second  blow  with  a  8  or  4  ft  swing.  Although  each  wedge  was  loosened  entirely 
within  2  or  3  minutes,  thus  lowering  the  centers  very  suddenly,  yet  on  account  of  tha 

*  This  bridge,  finished  without  accident,  in  1882,  reflects  much  credit  on  the  late  William  Lorenz, 
Esq,  Ch.  Eng;  on  Mr.  Charles  W.  Buchholz,  Assistant  in  Charge;  and  on  the  skilful  and  energetic 
contractors,  William  &  James  Nolan,  of  Reading,  Penna.  These  last  most  cordially  assisted  th«' 
"writer  in  making  observations  during  the  entire  progress  of  the  work. 


CENTERS  FOR  ARCHES.  o41 

good  character  of  the  masonry,  not  the  slightest  cracK  of  a  mortar  joint  could  after* 
wards  be  detected  in  any  part  of  the  work.  After  three  days  the  average  sinking  of 
the  keystones  was  only  .85  of  an  inch  ;  the  least  was  }^\  and  the  greatest  %  of  an 
inch.  The  heads  and  feet  of  the  posts  p  compressed  the  hemlock  caps  c,  and  the 
sills,  about  %  of  an  inch  each,  showing  that  for  arches  of  this  size  the  caps  and  sills 
had  better  be  of  some  harder  wood,  as  yellow  pine  or  oak;  although  probably  the 
oompreaeion  was  facilitated  by  the  large  mortices,  3  by  12  ins,  and  6  ins  deep. 


41 


642 


DAMS. 


TIMBER  DAMS. 


Primary  reqnisites,  in  the  erection  of  dams,  are,  a  foundation  suffi- 
ciently firm  to  prevent  tbem  from  settling,  and  thus  leaking;  the  prevention 
of  leaks  through  their  backs,  or  under  their  bases;  and  the  prevention  of  wear 
of  the  bottom  of  the  stream  in  front  of  the  dam,  by  the  action  of  the  falling 
water.  For  the  first  purpose,  hard  level  rock  bottom  is  of  course  the  best ;  and 
should  be  chosen,  if  possible.  In  that  case,  thick  planks,  ti.  Fig  6,  (single  or 
double,  as  the  case  may  be,)  closely  jointed,  and  reaching  from  the  crest,  c,  to 
the  back  lower  edge  w,  (where  they  should  be  scribed  down  to  the  rock ;)  with  a 
good  backing,  6,  of  gravel,  will  suffice  to  prevent  leaks.  Gravel,  or  rather  very- 
gravelly  soil,  is  far  better  than  earth  for  this  purpose ;  for  if  the  water  should 
chance  to  form  a  void  in  it,  the  gravel  falls  and  stops  it.  To  prevent  this  back- 
ing from  being  disturbed  near  the  crest  of  the  dam,  by  floating  bodies  swept 
along  by  freshets,  a  rough  pavement  of  stones,  about  15  to  18  inches  deep,  as 
shown  in  Fig  7,  should  be  added  for  a  width  of  about  10  to  20  feet;  or  until  its 
top  becomes  3  to  5  feet  below  the  crest  c  of  the  dam,  according  to  circumstances. 


In  Fig  1,  (a  dam  on  the  Schuylkill  navigation,)  the  upper  timbers,  e,  are  all 
close  jointed,  and  laid  touching,  so  as  not  to  require  planking  in  addition. 

But  if  the  bottom  of  the  stream  is  gravel  or  earth,  there  must  in  addition  to 
these  be  used  two  thicknesses  of  sheet  piles,  p,  Fig  2,  &c,  close  driven,  breaking 
joint,  to  a  depth  of  several  feet,  to  prevent  leaking  through  the  soil  beneath 
the  base  of  the  dam.  Frequently  but  one  thickness  is  used.  If  the  bottom  is 
soft  or  open  for  a  depth  of  only  a  few  feet,  it  is  at  times  better  to  remove  :t,  and 
base  the  dam  on  the  firmer  stratum  below  ;  still,  however,  using  the  sheet  piles. 
Old  decayed  timber  and  other  rubbish  should  be  removed  from  the  base.  In 
very  bad  soils  of  greater  depth,  it  may  be  necessary  to  support  the  dam  entirely 
upon  a  platform  resting  on  bearing  piles.  Here  great  precautions  are  neces- 
sary against  leaks ;  but  the  case  occurs  so  rarely,  that  we  shall  not  stop  to  con- 
sider it. 

As  to  the  wearing  away  of  the  bottom  of  the  stream  by  the  water  falling  over 
the  front  of  the  dam,  precautions  should  be  used  in  all  cases  except  that  of  very 


oard  rock,  or  of  medium  rock  protected  by  a  considerable  depth  of  water.  The 
dam  Fig  1,  was  built  upon  a  tolerably  firm  micaceous  gneiss  in  nearly  vertical 
strata,  co?ered  by:about  2  feet  of  water  in  ordinary  stages.  In  39  years  the  rock  was 


DAMS. 


643 


Fig.  3. 


worn  away  in  front  of  the  dam,  as  shown  in  the  fig,  to  the  average  depth  of  3  feet;  or  very  nearly  1 
inch  per  year.  The  depth  of  water  on  the  crest  c,  was  usually  from  6  to  18  ins ;  rarely  5  or  6  ft  dur- 
ing freshets ;  and  but  a  few  times  during  the  whole  period,  8  or  9  ft. 

At  Jones's  dam,  on  Cape  Fear  River;  height  of  dam,  16  ft;  front  vert; 

fall,  usually  10  ft,  into  6  ft  depth  of  water ;  the  soft  shale  rock,  in  vert  strata,  was,  in  the  course  of 
a  few  years,  worn  away  16  ft ;  and  the  dam  was  undermined  to  such  an  extent  as  to  fall  into  the  cavity. 
In  another  case,  dam  36  ft  high  ;  front  vert;  the  water  falling  upon  nearly  vert  strata  of  hard  shala 
rock,  usually  covered  by  but  about  2  ft  of  water ;  in  about  20  years  wore  it  to  an  irregular  depth  of 
from  10  to  20  ft;  and  extending  from  the  very  face  of  the  dam,  to  70  or  80  ft  in  front  of  it. 

In  Fig  2,  upon  a  stream  subject  to  very  violent  freshets,  the  gravel  was  washed  away  for  a  consid- 
erable width  and  depth  beyond  the  apron,  as  at  A.  To  prevent  a  repetition,  the  cavity  was  filled 
with  cribwork  full  of  stone,  clear  across  the  river. 

A  deposit  of  blocks  of  loose  stone,  of  even 
a  ton  weight  or  more,  will  not  serve  as  a  pro- 
tection in  front  of  a  dam  exposed  to  high 
freshets;  but  will  soon  be  swept  away.  A 
common  precaution  against  this  wear,  in  low 
dams,  is  an  apron,  a  a,  Pig  2  ;  or  dd,  Fig  3 ; 
of  either  rough  round  tree  trunks,  or  of  hewn 
timber,  laid  close  together ;  extending  under 
the  entire  base  of  the  dam,  and  from  15  to'30 
ft  in  front  of  its  face.  These  are  sometimes 
bolted  to  pieces,  ss,  Fig  2 ;  or  yy.  Fig  3  ;  laid 
under  them*  across  the  stream.  In  Fig  3, 
withvery  soft  bottom,  these  pieces  yy  are 
supposed  to  be  bolted  to  short  piles  1 1,  driven 
for  that  purpose. 

At  times  a  distinct  wide  low  timber  crib,  filled  with  stone,  and  covered  on  top  with  stout  plank, 
has  been  placed  in  front  of  the  dam,  to  receive  the  fall  of  the  water;  and  is  effective  in  protecting 
the  bottom.  Also,  in  some  cases,  a  dam  of  less  height,  and  of  cheap  character,  has  been  built  at  a 
short  distance  down  stream  from  the  main  one,  in  order  to  secure  at  all  times  a  deep  pool  in  front 
of  the  latter  for  breaking  the  force. 

O,  Another  precaution    is    to 

substitute  a  sloping  front  like 
cl,  Pig  4,  or  such  as  Figs  1 
and  2  would  form  if  reversed, 
for  the  nearly  vert  one  of  the 
other  figs ;  thus  to  some  extent 
reducing  the  force  of  the  wa- 
ter. This,  however,  is  but  a 
partial  remedy,  especially 
for  soft  bottoms  in  shallow 
water ;  for  the  sliding,  sheet 
still  descends  with  great  force. 
The  best  form  of  dam,  per- 
haps, in  such  cases,  is  that 
shown  in  Fig  5.  in  which  the 
front  consists  of  a  series  of  steps  of 
about  1  vert,  to  3  or  4  hor.  These  ef- 
fectually break  the  force  of  the  water; 
and,  with  the  addition  of  an  apron  o  o, 
secure  a  satisfactory  result.  It  i«  ob- 
jected against  this  form,  as  also 
against  Figs  4  and  6,  that  their  fronts 
are  liable  to  be  torn  by  descending 
trees,  ice,  and  other  bodies  swept 
along  during  freshets  ;  but  experience  shows  that  this  objection  has  but  little  weight;  for  when  suok 
bodies  pass,  the  sheet  of  water  is  thicker  than  usual ;  and  protects  the  front  timbers.  On  the  Sch 
Kar,  the  timbers  cl,  Fig  6,  scarcely  wear  thin  at  the  rate  of  an  inch  in  10  to  15  years. 


Fig.  6. 


Fig.  5. 


ROCK 


The  forms  of  \¥Ooden  dams  are  many ;  (see  the  figs,  which  show 

those  most  used ;)  varying  with  the  circumstances  of  the  case,  and  with  the  fancy  of  the  designer. 
In  the  United  States  they  are  usually  of  cribwork,  of  either  rough  round  logs  with  the  bark  on,  or  of 
hewB  timber ;  in  either  case  abant  a  foot  through.    These  timbers  are  merely  laid  on  top  of  each 


644 


DAMS. 


Figr.T. 


Fig.  8. 


other,  rorming  in  plan  a  series  of  rectangles  with  sides  of  about  7  to  12  ft.    They  are  not  rotched 

together,  but  simply  bolted  by  1  inch  square  bolts  (often  ragged  or  jagged)  about  2  to  2}4  feet  long, 

through  two  timbers  at  every  intersection.  These  are  not  found  to  rust  or  wear  seriously,  even  when 

exposed  to  a  current.    Square  bolts  hold  best.    Round  logs  are  tiatteued  where  they  lie  upon  eacn 

other.     Experience  shows  that  firmer  but  more  expensive  connections  are  entirely  unnecessary.   Th* 

cribs  are  usually,  but  not  always,  filled  with 

rough  stone.    In  triangular  dams,  disposed 

as  in  Figs  1,  2,  and  7,  this  stone  filling  is 

not  so  essential  as  in  other  forms  ;  because 

the  weight  of  the  water,  and  of  the  gravel 

backing,  tends  to  hold  the  dam  down  on  its 

base.     Still,  even  in  these,  when  the  lower 

timbers  are  not  bolted  to  a  rock  bottom,  or 

otherwise  secured  in  place,  some  stone  may 

be  necessary  to  prevent  the  timbers  from 

floating  away  while  the  work  is  unfinished, 

and  the  gravel  not  yet  deposited  behind  it. 

On  rock,  the  lowest  timbers  are  often  bolted 

to  it,  to  prevent  them  from  floating  away 

during  construction;  and  when  the  water 

is  some  feet  deep,  this  requires  cofifer-dams.     Or.  the  cribs  may  be  built  at  first  only  a  few  feet  high ; 

then  floated  into  place,  and  sunk  by  loading  them  with  stone;  for  the  reception  of  which  a  rough 

platform  or  flooring  will  be  reqd  in  the  cribs,  a  little  above  their  lowest  timbers.     The  bolting  to  the 

rock  may  then  be  dispensed  with.     The  water  may  flow  through  the  open  cribwork  as  the  building 

higher  goes  on  ;  attention  being  paid  to  adding  stone  enough  to  prevent  it  floating  away  if  a  freshet 

should  happen.     Or,  cribs  shown  in  plan  at  c  c.  Fig  8,  loaded  with 

stone,  may  be  sunk,  leaving  one  or  more  intervals,  like  that  at  o  o  o  o, 

between  them,  for  the  free  escape  of  the  water.     These  openings  to 

be  finally  closed  by  floating  into  them  closing-cribs  shaped  like  n. 

The  workmanship  of  a  dam  in  deep  water  can  of  course  be  much 
better  executed  in  coffer-dams,  than  by  merely  sinking  cribs.  The 
joints  can  be  made  tighter :  the  stone  filling  better  packed  ;  the  sheet 
piling  more  closely  fitted,  &c. 

When  a  very  uneven  rock  bottom  in  deep  water,  or  the  introduc- 
tion of  sluices  in  the  dam.  or  any  other  considerations,  make  it  ex- 
pedient to  build  dams  within  coffer-dams,  both  should  be  carried  on 
in  sections ;  so  as  to  leave  part  of  the  channel-way  open  for  the  es- 
cape of  the  water.  Commencing  at  one  or  both  shores,  the  first  section  of  the  coffer-dam  may  reach 
say  quarter  way  or  more  across  the  stream.  In  the  section  of  the  dam  itself  built  within  this  enclos- 
ing coffer-dam,  ample  sluices  should  be  left  for  the  water  to  flow  through  when  we  come  to  build  the 
closinff  section  of  the  coffer-dam.  When  the  dam  has  been  finished,  these  sluices  may  be  closed 
by  drop-timbers*.  Before  removing  one  section  of  coffer-dam.  the  outer  end  of  the  enclosed 
section  of  dam'  itself  must  be  firmly  finished  in  such  a  manner  as  to  constitute  a  part  of  the  inner 
end  of  the  next  section  of  coffer-dam.  It  is  impossible  to  give  details  for  every  contingency ;  the  en- 
gineer must  rely  upon  his  own  ingenuity  to  meet  the  peculiarities  of  the  case  before  him.  In  some 
cases  of  shallow  water,  mere  mounds  of  earth  may  answer  for  coffer-dams;  or  rough  stone  mounds, 
backed  with  earth  or  gravel. 

After  the  water  has  passed  beyond  the  crest,  c  in  the  figs,  there  is  no  necessity  for  preventing  its 
leaking  down  among  the  crib  timbers :  on  the  contrary,  the  thick  sheeting  planks,  (or  squared  tim- 
bers, as  occasion  may  require.)  ci,  Figs  4  and  6,  which  form  the  slopes  along  which  the  water  then 
flows  in  some  dams,  are  usually  not  laid  close  together,  but  with  open  joints  of  about  ^  inch  wide  be- 
tween them,  for  the  express  purpose  of  allowing  part  of  the  water  to  fall  through  them,  so  as  to 
keep  the  timbers  beneath  them  partially  wet;  which,  to  some  extent,  renders  them  more  durable.  In 
Figs  1,  4,  6,  and  7,  the  water  of  the  lower  pool  flows  freely  back  among  the  crib  timbers,  and  rough 
quarry  stones  with  which  the  cribs  are  filled  either  partly  or  entirely.  In  Figs  4  and  6,  these  stones 
are  not  shown.     In  the  dam,  Fig  1,  none  were  used.     In  Fig  2,  they  were  as  shown. 

A  substantial,  and  not  very  expensive  dam  of  the  form  of  Fig  7.  may  be  built  of  rough  stone  in 
cement.  Some  hewn  limbers  should  be  firmly  built  horizontally  into  the  masonry  of  the  sloping 
back  c  n  «;,  at  a  few  feet  apart,  with  their  tops  level  with  the  surf  of  the  masonry.  To  these  must  be 
well  spiked  close-jointed  sheeting-plank  cnw,  for  protecting  the  masonry  from  the  action  of  the 
water,  and  of  floating  bodies.  The  gravel  backing  fe,  may  be  omitted  ;  but  the  sheet  piles  p,  and  an 
apron  in  front  of  the  dam,  will  be  as  indispensable  in  yielding  soils,  as  if  the  dam  were  of  timber. 

Figs  1,  2,  4,  6.  and  7,  are  sections  drawn  to  a  scale,  of  existing  dams  in  Pennsylvania,  that  have 
stood  successfully  the  force  of  heavy  freshets  for  a  long  series  of  years,  t  These  freshets  at  times  carry 
along  large  bodies  of  ice,  trees,  houses,  bridges,  &c. ;  and  have  risen  to  11  ft  above  the  crests.  Fig  1, 
on  the  Sch  Nav,  was  built  in  1819,  and  served  perfectly  for  39  years,  until  in  1858  the  decay  of  much 
of  its  timber,  especially  of  the  close-laid  top  ones,  e,  rendered  it  necessary  to  build  a  new  one  just  in 
front  of  it.  It  was  of  extremely  simple  construction  ;  and  was  never  filled  with  stone.  The  bottom  tim- 
bers, o  o,  10  ft  apart,  were  bolted  to  the  rock  ;  and  immediately  over  each  of  them,  was  such  a  series  of 
inclined  timbers  as  is  shown  in  the  fig.  The  top  ones,  c,  however,  were  close-jointed,  and  laid  touching 
so  as  to  form  the  top  sheeting,  instead  of  thinner  planks.  The  short  pieces  at  t  were  laid  in  the  same 
way.  No  coffer-dam  was  used  ;  but  the  bottom  pieces  were  first  bolted  to  the  rook  ;  10  ft  apart ;  then 
the  stringers  and  the  sloping  pieces  were  added.  The  close  covering  (e)  was  carried  forward  from 
each  end  of  the  dam,  until  at  last  a  space  o'  only  about  60  ft  was  left  in  the  center,  for  the  water  t» 
pass.  The  close  covering  for  this  space  being  then  all  got  ready,  a  strong  force  of  men  was  set  to 
work,  and  the  space  was  covered  so  rapidly  that  the  river  had  not  time  to  rise  sufficiently  high  to 
impede  the  operation. 

*  Timbers  ready  prepared  for  closing  an  opening  through  which  water  is  flowing ;  and  suddenly 
dropped  into  place  by  means  of  grooves  or  guides  of  some  kind  for  retaining  them  in  position.  Sev- 
eral such  timbers  may  at  times  be  firmly  framed  together,  and  then  be  all  dropped  at  once ;  closing 
the  opening  or  sluice  at  one  operation  ;  especially  when  it  is  of  small  size.  In  some  cases,  a  crib 
may  be  sunk  on  the  up-stream  side  of  such  an  opening,  for  closing  it. 

t  Those  on  the  Schuylkill  Navigation  were  obligingly  furnished  by  James  F.  Smith,  Esq,  chief 
engineer  and  s'lperintendent  of  that  work.  Other  valuable  information  from  the  same  source  wiU 
be  found  in  different  parts  of  this  volume.  * 


DAMS.  645 


Fig  2  is  a  canal  feeder  dam  on  the  Juniata.  Here  s  s  are  timbers  stretching  clear  across  the  stream, 
(about  300  ft,)  and  sustaining  the  apron  a  a,  of  stout  hewn  timbers  laid  touching.  This  dam  was  filled 
witk  stone,  for  the  retention  of  which  the  front  sheeting  planks  were  added. 

Fig  6  is  on  the  Sch  Nav  ;  was  built  in  1855.  It  is  a  form  much  approved  of  on  that  work,  for  such 
situations;  namely,  firm  rock  foundation,  with  a  considerable  depth  of  water  in  front.  The  highest 
dam  (32  ft)  on  the  Sch  Nav,  is  very  similar  to  it ;  built  in  1851.  All  the  dams  on  this  work  are  of 
hewn  timber,  chiefly  white  and  yellow  pine.  The  water  occasionally  runs  from  8  to  12  feet  deep  over 
their  crests ;  and  then  overflows  and  surrounds  many  of  the  abuts.  The  vertical  back  allows  the 
overflowing  water  to  leak  down  among  all  the  lower  timbers  of  the  dam,  and  thus  tend  to  their 
preservation. 

Fig  4  shows  the  dams  on  the  Monongahela  slackwater  navigation  ;  W.  Milnor  Roberta,  eng.  They 
are  of  round  logs,  with  the  bark  on ;  flattened  at  crossings.  The  longest  ones  in  the  fig  are  10  feet 
apart  along  the  length  of  the  dam.  Experience  shows  that  such  dams  possess  all  the  strength  neces- 
sary for  violent  streanls.     On  rock,  the  lowest  timbers  are  bolted  to  it. 

P'ig  7  has  been  successfully  used  to  heights  of  40  ft.* 

Fig  3  is  intended  merely  as  a  hint  for  a  very  low  dam  on  yielding  bottom.  Its  main  supports  are 
piles  it,  from  4  to  8  ft  apart,  according  to  the  height  of  the  dam;  and  other  circumstances  ;  and  tt 
are  short  piles  for  sustaining  the  apron  dd.  It  may  be  extended  to  greater  heights  by  adding  braces 
in  front ;  which  may  be  covered  by  stout  planks,  to  form  an  inclined  slide  for  the  overfalling  water. 
Many  effective  arrangements  of  piles,  and  sloping  timbers  for  dams  on  soft  ground,  will  suggest  them- 
selves to  the  engineer.  Thus,  at  intervals  of  several  feet,  rows  of  3  or  more  piles  may  be  driven  trans- 
versely of  the  dam ;  the  top  of  the  outer  pile  of  each  row  being  left  at  the  intended  height  of  the  crest, 
while  those  behind  are  successively  driven  lower  and  lower;  so  that  when  all  are  afterward  con- 
nected by  transverse  and  longitudinal  timbers,  and  covered  by  stout  planking,  and  gravel,  they  will 
form  a  dam  somewhat  of  the  triangular  form  of  Fig  7.  It  would  be  well  to  drive  the  piles  with  an 
inclination  of  their  tops  up  stream. 

There  is  much  scone  for  ingenuity  both  in  designing,  and  in  constructing  dams  under  various  cir- 
cum«tances  ;  and  in  turning  the  course  of  the  water  from  one  channel  to  another,  by  means  of  ditches, 
pipes,  or  troughs,  Ac,  at  diCf  heights;  aided  at  times  by  low  temporary  dams  or  mounds  of  earth;  or 
of  sheet  piles,  &c ;  or  by  coffer-dams  ;  so  as  to  keep  it  away  from  the  part  being  built.  Bach  locality 
will  have  its  peculiar  features ;  and  the  engineer  must  depend  on  his  judgment  to  make  the  most  of 
them. 

Abntments  of  dams  as  a  general  rule  should  not  contract  the  natural 
width  of  the  itream  ;  or,  if  they  must  do  so,  as  little  as  possible ;  for  contractions  increase  the  height, 
and  violence  of  the  overflowing-water  in  time  of  freshets  ;  during  which  a  great  length  of  overfall  is 
especially  desirable.  They  should  be  very  firmly  connected  with  the  ends  of  the  dams ;  and  should, 
if  the  section  of  the  valley  admits  of  it,  be  so  high,  and  carried  so  far  inland,  that  the  high  water 
of  freshets  will  not  sweep  either  over  them,  or  around  their  extremities ;  and  thus  endanger  under- 
mining, and  destruction.  In  wide,  flat  valleys  they  cannot  be  so  extended  without  too  much  ex- 
pense; and  the  only  alternative  is  to  found  them  so  deeply  and  securely  as  to  withstand  such 
action ;  making  their  height  such  that  they  will,  at  least,  be  overflowed  but  seldom.  Their  ends 
adjacent  to  the  dam,  should  be  rounded  off,  so  as  to  facilitate  the  flow  of  the  water  over  the  crest. 

They  are  best  built  of  large  stone  In  cement;  for  although  sufficient  strength  may  bs  secured  by 
timber,  that  material  decays  rapidly  in  such  exposures.  If  of  earth  only,  they  are  very  apt  to  be 
carried  away  if  a  freshet  should  overtop  them. 

Sluices  should  be  placed  in  every  important  dam,  in  order  that 

all  the  water  may  be  drawrf  off,  if  necessary,  for  the  purpose  of  repairs ;  or  of  removing  mud  deposits ; 
or  finding  lost  articles  of  importance,  &c.  They  may  be  merely  strong  boxings,  with  floor,  sides,  and 
top  of  squared  timbers ;  and  passing  through  the  breadth  of  the  dam,  just  above  the  bottom.  To  pre- 
vent trees,  &c,  from  entering  and  sticking  fast  in  them,  some  kind  of  strong  screen  is  expedient.  In 
common  cases  a  sluice  should  not  exceed  about  3}^  ft  by  5  ft  in  cross-section ;  otherwise  it  becomes 
hard  to  work.  Two  or  more  such  openings  may  be  used  when  much  water  is  to  be  voided.  They 
should  be  near  the  abutments.  The  gates  or  valves  for  opening  and  shutting  them,  should  be  at  the 
up-stream  end ;  for  if  at  the  lower  one,  accumulations  of  mud,  &c,  will  fill  the  sluices,  and  prevent 
them  from  working.  They  are  usually  of  timber;  and  slide  vertically  in  rebates;  being  raised  an6 
lowered  by  rack  and  pinion  ;  but  in  very  important  dams  they  may  be  of  cast  iron.  Two  sets  of  sluices 
are  desirable ;  that  one  may  be  always  ready  for  use  if  the  other  is  stopped  for  repairs. 

The  part  of  the  apron  in  front  of  the  sluice  should  be  particularly  firm,  so  as  not  to  be  deranged  by 
the  water  rushing  out  under  a  high  head. 

Dams  [are  sometimes,  bat  rarely,  bnilt  in  the  form  of  an 
arch  ;  convex  up  stream.  This  form  is  strong ;  and  when  the  shores  are  of  rock, 
it  may  be  expedient  to  use  it ;  but  if  the  banks  are  soft,  they  wUl  be  exposed  to  wear  by  the  curreni 
thrown  against  them  at  the  abuts  of  the  arch. 

At  times  dams  are  built  obliquely  across  the  stream,  with 

the  object  of  increasing  the  length,  and  consequently  reducing  the  depth  of  water  over  the  crest  in 
times  of  freshets.  The  argument,  however,  appears  to  the  writer  to  be  of  but  little  weight,  inasmuch 
as  the  reduction  of  depth  would  extend  but  a  trifling  distance  up  stream  from  the  dam ;  and  would 
therefore  scarcely  have  an  appreciable  effect  in  diminishing  the  injury  to  the  overflowed  district  above 
Moreover,  the  increased  expense  is  probably  always  more  than  commensurate  with  any  advantag* 
gained. 


*  Cost  of  crib  dams.  With  common  labor  at  $1.50  per  day ;  lumber,  $20  per 
1000  ft.  board  measure,  delivered  ;  stone  for  filling,  $1  per  cub,  yard  ;  gravel  50  cents  per  cub.  yd. ; 
iron  for  bolts,  etc.,  4cts.  per  lb., — such  dams  in  shallow  water  usually  cost,  complete  ''rom  9  to  12 
cc&ts  per  cubic  foot,  or  $2.43  to  $3.24  per  cubic  yard  of  crib. 


646 


DAMS. 


Figs.  9  and  10  are  designs  for  small  ineasuriiijgi:  weirs,  suitable  for 
shallow  streams  up  to  say  100  feet  wide ;  Figs.  9  for  earth  or  gravel  bottom,  and 
Figs.  10  for  rock. 

In  the  former,  the  8  X  10  inch  hemlock  sills  Si  and  S3  are  first  laid  across  the 
bottom  of  the  stream,  which  is  trenched  where  necessary ;  care  being  taken  to 
lay  Si  in  a  true  line.  The  sills  should  extend  say  from  5  to  10  feet  into  each 
bank  ef  the  stream.    Tongued  and  grooved  sheet  piling  P,  of  3  X  10  inch  hem- 


(A).  Section, 

12  inches  0  i 

l.lllllllllHl 1 


(.IB).  Elevation,  looking  xlotcn  stream. 

4  5  6  T  S  9 

— I 1 1 — -f i 1 


Wigs,  0, — Measuring  Weir  on  Earth  or  Gravel  Bottom, 

lock,  is  then  driven  close  behind  the  upper  sill  Sj  to  a  depth  of  from  two  to  four 
feet,  and  spiked  to  Si.  A  third  sill,  S2,  of  the  same  length  as  Sj  and  S3,  is  then 
laid  behind  the  sheet  piling;  and  the  two  sills  Si  and  Sg  and  the  sheet  piling  P 
are  then  secured  together,  as  shown,  by  1  inch  bolts,  spaced  about  2  feet  apart. 
The  tops  of  the  sheet  piling  project  about  a  toot  above  the  sills,  and  are  stiffened 
by  4  X  4  inch  timbers  iv,  bolted  in  front,  of  them  and  resting  upon  the  flooring 
/  of  2  X  10  inch  spruce.  This  flooring,  like  the  sills,  extends  several  feet  beyond 
each  end  of  the  weir  into  the  bank,  and  is  there  loaded  to  its  full  capacity  with 
heavy  stones.  Any  spaces  left  underneath  it  by  uneven ness  of  the  bottom  should 
also  be  leveled  up  with  stones  or  gravel. 

A  10  X  10  inch  yellow  pine  post  M,  3  feet  high,  is  tenoned  between  sills  Sg  and 
Si  at  each  end  of  the  overflow,  and  braced  by  an  g  X  10  inch  vellow  pine  strut 
N,  tenoned  to  it  and  to  the  sill  S3.  Beyond  these  posts  the  sheet  piling  P  extends 
as  high  as  the  top  of  the  posts,  and  is  carried,  at  that  height  into  the  bank ;  the 
tops  of  the  piles  being  held  in  line  by  two  2X8  inch  waling  pieces  u  u  bolted  to 
them,  one  on  each  side. 

In  Figs.  10,  the  hemlock  sills,  Si  of  10  X  10  inch,  and  Sg  of  6'x  8  inch,  rest  upon 
a  Portland  cement  masonry  wall,  of  varying  height  to  accommodate  the  in- 
equalities of  the  rock  bottom;  and  are  secured  to  it  by  1  inch  bolts  spaced  about 
4  feet  apart.  These  bolts  pass  down  through  the  masonry,  as  shown,  and  a  foot 
or  more  into  the  rock  below. 

Between  the  two  sills  are  bolted  upright  3  X  10  inch  tongued  and  grooved  hem- 
lock planks  P,  15  inches  long.  At  each  end  of  the  weir,  a  10  X  10  inch  yellow  pine 
post  M  is  tenoned  between  the  sills,  as  in  Figs.  9,  and  built  into  the  masonry 
ends  of  the  dam,  which  last  extend  well, into  the  banks  of  the  stream 

In  both  Figs,  the  crest-piece  a,  is  of  2  X  8  inch  oak,  beveled  so  as  to  leave  a 
horizontal  top  face  34  inch  wide.  The  crest-piece  is  let  in  flush  with  the  back  of 
the  piles  or  boards  P^to  which  it  is  bolted,  and  is  let  into  the  end  posts  M  about 
2  or  3  inches.  At  low  stages  of  water,  the  flow  may  be  confined  to  a  portion  of 
the  length  of  the  overfall  by  flash-boards  placed  along  the  rest  of  the  dam. 

A  crest-piece  made  of  8  X  M^  inch  bar  iron  is  preferable  to  one  of  wood.  It  re- 
quires of  course  much  less  cutting  away  of  the  sheet-piling,  and  its  upper  edge  is 
less  subject  to  abrasion  by  drift  passing'over  the  weir.  The  top  edge,  and  the  abut- 
ting ends  of  the  several  lengths,  should  be  planed  smooth  and  square ;  the  former 


DAMS. 


647 


to  insure  a  sharp  inner  corner  at  a  for  the  water  to  pass  over,  and  the  latter  in 
order  to  avoid  leakage.  As  a  further  precaution  against  leakage,  a  strip  or  butt- 
strap  of  8  X  3^  inch  iron,  about  a  foot  long,  may  be  let  in,  between  the  a-est-pieca 
and  the  sheet  piling,  opposite  each  joint  of  the  former,  and  overlapping  both  the 
adjoining  ends,  the  piling  being  cut  away  '^  inch  deeper  at  those  points,  in  order 
to  accommodate  them.  Such  butt-straps,  if  placed  on  the  w/j-stream  side  of  the 
crest-piece,  -would  break  the  continuity  of  the  sheet  of  water  passing  over  the 
weir,  and  thus  interfere  somewhat  with  the  correctness  of  the  gauging.  Such 
iron  is  obtainable  in  any  commercial  center,  in  lengths  of  about  16  feet.  8  X  /^ 
weighs  6%  pounds  per 'running  foot ;  8X3^,  3^^  pounds. 

All  the  joints  should  be  caulked  with  oakum.  To  apply  the  usual  weir  formulae 
(see  Art.  14/,  p.  548)  the  back  of  the  weir  should  be  vertical  for  a  depth  p  below 


Figs.  10. — Measuring  Weir  on,  Mock  Bottom^ 

the  crest  a  equal  at  least  to  twice  the  head  H  on  the  weir.  It  is  therefore  better 
k»  protect  the  back  of  the  weir  by  tarpaulin  rather  than  resort  to  puddling,  ex- 
cept close  to  the  bottom. 

In  a  long  weir  with  a  low  fall,  it  is  difficult  to  secure  a  sufficiently  free  access 
of  air  to  the  space  behind  the  falling  sheet  of  water,  especially  when  tlie  stream 
is  low  and  the  sheet  tends"  to  hug  the  face  of  the  dam.  In  such  cases  a  partial 
vacuum  *  forms  between  the  falling  sheet  and  the  face  of  the  dam,  and  increases 
the  discharge,  thus  vitiating  the  results.  It  is  therefore  important,  in  designing 
measuring  weirs,  to  arrange  (as  far  as  possible)  so  that  the  sheet  of  water  may 
fall  clear  through  the  entire  distance  between  the  up-stream  and  down-stream 
levels  without  striking  any  portion  of  the  weir  itself,  for  such  striking  would 
diminish  the  clear  space  behind  the  sheet  and  increase  the  difficulty  of  pre- 
venting a  vacuum  there. 


*  Such  a  vacuum  causes  the  down-stream  water  near  w,  Figs.  9,  to  rise  behind  the 
sheet.  When  the  rarefaction  of  the  air  behind  the  sheet  has  proceeded  to  a  certain 
extent,  the  external  air  breaks  in  and  relieves  the  vacuum.  Then  another  vacuum 
forms,  and  is  in  turn  relieved,  and  so  on,  alternately.  At  such  times  it  has  been 
noticed  that  lieht  bodies,  such  as  chips,  etc.,  floating  in  the  down-stream  water  near 
the  ends  of  the  weir,  are  drawn  into  the  space  behind  the  sheet  and  carried  toward 
the  middle  of  its  length,  and  then  in  turn  ejected  at  the  point  where  they  entered, 
thus  traveling  back  and  forth  along  the  space  behind  the  sheet. 


648 


DAMS. 


Tremblings  in  I>anis.  Dams  over  which  the  water  falls  in  a  lon^ 
smooth,  unbroken  sheet  of  considerable  height,  are  more  or  less  subject  to 
tremblings,  caused  apparently  by  alternate  compression  and  rarefaction  of  the 
air  by  the  falling  sheet,  especially  in  the  space  (W,  Fig.  20,  p.  547)  behind  the 
sheet,  where  a  partial  vacuum  is  often  formed,  because  the  air  there  is  entangled 
in  the  falling  water  and  given  off  again  by  it  down  stream  in  the  shape  of  foam. 

Such  tremblings  sometimes  cause  a  rattling  of  windows  half  a  mile  or  more 
away.  We  have  known  this  to  be  stopped  (in  one  case  unintentionally)  by  build- 
ing a  well-covered  wide  crib  apron,  a  lew  feet  high,  against  the  front  of  the  dam, 
for  preventing  the  abrasion  of  the  bottom.  In  other  cases  a  series  of  oblique 
timbers  placed  against  the  front  of  the  dam,  and  part  way  up  it,  at  a  slope  of 
about  \}4  to  1,  and  covered  with  plank,  has  been  perfectly  effective  in  stopping 
it.  In  short,  any  device  which  admits  air  more  freely  behind  the  falling  sheet, 
or  destroys  the  continuity  of  the  latter  (such  as  flash  boards  of  different  heights 
or  placed  at  intervals  along  the  crest),  or  which  reduces  its  height  and  its  con- 
tinuous length,  ought  to  diminish  or  obviate  the  trouble. 

The  proper  time  for  building:  dams  is  of  course  at  the  longest 
period  of  low  stage  of  water. 


Table  of  tliickncss  of  i¥liite  pine  plank  required  not  to  bend 
more  tban  ^^^^  part  of  its  clear  liorizontal  stretcb,  unde» 
dilferent  beads  of  water,    (Original.) 


Stretch 
in  Ft. 


3 

4 
6 

8 

10 
12 
15 
20 


40 


Heads  in  feet. 

30      I      20      I      10 


Thickness  in  Inches. 

3 

2^ 

214 

4 
6 

l§ 

48 

8 

7 

b%(y 

10 

7 

1214 

10^ 

15 

13 

20 

17>^ 

14 

WATER   SUPPLY.  649 

WATEE  SUPPLY. 


Consumption  of  water.  Owing  largely  to  the  proper  extension  of  the 
use  of  water  in  dwellings,  the  quantity  required  in  cities  increases  faster 
than  the  population.     In  other  words,  the  per  capita  consumption  increases. 

Use.  Abundant  experience  shows  that  a  supply  of  50  gallons  (or  say  7  cubic 
feet)  per  capita  per  day  is  abundant  for  all  the  needs  and  luxuries  oi  well-to-do 
families  in  American  cities.  The  manufacturing  consumption,  of  course,  bears 
no  fixed  relation  to  the  population.  In  cities  it  is  generally  much  less  than 
the  domestic  consumption. 

Waste.  In  American  cities,  the  waste  often  amounts  to  two  or  three  times 
the  quantity  really  used.  Of  the  116  gallons  per  capita  per  day,  delivered  in 
New  York  in  1899,  Mr.  Freeman*  estimates  that  from  31  to  56  gallons  were  used, 
10  unavoidably  wasted,  and  from  50  to  75  avoidably  wasted. 

In  Philadelphia,  investigations  by  means  of  the  Deacon  waste-water  detector, 
on  142  modern  seven-room,  two-story  dwellings,  with  bath,  etc.,  on  two  inter- 
mediate streets,  showed  that,  of  222  gallons  per  capita  per  day,  furnished  through 
782  fixtures,  192  gallons,  or  86.5  per  cent,  were  wasted,  and  only  30  gallons,  or 
13.5  per  cent.,  were  used.  The  City  is  now  building  enormous  works  for  the 
purpose  of  pumping,  filtering,  conveying,  repumping,  storing,  and  distributing 
the  water  wasted,  as  well  as  the  smaller  quantity  used.  Of  the  total  cost,f  less 
than  half  would  have  sufficed  for  the  water  used  and  unavoidably  wasted. 

Sources  of  waste.  The  waste  is  caused  by  heedlessness ;  by  allowing 
water  to  run  to  waste  in  order  to  prevent  it  from  freezing  in  winter  and  in  order 
to  get  cooler  water  in  summer;  by  leaky  and  otherwise  defective  fixtures;  by 
unsuspected  leaks  in  mains  and  service  pipes,  etc. 

As  a  "guess,  tempered  by  judgment,"  Mr.  Freeman*  classifies  the  50  to  75 
gallons  per  capita  per  day,  wasted  in  New  York,  as  follows : 

Leaks  in  mains 10  to  15  gals  per  capita  per  day. 

"        service  pipes 10  to  15     "  "  " 

"        defective  plumbing 15  to  25     "  "  *' 

Careless  and  wilful  waste 14  to  17     *'  "  '* 

The  avoidable  waste  is  usually  perpetrated  by  a  small  fraction  (say  from  one- 
fifth  to  one-third)  of  the  population,  the  remainder  using  water  reasonably.  In 
the  Philadelphia  case,  above  cited,  of  the  782  fixtures,  22  were  found  to  be 
"  leaking  slightly,"  and  32  "  turned  on  continually." 

Waste  restriction.  Water  meters.  Waste  is  best  restricted  by 
making  its  avoidance  a  pecuniary  object  to  the  consumer;  and  this  is  best 
accomplished  by  the  use  of  the  water  meter,  at  least  on  all  services  (domestic 
industrial,  and  public)  where  waste  is  fou;id  to  be  going  on.  The  meters  should 
be  owned  and  maintained  by  the  corporation  supplying  the  water. 

Minimum  charg'e.'  In  order  to  encourage  the  liberal  use  of  water,  while 
discouraging  ii^waste,  and  thus  avoid  undue  economy  (tending  to  uncleanliness) 
each  consumer  should  be  charged  a  minimum  periodical  rate,  sufficient  to  cover 
amply  all  the  water  he  can  possibly  use  and  enjoy. 

Mr.  Freeman*  estimates  the  average  cost  of  domestic  meters,  for  New  York 
and  Brooklyn,  mostly  5-8  inch  and  3-4  inch,  with  a  few  of  larger  sizes,  at  S12.50 
each,  and  the  cost  of  installation  by  the  city,  working  systematically  and  on  a 
large  scale,  at  S2.50  each,  or  a  total  of  815.00  each.  He  assumes  "the  average 
life  of  the  ordinary  domestic  meter,  of  a  good  type,  well  cared  for,  and  with 
occasional  repairs  and  renewal  of  worn  parts,"  at  "not  far  from  20  years";  and 
annual  expenses  as  follows  : 

Providence,  R.  I.        New  York. 
Actual,  approx.  Assumed, 

Interest  on  cost  of  meter  and  setting ^0.50  3=0.45 

Depreciation  and  renewal  of  meter  (life  assumed 

20  years)  0.75  0.75 

Maintenance  and  repairs,  testing  and  resetting 0.46  0.70 

Reading  meters  and  computing  bills 0  42  0.60 

Total  annual  cost,  per  meter §2.13   •  S2.50 

*  Report  upon  New  York's  Water  Supply,  made  to  Bird  S.  Coler,  Comptroller, 
by  John  R.  Freeman,  Civil  Engineer,  1900. 
fThe  total  cost  may  reach  $20,000,000. 


650  WATER    SUPPLY. 

Free  water  for  fire  protection.  Cities  sometimes  give  to  manufac- 
turers a  free  supply  of  water  through  special  connections,  to  be  used  for  fire 
protection  only ;  the  manufacturer  giving  bond  not  to  use  such  connection  for 
any  other  purpose,  and  the  city  placing  a  meter  on  the  connection  for  the  detec- 
tion of  any  illicit  use  of  the  water  for  other  purposes. 

Water  for  city  use  shonld  not  be  drawn  from  the  very  bot- 
tom of  tbe  reservoir,  because  it  will  then  be  apt  to  carry  along  the  sedi- 
ment ;  which  not  only  injures  the  water,  but  creates  deposits  within  the  pipes; 
thus  obstructing  the  flow.  In  fixing  upon  the  necessary  capacity  of  a  reservoir, 
this  must  be  taken  into  consideration  ;  inasiuucii  as  all  the  water  below  the  level 
for  drawing  off,  must  be  regarded  as  lost.  When  circumstances  justify  the  ex- 
pense, it  is  well  to  curve  up  the  reservoir  end  of  the  service  main,  so  as  to  pro- 
vide it  with  valves  at  different  heights ;  for  drawing  off  only  the  purest  stratum 
that  may  be  in  the  reservoir.    With  this  view,  the  valve-tower  gen- 

erally has  such  valves  communicating  with  the  water  in  the  reservoir;  and  by 
this  means  only  the  purest  is  admitted  into  the  tower:  and  from  it,  into  the 
city  pipes.  This  refinement,  however,  is  rarely  practicable.  Such  valves  must 
of  course  be  worked  by  watchmen. 

Art.  1.  Reservoirs.  In  important  reservoirs  of  earth,  for  storing  water 
to  moderate  depths  for  cities,  experience  appears  not  to  sanction  dimensions 
bolder  than  10  feet  thick  at  top  ;  inner  slope  2  to  1 ;  outer  slope  \]4  to  1.*  A  top 
width  of  15  feet  to  20  feet,  and  inside  slopes  of  3  to  1,  are  adopted  in  some  im- 
portant cases;  with  outer  slopes  of  2  to  1.  Both  slopes,  however,  are  at  times 
made  only  13^  to  1.  The  level  water  surface  should  be  kept  at  least  3  or  4  feet 
below  the  top  of  the  embankment;  or  more,  if  liable  to  w^aves.  In  a  large 
reservoir,  a  quite  moderate  breeze  will  raise  waves  that  will  run  3  feet  (measured 
vertically)  up  the  inner  slope.  A  low  wall,  or  close  fence,  w,  Fig.  37,  is  some- 
times used  as  a  defence  against  them.  The  top  and  the  outer  slopes  shoiild  be 
protected  at  least  by  sod  or  by  grass.  To  assist  in  keeping  the  top  dry,  it 
should  be  either  a  little  rounding,  or  else  sloped  toward  the  outside.f  The  soft 
soil  and  vegetable  matter  should  be  carefully  removed  from  under  the  entire 
base  of  the  embankments ;  which  should  be  carried  down  to  soil  itself  imper- 
vious to  water,  in  order  that  leakage  may  not  take  place  under  them.  To  aid  in 
this,  a  double  row  of  sheet  piles,  or  a  sunk  wall  of  cement  masonry,  carried  to 
a  suitable  depth  below  the  bottom,  may  be  placed  along  the  inner  toe  in  bad 
cases.  If  there  are  springs  beneath  the  base,  they  must  either  be  stopped,  or 
led  away  by  pipes.  The  embankment  should  be  carried  up  in  layers,  slightly 
hollowing  toward  the  center,  and  not  exceeding  a  foot  in  thickness;  and  all 
stones,  stumps,  and  other  foreign  material,  such  as  clean  gravel,  sand,  and  de- 
composed mica  schists,  Ac,  that  may  produce  leakage,  carefully  excluded.  These 
layers  should  be  well  consolidated  by  the  carts;  and  the  easier  the  slopes  are, 
the  more  effectively  can  this  be  done.  The  layers,  however,  should  not  be  dis- 
tinct, and  separated  by  actual  plane  surfaces;  but  each  succeeding  one  should 
be  well  incorporated  with  the  one  below.  This  has  sometimes  been  done  by 
driving  a  drove  of  oxen,  or  even  sheep,  repeatedly  over  each  layer  ;  in  addition 
to  the  carting.  Rollers  are  not  to  be  recommended,  as  they  tend  to  produce 
seams  between  the  layers.  This  might  possibly  be  obviated  by  projections  on 
the  circumference  of  the  roller. 

Gravelly  earth  is  an  excellent  material,  perhaps  the  best.  The  choicest 
material  should  be  placed  in  the  slope  next  to  the  water;  and  should  be  de- 
posited and  compacted  with  special  care  in  that  portion,  so  as  to  prevent  the 
water  from  leaking  into  the  main  body  of  the  dam,  and  thus  weakening  it.  It 
is  not  amiss  to  introduce  a  bench,  6,  Fig  37,  in  the  outer  slope,  to  diminish 
danger  from  rainwash  by  breaking  the  rapidity  of  its  descent. 

If  the  bottom  of  the  reservoir  itself  is  on  a  leaky  soil,  or  on  fissured  rock, 
through  the  seams  of  which  water  may  escape,  it  must  be  carefully  covered 
with  from  V/2  to  3  feet  of  good  puddle ;  which,  in  turn,  should  be  protected  from 
abrasion  and  disturbance,  by  a  layer  of  gravel ;  or  of  concrete,  either  paved  or 
not,  according  to  circumstances. 

*  The  writer  suggests  that  a  top  width  equal  to  2  feet  +  twice  the  square  root 
of  the  height  in  feet,  will  be  safe  for  any  height  whatever  of  reservoir  properly 
constructed  in  other  respects. 

tSome  engineers  slope  the  top  toward  the  inside. 


RESERVOIRS.  651 

Reservoirs  constructed  with  the  foregoing  dimensions,  and  with  care,  may 
remain  safe  for  an  indefinite  period;  but  where  serious  damage  would  result 
from  failure,  the  following  additional  precautions  should  be  taken. 
The  inner  slopes  should  be  carefully  faced  up  to  the  very  top,  with  at  least  a 
close  drv  rubble-stone  pitching,  not  less  than  15  to  18  inches  thick  ;  as  a  protec- 
tion against  wash,  and  against  muskrats.  These  animals,  we  believe,  always 
commence  to  burrow  under  water.  If  the  slopes  are  much  steeper  than  2  to  I, 
this  drv  pitching  will  be  apt  to  be  overthrown  by  the  sliding  down  of  the  soft- 
ened earth  behind  it,  if  the  water  in  the  reservoir  should  for  any  cause  be 
drawn  down  rather  suddenlv.  It  will  be  much  more  effective,  but  of  course 
more  costlv,  if  laid  in  hydraulic  cement;  and  still  more  so  if  laid  upon  a  layer 
a  few  inches  thick  of  cement-and-gravel  concrete ;  especially  if  this  last  be 
underlaid  bv  a  layer  about  1^^  to  3  feet  thick  of  good  puddle,  spread  over  the 
face  of  the  slope  ;  the  great  object  being  to  protect  the  inner  slope  from  actual 
contact  with  the  water.  If  this  can  be  effectually  accomplished,  slopes  as  steep 
as  1><  to  1  will  be  perfectly  secure  ;  for  the  danger  does  not  arise  from  any  want 
of  weight  of  the  earth  for  resisting  overthrow.  (Special  care  sliowld  be 
bestowed  upon  the  inner  toe  of  the  slope,  to  prevent  water  from 
finding  its  wav  beneath  it,  and  softening  the  earth  so  as  to  undermine  the  stone 
pitching.  Near  the  top,  reference  should  be  had  to  danger  of  derangement  by 
ice,  frost,  rain,  and  waves.  Flat  inner  slopes  tend  not  only  to  prevent  the  dis- 
placement of  the  pitching;  but  increase  the  stability  of  the  embankment,  by 
causing  the  pressure  of  the  water  (which  is  always  at  right  angles  to  the  slope) 
to  become  more  nearly  vertical ;  and  thus  to  hold  the  embankment  more  firmly 
to  its  base  than  if  there  were  no  water  behind  it.  Sometimes  the  toes  of  both 
the  inner  and  outer  slopes  abut  against  low  retaining-walls  in  cement.  This 
gives  a  neat  finish,  and  tends  to  preservation  from  injury. 

Many  engineers,  in  order  to  prevent  leaking,  either  through  or  beneath  the 
embankment,  construct  a  puddle-wall,  p,  .Fig.  37,  of  well-rammed  imper- 
vious soil,  (gravelly  clay  is  the 
best,)  reaching  from  the  top 
to  several  feet  Ijelow  the  base. 
This  wall  should  not  be  less 
than  6  or  8  feet  thick  on  top, 
for  a  deep  reservoir;  and 
should  increase  downward  by 
q^vefe  (and  not  by  slopes,  or 
Fig.  37.  batters)  at  the  rate  of  about 

1  in  total  thickness,  to  3  or  4 
in  depth.  Other  engineers  object  to  these  puddle-walls ;  and  contend  that  leak- 
age should  be  prevented  by  making  both  the  inner  slopes,  and  the  bottom  of  the 
reservoir,  water-tight,  by  means  of  puddle,  concrete,  and  stone  facing  in  cement, 
as  just  alluded  to.  They  argue  that  if  the  embankment  is  well  constructed,  it 
is  itself  a  puddle-wall  throughout. 

Near  San  Francisco,  Cal,  are  two  earthen  reservoir  dams 
built  about  1864.  one  95  feet  high,  26  on  top,  inner  slope  2.75  to  1,  outer  2.5  to  1. 
The  other  93  high,  25  on  top,  inner  slope  3.5  to  1,  outer  3  to  1.  In  each  the  pud- 
dle-wall is  carried  47  feet  deeper  than  the  base.    No  stone  facing. 

It  is  difficult  to  prevent  water  under  high  pressure  from 
finding  its  way  throug-h  considerable  distances  along-  seams 
where  earth  is  in  contact  with  smooth  rock,  wood,  or  metal ;  as,  for  instance, 
along  the  surfaces  of  iron  pipes  laid  under  reservoir  embankments ;  or  along 
the  tie-rods  sometimes  used  through  the  puddle  of  coffer-dams ;  and  the  same 
is  apt  to  occur  under  the  bases  of  embankments  which  rest  on  smooth  rock. 
Special  care  should  be  taken  that  the  earth  used  in  such  positions  is  not  of  a 
porous  nature  ;  and  that  it  is  thoroughly  compacted  all  along  the  seam  ;  and  the 
straight  continuity  of  the  Sf^am  should  be  interrupted  or  broken  as  frequently 
as  possible  by  pr(yections.  Faucets  or  flanges  do  this  to  a  limited  extent  in  the 
case  of  iron  pipes  ;  and  something  similar,  but  on  a  larger  scale,  should  at  short 
intervals  be  constructed  in  the  shape  of  collars  or  yokes  of  cement  stonework, 
in  the  case  of  rock  or  masonry. 

It  is  usually  advisable  to  divide  reservoirs  into  two  parts,  so  that 
while  the  water  in  one  part  is  being  drawn  off  for  use,  that  in  the  other  may 
purify  itself  by  settling  its  sediment.  Also,  one  part  may  remain  in  use,  while 
the  other  is  being  cleaned  or  repaired.  Many  days,  or  even  two  or  three  weeks, 
sometimes,  are  required  for  the  complete  settlement  of  the  very  fi«e  clayey  par- 
ticles in  muddy  water;  depending  on  the  depth  of  the  reservoir.  One  or  more 
flights  of  steps  to  the  bottom  of  the  reservoir  should  be  provided. 

Mud  in  Reservoirs.  The  reservoirs  of  the  New  River  Water  Co,  Lon- 
don, England,  were  uncleaned  for  100  years,  during  which  mud  8  feet  deep  wag 


652  EESERVOIRS. 

deposited,  or  about  an  inch  annually.  At  Philadelphia  it  is  about  .25  inch  per 
annum  from  the  Schuylkill,  and  1  inch  from  the  Delaware  River.  At  St.  Louis, 
Missouri,  about  3  to  4  feet  per  year!  Vegetation  is  apt  to  take  place  in  shallow 
reservoirs  and  near  the  edges  of  deep  ones,  especially  in  very  warm  weather; 
and  the  plants,  on  decaying,  injure  the  water. 

Water  flouring'  throug'ti  iiiar»^li  lands  is  sometimes  unfit  for  drink- 
ing purposes.  That,  for  instance,  in  some  sections  of  the  Concord  River,  Massa- 
chusetts, was  reported  by  the  eminent  hydraulic  engineer,  Loammi  Baldwin,  of 
Boston,  to  be  aLbsolutelj  poisonous  from  this  cause. 

The  construction  of  a  large  deep  reservoir  is  not  only  a  very  costly,  but  a 
very  hazardous  undertaking.  With  every  watchfulness  and  care,  it  is  almost 
impossible  entirely  to  prevent  leaking;  although  this  may  not  manifest  itself 
for  months,  or  even  years.  Should  a  break  occur,  especially  near  a  city,  it 
would  probably  be  attended  by  great  loss  of  life  and  property.  If  the  water 
once  finds  its  way  in  a  stream,  either  across  the  unpaved  top,  or  through  the 
body  of  the  embankment,  the  rapid  destruction  of  the  whole  becomes  almost 
certain. 

Art.  la.  IStorin^  Reservoirs.  The  entire  annual  yield  of  a  stream 
may  be  much  more  than  sufficient  for  supplying  a  certain  population  with 
water;  and  yet  in  its  natural  condition  the  stream  may  not  be  available  for  this 
purpose,  because  it  becomes  nearly  dry  in  summer,  when  water  is  most  needed; 
while,  at  other  seasons,  the  rains  and  melted  snows  produce  floods  which  supply 
vastly  more  than  is  required;  and  which, must  be  allowed  to  run  to  waste.  A 
storing  reservoir  is  intended  to  collect  and  store  up  this  excess  of  water,  so  that 
it  may  be  drawn  oflT  as  required  during  the  droughts  of  summer,  and  thus 
equalize  the- supply  throughout  the  entire  year.  This,  when  the  locality  per- 
mits, is  eflfected  by  building  a  dam  across  the  stream,  to  form  one  side  of  the 
reservoir;  while  the  hill-slopes  of  the  valley  of  the  stream  form  the  other  sides. 
The  stream  itself  flows  into  this  reservoir  at  its  up-stream  end.  When  the 
stream  is  liable  to  become  nearly  dry  during  long  summer  droughts  experience 
shows  that  the  capacity  of  the  reservoir  should  be  equal  to  from  4  to  6 
months'  supply,  according  to  circumstances.  During  the  construction  of  the 
dam,  a  free  channel  must  be  provided,  to  pass  the  stream  without  allowing  it 
to  do  injury  to  the  work.  If  the  dam  were  built  precisely  like  Fig  37,  entirely 
of  earth,  it  would  plainly  be  liable  to  destruction  by  being  washed  away  in  case 
the  reservoir  should  become  so  full  that  the  water  would  begin  to  flow  over  its 
top.  To  provide  against  this  we  may,  by  means  of  masonry,  or  of  cribs  filled 
with  broken  stone,  or  otherwise,  construct  either  the  whole,  or  part  of  the  dam, 
to  serve  as  an  overfall,  or  a  ivaste-weir.  Or  a  side  channel  (an  open  cut, 
pipes,  or  a  culvert,  &c)  may  be  provided  at  one  or  both  ends  of  the  dam,  and  in 
the  natural  soil,  at  such  a  level  as  to  carry  away  the  surplus  flood  water  before 
it  can  rise  high  enough  to  overtop  the  earthen  dam.  Besides  these,  and  the 
pipes  for  carrying  the  water  to  the  town,  there  should  be  an  outlet,  with  a  valve 
or  gate,  at  the  level  of  the  bottom  of  the  reservoir;  in  order  that,  if  necessary 
for  repairs,  or  for  cleaning  by  scouring,  all  the  water  may  be  drawn  off.  The 
entrances  to  the  city  pipes  should  be  protected  by  gratings,  to  exclude  fish,  &c. 

To  facilitate  repairs  or  renewals  of  all  valves,  «fec,  nrliicb 
are  lender  water,  the  reservoir  ends  of  the  pipes  or  culverts  to  which  they 
are  attached,  may  be  surrounded  by  a  water-tight  box  or  chamber,  which  will 
usually  be  left  open  to  the  reservoir ;  but  may  be  closed  when  repairs  are  re- 
quired. Access  may  then  be  had  to  them  by  entering  at  the  outer  end,  after 
the  water  has  flowed  away  from  inside.  In  case  the  outlet  is  through  a  long 
line  of  pipes  which  cannot  thus  be  entered,  a  special  entry  for  this  purpose  may 
be  cast  in  the  pipe  itself,  near  the  outer  toe  of  the  embankment;  to  be  kept 
closed  except  in  case  of  repairs.  Sometimes  a  better,  but  more  .expensive  means 
of  access  to  such  valves,  is  secured  by  enclosing  them  in  a  valve-toirer  of 
masonry.  This  is  a  hollow  vertical  water-tight  chamber,  like  a  well ;  but  near 
the  toe  of  the  inner  slope;  having  its  foundation  at  the  bottom  of  the  reservoir; 
whence  the  tower  rises  through  the  water  to  above  its  surface.  This  chamber 
is  provided  with  valves  or  gates  usually  left  open  to  the  reservoir;  but  which 
may  be  closed  when  repairs  are  needed  ;  and  the  water  in  the  tower  allowed  to 
escape  from  it  through  the  open  valves  of  the  outlets.  This  done,  workmen  can 
descend  through  the  tower  by  ladders  from  the  aperture  at  its  top. 

At  times  the  outlets  for  the  discharge  of  surplus  flood  water  are,  like  those  for 
scouring,  placed  at,  or  just  above,  the  level  of  the  bottom  of  the  reservoir.  In 
order  that  these  may  work  in  case  of  a  sudden  flood  at  night,  &c,  they  must  be 
furnished  with  self-acting  valves,  which  will  open  of  their  own  accord  when  the 
flood  is  about  to  rise  too  high.  This  may  be  effected  by  attaching  them  to  floata, 
the  rising  of  which,  when  the  water  is  high,  will  pull  them  open.  All  such  out- 
lets should  be  large  enough  to  let  men  enter  them  for  repairs.    They  should  by 


WATER-PIPES. 


653 


no  means  be  laid  through  the  artificial  earthen  body  of  the  dam  itself,  without 
being  supported  upon  masonry  reaching  down  to  a  firm  natural  foundation; 
otherwise  they  are  very  apt  to  be  broken  by  the  subsidence  of  the  embankment. 
It  is  usually  safer  to  carry  them  through  the  firm  natural  soil  near  one  end  of 
the  dam.  Their  valves,  if  only  single,  should  be  at  their  inner  or  reservoir  end, 
so  as  to  leave  the  outlets  themselves  usually  empty,  for  inspection ;  but  it  is 
better  to  have  two  valves,  so  that  one  may  be  used  when  the  other  needs  repair; 
and  "in  this  case  one  may  be  placed  at  each  end.  Reservoirs  which  are  supplied 
by  pumps,  need  no  precautions  against  overflow ;  because  the  pumping  is 
stopped  when  they  are  filled  to  the  proper  height.  Large  storing  reservoirs 
necessarily  submerge  more  or  less  land,  which  has  therefore  to  be  purchased. 
By  intercepting  the  descending  water,  they  frequently  prevent  spring  floods 
from  injuring  low  lands  farther  down  stream.  If  there  are  mills  down  stream 
from  the  reservoir,  they  would  evidently  be  deprived  of  water  for  driving  them, 
unless  a  portion  of  that  stored  in  the  reservoir  be  devoted  to  that  purpose. 
Water  thus  applied  to  compensate  for  the  loss  of  the  natural  stream,  is  called 
eompensation  urater;  and  the  reservoir,  a  compensating  one. 

Art.  lb,  Distributing-  reservoirs.  Frequently  a  valley  fit  for  a  storing 
reservoir  can  be  found  only  at  a  long  dist  (sometimes  many  miles)  from  the  town ;  and  it  then  be- 
comes expedient  to  construct  also  an  additional  one  of  smaller  size  than  the  storing  one,  near  the 
town;  and  at  as  great  an  elevation  above  it  as  circumstances  will  permit ;  but  lower  than  the  storing 
one.  This  is  called,  by  way  of  distinction,  a  distributing  reservoir,  because  from  it  the  water,  after 
having  flowed  into  it  from  the  storing  reservoir,  through  the  long  supply  pipe  which  connects  them,  is 
distributed  in  various  directions  through  the  town,  by  means  of  the  street  mains,  or  pipes.  This 
small  reservoir  should  hold  a  supply  suflBcient  at  least  for  a  few  days;  a  few  weeks  would  be  better; 
and  the  end  of  the  supply  pipe  which  terminates  in  it,  should  be  provided  with  a  valve  for  shutting 
ofifthe  supply  from  the  storing  reservoir.  These  precautions  permit  repairs  to  be  made  along  the  lin« 
Of  supply  pipe  without  depriving  the  town  of  water  in  the  mean  time.  With  a  view  to  such  repairs ;  as 
well  as  to  scouring  out  sediment  from  the  supply  pipe,  this  last  should  be  provided  with  Ontlet 
valves  at  various  low  points  along  the  entire  interval  between  the  two  reservoirs  ;  especially  at 
those  at  which  the  valves  may  disch  into  natural  watercourses.  On  opening  these  valves,  the  oui* 
rush  of  the  water  carries  a\say  sediment;  and  leaves  the  pipe  empty  for  inspection. 

In  fixing*  npon  the  diams  of  pipes  for  supplying  cities,  it  is  necessary 

to  bear  in  mind,  that  by  far  the  greater  portion  of  the  24  houra'  yield  is  actually  drawn  from  them 
during  only  8  to  12  hours  of  daylight;  and  therefore  the  capacity  of  the  pipes  must  be  sufficient  to 
furnish  the  daily  supply  in  much  less  than  24  hours.  Again,  during  the  hot  summer  months,  much 
more  water  is  used  than  during  the  winter  ones;  and  this  consideration  necessitates  a  still  larger  diam. 

Art.  2.      Systems  or  street  pipes  for  supplying  cities.    The 

writer  knows  of  no  practical  rules  for  proportioning  the  diams  for  such  systems.  The  various  com- 
plications involved,  render  a  purely  scientific  investigation  of  little  or  no  service.  With  much  hesi- 
tation, he  ventures  the  following  purely  empirical  rules  of  his  own  ;  based  on  such  limited  observa- 
tions as  have  casually  fallen  under  his  notice. 

Rule  I.  When,  at  no  point  in  a  system  of  city  pipes,  is  the  head,  or  vert  dist  helofo  the  surface 
of  the  reservoir,  compared  with  the  hor  dist  from  the  reservoir,  less  than  at  the  rate  of  5Q  ft  per  mile, 
then  the  population  in  the  last  column  of  the  following  Table  A,  may  be  abundantly  supplied,  for  all 
city  purposes,  by  either  one  pipe  of  the  inner  diam  or  bore  in  the  1st  col ;  or  by  2,  3,  <tc,  pipes  of  tha 
diamsdn  the  other  cols.  These  diams  are  given  to  the  nearest  safe  %  inch.  The  supply  is  assumed 
to  be  about  60  galltns  per  day  to  each  inhabitant. 

TABIiE  A.    (Original.) 


KUMBBR  OF  PIPES. 

1 

2 

3          I         4 

6 

8 

12 

24 

Population 

Diam. 

Diam. 

Diam. 

Diam. 

Diam. 

Diam. 

Diam. 

Diam. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

6 

4^ 

3% 

33^ 

3 

2% 

2% 

IJi 

1647 

8 

GH 

5J4 

4% 

4 

3% 

3% 

2% 

3465 

10 

1% 

QH 

5% 

5 

4% 

3% 

3 

5908 

12 

9« 

1H 

7 

5X 

5>i 

4% 

3% 

9324 

14 

\o% 

^H 

^% 

7 

6% 

5% 

4% 

13706 

16 

12^   ' 

10% 

9}i 

1% 

6% 

6 

4% 

19141 

18 

\m 

11% 

103^ 

8% 

7% 

6% 

5% 

25677 

20 

\b% 

13 

11% 

9% 

8% 

73^ 

5% 

33426 

22 

16H 

U% 

12% 

10% 

93^ 

83^ 

6% 

42433 

24 

18^ 

15  H 

13% 

11% 

10% 

9 

6% 

52671 

26 

19% 

16?i 

15 

12% 

IIH 

9% 

7% 

64447 

28 

21 M 

18^ 

16H 

13% 

UH 

103^ 

8 

77565 

80 

22K 

19% 

17% 

14% 

13% 

1134 

8% 

91580 

32 

24% 

20% 

18}4 

15% 

14 

11% 

9 

108160 

34 

25  J^ 

22 

19% 

16% 

15 

12% 

9% 

125840 

36 

27% 

23  Ji 

20% 

17% 

15% 

13% 

10% 

144480 

40 

30  ^ 

25  Ji 

23% 

19% 

17% 

15 

11% 

188320 

44 

33  J^ 

28% 

25% 

21% 

19% 

163^ 

12% 

239600 

48 

36)^ 

31 

27% 

23% 

21% 

18 

13% 

297600 

54 

41 

34  Ji 

31  >i 

26% 

23% 

20% 

15% 

391200 

60 

*5% 

38% 

34% 

293^ 

26% 

22% 

17% 

511200 

66 

50% 

42% 

38% 

32% 

29% 

24% 

18% 

650400 

72 

46^ 

41% 

35% 

31% 

26% 

2034 

800000 

80 

60;^ 

51% 

46>i 

3934             353^ 

29% 

22% 

1064000 

654 


WATER-PIPES. 


It  is  well  to  allow  in  addition  from  "%  inch  to  1  inch,  or  more,  (depending  on 
the  character  of  the  water,)  to  each  diameter ;  for  deposits  and  concretions. 

The  water,  after  reaching  the  city  through  one  or  more  large  main  pipes  from 
the  reservoir,  must  be  distributed  through  the  streets  by  means  of  smaller 
mains  branching  from  the  larger  ones.  The  diameters  of  these  smaller  ones 
also  may  be  found  by  Table  A.  Thus,  if  a  street,  with  its  alleys,  &c,  contains 
about  6000  persons,  (Ihe  rate  of  head  being,  as  before,  not  less  than  50  feet  to  a 
mile  at  any  point  of  the  system,)  then  we  see  by  the  table  that  a  10-inch  pipe 
Till  answer.  It  would  be  well  to  lay  no  city  street  pipes  of  less  than  6  inches 
diameter. 

Mains  wtiich  cross  each  other  should  he  connected  at  some 
of  their  intersect  ions,  to  allow  the  water  a  more  free  circulation  through- 
out the  entire  system  ;  so  that  if  the  supply  at  any  point  is  temporarily  cut  off 
from  one  direction  by  closing  the  valves  for  repairs,  or  is  diminished  by  exces- 
sive demand,  it  may  be  maintained  by  the  flow  from  other  directions. 

Avoid  dead  ends  when  possible,  as  the  water  in  them  becomes  foul  and 
unwholesome. 

Rule  2.  With  the  same  diameters,  different  rates  of  head  will  supply  ihe  propor- 
tionate populations  in  column  3  of  Table  B.  Or,  to  find  ihe  diameters  which  at  different 
rates  of  head  will  supply  the  same  populations  given  in  ihe  la^t  column  of  Table  Aj 
multiply  the  diameter  given  in  Table  A,  by  the  corresponding  number  in  col- 
umn 4  of  Table  B;  or  (approximately)  do  as  directed  in  column  5. 

TABIiE  B.    (Original.) 


Col.  1. 

Col.  2. 

Col.  3. 

Col.  4. 

Col.  5. 

Rate  of  Head, 

Rate  of  Head, 
compared  with 
that  in  Table  A. 

Proportionate 

Proportionate 
Diam.  to  supply 

Remarks. 

in  Feet  per  Mile. 

Populations. 

the  Populations 
in  Table  A. 

5 

.1 

.32 

1.58 

10 

.2 

.45 

1.37 

im 

.25 

.50 

1.32 

Add  one-third. 

15 

.3 

.55 

1.27 

Add  full  one-fourth. 

20 

.4 

.64 

1.20 

Add  one-fifth. 

25 

.5 

.71 

1.14 

Add  one-seventh. 

30 

.6 

.78 

1.11 

Add  one-ninth. 

35 

.7 

.84 

1.07 

Add  one-fourteenth. 

37H 

.75 

.87 

1.06 

Add  one-sixteenth. 

40 

.8 

.90 

1.05 

Add  one-twentieth. 

45 

.9 

.95 

1.02 

Add  one-fiftieth. 

50 

1.0 

1.00 

1.00 

75 

1.5 

1.23 

.92 

Deduct  one-thirteenth. 

100 

2.0 

1.41 

.88 

Deduct  one^eighth. 

125 

2.5 

1.59 

.83 

Deduct  full  one-sixth. 

150 

3.0 

1.73 

.80 

Deduct  one-fifth. 

200 

4.0 

2.00 

.76 

Deduct  nearly  one-fourth. 

250 

5.0 

2.25 

.73 

Deduct  nearly  two-sevenths. 

300 

6.0 

2.46 

.69 

Deduct  three-tenths. 

400 

8.0 

2.83 

.66 

Deduct  full  one-third. 

500 

10.0 

3.18 

.63 

Example.  By  Table  A  we  see  that  with  the  rate  of  head  of  50  leet  per 
Mile,  a  30inch  pipe  will  supply  a  population  of  91580  ;  but*  with  three  times  that 
rate  of  head,  or  150  feet  per  mile,  we  see  by  column  3,  Table  B,  that  the  same 
pipe  will  supply  1.73  times  as  many  persons,  or  91580X1.73=158433  persons. 
But  if,  at  this  greater  rate  of  head,  we  stil!  wish  to  supply  only  91580  persons, 
then  we  find  in  column  4,  Table  B,  that  we  may  diminish  the  diameter  of  the  pipe 
from  30,  down  to  30  X  -80  =  24  inches;  or,  by  column  5,  we  have  30  —  6  =  24 
inches. 

Again,  after  the  water  has  reached  the  city  by  the  30-inch  pipe  of  Table  A, 
if  we  wish  to  distribute  it  through  the  city  "by  say  eight  branches  or  smaller 
mains,  we  see  by  column  6,  Table  A,  that  each  of  them  must  have  at  least  IS}/^ 
inches  diameter.  From  these  eight,  other  smaller  ones  may  branch  off  into  the 
cross  streets,  alleys,  &c;  and  in  estimating  the  supply  required  for  any  partic- 
ular street  main,  we  must  evidently  add  what  is  required  also  for  such  cross 
streets,  &c,  &c,  as  are  to  be  fed  from  said  main. 

If  certain  limited  parts  of  a  city  pipe  system  have  considerably  less  rates  of 
head  than  most  of  the  remainder,  it  may  become  expedient  to  supply  the  former 
by  a  special  separate  main  of  larger  diameter;  which  may  start  either  directly 


WATER-PIPES.  600 

from  the  reservoir;  or  as  a  branch  from  the  grand  leading  main  which  feeds  tli« 
lower  parts,  according  to  circumstances. 

It  must  be  remembered,  tliat  although  by  increasing  the  diameters,  an  abun- 
dant supply  may  be  obtained  under  a  small  rate  of  head,  as  well  as  under  a  great 
one,  yet  the  water  will  not  rise  to  as  great  a  height  in  the  service  pipes  for  sup- 
plying the  different  stories  of  dwellings,  &c.  Even  with  the  diameters  in  Table 
A,  the  water,  under  ordinary  use,  will  not  rise  in  these  pipes  to  the  full  height 
of  the  surface  of  the  reservoir;  and  if  an  unusual  drawing-off  is  going  on  at 
the  same  time  at  many  parts  of  the  system,  as  in  case  of  an  extensive  fire,  or 
frequently  during  the  hot  summer  months,  it  may  not  rise  to  even  one-half  of 
^hat  height. 

A-rt.  3.  Tlie  following:  lias  been  found  very  effective  for 
preventing  concretions  in  water  pipes.  Formerly  in  Boston,  cast- 
Aron  city  pipes,  4  inciies  diameter,  became  closed  up  in  7  years ;  and  those  of 
Jarger  diameter  became  seriously  reduced  in  the  same  lime.  But  later,  during 
8  years,  in  which  this  varnish  was  used,  no  concretions  formed.*- 

Coal-pitch  varnish  to  be  applied  to  pipes  and  castings, 
made  for  the  Water  Department  of  Philadelphia,  under 
the  following  conditions: 

First.  Every  pipe  must  be  thoroughly  dressed  and  made  clean,  free  from  the 
earth  or  sand  which  clings  to  the  iron  in  the  moulds :  hard  brushes  to  be  used 
in  finishing  the  process  to  remove  the  loose  dust. 

Second.  Every  pipe  must  be  entirely  free  from  rust  when  the  varnish  is  ap- 
plied. If  the  pipe  cannot  be  dipped  immediately  after  being  cleansed,  the  sur- 
face must  be  oiled  with  linseed  oil  to  preserve  it  until  it  is  ready  to  be  dipped: 
DO  pipe  to  be  dipped  after  rust  haa  set  in. 

!niird.  The  coal-tar  pitch  ia  made  from  coal  tar,  distilled  until  the  naphtha 
Is  entirely  removed,  and  the  material  deodoriaed.  It  should  be  distilled  until  it 
has  about  the  consistency  of  wax.  The  mixture  of  five  or  six  per  cent  of  linseed 
oil  is  recommended.  Pitch  which  becomea  hard  and  brittle  when  cold,  will  not 
answer  for  this  use. 

Fourth.  Pitch  of  the  proper  quality  having  been  obtained,  it  must  be  care- 
fully heated  in  a  suitable  vessel  to  a  temperature  of  300  degrees  Fahrenheit,  and 
must  be  maintained  at  not  lesa  than  this  temperature  during  the  time  of  dip- 
ping. The  material  will  thicken  and  deteriorate  after  a  number  of  pipes  have 
been  dipped ;  fresh  pitch  must  therefore  be  frequently  added;  and  occasionally 
the  vessel  must  be  entirely  emptied  of  its  old  contents,  and  refilled  with  fresh 
pitch ;  the  refuse  will  be  hard  and  brittle  like  common  pitch. 

Fifth.  Every  pipe  must  attain  a  temperature  of  300  degrees  Fahrenheit,  before 
it  is  removed  from  the  vessel  of  hot  pitch.  It  may  then  be  slowly  removed  and 
laid  upon  skids  to  drip. 

All  pipes  of  20  inches  diameter  and  upward,  will  require  to  remain  at  least 
thirty  minutes  in  the  hot  fluid,  to  attain  thia  temperature;  probably  more  in 
cold  weather. 

Sixth.  The  application  must  be  made  to  the  satisfaction  of  the  Chief  Engineer 
of  the  Water  I)epartroent :  and  the  material  be  subject  at  all  times  to  his  ex- 
amination, inspection,  ana  rejection. 

Seventh.  Payment  for  coating  the  pipes  will  only  be  made  on  such  pipes  as 
are  sound  and  sufficient  according  to  the  specifications,  and  are  acceptable  inde- 
pendent of  tiie  coating. 

Eighth.  No  pipe  to  be  dipped  until  the  authorized  inspector  has  examined  it 
as  to  cleaning  and  rust;  and  subjected  it  thoroughly  to  the  hammer  proof.  It 
may  then  be  dipped,  after  which,  it  will  be  passed  to  the  hydraulic  press  to  meet 
the  required  water  proof. 

Ninth,  The  proper  coating  will  be  tough  and  tenacious  when  cold  on  the 
pipes,  and  not  brittle  or  with  any  tendency  to  scale  off.  When  the  coating  of 
any  pipe  has  not  been  properly  applied,  and  does  not  give  satisfaction,  whether 
from  defect  in  material,  tools,  or  manipulations,  it  shall  not  be  paid  for;  if  it 
scales  off  or  shows  a  tendency  that  way,  the  pipe  shall  be  cleansed  inside  before 
it  can  be  recoated  or  be  receivable  as  an  ordinary  pipe. 

*  Mr.  Dexter  Brackett,  of  Boston,  informs  us,  1892,  that  while  tubercles  form 
there  in  uncoated  pipes  to  a  thickness  of  about  three-quarters  of  an  inch,  rendering 
4-iiich  pipes  of  'ittle  or  no  value  for  fire  supply,  yet  no  actual  stoppage  has  been 
known  to  occur  from  this  cause  during  the  twenty-three  years  of  his  connection 
with  the  City  Engineering  Depat  tmeiit.  He  states  also  that  even  their  coated  pipes, 
taken  up  after  being  in  the  ground  for  ten  or  fifteen  years,  are  generally  found  to  be 
pitted  on  their  inner  surfaces. 


656 


WATER-PIPES. 


Art.  4.  The  pipes  are  laid  to  conform  to  the  vertical  undulations  of  the  street 
surfaces.  The  tops  of  the  pipes  are  laid  not  less  than  3^/^  feet  below  the  surface  of 
the  street;  but  in  3-inch  pipes  the  water  has  at  times  been  frozen  at  that  depth. 

In  Pliilada.,  in  1885,  tliere  were  about  784  miles  of  street 
pipes;  or  about  1  mile  to  every  llOU  inhabitants.  The  population  was  about 
860,000;  residing  in  about  150,000  dwellings.  Berlin,  1887-8;  1,400,000  inhab- 
tants,  in  20,000  houses  (average  70  persons  per  house).  Mean  consumption  per  head, 
17  U.  S.  gallons  per  day;  maximum,  24;  minimum,  12}^;  all  approximate.  25,000 
wheel  meters  in  use. 

No  sralvanic  action  has  been  observed  where  lead  pipes  or  brass  unite  with 
cast-iron  ones.  No  pipe  less  than  6  inches  diam  should  be  laid  in  cities;  and 
even  they  only  for  lengths  of  a  few  hundred  feet.  Tlieir  insuflBciency  is  chiefly  felt  in 
case  of  fire.  8  ins  would  be  a  better  minimum.  No  more  leakage  occurs  in  winter 
than  in  summer;  except  from  the  bursting  of  private  service-pipes  by  freezing. 

To  compact  the  earth  thoroughly  against  the  pipes  excludes  air,  and  greatly  im« 
pedes  rust.  Pipes  may  be  corroded  by  the  leakage  of  gas  through  the  body  as  well  ai 
through  the  joints  of  adjacent  g-as-pipes. 


WEIGHT  OF  CA8T.IRON  WATER-PIPES, 

As  used  in  Pliila,and  tested  by  hydraulic  press  before  delivery  to  an  internal 
pres  of  300  lbs  per  sq  inch.  This  table  includes  spigots,  and  faucets  or  bells.  Th« 
pipes  are  required  to  be  made  of  remelted  strong  tough  gray  pig  iron,  such  as  may 
be  readily  drilled  and  chipped;  and  all  of  more  than  3  ins  diam  to  be  cast  vertically, 
with  the  bell  end  down.  Deviations  of  5  per  cent  above  or  below  the  theoreti- 
cal weights,  are  allowed  for  irregularities  in  casting,  which  it  seems  impossible  to 
avoid. 

The  pipes  are  in  lengths  from  3  to  3^  ins  longer  than  12  ft ;  so  that  when  laid  they 
measure  12  ft  from  the  mouth,/,  Fig  38,  of  one  bell  to  that  of  the  next. 


Diam. 

Thick- 
ness. 

Wt  par 
length. 

Diam. 

Thick- 
ness. 

Wt  per 
length. 

Diam. 

Thick- 
ness. 

Wtper 
length. 

Ini. 

Ids. 

Lbs. 

Ins. 

Ina. 

Lb». 

Ins. 

Ins. 

Lbs. 

3 

^ 

158 

16 

% 

1322 

36 

ii 

4334 

4 

211 

20 

1654 

36 

l5s 

4862 

6 

l'. 

385 

20 

1 

1798 

36 

Ip 

5366 

8 

460 

30 

3313 

48 

7282 

10 

'i 

667 

30 

3010 

48 

ii 

8667 

12 

899 

30 

1 

3964 

48 

9378 

The  following  sizes  of  lap-welded  wrong'lit-iron  water-pipe  are 

made  by  the  National  Tube  Works  Co.,  McKeesport,  Pa.,  and  fitted  with  their 
**  Converse  patent  lock. -joint."  One  end  of  each  length  of  pipe  has  the 
lock-joint  permanently  attached  (leaded)  to  it  at  the  works  before  shipping.  The 
"weights  per  foot  "include  these  joints.  The  weight  of  "lead  per  joint"  given  is 
that  required  to  be  poured  in  laying  the  pipe,  or  that  for  one  side  only  of  the  joint. 

Outer  cliam,  ins 2         3         4         5  6  8  10  12  18 

Wei^bt  per  ft,  lbs 1.86  3.48  5.26  7.33  8.76  13.20  17.08  2o.l2  47.70 

I.ead  per  joint,  lbs ^      1^      2%  3%  3^  b%  6  %%  14 

Average  car  load : 

Number  of  lengths 800      880      275  145  126  128  80  56  40 

*'        "  feet 11500  5600  4500  2600  2000  2000  1200  800  630 

The  pipes  are  tested  for  a  bursting  pressure  of  500  lbs  per  square  inch,  or  higher 
if  desired.  They  are  furnished  either  coated  with  asphaltum,  or  ''kala* 
meined;"  or,  if  desired,  first  kalameined  and  then  coated  with  asphaltum. 
Kalameining  consists  in  "incorporating  upon  and  into  the  body  of  the  iron  a  non- 
eorrosive  metal  alloy,  largely  composed  of  tin."  The  surface  thus  formed  is  not 
eracked  by  blows,  or  by  bending  the  pipe,  either  hot  or  cold. 


WATER-PIPES.  />-y 

Th»  joint,  or  coupling,  is  of  cast-iron,  and  has  internal  recesses  which  receive  anl 
hold  lugs  on  the  outside  of  each  length  of  pipe,  near  each  of  its  ends.  The  joint  is 
then  poured  with  lead  in  the  usual  way  (see  next  page),  eitl^er  with  clay  collars,  or 
with  a  special  pouring  clamp  furnished  by  the  Co,  This  clamp  resembles  the 
*'jointer,"  Figs  39  &c,  except  that  it  is  in  two  rigid  semi-circular  pieces,  connected 
together  by  a  hinge-joint,  and  furnished  with  handles  like  those  of  a  lemon-squeezer, 
and  has  a  hole  in  one  side  for  pouring.  The  coupling  forms  a  flush  inner  surface 
with  the  pipe  at  the  joint,  thus  avoiding  much  of  the  resistance  of  cast-iron  pipes 
to  flow.  For  cases  where  it  may  be  necessary  to  make  frequent  changes,  the  coup- 
lings are  made  in  two  pieces,  which  are  bolted  together  by  flanges. 

Wrong^ht-iron,  for  pipes.,  has  the  great  advantag-es  over  cast-iron 
of  lightness,  toughness,  and  pliability.  The  lightness  of  wrought-iron  pipes  ren- 
ders them  easier  to  handle,  and  cheaper  per /ooi  notwithstanding  that  their  cost  per 
ton  is  about  25  per  cent  greater.  They  are  not  liable  to  breakage  in  transportation 
or  from  rough  handling,  and  they  may  be  bent  through  angles  up  to  about  25°. 
They  therefore  require  no  special  bend  castings  for  such  angles.  The  National  Co 
supply  bending  machines,  to  be  worked  by  two  men.  One  machine  can,  by  changing 
the  dies,  be  used  in  bending  all  sizes  of  pipe.  The  pipes  are  in  lengths  of  from  15  to 
18  feet,  instead  of  12  feet,  as  in  the  case  of  cast-iron,  so  that  fewer  joints  are 
required  per  mile. 

The  Co  furnish  special  "service  clamps"  and  tapping  machines  for  attaching: 
service  pipes  to  mains.  This  may  be  done  (as  in  the  case  of  the  Payne 
machine,  while  the  main  is  under  pressure.    The  service  clamp  is  a  cast- 

iron  saddle,  which,  before  the  main  is  tapped,  is  attached  to  it  by  means  of  a  U 
bolt,  and  which  remains  permanently  so  attached  after  the  tapping.  A  sheet-lead 
gasket  is  placed  between  clamp  and  main.  The  clamp  has  a  tapped  cylindrical 
opening  through  it,  into  which  the  corporation  stop  is  screwed  before 

the  pipe  is  tapped.  The  drill  of  the  tapping  machine  passes  through  the  stop,  and 
through  the  cylindric'kl  opening  in  the  clamp,  and  drills  through  the  lead  gasket 
and  through  the  side  of  the  main. 

The  Co  furnish  also  pipe-cutting  machines,  and  special  castings  (reducers, 
crosses,  «fec,  &c)  fitted  with  the  Converse  joint. 

Art.  5.    Wrongtit-iron  pipes  corrode  much  more  rapidly  than  cast. 

A  grutta-perelia  pipe^  14  i"ch  thick,  and  %  inch  bore,  has  sustained  safely 
an  internal  pres  of  more  than  250  fts  per  sq  inch;  equal  to  nearly  600  feet  head.  It  merely  swelled 
slightly  at  337  lbs.  In  1851  a  tube  of  that  material,  2>^  ins  bore,  about  34  inch  thick,  and  1350  ft  long, 
was  sunk  iq  the  East  River,  New  York,  to  carry  the  Croton  water  to  Blackwell's  Island.  It  was  held 
down  by  weights.  It  proven  unsatisfactory  owing  to  abrasion  caused  by  tidal  currents,  and  injury 
from  the  anchors  of  dragging  vessels.  A  wrapping  of  canvas,  conQned  by  spun  yarn,  was  useful  ia 
preventing  the  former,  but  not  the  latter.    This  pipe  was  replaced  in  1870  by  wrought-iron  pipes. 

Ball's  patent  iron  and  cement  pipe,  is  made  by  The  Patent  Water 
and  Gas  Pipe  Co,  of  Jersey  City,  N.  J.  It  is  formed  of  riveted  sheet-iron,  and  each  length  is  dipped 
into,  and  coated  with,  a  hot  mixture  of  coal  tar  and  asphalt.  The  lining  of  hydraulic  cement  is  thea 
applied.  This  ranges,  in  thickness,  from  %  inch  for  12-inch  pipes  to  1  inch  for  20-inch  pipe.  This 
pipe  is  made  up  to  diams  of  36  ins.  It  is  laid  in  a  bed  of  cement  mortar,  and  completely  covered  witli 
the  same.  Suitable  means  are  provided  for  making  all  the  attachments,  Ac,  required  in  city  pipes 
for  water  and  gas.  More  than  1300  miles  of  it  are  in  use  in  various  towns,  some  of  it  for  35  years ; 
and  it  appears  to  give  general  satisfaction.  Tubercles  do  not  form  in  these  pipes,  as  they  are  apt  to 
do  in  cast-iron  ones.  There  is  every  reason  to  suppose  that  they  are  durable.  The  trenches  being 
dug,  the  Jersey  City  Co  furnish  pipes  and  lay  them  (including  the  cement). 

A.    Wyckofif  &    Son,  Elmira,  N.  Y.,    make  wooden  water  pipes.    For 

pressures  of  15  to  20  fts  per  sq  Inch,  they  furnish  either  plain  pipes,  3}4  to  7  ins  square  externally, 
and  from  1^4  to  4  ins  internal  diam ;  or  round  pipes,  1  inch  to  16  ins  bore,  coated  externally  witb 
asphaltum  cement.  At  their  ends,  both  the  square  and  the  round  pipes  are  banded  with  iron.  For 
pressures  from  iO  to  160  fts  per  sq  inch,  the  round  wooden  pipes,  before  being  coated  with  cement, 
are  spirally  wrapped,  by  steam  power,  with  hoop  iron,  which  is  first  passed  through  a  preparation 
of  coal-tar.  The  iron  is  wound  so  tightly  as  to  be  imbedded  in  the  pipe,  leaving  its  outer  surface 
flush  with  that  of  the  wood.  The  ends  of  each  length  of  pipe  receive  extra  banding.  The  asphaltunj 
cement  coating  is  then  applied.  These  pipes  have  been  extensively  and  successfully  used  for  both 
water  and  gas.     Suitable  arrangements  are  provided  for  joints  and  connections. 

Water  pipes  of  bored  oak  and  pine  logi§i,  laid  in  Philada  50  to  60 

years  ago,  are  frequently  quite  sound,  and  still  fit  for  use,  except  where  outer  sap  wood  is  decayed. 
"When  this  is  removed,  many  of  these  old  pipes  have  been  relaid  in  factories,  &c.  Clay  well  packeot 
around  wooden  pipes,  excludes  the  contact  of  air,  and  thus  contributes  greatly.to  their  durability. 
Loose  porous  soils,  such  as  gravel,  &c,  on  the  contrary,  are  unfavorable. 

Pipes  made  of  bituminize<l  paper,  prepared  under  great  pressurt^ 
have  been  used  for  both  water  and  gas.  They  are  much  less  liable  to  break  than  cast-iron,  and  ds 
not  weigh  or  cost  more  than  about  half  as  much.  Pipes  of  5  ins  bore  and  J^  inch  thick,  have  resisted 
test  strains  of  220  E)s  per  sq  iuok;  equal  to  a  water  head  of  601  ft. 

42 


658 


WATER    PIPES. 


Costs  of  "water  pipe  and  laying*.  The  following  figures  are  deduced 
from  a  table  kindly  furnished  by  Mr.  Allen  J.  Fuller,  General  Superintendent, 
Bureau  of  Water,  Philadelphia.  They  represent  average  conditions  for  straight 
pipe,  laid  in  earth,  in  that  city.  The  cost,  in  any  given  case,  may  differ  materially 
from  these  figures,  according  to  circumstances. 

"Laying"  includes  all  handling  of  materials,  after  their  delivery  on  the 
ground,  for  laying  them  in  the  trench,  making  joints,  calking,  etc.  Calkers 
receive  $2.50,  lead  men  $2.00,  and. laborers  $1.75  per  day  of  8  hours. 

The  costs  of  materials  are  taken  as  follows  :  Pipe  castings,  1.2  cts.;  lead,  5  cts.; 
gasket,  33^  cts.;  coke,  0.27  cts.,  per  pound  ;  blocking,  1.7  cts.  per  ft.  B.  M. 

Add  for  stops,  branches,  fire  hydrants,  special  castings,  repaving,  damages, 
foremen's  wages,  cost  of  tools,  etc.  For  rough  estimates,  to  cover  fixtures,  rock 
excavation,  additional  depth  required  for  trench,  wear  and  tear  of  tools,  and 
ordinary  repaving,  but  not  including  damages,  asphalt  repaving,  or  trestling, 
the  costs  in  the  taole  may  be  increased  as  follows : 

Diameter  of  pipe         4         6         8        10        12        16  to  48  inches. 
Add 80        70        65        60        50  40      percent. 


Materials. 


Pipe. 

Per  length  of  12  feet. 

Per 

Diam. 

Thick- 
ness. 

Iron. 

Lead. 

Gasket. 

Coke. 

Block' g. 

Total. 

lineal 
foot. 

Ins. 

Ins. 

Rs. 

S 

fts. 

s 

Bbs. 

s 

lbs. 

s 

Ft. 
BM 

$ 

$ 

« 

4 

% 

214 

2.57 

6 

0.30 

0.2 

0.01 

4 

0.01 

2 

0.03 

2.92 

0.24 

6 

367 

4.40 

w 

0.50 

0.2 

0.01 

4 

0.01 

2 

0.03 

4.96 

0.41 

8 

IS 

490 

5.88    R 

0.62 

0.3 

0.01 

4 

0.01 

2 

0.03 

6.56 

0.55 

12 

n 

•918 

11.02    18 

0.90 

0.6 

0.02 

5 

0.01 

2 

0.03 

11.98 

1.00 

18 

1510 

18.12    30 

1.50 

0.8 

0.03 

7 

0.02 

2 

0.03 

19.70 

1.64 

24 

2458 

29.50    40 

2.00 

1.0 

0.03 

8 

0.02 

3 

0.05 

31.60 

2.63 

30 

\i 

3325 

39.90    75 

3.75 

1.3 

0.05 

9 

0.02 

4 

0.07 

43.79 

3.65 

1 

4009 

48. Hi  75 

3.75 

1.3 

0.05 

9 

0.02 

4 

0.07 

52.00 

4.33 

36 

1j 

4610 

55.32  115 

5.75 

2.0 

0.07 

10 

0.03 

5 

0.09 

61.25 

5.10 

5680 

68.16  115 

5.75 

2.0 

0.07 

10 

0.03 

5 

0.09 

74.09 

6.17 

48 

11/ 

80381  96.46  150 

7.50 

2.4 

0.08 

12    0.03 

8 

0.14 

104.21 

8.68 

i>l 

9455  113.46  150    7.50 

2.4 

0.08,  12  |0.03 

« 

0.14 

121.21 

10.10 

Earthwork. 


Pi] 

-»e. 

Trench. 

Earthwork, 

per  lineal  foot. 

Width,  feet. 

Excavation. 

Back  fill 

Diam. 

Thick- 
ijess. 

Depth, 

and 
removing 

Total, 

Top. 

Bot- 

feet. 

Cubic 

surplus. 

Ins. 

Ins. 

tom. 

yards. 

s 

% 

% 

4 

% 

2.50 

2.25 

4.50 

0.40 

0.09 

0.02 

0.11 

6 

2.50 

2.25 

4.50 

0.40 

0.09 

0.02 

0.11 

8 

/s 

2.50 

2.25 

4.50 

0.40 

0.09 

0.02 

0.11 

12 

9 

2.75 

2.50 

4.50 

0.47 

0.11 

0.06 

0.16 

18 

P 

2.75 

2.50 

4.50 

0.48 

0.11 

0.09 

0.20 

24 

3.70 

3.00 

4.75 

0.64 

0.14    • 

0.15 

0.30 

30 

it 

4.25 

3.25 

5.00 

0.76 

0.17 

0.23 

0.40 

1 

4.25 

3.25 

5.00 

0.76 

0.17 

0.23 

0.40  . 

36 

\% 

5.00 

3.75 

5.50 

0.97 

0.22 

0.30 

0.52 

iS 

5.00 

3.75 

5.50 

0.96 

0.22 

0.30 

0.52 

48 

h| 

7.00 

4.50 

6.50 

1.47 

0.44 

0.56 

0.99 

IK 

7.00 

4.50 

6.50 

1.47 

0.44 

•0.56 

0.99 

WATER   PIPES. 


659 


Hauling, 

Laying,  Recapitulation. 

Hauling 

Recapitulation 

of  Cost. 

Pipe. 

per  lineal  foot,  at 
75  cents  per  ton. 

Laying, 

per 

lineal  foot. 

per 
lineal 

, 

j^ 

,    . 

1 

s 

|i 

1 

foot. 

6-i 

u  o 

&I 

1 

Ins. 

Ins. 

% 

« 

% 

$ 

S 

.  s 

% 

« 

% 

4 

5^ 

0.01 

0.01 

0.02 

0.03 

0.24 

0.11 

0.02 

0.03 

0.40 

6 

t 

0.01 

0.01 

0.02 

0.04 

0.41 

0.11 

0.02 

0.04 

0.58 

8 

1^5 

0.01 

0.01 

0.02 

0.04 

0.55 

0.11 

0.02 

0.04 

0.72 

12 

0.03 

0.01 

0.04 

0.05 

1.00 

0.16 

0.04 

0.05 

1.25 

18 

n 

0.04 

0.01 

0.05 

0.06 

1.64 

0.20 

0.05 

0.06 

1.95 

24 

0.07 

0.01 

0.08 

0.08 

2.63 

0.30 

0.08 

0.08 

3.09 

30 

ii 

0.09 

0.02 

0.11 

0.08 

3.65 

0.40 

0.11 

0.08 

4.24 

1 

0.11 

0.02 

0.13 

0.08 

4.33 

0.40 

0.13 

0.08 

4.94 

36 

0.13 

0.02 

0.15 

0.08 

5.10 

0.52 

0.15 

0.08 

5.85 

IS 

0.16 

0.02 

0.18 

0.09 

6.17 

0.52 

0.18 

0.09 

6.96 

48 

IM 

0.22 

0.03 

0.25 

0.12 

8.68 

0.99 

0.25 

0.12 

10.04 

1>I 

0.26 

0.03 

0.29 

0.12 

10.10 

0.99 

0.29 

0.12 

11.50 

660 


WATER-PIPES. 


Art.  6.  Cast-iron  Pipe  Joints.  Philadelphia  standard.  The  clear 
distance,  d,  between  the  spigot  and  the  faucet,  is  nearly  uniform  for  all  sizes 
of  pipe,  varying  only  from  ^^  inch  for  4-inch 
pipe,  to  /s  inch  for  30-inch  pipe.  The  depth, 
m  w,  of  the  faucet  varies  from  3  ins  in  4-inch 
pipe,  to  4  ins  in  30-inch  pipe. 

The  small  beads  at  s  and  m,  s'  and  m'  on  the 
spigot  end  of  the  pipe,  project  about  34  i^^ch ; 
and  prevent  the  calking  material  from  entering 
the  pipe.  The  calking  consists  of  about  1  to  2 
ins  in  depth  of  well-rammed,  untarred  gasket, 
or  rope  yarn ;  above  which  is  poured  melted 
lead,  confined  from  spreading  by  means  of  clay 
plastered  around  the  joint.  The  lead  is  after- 
wards compacted  by  a  calking  hammer. 

The  lead  is  poured  through  a  hole  left  in  the 
clay  on  the  upper  side  of  the  pipe.  In  large 
.pipes,  two  additional  holes  are  left  in  the  clay, 
one  at  each  side  of  the  pipe,  and  lead  is  first 
poured  into  the  side  holes  by  two  men  at  once, 
one  man  pouring  into  each  side  hole  until  the 
oint  is  half  full.  The  side  holes  are  then 
stopped,  and,  after  the  lead  already  poured  has 
hardened,  the  two  men  finish  the  pouring  by 
means  of  the  top  hole.  This  course  is  necessary, 
because  the  great  weight  of  melted  lead  in  the 
entire  large  joint  would  press  away  the  clay  at  the  lower  side  of  the  joint,  and 
thus  escape. 

The  moisture  in  the  clay  is  liable  to  freeze  in  cold  weather,  and  to  render  it  too 
hard  to  be  used.  It  is  also  liable,  at  all  times,  as  is  also  any  dampness  in  the  pipe, 
to  be  converted  into  steam  by  the  heat  of  the  melted  lead.'  The  steam  sometimes 
breaks  out,  or  "blows  "  through  the  clay,  allowing  the  lead  to  escape. 

Art.  7.  The  Watkins  patent  "  Pipe  Jointer  "  avoids  these  difficulties  by 
dispensing  with  the  ring  of  clay.    It  consists  of  a  ring  R,  Figs  39  and  40,  of  square 


S^ 


Hm 


b- 


.SCALE  OF  INCHES. 
2      3^56 

Fig  38 


cross-section,  and  made  of  packing  composed  of  alternate  layers  of  hemp  cloth 
and  India  rubber.  This  ring  is  encircled  by  one  or  more  thin  strips  of  spring 
steel,  which  are  riveted  to  it  at  intervals,  as  shown.  E  E  are  iron-elbows  riveted 
outside  of  the  steel  bands.  After  the  gasket  has  been  rammed  into  its  place,  the 
ring  is  placed  around  the  spigot  near  the  faucet,  in  the  position  shown  in  Fig  40, 
and  is  held  loosely  by  the  clamp,  Fig  41,  one  point  of  which  enters  a  small  pit  in 
each  of  the  elbows,  E  E.  The  ring  is  then,  by  means  of  a  hammer,  driven  close 
up  against  the  end,/,  of  the  faucet,  Fig  38  ;  the  screw  of  the  clamp  is  tightened 
somewhat,  so  as  to  bring  the  ring  close  to  the  spigot;  a  small  dam  of  clay  is 
placed  in  front  of  the  aperture  between  the  two  elbows,  E  E;  and  the  joint  is 
ready  for  pouring.  After  the  lead  has  hardened,  the  "jointer"  is  removed,  and 
is  ready  for  use  at  another  joint.  Upon  its  removal  the  lead  is  found  smooth, 
requiring  no  chipping.  One  can  be  used  for  several  hundred  joints.  They  dis- 
pense with  the  services  of  the  men  who  prepare  the  clay  collars,  and  supply  them 
to  the  pourers.    Thos.  Watkins,  Johnstown,  Pa. 


WATER-PIPES. 


661 


Fig.  42. 


•  Art.  8.  As  a  further  preventive  against  the  escape  of  any  of  the  g-as- 
feet  into  the  pipe,  a  ring  of  lead  pipe  is  sometimes  placed  in  the  joint 
before  the  gasket  is  inserted.  This  lead  pipe  is  of  such  diameter  that  it  can  just 
be  pushed  through  the  space,  d,  Fig  38,  between  the  spigot  and  the  faucet ;  and 
of  such  length  as  just  to  encircle  the  water-pipe.  It  is  driven  as  closely  as  pos- 
sible into  the  narrow  annular  space  at  o  o,  Fig  38.  The  gasket  is  then  rammed 
in,  and  the  lead  poured,  as  usual. 
Art.  9.    In  John  F.  Ward's  flexible  joint.  Fig.  42,  for  cast-iron 

Eipes  laid  across  the  irregular  beds  of  streams,  a  portion,  a  o,  of  the  inside  of  the 
ell  B  is  accurately  turned  to  form  the  middle  zone  of  a  sphere,  with  center  at  C, 

and  the  narrow  surface  m  m,  on  the 
outside  of  the  spigot  S,  is  turned  to 
form  a  segment  of  a  sphere  accurately 
fitting  the  former.  The  lead  is  poured 
while  the  two  adjacent  lengths  of  pipe, 
resting  on  suitable  vessels  or  floats, 
are  in  a  straight  line,  or  nearly  so.  The 
lead  occupies  the  space  myi,  shown  black, 
and  is  held  in  place  on  the  spigot  by  the 
depression  d  d.  As  fast  as  the  joints  are 
thus  filled,  the  floats  are  moved  for- 
ward, and  the  pipes,  if  small,  are  passed 
into  shallow  water  without  further  care. 
Suitable  apparatus  is  used  for  lowering 
large  pipes  into  deep  water  without  un- 
due strain  on  the  joints.  The  joint  per- 
mits a  deflection  of  15°,  as  shown  ;  but 
further  deflection,  which  would  be  lia- 
ble to  split  the  bell,  is  prevented  by  the 
stops  at  0  0  on  the  bell  and  r  r  on  the 
spigot.  In  some  casespreliminary  dredg- 
ing may  be  expedient,  to  diminish  abrupt  irregularities  of  the  bottom.  Over 
thirty  lines  of  pipe  furnished  with  this  joint  have  been  successfully  laid,  of 
diameters  up  to  3  feet. 

Art.  10.  In  Figs  43,  A  is  a  double  branch;  which  is  a  pipe  having,  in 
addition  to  the  faucet,  c,  at  one  end,  two  others,  s  and  t,  to  which  pipes  leading  in 
opposite  directions  (as  at  cross-streets)  may  be  attached.  If  either  4*  or  i  be  omitted, 
the  pipe  is  a  single  branch.  The  pipe  is  stronger  when  these  extra  faucets 
are  near  its  end,  than  if  they  were  at  its  middle.  In  a  lougline  of  pipes,  for  the  sake 
of  expedition,  difierent  gangs  of  men  are  frequently  laying  detached  portions  some 
distance  apart ;  and  when  two  ends  of  difierent  portions  are  brought  near  enough 
together  to  be  united,  as  h  and  7%  Fig  C,  their  junction  cannot  be  eff"ected  by  the 
usual  spigot-and-faucet  joint.  In  this  case  a  cast-iron  sleeve,  1 1,  is  used,  which  is 
first  slid  upon  one  of  the  pieces  of  pipe  ;  and  (after  the  other  piece  also  is  laid)  is 
slid  back  into  the  position,  in  the  fig,  so  as  to  cover  the  joint.    Sleeves  are  usually 

about  a  foot  long ;  as  thick  as 
.  the  pipe ;  and  their  diam  is 
sufficient  to  allow  the  usual 
joint  of  gasket  and  lead. 
There  is  of  course  such  a  joint 
_  at  each  end  of  the  sleeve. 
Art.  11.  When  a 
crack  occurs  in  a  pipe, 
a  a.  Fig  B,  already  in  use,  it  is 
repaired  by  means  of  a  cast- 
iron  sleeve,  g  g,  made  in  two 
parts,  bolted  together  by 
means  of  flanges  as  at  n  n.  In 
other  respects  it  is  like  the 
preceding  sleeve.  The  inter- 
mediate white  ring  is  the  lead 
joint.  If  the  crack  is  too  long,  or  otherwise  too  bad  to  be  remedied  by  a  sleeve, 
the  pipe  is  broken  to  pieces  ;  and  the  lead  joints  at  its  ends  melted  out,  so  as  to 
allow  of  its  removal.  Then,  since  an  entire  new  pipe  cannot  now  be  inserted, 
owing  to  the  overlapping  of  the  spigot-and-faucet  ends,  two  short  pieces  must  be 
substituted  for  it.  One  end  of  each  of  these  is  lead-jointed  to  the  pipes  already 
laid ;  while  the  other  two  ends,  which  will  probably  be  a  few  inches  apart,  are 
covered  by  a  sleeve,  1 1,  Fig.  C. 


Figs, 


662 


WATER-PIPES. 


Cracks  may  at  times  be  temporarily  repaired  in  an  emergency,  by  a  wrapping 
of  folds  of  canvas  thoroughly  saturated  with  white-lead  paint;  and  tightly  con- 
fined to  tlie  pipe  by  a  spiral  banding  of  thin  hoop- 
iron  or  wire.  Or,  by  an  iron  band,  made  in  two 
parts,  B  B,  Fig  44,  and  clamped  together  by  screw- 
bolts,  S  Si  Such  bands  are  useful,  also,  for  strength- 
ening pipes  that  are  considered  to  be  in  danger  of 
bursting. 

Art.  13.  To  attach  a  pipe,  e,  Fi^  43, 
to  one,  /',  already  in  use,  but  in  which  no 
provision  has  been  made  for  such  attachment,  a 
piece  may  be  cut  out  of/,  as  at  v  v,  and  a  casting,  e, 
furnished  with  flanges,  m  m,  bolted  over  the  opening,  by  screw-bolts  passing 
through  female  screws  tapped  in  the  thickness  of  the  pipe.  If  the  new  pipe  is 
so  large  that  the  opening,  v  v,  if  circular,  would  be  inconveniently  wide,  it  may 
be  made  oval^  witn  the  longest  diameter  in  the  direction  of  the  length  of  the 
pipe,/.    In  that  case  the  casting  e  will  be  oval  at  its  flanges  ;  and  circular  at  c  c. 

Art.  13.  Air  valves.  Air  is  apt  to  collect  gradually  at  the  high  points 
of  vertical  curves  along  the  supply  pipes;  and,  unless  removed,  obstructs  the 
flow.     This  may  be  prevented  by  an  air  valve,  Fig  44A.     This  consists  of  a  cast- 


Fiff.  44A. 


iron  box,  c  cd  d,  confined  to  the  main  pipe  mm,  by  screw-bolts  passing  through 
its  flange  d  d.  It  has  a  cover  g  n  g,  confined  to  it  by  screws  1 1;  and  at  the  top 
of  which  is  an  opening  n,  for  the  escape  of  air  from  within.  In  this  box  is  a 
float/,  which  may  be  a  close  tin  or  copper  vessel,  or  of  layers  of  cork,  as  sup- 
posed in  the  fig ;  or  &c.  This  float  has  a  spindle  or  stem  s  s,  fast  to  it ;  which 
passes  through  openings  in  the  bridge-bars  a  a,  and  o;  thereby  allowing  the  float 
to  rise  and  fall  freely,  but  preventing  it  from  moving  sideways.  When  the  pipe 
m  m  is  empty,  the  float  is  down;  its  base  y  resting  on  the  cross-bar  a  a.  The 
stem  s  s  has  fixed  to  it  a  valve  v,  which  rises  and  falls  with  it  and  the  float. 
Suppose  the  pipe  m  m  to  be  empty,  and  consequently  the  float  and  the  valve  v 
down.  Then,  if  water  be  admitted  into  the  pipe,  it  will  rise  and  fill  also  the  box 
as  far  up  as  e ;  and  in  doing  so  will  lift  the  float/,  and  the  valve  v,  to  the  position 
in  the  fig  ;  thus  "preventing  egress  to  the  outer  air  by  closing  the  opening  at  < , 
Now,  air  carried  along  by  the  water,  will,  on  account  of  its  lightness,  ascend  to 
the  highest  points  it  meets  with. 


WATER-PIPEB.  663 


Hence,  when  such  air  arrives  under  the  opening  a o,  it  will  rise  thrt)ugh  it, 
and  ascend  to  e;  the  closed  valve  preventing  it  from  going  farther.  Thus 
successive  portions  of  air  ascend,  and  in  time  accumulate  to  such  an  extent 
as  gradually  to  force  much  of  the  water  downward  out  of  the  box.  When 
tkis  takes  place,  the  float,  which  is  held  up  only  by  the  \fater,  of  course  de- 
scends also ;  and  in  doing  so,  pulls  down  with  it  the  valve  v.  The  accumulated 
air  then  instantly  escapes  through  the  openings  at  v  and  n,  into  the  atmosphere ; 
andthe  water  in  the  pipe  wim,  immediately  ascends  again  into  the  box,  carry- 
ing with  it  the  float;  and  thus  again  closing  the  valve  v.  The  valve,  and  the 
valve-seat  e,  are  faced  with  brass,  to  avoid  rust,  and  consequent  bad  fit.  The 
whole  is  protected  by  an  iron  or  wooden  cover,  reaching  to  the  level  of  the  street. 

Air  Talves  are  no  long-er  used  in  city  pipes ;  their  place  being 
•upplied  by  the  fireplugs  at  average  distances  of  about  150  yards  apart.  These^ 
being  placed  as  much  as  possible  at  the  summits  of  undulations  in  the  lines  of 
pipes,  for  convenience  of  washing  the  streets,  and  being  frequently  opened 
for  that  purpose,  permit  also  the  escape  of  accumulated  air. 

The  escape  of  compressed  air  thron^ti  an  air  valve,  or 
•tlier  opening^,  has  been  known  to  produce  bursting-  of  the 
main  pipes;  for  the  escape  is  instantaneous,  and  permits  the  columns  of 
water  in  the  pipes  on  both  aides  of  the  valve,  to  rush  together  with  great 
forces,  which  arrest  each  other,  and  react  against  the  pipes. 

Air-Tessels.  Motion  is  imparted  to  the  water  in  a  line  of  pipes,  by  the 
forward  stroke  of  the  piston  of  a  single-acting  pump;  but  during  the  backward 
stroke,  this  motion  is  stopped ;  and  the  water  in  the  pipes  comes  to  rest.  There- 
fore, at  the  next  forward  stroke,  all  the  water  has  to  be  again  set  in  motion ; 
and  the  force  that  must  be  exerted  by  the  pump  to  do  this  is  much  greater  than 
would  be  required  if  the  motion  previously  imparted  had  been  maintained 
during  the  time  of  the  backstroke.  The  addition  of  an  air-vessel  secures  this 
maintenance  of  motion,  and  thus  eflfects  a  great  saving  of  power ;  besides  dimin- 
ishing  the  danger  of  bursting  the  pipes  at  each  forward  stroke.  It  is  merely  a 
tall  and  strong  air-tight  iron  box,  usually  cylindrical,  strongly  bolted  on  top 
of  the  pipes  just  beyond  the  pump,  and  communicating  freely  with  them 
through  an  opening  in  its  base.  It  is  full  of  air.  The  forward  stroke  of  the 
pistoD  then  forces  water  not  only  along  the  pipes,  but  also  into  the  lower  part 
of  the  air-vessel,  through  the  opeming  in  its  base;  thus  compressing  its  con- 
tained air.  But  during  the  backstroke,  this  compressed  air,  being  relieved  from 
the  pressure  of  the  pump,  expands ;  and  in  so  doing  presses  upon  the  water  in 
the  pipes,  and  thus  keeps  it  in  motion  until  the  next  forward  stroke  ;  and  so  on. 
Xn  air-vessel  also  acts  as  an  air-cushion;  permitting  the  piston  to  apply  its  force 
to  the  water  in  the  pipes  gradually:  thus  preserving  both  the  pipes  and  the 
pump  from  violent  shocks.  The  air  in  the  vessel,  however,  becomes  by  degrees 
absorbed  and  taken  away  bv  the  water;  and  its  action  as  a  regulator  then 
ceases.  To  prevent  this,  fresh  air  must  be  forced  into  the  vessel  from  time  to 
time  by  a  condenser,  or  forcing  air-pump.  A  double-acting  pump  does  not  so 
much  need  an  air-vessel.  There  is  no  particular  rule  for  the  size  or  capacity  of 
ftir-vessels.  In  practice  it  appears  to  vary  from  about  5  to  50  times  that  of  tht 
pump;  with  a  height  equal  to  two  or  more  times  the  diameter.  A  stand-pip« 
(see  below)  is  sometimes  used  instead  of  an  air-vessel. 

A  stand-pipe  is  sometimes  used  for  the  same  purpose  as  an  air-vessel  (see 
mbove).  It  is  a  tall  pipe,  open  to  the  air  at  top;  and  communicating  freely  at 
its  foot  with  the  water-pipe,  in  the  same  manner  as  in  an  air-vessel.  Its  top 
must  be  somewhat  higher  than  that  to  which  the  pump  has  to  force  the  water 
through  the  system  of  pipes;  otherwise  the  water  would  be  wasted  by  flowing 
over  its  top.  The  area  of  its  transverse  section  should  be  at  least  equal  to  that 
of  the  pipe  or  pipes  which  conduct  the  water  from  it ;  but  it  is  at  times  better 
to  have  it  much  larger,  as  a  stand-pipe  may  then  answer,  especially  in  a  small 
town,  ag  a  reservoir,  if  the  pumping  should  cease  for  a  few  hours.  A  stand-pipe 
should  be  cylindrical,  not  conical;  for  if  thick  ice  should  form  on  top  of  the 
water  in  a  conical  one,  a  suddeu  forcing  of  it  upward  by  the  pump  might  strain 
the  stand-pipe  seriously.  The  stand-pipes  connected  with  the  Philadelphia 
Water-Works  are  from  125  to  170  feet  high  •  5  feet  diameter ;  and  made  of  riveted 
boiler-iron  about  %  inch  thick  near  the  base,  and  about  i^  inch  near  the  top. 
They  have  no  protection  from  the  weather;  nor  are  they  braced  in  any  manner; 
but  retain  their  positions  by  their  own  inherent  strength,  although  sxposed  at 
times  to  violent  winds. 


664 


WATER-PIPES. 


Art.  14.    The  serTice-pipes  for  supplying;  sing^le  dwellings, 

are  of  lead ;  and  of  3^  to  %  inch  bore.  They  are  connected  with  the  street  maing. 
n  n,  Figv45,  by  a  brass  ferrule,/,  here  shown  at 
V^  real  size.  The  dotted  lines  show  its  i^  inch  bore. 
The  tapering  ferrule  is  merely  hard  driven  into  a  cyl- 
indrical hole  reamed  out  of  the  main,  as  at  s.  The  lead 
pipe,  o,  is  attached  to  the  other  end  of  the  ferrule; 
overlapping  it  about  1^  ins ;  and  the  joint  soldered,  t. 
The  extra  thickness  near/,  is  for  giving  proper  shape 
and  strength  for  hammering  the  ferrule  into  the  main. 
The  pipe  and  solder  are  shown  in  section.  Besides  the 
stopcocks  attached  to  each  service-pipe,  and  to  its 
branches  through  the  house,  there  is  an  underground  one  by  which  the  city  authori- 
ties can  stop  off  the  water  in  case  of  delinquency  in  payment  of  dues  ;  and  another 
by  which  the  plumber  can  stop  it  off  when  so  required  during  indoor  repairs.  Oalo 
'Vanizecl  iron  tubes  are  being  much  used  for  service-pipes,  especially  for  hot 
water;  being  less  subject  to  contraction  and  expansion,  which  produce  leaks. 


Fig.  45. 


Art.  15.  The  so-called  '"corporation  stops"  or  "corporation  cocks'* 
are  inserted  into  the  pipe  by  a  special  machine,  Fig  46.  Their  great  advantage  over 
the  ferrule,  Fig  45,  is  that  they  can  be  inserted  into  a  pipe  when  the 
latter  is  full  of  water  under  pressure.  Besides, 
inasmuch  as  they  are  screwed  into  the  pipe,  they  are  in  no  danger 
of  being  forced  out  of  it  by  any  pressure  within  it.  As  their 
name  implies,  they  are  furnished  with  a  stop-valve,  which  is  kept 
closed  while  the  valve  is  being  inserted  into  the  pipe,  and  is  then 
opened,  and  generally  remains  open  permanently. 

Pipe-tapping:  machines,  for  drilling  and  tapping  the 

pipe,  and  for  attaching  these  stops,  are  made  in  a  variety  of  forms. 

Fig  46  shows  one  made  by 

Walter   S.  Payne    A 

Co.,  Fostoria,  Ohio.     Each 

TTT       of  these  machines  is  furnished 

with  a  number  of  malleable 

caat-iron  saddles,  which  fit  the  various  diams  of  pipe 

with  which  it  is  to  be  used.    The  saddle  is  not  shown 

in  the  fig.    It  is  fastened  to  the  pipe  by  a  chain  slung 

r  around  the  latter.    The  chain  is  tightened  by  a  bolt 

and  nut  at  each  end. 

The  brass  cylinder,  C  C  (into  which  a  tap-and-drill, 
T,  and  the  stop,  S,  have  first  been  inserted),  is  then 
screwed  into  the  saddle  by  means  of  the  thread  at  A, 
The  stop  is  temporarily  screwed  on  to  a  mandrel,  M. 
This  mandrel,  and  the  drill-shank,  K,  pass  through 
stuffing-boxes  cast  in  one  with  the  head  of  the  cyl. 
By  means  of  a  handle,  not  shown  in  the  fig,  this  head 
is  now  revolved  (while  the  body  of  the  cyl  remains 
stationary)  so  as  to  bring  the  drill,  T,  and  stop,  S,  into 
the  respective  positions  shown  in  the  fig.  When  the 
cyl  head  comes  to  the  proper  position,  it  is  stopped  by 
a  lug  inside  of  the  cyl.  The  drill  is  then  immediately  over  the  center  of  a  large 
circular  opening  in  the  base  of  the  cyl,  C  C.  and  over  a  similar  opening,  through  the 
saddle,  to  the  surface  of  the  pipe  to  be  tapped.  It  is  then  pushed  down  until  it 
touches  said  pipe.  The  ratchet-wrench,  W  W,is  then  set  on  the  square  head  of  the 
drill-shank,  K;  the  feeder-yoke,  Y,  with  feed  screw,  F,  is  put  in  position  as  shown ; 
and  the  pipe  is  drilled  and  tapped  by  working  the  wrenqh ;  whereupon  the  water  in 
the  pipe,  if  under  pressure,  rushes  out  through  the  hole  thus  made,  and  fills  the 
cylinder. 

By  reversing  the  position  of  the  switch  on  the  ratchet  in  the  wrench,  W,  and  by 
^rorking  the  latter,  the  tap  is  now  withdrawn  from  the  hole,  but  remains  in  the  cyl. 
The  cyl  head  is  now  revolved  so  as  to  reverse  the  positions  of  S  and  T ;  the  lug  in- 
side of  the  cyl  stopping  the  head  when  the  stop  is  immediately  over  the  hole.  By 
means  of  the  ratchet-wrench,  applied  to  the  square  head  of  the  mandrel,  M,  the  stop, 
S  {the  valve  of  which  must  be  closed),  is  now  screwed  into  the  hole,  but  only  far  enough 
to  hold  securely,  and  thus  prevent  the  further  escape  of  the  water  from  the  pipe 
when  the  machine  is  now  removed.  The  stop  is  now  screwed  firmly  into  place  by 
means  of  a  wrench  applied  to  a  square  on  the  stop  itself.  When  the  pres  in  the  pipe 
exceeds  about  200  lbs  per  sq  inch,  the  feeding  apparatus,  as  used  with  the  drill  (se« 
fig),  may  be  also  used  in  aiding  the  insertion  of  the  stop. 


WATER-PIPES, 


665 


The  mandrel,  M,  is  made  in  two  lengths  (one  of  which  screws  into  the  other)  i« 
order  that  the  upper  part  may  be  out  of  the  way  of  the  wrench-handle  while  drill* 
iug.  It  has  three  or  more  difif  threads  at  its  foot,  to  suit  diSf  sizes  of  stop.  Stops, 
made  to  suit  the  machine,  are  furnished  as  wanted. 

The  machine  can  work  in  any  direction  radial  to  the  pipe,  and  can  therefore  be 
used  for  tapping  a  pipe  in  anj"^  part  of  its  circumference. 

After  the  stop  is  inserted,  the  service-pipe  is  attached  to  its  outer  end  by  a  coup- 
ling nut  passing  over  the  thread  there  shown. 

The  machines  are  guaranteed  to  tap  under  a  pressure  of  600  lbs  per  square  inch. 

Art.  15  a.  The  pneumaUc  dome  Figs.  46  a,  and  46  6,  invented  by 
Mr.  N.  Monroe  Hopkins,  of  Washington,  D.  C,  is  designed  to  prevent  the  burst- 
ing of  water-pipes  in  freezing  weather. 

In  unprotected  pipes,  the  water,  in  freezing,  is  unable  to  expand  longitudi- 
nally, and  therefore  frequently  bursts  the  pipe  in  expanding  lat- 
erally. The  domes,  being  placed  in  the  pipe,  as  shown,  at  intervals 
of  about  12  feet,  where  freezing  is  to  be  apprehended,  permit  the 
longitudinal  expansion,  which  pushes  the  ice  in  both  directions 
toward  each  dome,  wher^  it  compresses  the  air-cushion  there  pro- 
vided. In  the  horizontal  dome.  Fig.  46  a,  the  double  inclined  planes, 
cast  in  its  lower  side  at  c,  compel  the  two  horizontal  columns  of  ice 
to  rise  into  the  dome,  instead  of  merely  abutting  endwise  against 
each  other.  • 

In  order  to  insure  that  the  domes  in  a  system 
(such  as  those  for  a  house  or  mill  or  on  a  bridge) 
shall  not  be  deprived  of  their  air  by  the  flow 
of  the  water  in  the  pipes  below  them,  an  in- 
spirator is  placed  in  the  pipe  at  the  entrance 
to  the  system.  The  inspirator  consists  essen- 
tially of  a  constriction  in  the  pipe,  which  in- 
creases the  velocity  of  flow  at  that  point,  and  thus  causes  an  indraft  of  air 
through  a  valve  provided  for  the  purpose  (see  Venturi  meter,  page  532).  The 
air,  thus  introduced,  is  carried  along  the  pipe  in  bubbles  between  the  surface 
of  the  water  and  the  top  of  the  pipe,  and  is  entrapped  by  the  domes.  When, 
by  closing  faucets,  etc.,  the  flow  in  the  pipe  is  checked,  the  increase  of  pressure 
closes  the  valve. 

Severe  tests  of  both  large  and  small  pipes  (4-inch  and  ^-inch),  protected  by 
these  domes,  have  shown  them  to  be  always  effective  in  preventing  rupture. 


Fig.  46  a. 


Fig.  46  6. 


666 


.  STOP- VALVES. 


Art,  16.  Stop-Talves,  or  g>ates,  opening  vertically  In  grooves,  are  placed 
across  the  street  pipes  at  intervals  of  from  100  to  300  yds.  Their  use  is  to  shut  oflf 
the  water  from  any  section  during  repairs ;  the  water  of  such  sections  being  allowed 
to  run  to  waste,  and  to  soak  into  the  ground. 

The  details  are  much  varied  by  difif  makers. 


Fia.  47. 


Fig.  48. 


Figs  47  and  48  show  such  a  gate  made  by  Chapman  Valve  Mfg  Co,  Indian 
Orchard,  Mass.  The  valve,  v,  is  cast  in  one  piece.  When  down,  as  in  the  figs,  it 
closes  the  pipe.  As  in  other  styles,  it  opens  vert  by  means  of  a  screw,  D,  the  valve 
rising  into  the  cast-iron  case  or  box,  B  B,  and  leaving,  when  all  the  way  up,  an 
opening  of  the  full  diam  of  the  pipe.  The  screw  is  turned  by  a  wrench  fitting  on 
its  square  head,  h.  The  screw,  D,  itself,  is  prevented  from  moving  vert  bj  the  col* 
lar,  C. 

The  two  principal  castings  which  compose  the  box  or  cover  are  bolted  together  by 
means  of  flanges, g.  The  joint  faces  of  the  castings  are  carefully  smoothed;  and  a 
thin  strip  of  lead  is  inserted  between  them,  as  a  precaution  against  leaks.  The  recess, 
B,  admits  small  particles  of  foreign  matter  which  might  otherwise  prevent  the  gate 
from  closing  perfectly.  The  valve  seats  are  faced  with  Babbitt  metal.  At  the  top 
of  the  cover,  the  screw  stem  passes  through  a  stufiing-box.  which  prevents  leaking 
at  that  point.     Very  careful  workmanship  is  required  throughout. 


STOP-VALYES. 


667 


Chapman  Bell-end  Water«g;ates. 


Bore. 

Wt. 

Bore. 

Wt. 

Bore. 

Wt. 

Bore. 

Wt. 

ins. 

lbs. 

ins. 

lbs. 

ins. 

lbs. 

ins. 

Tbs. 

2 

32 

6 

195 

12 

600 

20 

1700 

3 

55 

7 

245 

14 

843 

24 

2750 

4 

116 

8 

290 

16 

1080 

30 

6400 

5 

loO 

10 

439 

18 

1475 

36 

8300 

Art.  17.  Fig  49  shows  an  arrangement  i¥itli  outside  screw  for  raising 
and  lowering  tlie  valve.  Here  the  screw,  D,  does  not  revolve,  but  is  attached  to  the 
valve,  and  rises  and  falls  with  it,  being  raised  or  low- 
ered by  turning  the  wheel,  W,  at  the  center  of  which  is 
a  nut  through  whick  the  screw  passes.  The  nut  is  fixed 
in  the  wheel,  and  is  so  confined  that  it,  and  the  wheel, 
cannot  move  vertically. 

Art.  18.  A  four-way  stop,  or  four-way 
valve.  Figs  50  and  51.  is  placed  at  the  intersection  of 
two  mains ;  the  four  ends  of  which  are  attached,  respec- 
tively, to  the  four  openings,  M  M  M  M.  At  the  bottom 
is  an  additional  opening,  connecting,  by  means  of  an 
elbow,  H.  with  pipes  running  to  a  fire-hydrant  at  the 
street  curb.  See  Arts  20  and  21.  Two  or  more  of  such 
bottom  openings  may  be  made,  if  desired,  for  the  sup- 
ply of  as  many  fire-hydrants.  All  of  the  openings  are 
opened  or  closed  at  one  time  by  raising  or  lowering 
the  valve  or  plug,  P,  by  means  of  a  wrench  or  key  ap- 
plied to  the  square  head,  S,  of  the  screw  stem.  As  in 
Figs  47  and  48,  the  screw  turns,  but  is  prevented  from 
rising  and  falling,  and  the  plug  moves  up  and  down  on 
the  screw. 

Inasmuch  as  all  sediment  escapes  into  the  bottom 
opening  which  leads  to  the  fire-hydrant,  the  valve  is  not 
liable  to  clogging  through  this  cause.     The  fire-hy- 


Fig.  49 


Fig.  50 


:H'ie.  51 


(irant,  benig  fed  from  both  of  the  mains,  obtains  a  fuller  supply  than  would  be  possi' 
hie  if  it  were  fed,  as  usual,  through  only  one  main. 


668 


STOP-VALVES. 


Art.  19.  Whatever  the  «tyle  of  the  gate  may  be,  it  is,  when  attached  to  the  pipe, 
protected  by  a  surrounding^  box,  generally  of  plank  or  cast-iron,  with 
four  sides,  which  taper  so  that  the  box  is  of  smaller  hor  section  at  top  than  at  bottom. 
It  is  open  at  bottom,  but  has  a  movable  iron  top,  level  with  the  street.  This  top  is 
taken  off"  when  the  valve  is  to  be  opened  or  closed,  or  inspected.  Two  of  the  oppo- 
site sides  of  the  box  of  course  have  openings  for  tlte  passage  of  the  pipes  to  or  from 
the  valve. 

The  gates,  especially  of  large  mains,  must  be  closed  very  slowly.  Otherwise,  the 
too  sudden  arresting  of  the  momentum  of  the  flowing  water  would  be  apt  to  break 
either  them  or  the  covers;  or  burst  the  pipes.  As  a  precaution  against  this,  the 
covers  for  very  large  valves  are  cast  with  outside  strengthening  ribs. 

No  self-acting  air-valves  (Pig  44  A)  are  now  placed  at  street  summits,  to  allow 
confined  air  to  escape.  The  fire-plugs  answer  instead.  The  rad  for  hor  bends  in 
mains  is  if  possible  not  less  than  about  12  times  their  diams;  they  are  made  as  large 
as  the  widths  of  the  streets  will  admit ;  usually  about  50  ft.  Fire-plug^il^  Figs 
52,  &c,  are  placed  as  much  as  possible  at  summits,  so  as  to  serve  also  for  w**«liiug 
the  streets ;  and  for  the  escape  of  accumulated  air.  They  average  about  8  iia  iwv»- 
ber  to  each  mile  of  pipe ;  or  1  to  each  block  of  building*. 

confined  air  to  escape.  The  fire-plugs  answer  instead.  The  rad  for  hor  bends  in 
mains  is  if  possible  not  legs  than  about  12  times  their  diams  ;  they  are  made  as  large 
as  the  widths  of  the  streets  will  admit ;  usually  about  50  ft.  FJre-plng'S.,  Figs 
52,  &c,  are  placed  as  much  as  possible  at  summits,  so  as  to  serve  also  for  washing 
the  streets  ;  and  for  the  escape  of  accumulated  air.  They  average  about  8  in  num- 
ber to  each  mile  of  pipe ;  or  1  to  each  block  of  buildings. 


Fire  Hydrants. 


IF'iiS.Sa 


irifi.54r 


FIRE-HYDRANTS.  669 

Art.  20.  Fig,52  represents  a  common  street  fire-plug^,  or  fire-hydrant* 


The  valve,  v,  is  of  layers  of  well- 
hammered  sole-leather ;  and,  when 
closed,  shuts  against  a  brass  ring 
seat,  0,  which  is  confined  to  its  place 
by  a  lead  joint.  The  valve  is  opened 
by  working  down  the  screw,  «, 
which,  by  means  of  a  swivel  joint 
at  a,  can  revolve  without  turning 
the  valve-rod,  y.  When  the  valve, 
V,  is  closed,  after  the  plug  has  been 
in  use,  the  chamber,  c,  is  full  of 
water,  which,  if  allowed  to  remain, 
would  be  in  danger  of  freezing, 
and  of  bursting  the  plug.  But,  in 
closing  the  valve,  we  raise  the 
flange,  l,on  the  rod, and  thus  allow 
the  water  to  escape  through  the 
opening  at  I,  whence  it  runs  to 
waste  into  the  ground,  through  the 
open  lower  end  of  the  frost- 
jacket,  jj,  which  is  a  hollow 
cast-iron  cylinder  surrounding  the 
working  parts  of  the  hydrant. 
Being  free  to  slide  vert  it  rises  and 
falls  when  the  level  of  the  ground 
is  disturbed  by  frost,  and  the  hy- 
drant is  thus  protected  against  in- 
jury from  this  cause. 

The  top,  t,  of  the  hydrant  case,  is 
cast  in  one  piece  with  the  cham- 
ber, c. 

The  stopper,  e,  screws  on  over 
the  nozzle,  n. 

Art.  21.  In  the  Cliapmaii 
fire-liydrant.  Figs  63,  54,  and 
55,  made  by  the  Chapman  Valve  Mfg 
Co, 'Indian  Orchard,  Mass,  the 
valve,  V  V,  is  a  sliding  one. 
The  stem,  y,  Fig  53,  to  which  the 
screw,  s,  is  attached,  is,  like  that  in 
Figs  47  and  48,  prevented  from  mov- 
ing vert  by  a  collar,  fast  to  it  near 
its  top,  and  confined  in  a  circular 
groove.  When  the  rod  and  screw 
arft  made  to  revolve,  by  means  of  a 
wrench  applied  to  the  square  head 
of  the  former,  the  valve  slides  up 
or  down  on  the  screw,  admitting 
the  water  to,  or  shutting  it  ofi"  from, 
the  hydrant.  The  valve  slides  on 
the  two  guides,  g  g,  which  are  cast  in  one  piece,  respectively,  with  the  two'vert  sides 
of  the  lower  part  of  the  hydrant.  Its  circular  face,  where  it  comes  into  contact 
with  the  hydrant  case,  h,  is  faced  with  a  gun-metal  ring,  e  «,  which  bears  against  a 
iimilar  ring,  made  of  Babbitt  metal,  let  into  the  hydrant  case. 

The  irater,  left  in  the  hydrant  case  after  closing  the  valve,  escapes  through 
a  cylindrical  hole,  d,  about  %  inch  diam,  bored  through  the  guide,  g,  and  case,  h. 
This  hole  is  at  such  a  ht  as  to  be  just  above  the  top  of  the  loose  plate,  j),  when  the 
valve  is  closed,  as  in  Fig  53.  This  plate  lies  in  avert  groove  in  the  side  of  the  valve- 
casting,  and  is  pressed  against  the  side  of  the  case,  h,  by  two  spiral  springs  confined 
in  cavities,  c  c,  in  the  valve  casting.  When  the  valve  begins  to  rise,  the  hole,  d,  ia 
ekwed  by  the  plate,  p,  and  remains  so  until  the  valve  is  again  entirely  closed. 


670  TEST   BORINGS. 

Test  Borings.    Fic;s  1  and  2  show  a  tool  for  boring   into  soils, 

clay,  sand^  or  gravel,  even  when  quite  indurated,  or  when  frozen.      It  will 
not  bore  through  hard  rock,  or  through  large  boulders.     It 
consists  of  two  sheet-iron  cylindrical  segments  S  S,  called 
"  pods,"  having  their  lower  or  cutting  edges  shod  with  steel. 
These  edges  project  (as  shown  in  Fig  1)  be3'ond  the  sides  of 
the  auger,  and  thus  make  the  hole  larger  than  it,  so  that  it 
cannot  bind  or  stick.   The  two  cutting  edges  are  equidistant 
,  from  the  vert  cen  line  of  the  tool,  and  this  insures  a  straight 
^  and  vert  hole.     At  a  the  auger  is  attached  to  the  lower  end 
of  a  vert  boring  rod  composed  of  a  number  of  l^/^-inch  squar* 
iron  bars,  or  23^-inch  iron  tubes,  about  10  to  15  ft   long, 
jointed  together  at  their  ends  by  means  of  square  socket* 
joints.    At  the  top  of  this  boring-rod  is  a  swivel-hook,  by 
means  of  which  the  entire  apparatus  is  hung  to  the  end  of  a 
Fig  1.  Fio  2.      rope,  which  passes  over  a  pulley  at  the  top  of  a  derrick  or 

tripod,  and  down  to  a  drum  worked  by  a  windlass  and  gear- 
ing. By  means  of  this  drum  and  rope,  the  auger  and  boring-rod  (which  at  first  con« 
sists  of  only  one  bar)  are  lifted,  and  suspended  over  the  intended  hole.  The  auger 
is  then  lowered,  and  rotated  hor  by  two  men  or  one  horse,  working  at  the  ends  of 
levers  which  grip  the  boring-rod  a  few  ft  above  the  ground.  The  swivel  at  the  top 
of  the  boring-rod  permits  this  rotation  to  take  place  without  twisting  the  rope. 
The  shape  of  the  auger  is  such  that  its  rotation  feeds  or  screws  it  into  tlie  ground; 
and  the  man  at  the  windlass  has,  during  the  boring,  merely  to  keep  the  rope  tight, 
80  as  to  prevent  the  auger  from  boring  too  fast,  and  becoming  clogged.  In  about  S 
revolutions  the  auger  fills  with  earth.  By  means  of  the  windlass  it  is  then  raisetj 
to  about  2  ft  above  the  ground ;  and  by  unkeying  and  removing  the  band  b  the  auger 
is  opened  like  a  pair  of  tongs,  and  the  earth  emptied  into  a  wooden  box  which  has 
in  the  meantime  been  placed  over  the  hole.  The  box  is  then  removed  and  emptied, 
and  the  boring  proceeds  as  before.  When  the  boring  has  reached  a  depth  of  about 
10  ft,  a  second  bar  must  be  added  to  the  top  of  the  rod.  For  this  purpose  the  rod 
and  auger  are  raised  a  few  inches;  a  slight  frame-work  of  boards  is  placed  on  tho 
ground,  close  to  the  boring-rod  and  surrounding  it;  and  a  flange  is  clasped  tightly 
to  the  rod  just  above,  and  close  to,  the  framework.  The  framework  and  flange  now 
support  the  rod  and  auger;  the  swivel-hook  and  rope  are  removed,  and  attached  to 
the  upper  end  of  the  second  bar,  which  is  then  raised,  and  its  lower  end  is  fastened 
into  the  socket-joint  upon  the  top  of  the  first  one.  The  rope  is  then  drawn  tight; 
the  flange  removed;  the  auger  lowered  to  the  bottom  of  the  hole;  and  the  boring 
resumed.  Additional  lengths  of  boring-rod  are  attached  in  the  same  way  from  time 
to  time,  as  required  by  the  descent  of  the  auger. 

The  borers  may  be  made  from  6  to  18  inches  in  diameter,  or  larger.  If  desired, 
the  boring  may  be  made  from  24  to  36  ins  diani  by  attaching  a  reamer  to  the 
auger.  This  auger  will  bore  to  a  depth  of  100  ft  or  more  at  the  rate  of  from  5  to 
20  ft  per  hour.  It  removes  stones  as  large  as  half  the  diam  of  the  hole.  In  dry 
soils  a  bucketful  of  water  is  poured  into  the  hole  each  time  the  auger  is  raised. 

This  borer  may  be  advantageously  used  in  boring  the  holes  for  sand  piles, 
and  at  times,  instead  of  driving-  \¥Ooden  piles,  it  may  be  better  to 
plant  them  (butt  down  if  preferred)  in  holes  bored  by  this  auger  ;  ramming  the 
earth  well  around  them  afterwards.  This  will  save  adjacent  buildings  from  the 
jarring  and  injury  done  by  a  pile  driver. 

If  sand,  mud,  or  loose  gravel  is  readied  in  boring  with  this  tool, 
the  hole  is  reamed  out  4  ins  larger,  and  a  tubing  of  inch  boards  is  inserted 
into  the  hole,  and  driven  into  and  through  the  sand  or  gravel,  which  is  then 
removed  from  within  the  tubing  by  means  of  a  san<l-punip,  consisting  of  a 
hollow  iron  cylinder,  about  5  ins  diam  X  30  ins  long,  with  a  valve  at  its  foot, 
opening  upward.  The  sand-pump  is  lowered  to  the  bottom  of  the  hole  ;  covered 
with  water  to  a  depth  of  2  to  4  ft,  and  churned  quickly  up  and  down  4  to  6  ins, 
by  hand,  20  or  30  times,  during  which  the  sand  fills  the  pump,  which  is  then 
drawn  up  and  emptied.  From  10  to  20  ft  in  depth  of  sand,  mud,  &c,  per  hour 
can  thus  be  taken  from  a  6  to  18-inch  hole.  This  pump  is  also  used  for  removing 
broken  earth,  &c,  from  a  hole  bored  in  compact  earth  by  the  borer  first  described. 
The  cost,  with  derrick,  boring-rods,  rope,  sand-pump,  &c,  &c,  complete,  is 
about  $175.  Tbe  auger  Tveigbs  from  150  to  200  lbs,  according  to  size. 
Boring-rod  1%  ins  sq,  3%  ft>s  per  ft.     Derrick,  150  fibs. 

The  sand-borer.  Figs  3  and  4,  like  the  sand-pump  just  described,  is  used 
inside  of  tubing,  and  for  the  same  purpose.  The  hollow  iron  cylinder  C,  10  ins 
diam  X  30  ins  long,  slides  vertically  on  the  rod,  but  the  screw  is  fast  to  the  rod. 
While  boring,  the  sand  below  and  around  the  cyl  keeps  it  in  the  position  shown  in 


ARTESIAN   WELL   BORING. 


671 


fig  S.    Six  revolutions  of  the  rod  and  screw  fill  the  cyl  with  sand.    The  rod  is  then 
liftpi.    This  first  draws  the  screw  up  into  the  cyl,  as  in  Fig  4;  and  a  valve  at  tht 
foot  of  the  screw  closes  the  bottom  of  the  cyl,  and  prevents  the  sand 
from  falling  out  when  the  borer  is  lifted  from  the  hole.    The  rod  is 
kollow,  and  open  at  top  and  bottom.     This  allows  passage  of  the  air, 
»nd  thus  prevents  resistance  from  suction  in  withdrawing  the  borer. 
This  tool  is  rotated  and  withdrawn  in  the  same  way  as  the  earth  borer  ^.l 
first  described.     Cost  about  ^30. 

Steel  prospecting-  aty^ers,  from  2  to  4  ins  diam,  and  2  ft 
long,  are  used  for  boring  holes  from  23>^  to  6  ins  diam,  and  to  depths 
of  10  to  50  ft,  into  clay,  sand,  or  fine  gravel,  of  all  of  which 
they  bring  up  samples.  They  are  turned  by  wrenches,  and  by  man 
or  horse  power.    See  also  p  674. 


The  boring^  tool  shown  in  vert  section  by  Fig  6,     rias  3, 4 

and  in  hor  cross  section  by  Fig  5,  is  very  useful  for  boring  shal- 
low holes  by  hand  throngh  surface  soils,  clay,  and  gravel,  and 
bringing  up  samples.  The  borer  proper  consists  of  a  cylinder  of 
spring  steel,  3  or  4  ins  diam,  and  4  or  5  ins  high,  with  sides  J<^ 
inch  thick,  having  a  vert  slit  (see  cross  section)  throughout  its 
height,  and  beveled  to  a  cutting  edge  all  around  its  foot,  as 
shown  in  the  vert  section.  At  its  top  it  is  riveted,  as  shown,  or 
welded,  to  the  inverted-v-shaped  forging,  which,  by  means  of  the 
socket  at  its  top,  is  sci^wed  to  a  length  of  gas-pipe  which  serves 
as  a  handle,  and  to  which  other  pieces  are  joined  by  sockets  as 
toring  proceeds. 

The  boring  is  done  by  two  men,  who  grasp  the  handle,  and, 
holding  the  tool  vert,  drive  it  into  the  ground  by  repeatedly 
lifting  it  and  forcibly  bringing  it  down  upon  the  same  spot.  As 
the  tool  strikes  the  ground,  the  beveled  shape  of  its  cutting  edge 
causes  it  to  open  slightly,  and  when  the  downward  pres.  is  re- 
lieved in  lifting  it,  it  springs  back  and  grasps  the  earth  which 
h^  entered  it.  It  soon  fills,  and  the  men,  finding  that  it  ceases 
to  penetrate  readily,  lift  it  to  the  surface  and  empty  it.  The 
character  of  its  contents  from  different  depths,  measured  along 
the  handle,  is  noted  from  time  to  time. 

In  six  days  of  8  hours  each,  three  men  (one 
resting  at  intervals)  using  one  such  auger 
between  them,  bored  20  hoi es^  averaging  9}^ 
ft  each,  m  loam,  gravel,  claj',  and  decom- 
posed mica  schist,  at  a  cost  of  22  cts  per  foot.  ^| 
Wages  of  each  man,  $2  per  day. 

For  work  in  loam,  clay,  or  non-running 
•and,  an  effective  screw-anger  can 
be  made  by  any  good  blacksmith,  by  merely 
forming  a  one-inch  sq  bar  of  iron  or  steel  -piQ  5^ 

Into  corkscrew  shape  about  2  ft  long,  with 
6  complete  turns  6  ins  in  diam  ;  its  lower  end  sharpened  to  form  a  vertical  cutting 
edge,  which  should  project  say  .5  of  an  inch  beyond  the  spiral  of  the  screw,  in  ordef 
to  diminish  friction.  It  will  bring  up  full  samples.  Requires  a  derrick,  or  some 
other  simple  mode  of  lifting,  when  the  screw  is  full. 

Artesian  Well  Drilling*.  Deep  vert  holes  in  earth  and  rock,  6  and  8  ins 
in  diam,  such  as  are  reqd  for  artesian  wells  for  water  and  oil,  an^  for  mining  explora- 
tions, are  drilled  by  repeatedly  lifting  and  dropping,  in  the  same  vert  line,  a  heavy 
iron  bit,  Figl,p  673,  with  a  steel  cutting-edge.  The  bit  is  partly  revolved  horizon- 
tally after  each  blow,  to  insure  roundness  of  hole.  The  length  of  the  cutting-edge 
of  the  bit  is  a  little  greater  than  the  diam  of  the  bit,  and  the  hole  is  thus  made  suf- 
ficiently large  to  prevent  the  bit  from  binding  in  it. 

The  bit  is  the  lowest  one  of  a  series  of  iron  and  steel  bars,  &c,  screwed 

together  at  their  ends,  and  called  a'*  string  of  tools."  The  string  of  tools 
varies  in  length  from  25  to  60  ft,  according  to  the  size  and  depth  of  the  hole,  and  the 
hardness  of  the  rock;  and  its  diam  throughout  (above  the  cutting-edge)  is  an  inch 
or  two  less  than  that  of  the  hole.  Its  weight  is  from  80O  to  4000  lbs.  Its  upper 
member  is  always  a  "rope-socket,"  Fig  4  {without  a  swivel),  to  which  the  lower  end 
of  the  supporting  rope  cable  is  attached.  This  cable  passes  up  out  of  the  hole  to 
a  hor  lever,  which,  by  means  of  a  horse-power  or  steam-engine,  is  k?pt  con- 
stantlymoving  up  and  down  with  a  see-saw  motion.  The  string  of  tools,  with  th« 
cuttiag^edge  of  the  bit  at  its  lower  end,  is  thus  alternately  lifted  from  2  to  4  ft,  and 


F106. 


672  ARTESIAN   WELL   BORING. 

let  fall,  from  30  to  50  times  per  minute,  and  so  drills  its  way  into  the  rock  or  eartk, 
Prom  4  to  10  ft  in  depth  of  water  are  kept  in  the  hole,  to  facilitate  the  drilling  an4 
the  removal  of  debris.  After  water  is  reached,  the  drilling  may  be  continued,  eve« 
if  the  hole  is  full  of  water;  but  a  great  depth  of  water  of  course  diminishes  the  forc« 
of  the  blows  of  the  bit.  A  suitable  arrangement  must  be  provided  for  paying; 
ont  the  rope  as  the  boring  tool  descends.  A  clamp  is  attached  to  the  cable; 
and  the  man  in  charge,  by  turning  the  clamp,  twists  the  rope,  and  thus  turns 
the  bit  horizontally  about  one-fifth  of  a  revolution  after  each  stroke,  until 
six  or  eight  complete  revolutions  have  been  made  in  one  direction.  He  then  re- 
Terses  the  motion,  and  makes  an  equal  number  of  \urns,  at  the  same  rate,  in  the 
opposite  direction. 

After  drilling  a  few  feet,  the  string  of  tools  is  lifted  out  of  the  hole  by  means  of 
the  cable,  to  allow  the  removal  of  the  debris  which  has  accumulated  in  the 
hole.  This  is  done  by  means  of  a  sand-punip,  which  is  a  sheet-iron  cylinder, 
say  4  ins  diam,  and  4  to  6  ft  long,  provided,  at  its  foot,  with  a  valve  opening  upward. 
The  pump  is  lowered  to  the  bottom  of  the  hole,  and  filled  with  the  mixed  water  and 
debris  by  churning  it  up  and  down  a  number  of  times.  Sometimes,  in  addition  to 
the  valve,  the  pump  is  fitted  with  a  plunger,  which  is  at  the  foot  of  the  pump  when 
the  latter  is  let  down  to  the  bottom  of  the  hole.  The  plunger  is  then  drawn  up  into 
the  pump,  and  the  debris  follows  it.  In  either  case,  the  pump,  when  filled,  is  lifted 
out  of  the  hole  and  emptied;  the  string  of  tools  is  again  lowered  into  the  hole,  and 
the  drilling  resumed.  The  debris  must  be  removed  after  every  3  to  5  ft  of  drilling. 
Otherwise  it  would  interfere  too  greatly  with  the  action  of  the  bit. 

Wells  are  usually  drilled  from  6  to  8  ins  diam.  For  diams  less 
than  6  ins,  the  tools  are  so  slender  that  they  are  liable  to  be  broken  in  a  deep  hole. 

The  same  apparatus  is  used  for  drilling  through  the  earth  above 
the  rock,  before  the  latter  is  reached.  This  is  called  "spudding."  In  this  case 
the  sides  of  the  hole  must  be  prevented  from  caving  in.  For  this  purpose  a  wrought- 
iron  pipe  of  such  diam  as  to  fit  the  hole  closely,  and  %  inch  thick,  is  inserted  into 
the  hole,  and  is  driven  down  from  time  to  time  as  the  drilling  proceeds.  The  pipe 
is  driven  by  means  of  a  heavy  maul  of  oak,  or  other  hard  wood,  14  to  18  ins  square, 
and  10  to  16  ft  long.  This  maul  is  attached,  by  one  end,  to  the  lower  end  of  the 
same  cable  which,  during  drilling,  supports  the  string  of  tools.  It  is  thus  repeat- 
edly lifted,  ar.:*  dropped  upon  the  head  of  the  tube,  which  is  protected  by  a  cast-iron 
•'  driving-cap."  The  foot  of  the  tube  is  shod  with  a  steel  cutting-edge  ring,  or  "  steel 
shoe."  When  the  tube  has  been  driven  as  far  as  it  will  readily  go,  the  maul  is  re- 
moved from  the  end  of  the  rope;  the  string  of  tools  substituted;  and  the  drilling 
resumed  within  the  pipe. 

The  pipe  is  put  together  in  lengths  of  from  8  to  18  ft,  and  the  drilling  and  pipe- 
driving  proceed  alternately  until  the  rock  is  reached,  and  the  foot  of  the  pipe  forced 
into  it  to  a  depth  of  a  few  ins,  or  far  enough  to  shut  off  quicksand  or  surface  water. 

If  quieksand  is  encountered,  the  string  of  tools  is  removed,  and  the 
•and-pump  is  used  inside  of  the  pipe. 

For  reaming:  out,  or  enlarging:,  holes,  or  for  straig^htening 
crooked  ones,  <&c,  special  tools,  such  as  reamers,  &c,  are  substituted  in  place  of  the 
boring  bit. 

Special  care  must  be  taken  to  have  all  the  rubbing  surfaces  thor- 
OUg'hly  lubricated.  The  pulley  in  the  mast-head,  and  the  pinion-wheels 
of  the  horse  power  (if  such  be  used)  should  be  well  oiled  every  two  or  three  hours. 

In  very  cold  or  "wet  iRreather,  a  shed  of  roug'h  boards,  or  a  cover- 
ing of  canvas,  about  8  ft  high,  should  be  erected,  to  protect  the  men ;  and,  if  steam 
is  used,  2  or  3  boards  should  be  used  as  a  covering  for  the  belt,  which  will  slip  if  wet. 

The  following  description  is  based  upon  tlie  machines  made  by  the  Pierce 
Well  Engineering  wid  Supply  Co.* 

For  holes  from  200  to  1000  ft  deep,  portable  drilling:  ma- 
chines,! worked  by  horse  or  steam  power,  are  used.  In  these  machines,  the 
drill-rope,  extending  from  the  string  of  tools  up  out  of  the  hole,  passes  over  a  sheave 
at  the  top  of  a  wooden  mast;  down  to,  and  around,  a  pulley  fast  to  the  working 
lever;  and  thence,  by  way  of  a  pulley  fixed  at  the  foot  of  the  mast,  to  a  drum  upon 
which  it  is  wound.  To  this  drum  a  friction  and  ratchet  wheel  is  attached,  for  pay- 
ing out  the  cable  as  the  tools  descend. 

The  mast  is  hing^ed  six  feet  above  its  foot,  so  that  its  upper  part  may  be 
Jaid  hor  when  the  machine  is  to  be  moved.  When  at  work,  it  is  held  in  position  by 
two  timber  struts  or  braces,  bolted  to  it  near  its  top,  and  having  their  lower  ends 
fastened  to  the  "drill-jack,*'  which  is  a  light  and  strong  framework,  9  ft  long, 
3  ft  wide,  and  4  ft  high,  at  the  foot  of  the  mast,  containing  the  working  lever  which 

*See  Business  Directory,  No.  484.  fSee  Price-list,  3.67. 


ARTESIAN   WELL   BORING. 


673 


Fig.  4. 


Uj 


raises  the  rope  and  lets  It  fall,  the  drum  ou  which  the  rope  is  wound,  the  shaft  and 

cam  which  woik  the  lever,  &c.    The  operator  stands  at  the  foot  of  the  mast,  andL 

by  means  of  foot-  and  hand-levers  within  his  reach,  regulates 

all  the  movements  of  the  machine.     One  of  these  governs 

the  pawl  and  ratchet  wheel  regulating  the  paying  out  of  the 

cable.    By  letting  the  ratchet-wheel  of  the  drum  move  one 

notch,  the  bit  is  fed  down  quarter  of  an  inch. 

The  operator,  by  moving  a  slide  with  his  foot,  holds  the 
working  lever  down,  out  of  reach  of  the  cam,  thus  stopping 
the  up-and-down  motion  of  the  rope  and  tools.  By  means 
of  another  lever  he  can  now  put  the  rope-drum  in  gear  with 
the  main  driving-shaft,  so  that  the  rope  is  wound  up  on  the 
drum,  and  the  tools  drawn  up  out  of  the  hole.  Another 
lever  controls  the  separate  reel  on  which  the  light  rope,  car- 
rying the  sand-pump,  is  wound.  All  these  operations  are 
performed  by  the  same  power  (horse  or  steam,  as  the  case 
may  be),  which  works  on  without  stopping;  the  various 
changes  being  made  by  merely  throwing  the  different  pai'ts 
into,  or  out  of,  gear  with  the  main  driving-shaft. 

One  of  these  portable  machines  requires 
two  horses  or  a  small  steam-engine,  a  man  to  attend  the 
Bame,  and  another  man  to  operate  the  machine,  empty  the 
sand-pump,  change  the  tools,  &c.  It  can  be  transported  on 
a  farm  wagon  over  any  common  road.  Two  men  can  unload 
it,  set  it  up,  and  commence  drilling,  in  two  hours;  and,  un- 
less steam  is  preferred,  the  two  horses  used  for  its  transpor- 
tation furnish  the  motive  power.  The  machine  can  be  taken 
down  and  reloaded  in  the  wagon  in  two  hours. 

Figs  1  to  4  show  the  tools  used  with  these  ma- 
chines. For  the  different  sizes  of  machine  they  differ 
chiefly  in  their  dimensions  and  weights. 

Fig  1  shows  the  drilling  bit,  which  is  30  to  36  ins 
long,  and  weighs  about  100  lbs.  Its  lower  or  cutting 
edge  is  6  ins  long.  Its  top  is  screwed  into  the  foot 
of  the  "  aug-er-stem,''  Fig  2,  which  is  of  3-inch 
round  iron,  12  ft  long,  and  weighs  350  lbs.  Its  use  is  that 
of  a  weight,  giving  additional  force  to  the  blows  of  the  bit. 
Its  top  is  screwed  into  the  foot  of  the  '*  drill-jars,"  Fig 
3 ;  and  to  the  top  of  these  is  screwed  the  '*  rope-socket," 
Fig  4,  to  which  the  drilling  cable  is  attached.  If  the  bit, 
or  auger-stem,  becomes  u'edg-ed  in  the  hole 
by  any  means,  the  operator  stops  the  churning  motion 
of  the  tools,  and  the  rope  is  let  out  about  12  ins.  This  per- 
Inits  the  upper  link  U  of  the  drill-jars.  Fig  3,  to  slide  down 
about  12  ins  in  the  slot  S  in  their  lower  link.  The  churn- 
ing motion  is  then  started  a^ain,  and  the  upward  jerk  of 
the  link  U  against  the  upper  end  of  the  slot  loosens  the 
tools. 

These  machines  are  made  in  a  number  of  sizes,  to  drill 
holes  from  200  to  1000  feet  deep.  The  string  of  tools  weighs 
from  800  to  1800  lbs. ;  and  the  machine  complete,  including 
tools,  rope,  mast,  etc.,  but  exclusive  of  power,  from  1800  to 
4600  lbs.  They  cost  from  $700  to  $1500  exclusive  of  power. 
The  smaller  sizes  may  be  worked  by  horse  power.  A  horse 
power  weighs  about  800  lbs.,  and  costs  about  $75.  Steam 
engine,  1600  to  3600  lbs.,  $150  to  S300. 

For  \irell9  from  1000  to  3000  feet  deep,  a 
stationary  machine,  with  a  walking-beam,  is  used,  similar 
to  thoM  employed  in  the  oil  regions  of  Pennsylvania.  A 
square  pyramidal  derrick  is  erected,  74  feet  high,  20  feet 
square  at  base,  4  feet  square  at  top.  Each  of  its  4  corner  legs 
is  of  2  inch  X  8  inch  and  2  inch  X  10  inch  planks,  spiked 
together  so  as  to  form  a  10  inch  X  10  inch  angle-piece,  2 
inches  thick.  The  legs  are  braced  together  by  horizontal 
and  diagonal  timbers.  The  walking-beam  is  of  timber,  26 
feet  long,  12  inpbes  wide,  and  26  inches  deep  at  the  middle 

of  its  length,  where  it  is  pivoted  to  the  top  of  a  wooden  post       Fig.  1.  FiG.  3. 

18  inches  square  and  12  feet  high,  called  a  "  Samson  post." 

This  post,  at  its  foot,  is  dovetailed  into  the  main  sill  of  the  machine,  which  is  18 
inches  wide  X  24  inchep  deep. 
43 


FlG.J 


\ 


I;. 


r 


674  ARTESIAN   WELL   BORING. 

The  motive  power  is  a  15-hp  steam-engine,  which,  by  means  of  a  belt  and  pulley, 
crank  and  pitman,  working  at  one  end  of  the  walking-beam,  gives  to  the  latter  its 
see-saw  motion.  To  the  other  end  of  the  beam,  and  immediately  over  the  well,  is 
suspended,  by  means  of  a  hook,  a  "  temper-screw."  This  last  is  composed  of  two 
bars  of  iron,  about  %  X  2  ins,  6  ft  long,  hung  2  ins  apart,  fastened  together  at  their 
top  ends,  at  which  point  there  is  an  eye,  which  is  suspended  on  the  walking-beam 
hook.  At  the  bottom  of  the  two  bars  there  is  a  sleeve-nut,  and  between  the  two 
bars  and  passing  through  the  nut,  is  a  screw  5  ft  long,  at  the  bottom  of  which  there 
is  a  head,  which  carries  a  swivel,  set-screw,  and  a  pair  of  clamps.  These  grasp  the 
cable,  2  or  2^4  i»8  diam,  which  carries  at  its  lower  end  the  string-  of  tools. 
This,  for  a  2000-ft  hole,  consists  of  a  steel  bit,  3  or  4  ft  long,  weighing  200  to  400  lbs; 
an  auger-stem  of  4  or  5-inch  round  iron,  from  24  to  30  ft  long,  and  weighing  from 
1200  to  2100  lbs;  steel-lined  drill-jars  8  ft  long,  weighing  600  to  700  lbs  ;  a  sinker-bai 
of  round  iron  of  same  diam  as  the  auger-stem,  12  to  15  ft  long,  and  weigliing  from  600 
to  1100  lbs  ;  and  a  rope-socket,  '1]A  ft  long,  weighing  200  lbs.  Total  length  of  string 
of  tools,  50  to  60  ft,  total  weight,  3000  lbs  ;  or,  for  an  8-inch  hole  in  the  hardest  rock, 
4000  lbs.  The  sinlier-bar  is  added  to  give  additional  wt,  and  thus  to  assist  in 
pulling  the  cable  down  through  the  water,  either  in  lowering  the  string  of  tools  or 
in  working  the  drili-jars.  The  shapes  of  the  other  tools  are  given  by  Figs  1  to  4. 
Special  tools  are  used  for  recovering  articles  that  may  be  accidentally  dropped 
into  the  hole. 

The  drillingr  cable  is  ivound  on  a  drum,  called  a  bull-wheel  shaft,  at  the 
foot  of,  and  inside  of,  the  derrick.  While  drilling  is  going  on,  it  passes  from  the 
bull-wheel  shaft  loosely  over  the  sheave  at  the  top  of  the  derrick,  and  down  to  the 
clamps  at  the  lower  end  of  the  temper-screw  on  the  end  of  the  walking-beam.  As 
the  drilling  progresses,  the  temper-screw  is  turned  or  fed  out  by  the  man  in  charge, 
who  also,  by  means  of  a  clamp,  twists  the  rope,  so  as  to  change  the  position  of  the 
bit  after  each  stroke. 

When  the  tools  are  to  be  lifted  out  of  the  hole,  the  cable  is  disengaged  from  the 
clamps  on  the  temper-screw,  and  is  wound  upon  the  bull-wheel  shaft,  which,  for  this 
pux-pose,  is  thrown  into  gear  with  the  steam-engine;  the  pitman  being  at  the  same 
time  removed  from  the  crank-pin,  so  that  the  walking-beam  is  at  rest.  As  in  the 
portable  machines,  the  sand-pump  is  also  raised  by  the  same  power  which  does  the 
drilling. 

About  10000  ft  b  m  of  rough  lumber  are  reqd  for  the  derrick,  walk- 
i;iK-beam>  sills,  <fec,  and  about  3000  ft  more  for  sheds  over  the  boiler,  engine,  and  belt. 

In  ordinary  hard  limestone  rock,  such  a  machine  will  flrill  about  l^^^ft 
1>QY  hour  under  the  most  favorable  circumstances.  Two  men  are  required  ; 
one  to  attend  to  the  boiler,  sharpen  the  bits,  Ac,  and  one  to  operate  the  machine. 

In  Pierce's  machine  =^  for  test-boring,  mineral  prospecting  and  well 
boring,  the  pipes  are  driven  by  an  iron  ram,  like  that  of  a  pile  driver,  but 
bushed  with  hard  wood  on  its  lower  or  striking  end.  The  ram  is  worked  by  a 
hand  winch.  The  pipes  are  in  lengths  of  5  to  10  feet.  After  each  length  is 
driven,  water,t  under  pressure,  is  forced,  by  a  hand  pump,  through  a  hollow 
drill  rod,  into  the  bottom  of  the  hole,  while  the  drill  rod  is  churned  up  and 
down  by  hand.  The  water  forces  the  drillings  (mud,  sand,  gravel,  etc.)  to  the 
surface.  The  smallest  machine  drives  2  to  3  inch  pipes;  the  largest,  2  to  8  inch. 
The  machines  are  in  detachable  parts,  weighing  from  10  to  65  lbs  each.  Four 
upright  iron  pipes,  which  carry  the  head  casting  and  crown  pulley,  act  as  guides 
for  the  ram,  their  ends  fitting  into  sockets  in  castings  at  their  heads  and  feet. 
The  driving  rams  are  made  in  sections  which  are  bolted  together.  In  the 
smaller  machines  the  weight  of  the  ram  may  thus  be  made  from  100  to  200 
pounds,  and,  in  the  larger  machines,  from  100  to  2,000  pounds,  as  required. 
Borings  can  be  made  to  depths  of  100  to  400  feet.  These  machines  have  been 
extensively  used  in  Nicaragua  by  the  Isthmian  Canal  Commission.  If  desired, 
the  machines  can  be  furnished  with  special  tools  for  boring  in  rock  and  for 
taking  out  solid  cores  (as  with  the  diamond  drill),  with  others  for  taking  out 
dry  cores  in  earth,  and  with  sand-pumps  and  mud  sockets  for  bringing  up  mud, 
fine  sand,  gravel,  and  detached  pieces  of  rock  and  minerals. 

♦Business  Directory,  No.  484. 

fin  Alaska,  where  the  frost  extends  to  great  depths,  boiling  or  hot  water  is 
used.  This  is  obtained  by  melting  ice  or  snow  in  iron  tanks  about  4  ft  square 
and  2  ft  deep. 


MACHINE   ROCK-DRILLS.  675 

MACHINE  ROOK-URILLS. 


Art.  1.  Machine  Rock-drills  bore  much  more  rapidly  than  hand  drills,; 
and  more  economically,  provided  the  work  is  so  great  as  to  justify  the  preliminary 
outfit.  They  drill  in  any  direction,  and  can  often  be  used  in  boring  holes  so  located 
that  they  could  not  be  bored  by  hand.  They  are  worked  either  by  steam  directly  ; 
or  by  air,  compressed  by  steam  or  water  power  into  a  tank  called  a  "receiver,"  and 
thence  led  to  the  drills  through  iron  pipes.  The  air  is  best  for  tunnels  and  shfcfte, 
because,  after  leaving  the  drills,  it  aids  ventilation. 

Art.  2.  Such  drills  are  of  two  kinds:  rotating-  drills  and 
percussion  drills.  In  the  former,  the  drill-rod  is  a  long  tube,  revolving  about 
its  axis.  The  end  of  this  tube,  hardened  so  as  to  form  an  annular  cutting-edge,  is 
kept  in  contact  with  the  rock,  and,  by  its  rotation,  cuts  in  it  a  cylindrical  hole,  gen- 
erally with  a  solid  core  in  the  center.  The  core  occupies  the  core-barrel.  Art  8. 
The  drill-rod  is  fed  forward,  or  into  the  hole,  as  the  drilling  proceeds.  The  debris 
is  removed  from  the  hole  by  a  constant  stream  of  water,  which  is  led  to  the  bottom 
of  the  hole  through  the  hollow  drill-rod,  and  which  carries  the  debris  up  through 
the  narrow  space  between  the  outside  of  the  drill-rod  and  the  sides  of  the  hole. 

In  percussion  drills,  the  drill-rod  is  solid,  and  its  action  is  that  of  the 
churn  drill. 

Art;,  3.  In  the  Brandt  (European)  rotary  drill,  the  cutting-edge  at  the 
end  of  the  tubular  drill-rod  is  armed  with  hardened  steel  teeth.  It  is  pressed  against 
the  rock  under  enormous  hydraulic  pressure,  and  makes  but  from  5  to  8  revolutiona 
per  minute. 

Art.  4.  The  Diamond  drill  is  the  only  form  of  rotary  rock-drill  exten- 
iively  used  in  America.  In  it,  the  boring-rod  consists  of  a  number  of  tubes  jointed 
rigidly  together  at  their  ends  by  hollow  interior  sleeves. 

Art.  5.  The  borinjf-bit,  Fig  i,  is  called  a  "core-bit.*'  Its  cutting-edge 
has  imbedded  in  it  a  number  of  diamonds  as  shown.  These  are  so  arranged  as 
to  project  slightlj'  from  both  its  inner  and  outer  edges.  Annular  spaces  are  thua 
left  between  core  and  core-barrel,  and  between  the  latter  and  the  walls  of  the  hole. 
These  spaces  permit  the  ingress  and  egress  of  the  water  used  in  removing  the  debris 
from  the  hole,  and,  at  the  same  time,  prevent  the  core  from  binding  in  the  barrel,  or 
the  latter  in  the  hole. 

Fig.l  "Fig, 


COKE  BIT?  COBE  ilFTEB.  BORII^G  HiCAD. 


Art.  6.  Just  above  the  "core-bit,"  the  ^'core-lifter,"  Fig  2,  is  screwed  to 
the  barrel,  ^is  is  a  tube  about  8  Ins  long  and  of  the  same  outer  diam  as  tht 
barrel.  Inside  it  is  slightly  coned,  with  the  base  of  the  cone  upward,  and  fuTi 
nished  with  a  loose  split-ring,  R,  toothed  inside,  and  similarly  coned.  While  the 
drilling  is  going  on,  this  ring  encircles  the  core  closely,  and  remains  loose  from  the 
outer  cylinder;  but  when  the  drilling  is  stopped,  and  the  drill -rod  begins  to  be 
raised,  the  ring  is  caught  and  raised  by  the  outer  cylinder;  and,  by  reason  of  its 
beveled  shape,  is  pressed  hard  against  the  core  of  rock,  which  is  pulled  apart  close 
to  its  foot  by  the  power  which  lifts  the  drill-rod. 

Art,  7,  This  power  is  supplied  by  a  rope-drum,  fastened  to  the  top  of  tV>« 
frame  'ffr-ii;h  supports  the  drill  and  worked  by  the  same  (  ngime  wAfcu  rotates  the 
drill-rod.  The  rope  from  the  drum  passes  up  to  a  pulley  at  the  top  of  a  derrick, 
and  thence  down  to  the  upper  end  of  the  drill-rod.  The  considerable  height  of  the 
ierricb  enables  ftwm  40  to  60  feet  of  the  drill-rod  to  be  remored  In  one  piece. 

Art.  8.  Above  the  "  core-lifter  "  is  the  **  CO  re- barrel,**  This  is  a  wronglit- 
iron  tube  from  8  to  16  ft  long.  It  is  spirally 

grooved  outside,  to  permit  the  ascent  of  the  water  and  debris  from  the  hole;  and  is 
sometimes  set  with  diamonds  on  its  outer  surface,  to  prevent  wear.  The  bit,  lifter, 
and  barrel  are  of  uniform  outer  diam,  a  little  less  than  the  diam  of  the  hole.  The 
outer  d'.am  of  the  driU*red  Tariee  from  about  1^  ins  for  2-inch  barrel  to  5^  ins  for  1& 
inch  barreL 


676 


MACHINE   ROCK-DRILLS. 


Art.  9.    Where  it  is  not  desired  to  preserve  the  core  intact,  a  ^'boring** 

head,"  Fig  3,  may  be  used  instead  of  the  "core-bit."  Fig  1.  This  is  a  solid  bit 
(excei)t  that  it  is  perforated  with  holes  which  allov/  the  water  to  pass  out  from 
the  drill-rod),  and  is  armed  with  diamonds,  some  of  which  project  beyond  its  circum- 
ference. 

Art.  10.  The  drill-rod  revolves  at  a  speed  of  from  200  to  400  revolutions 
per  minute.  The  eiig^ine,  by  which  it  is  rotated,  consists  usually  of  two  cylin- 
ders, either  fixed  or  oscillating,  operated  by  steam  or  compressed  air,  and  working 
at  right  angles  to  each  other.  By  means  of  cranks  they  turn  a  shaft,  which  com- 
municates its  motion,  through  bevel  gearing,  to  the  drill-rod.  The  latter  is  fed 
down,  as  the  hole  progresses,  either  by  other  bevel  g'earing^  driven  by  the 
same  engine  ;  or  by  being  attached  to  a  cross-head  which  connects  the  piston  rods  of 
2  hydranlic  cylinders,  the  piston  rods  being  parallel  with  the  drill  rod. 

Art.  11.  The  diamond  drill  bores  perfectly  circular  holes,  in  strai^^ht 
lines  and  in  any  direction,  to  great  depths;  from  300  to  1500  feet 
being  not  uncommon.  This,  with  the  fact  that  it  brin^  up  unbroken 
cores,  from  8  to  16  ft  long,  which  show  the  precise  nature  and  stratification  of  the 
rock  penetrated,  renders  it  very  valuable  in  test-boring,  prospecting  of  mines,  Ac. 
They  are  also  furnished  of  suflicient  size  to  bore  holes  from  6  to  15  ins  diam,  for 
artesian  wells.  The  roundness  of  the  holes  bored  enables  the  use  of  casing  of 
nearly  as  great  diam  as  that  of  the  hole ;  and  their  straightness  is  advantageous  in 
case  a  pump  has  to  be  used. 

Art.  12.  In  soft  rock  a  bit  may  drill  through  200  ft  or  more  without  resetting. 
On  the  other  hand,  in  very  hard  rocks,  similar  drills  will  wear  out  in  10  ft  or  less. 
In  1883-4,  a  diamond  drill  by  the  Am'n  IHamond  Rock  Boring  Co,  weighing  com- 
plete about  1400  flbs,  and  costing  about  $2800,  bored,  in  1428  hours  of  actual  boring, 
63  holes  of  2  ins  diam,  and  aggregating  9141  lineal  ft.  Average  length  of  hole  172.S 
ft.  Averag-e  rate,  6.4  lin  ft  per  hour;  greatest,  12.8.  Average  total  cost. 
about  96  cts  per  lin  ft.  The  rock  was  principally  limestone,  with  some  quartz  and 
sandstone.  The  holes  were  bored  at  angles  varying  from  0°  to  45°  with  the  vertical. 

As  a  rough  average  we  may  say  that  in  ordinary  rocks,  as  granite,  lime- 
stone, and  hard  sandstone,  these  drills  will  bore  deep  holes,  2  to  3  ins  diam,  at  from 
1  to  3  ft  per  hour,  and  at  a  cost  of  from  ^1  to  $2  per  ft. 

Art.  13.  These  drills  are  made  of  many  widely  different  sizes^  and  with 
different  mountings,  depending  upon  the  nature  of  the  work  to  be  done. 

They  are  sold  under  restrictions  as  to  the  location  and  extent  of  the  territory 
in  which  they  are  to  be  used.  The  prices  depend,  to  a  great  extent,  upon  the 
nature  of  these  restrictions.  The  card  prices  for  some  of  the  leading  sizes,  are 
as  follows :      Discount,  see  price  list. 


Diam 

of 
hole. 

Diam 

of 
core. 

Depth 

of 
hole. 

Boiler 
H.  P. 

required. 

Card  price. 

Drill. 

Pump 

Ins. 

Ins. 
2 

1 

Feet. 
4000 
1500 
1000 
600 
400 

H.  P. 

25 

15 

12  to  15 

10 

hand 

4000 
2500 
1900 
1400 
425 

« 

3400 
2800 
1900 

Art.  14.  In  percussion  drilling  machines,  the  drill-bar  is  driven 
forcibly  against  the  rock  by  the  pressure  of  steam  or  of  compressed 

air,  acting  upon  a  piston,  P,  Fig  4,  moving  in  a  cylinder,  C  C,  Figs  4  and  5 ;  and 
makes  about  300  strokes  per  minnte.  The  rotation  of  the  drill-bar  is  accomplished 
automatically,  as  explained  in  Art  27. 

Art.  15.  The  cylinder,  C  C,  is  free  to  slide  longitudinally  in  the  fixed 
frame  or  shell,  S  S,  Fig  5,  to  which  it  is  attached,  and  which,  in  turn",  is  fixed  to  the 
tripod  or  other  stand  (see  Arts  18  and  19)  upon  which  the  machine  is  supported. 

Art.  16.     The  drill-rod,  R,  corresponding  to  the  churn  drill,  is 

fastened,  by  an  appropriate  chuck,  K,  to  the  end  of  the  piston-rod,  0.  The  drilling 
is  begun  with  a  short  drill-rod,  and  with  the  cylinder  as  far  from  the  hole  as  the 
length  of  the  shell,  S,  will  permit.  As  the  bit  penetrates  the  rock,  the  cylinder  is 
fed  forward,*  either  automatically  or  by  hand  (see  Art  28),  as  far  as  the  length  of 

*  By  forward,  or  downward*  we  mean  toward  the  hole  which  is  being  drilled.  By  baok« 
tvardt  01"  npw»rd,/»*o»i  the  hole. 


MACHINE   ROCK-DRILLS.  677 

the  shell  permits.  The  drilling  is  then  stopped,  by  shutting  ofif  the  steam,'*'  and  th« 
cylinder  is  run  back,  by  reversing  the  motion  of  the  feeding  apparatus.  The  short 
drill-bar  is  then  removed,  and,  if  the  drilling  is  to  be  continued,  a  longer  one  is  sub* 
8tituted  in  its  place,  and  the  process  repeated. 

Art.  IT.  Inasmuch  as  the  act  of  drilling  wears  the  edges  of  the  bit,  thus  reduc- 
ing its  diam  8omev»rhat,  the  hole  i¥ill  of  course  be  tapering,  or  of 
slightly  less  diam  at  bottom  than  at  top.  The  second  bit  must  therefore  be  of 
slightly  less  diam  than  the  tirst;  say  from  j^  to  }/g  inch  less;  the  third  must  be  less 
than  the  second,  and  so  on.  On  the  other  hand,  in  long  holes,  the  drill-bar  will 
seldom  be  in  a  perfectly  straight  line,  so  that  the  bit,  instead  of  striking  always  in 
the  same  spot,  will  describe  a  circle,  and  thus  enlarge  the  hole. 

Art.  18.  The  shell,  S,  in  which  the  cylinder  slides,  is  provided  with  an  arrange- 
ment by  which  it  may  be  clamped,  either  to  a  tripod,  as  in  Fig  6,  or  to  a  long 
bar  or  column,  along  which  it  may  slide.  The  column,  if  hor,  may  rest  upon 
two  pairs  of  legs ;  or  it  may  be  braced.  In  any  position,  against  the  opposite  sides  of 
a  narrow  cut,  or  agjiinst  the  floor  and  ceiling  of  a  tunnel-heading,  &c,  in  which  case 
one  of  its  ends  is  provided  with  a  screw  which  is  run  out  so  as  to  cause  the  two  ends 
of  the  col  to  press  firmly  against  the  opposite  rock  walls ;  or  rather  against  wooden 
blocks  which  are  always  placed  between  each  end  of  the  col  and  the  rock.  In  any 
case,  the  supports  of  the  drill  are  so  jointed  that  it  can  bore  in  any  direction. 

Art.  19.  Frequently  the  drill  is  clamped  to  a  short  arm,  which,  in 
turn,  is  clamped  to  the  column,  and  projects  at  right  angles  from  it.  The  arm  may 
be  slid  lengthwise  of  the  column,  and  may  be  revolved  around  it,  and  thus  may  be 
placed  in  any  desired  position,  and  there  clamped.  This  gives  the  drill  a  greater 
range  of  motion,  and  enables  it  to  bore  holes  over  a  greater  space  than  would  other- 
wise be  possible  without  moving  the  column. 

Art.  20.  In  tunnels,  one  or  more  drills  may  be  mounted  upon  a  drill-car- 
riage, travelling  upon  a  railroad  track  running  longitudinally  of  the  tunnel. 
Upon  this  track  the  carriage  is  moved  up  to  the  work,  or  run  back  f'om  it  when  a 
blast  is  to  be  fired.  The  gauge  of  the  track  may  be  made  wide  enough  to  admit  of 
a  second  track,  of  narrower  gauge,  running  underneath  the  drill-carriage.  Upon 
said  narrower  track  the  cars  are  run  which  carry  away  the  debris.  Drill-carriage4 
are  less  commonly  used  in  this  country  than  in  Europe. 

Art.  21.  The  pressure  used  in  the  cylinders  of  percussion  drills  is 
usually  from  about  60  to  70  lbs  per  sq  inch.  In  an  hour,  one  will  drill 
a  hole  from  1  to  2  ins  diam,  and  from  3  to  10  ft  deep,  depending  oa  the  character  of 
the  rock  and  the  size  of  the  machine  at  from  10  to  25  cts  per  lin  ft  with  lal)or  at 
$1  per  day.  A  bit  requires  sharpening;:  at  about  every  2  to  4  ft  depth  of 
hole.    One  blacksmith  and  helper  can  sharpen  drills  for  5  or  6  machines. 

Art.  22.  The  bits  are  of  many  different  shapes,  varying  with 
the  nature  of  the  work  to  be  done.  For  uniform  hard  rock,  the  bit  has  two  cutting- 
«idges,  forming  a  cross  with  equal  arms  at  right  angles  to  each  other.  For  seamy 
rock,  the  arms  of  the  cross  are  equal,  but  form  two  acute  and  two  obtuse  angles  with 
each  other,  as  in  the  letter  X.  For  soft  rock,  the  cutting-edge  sometimes  has  the 
shape  of  the  letter  Z. 

Art.  23.  Each  drill  requires  one  man  to  operate  it.  Two  or  three  men 
are  required  for  moving  the  heavier  sizes  from  place  to  place.  One  man  can  attend 
to  a  small  air-compressor  and  its  boiler. 

Art.  24.  Figs  4  and  5  represent  the  '*  Eclipse  "  percussion  drill  of  the  Inger- 
soll-Sergeant  Drill  Co,  Havemeyer  Building,  New  York.  Fig  5  shows  the  drill, 
mounted  (as  is  most  frequently  the  case)  upon  a  tripod.  Fig  4  is  a  longitudinal  sec- 
tion through  the  cylinder,  valve-chest,  and  piston. 

Art.  25.  The  cylinder,  C,  is  provided  at  each  end  with  a  rubber  cushion, 
N,  for  deadening  the  blows  of  the  piston,  which,  in  all  percussion  drills,  is  liablft,  at 
times,  to  strike  either  cylinder-head.  The  side  of  each  cushion  nearest  the  piston  is 
protected  by  a  thin  iron  plate.    The  cushions  have  to  be  renewed  from  time  to  time. 

Art.  26.  The  valve,  V,  is  sliaped  somewhat  like  a  spool.  The  bolt,  B, 
passes  loosely  through  its  center  and  guides  it.  Steam  is  admitted  from  the  boiler 
to  the  steam-chest,  and  occupies  all  of  the  space  between  the  two  end  flanges  of  the 
valve,  except  u.  It  drives  the  valve  alternately  from  one  end  of  the  valve-chest  to 
the  other,  and  back,  according  as  one  end  or  the  other  is  relieved  from  opposing 
pressure  by  being  put  into  communication  with  the  exhaust.  U,  by  way  of  the  pas- 
sages, D  D'  and  F  F'.  D  and  D'  communicate  with  the  ends  of  the  steam-chest 
through  passages  not  shown ;  while  F  and  F'  communicate,  through  similar  pas- 
sages, with  the  exhaust,  E.  The  piston  has  an  annular  channel,  L  L',  encircling  it. 
Whatever  the  position  of  the  piston,  one  of  the  passages.  D  or  D',  is  always,  by  means 
of  this  channel,  in  communication  with  its  corresponding  passage,  F  or  F',  leading 

^Hi'^^  ^^°l^  repetitions,  we  wUi  use  the  word  steam  to  signify  either  steam  or  compressed  ain 
Whichever  happeos  to  be  used.  ^ 


678 


MACHINB  ROCK-DRILUS. 


to  the  exhaust.  Thus,  one  or  the  other  end  of  the  valve-chest  is  always  in  com 
muuication  with  the  open  air;  and  to  that  end  the  valve  is  driven  by  the  pres  c^ 
the  steam  surrounding  it,  admitting  steam  to  the  cyl,  C,  from  the  other  end. 

Art.  27.  The  rotation  of  the  piston,  and,  with  it,  that  of  the  drilj* 
bar,  is  eflfected  thus:  The  spirally-grooved,  cylindrical  steel  bar,  A,  called  a  rifle- 
bar,  passes  through  and  works  in,  the  rifle-nut,  H,  which  is  firmly  fixed  in 
the  end  of  the  piston,  and  has  spiral  grooves  corresponding  with  those  on  the  rifle- 
bar.  Said  bar  is  fixed,  at  its  upper  end,  to  the  ratchet-wheel,  J,  the  pawls  of  which 
are  so  arranged  that,  on  the  down  stroke  of  the  piston,  the  rifle-nut,  H,  acting  upon 
the  grooves  on  the  rifle-bar,  causes  it,  and,  with  it,  the  ratchet-wheel,  to  revolve 
about  their  common  axis.  The  weight  and  mo- 
mentum of  the  piston,  &c,  are  such  that  it  thus 
readily  turns  the  rrttchet-wheel  without  itself 
turning.  Thus  the  bit  is  prevented  from  rotating 
while  delivering  its  blow.  But,  on  the  itp  stroke, 
the  tendency  of  the  rifle-nut  is  to  turn  the  rifle- 
bar  and  ratchet-wheel  in  the  opposite  direction; 
and  as  this  is  prevented  by  the  pawls,  the  rifle- 
har  remains  stationary,  while  the  piston,  piston- 
rod,  and  di'ill  are  made  to  revolve  about  their 
common  axis. 

Art.  28.  The  feed-screw,  M,  is  col- 
lared, at  its  upper  end,  to  the  fixed  fi-ame,  Q,  It 
is  thus  prevented  from  moving  longitudinally 
when  revolved  by  means  ot  the  crank  fixed  to  its 
top.  Its  lower  end  works  in  a  nut,  T,  fixed  to  the 
cylinder,  which  last  is  thus  moved  longitudinally 
backward  or  forward  as  the  crank  is  turned. 


B'ig.  5. 


Large  drills  are  frequently  furnished  with  an  utltomatic  feeding"  arrang^e« 

tiient  in  addition  to  the  hand-crank.    In  this  arrangement,  when  the  cylinder 
requirea  feeding  forward,  and  when,  consequently,  the  piston  is  running  nearly  ta 


MACHINE   ROCK-DRILLS. 


679 


the  forward  limit  of  its  stroke,  the  piston  presses  against  a  cam  projecting  into  th« 
eyl  near  the  forward  end,  and  presenting  an  inclined  plane  to  it.  The  motion  of 
this  cam,  by  means  of  an  exterior  axle,  running  alongside  of  the  cyl  and  furnished 
at  its  top  with  a  dog,  turns  a  ratchet-wheel  fixed  to  the  feed-screw.  When  desired, 
the  automatic  feed  may  be  thrown  out  of  gear,  and  the  feed  moved  by  hand. 

Art.  29.  The  tripod  leg-S  consist  of  wi-ought-iron  tubes,  W  W.  These  are 
screwed  at  tlieir  upper  ends  into  sockets,  X  X.  At  their  lower  ends,  they  receive 
the  pointed  and  tapering  steel  bars,  Y  Y,  about  2  or  3  ft  long.  The  legs  may  be 
lengthened  or  shortened  by  turning  tbe  set-screws,  Z  Z,  thus  regulating  the  distance 
to  which  the  bars,  Y  Y,  can  enter  the  legs.  The  clamps,  b  b,  have  L-shaped  hooks 
of  3^  inch  to  1  inch  round  iron  forged  to  them.  On  these  hooks  tlie  'weig^bts, 
d  d,  are  hung,  which  hold  the  machine  down  against  the  upward  reaction  of  its 
blows. 

Art.  30.  The  following  table  gives  the  principal  dimensions  of  these 
drills,  with  the  diams  and  leng-tbs  of  boles  to  which  each  is  adapted. 
Size  H  is  used  for  submarine  work,  heavy  tunneling,  and  deep  rock  cutting,  G 
and  F  for  tunneling,  street  grading,  quarrying,  and  sewer  work.  E,  D,  and  C 
for  general  mining  purposes.  B  is  adapted  only  for  very  light  work.  In  asking 
for  estimates  on  drills  and  compressors,  give  the  fullest  possible  description 
(accompanied  by  a  sketch)  of  the  work  ta  be  done,  stating  its  present  and  pro- 
posed extent.  State  whether  the  work  is  on  the  surface  or  underground.  State 
hcrw  far  the  steam  or  compressed  air  will  be  carried.  Give  depth  of  holes  to  be 
drilled,  nature  of  rock,  &c.  Percussion  drills  are  sold  without  restriction  as  to 
the  puipose  or  extent  to  which  they  are  to  be  used. 


Letter  designat 

ing  the  size  of  the  machine. 

A 

B 

^ 

I> 

£ 

F 

« 

H 

Inner  diameter  of 

cylinder ins. 

.   1% 

2K 

2% 

3 

3% 

3>^ 

4^ 

5 

Length  of  full 

stroke  " 

3 

4 

5 

6 

6 

6X 

7 

7 

Length  of  feed  " 

12 

20 

24 

24 

24 

26 

34 

34 

Length  of 

machine* '* 

36 

34 

36 

40 

42 

53 

60  . 

60 

Wt  of  machine, 

unmounted,  lbs. 

80 

155 

195 

230 

250 

345 

605 

670 

Wt  of  tripod, 

without  wts.  •* 

125 

125 

125 

125 

150 

275 

275 

Wtof3wtsfor 

X    ' 

tripod  legs,    " 

250 

250 

250 

250 

350 

400 

400 

Wt  of  column, 

arm  &  clamp  " 

200 

280 

280 

280 

420 

420 

420 

Diam  of  hole 

drilled ins. 

^to% 

%io\% 

1  to  2 

1  to2 

1  to  2 

l%io2% 

2  to  4 

3  to  6 

Max  depth  of 

vert  t  hole ft. 

X 

4 

8 

10 

12 

16 

30 

40 

*  From  top  of  handle  of  feed-crank  to  lower  end  of  piston  at  the  end  of  the 
down  stroke. 

t  For  greatest  ad  visable  Igth  of  hor  holes,  deduct  one-fourth  from  these  depths, 

J  Machine  A  is  mounted  on  a  small  frame. 

Art.  31.  The  drills  of  different  makers  differ  cbiefly  in  the 
methods  by  which  the  valve  is  operated.  In  some  this  is  done,  as  in  the  Ingersoll 
*'  Eclipse  "drill,  Art  26,  by  the  pres  of  steam.  In  others,  the  valve  is  moved 
by  a  lever  or  tappet,  which  projects  into  the  cylinder  so  as  to  come  into 
contact  with,  and  be  moved  by,  the  piston  at  each  stroke.  As  these  strokes  are 
made  with  great  force  some  300  or  more  times  per  minute,  such  valve-gear  is 
necessarily  subject  to  great  wear. 

Art.  32.  In  the  "  L,ittle  Oiant  Brill,"  made  by  the  Rand  Brill 
Co.,  the  valve,  V,  Fig  6,  is  slid  backward  and  forward,  in  the  same  direction  in 
which  the  piston  is  moving,  by  the  tappet,  T,  which  is  pivoted  at  p.  The 
inclined  lower  corners  of  tbis  tappet  ride  up  as  they  come,  alternately,  in  contact 
with  the  shoulders,  s  s,  of  tbe  piston. 

Art.  33.  In  the  "Economizer"  and  the  "Sluffffer"  (Rand  Drill 
Co.)  the  valve,  as  in  the  Ingersoll  "  Eclipse"  drill,  is  moved  by  steam,  but  upon 


680 


MACHINE   ROCK-DRILLS. 


a  quite  different  principle.  In  these  two  drills,  there  is  no  steam  cushion  for  the 
piston  to  strike  against  on  the  down  stroke,  the  force  of  which  is  thus  more 
completely  expended  upon  the  rock.  The  cushion  behind  or  above  the  piston, 
on  the  return  stroke,  is  formed  by  exhaust  steam.  Both  of  these  drills  cut  off 
steam  before  the  completion  of  either  stroke,  thus  using  the  steam  expansively. 
On  the  down  stroke,  the  "Economizer"  cuts  off  earlier  than  the  "Slugger.'* 
Hence  its  name.  In  both  machines 
the  point  of  cut-off  is  fixed  when 
the  machine  is  made. 

Art.  34.  In  the  improved 
Burleig-h  drill,  the  valve,  V, 
Fig  7,  is  moved  by  two  tappets, 
T  T',  which  are  alternately  struck 
by  the  ends  of  the  piston,  P. 

Art.  35.    In  the  "Dynamic" 
rock-drill,  invented  by  Prof  De 
Volson    TVood,    the   valve    is 
attached  to  a  valve-piston,  V,  Fig 
8,  which  is  moved  backward  and 
forward  by   steam,  which  is  ad- 
mitted   so  as    to  act   alternately  _,. 
upon  its  two  ends.     The  admission  Jig",  fo. 
of  this  steam  is  controlled  by  a  small  auxiliary  valve,  a.     A  hub  on  the  back  ol 
the  auxiliary  valve  fits  in  the  spiral  groove  shown  on  the  plug,  n.     This  plug  ia 
constantly  pressed  downward  (as  the 
Fig  stands)  by  steam  pressing  upon 
its  upper  shoulder,  but  it  is  lifted  at 
each  forward  stroke  by  the  conical 
surface  of  the  piston,  P,  pressing 
against  its  foot.     It  thus  moves  con- 
stantly up  and  down,  carrying  the 
valve,  a,  with  it.     By  turning  the 
plug,  n,  by  means  of  the  adjusting- 
stem,  5,  the  hub  of  the  valve  is  made  to 
occupy  a  higher  or  lower  point  in  the 
spiral  groove,  and  thus  the  stroke  of 
the  piston  may  be  varied,  or  may  be  Fig".  7. 
confined  to  any  part  of  the  cylinder. 

In  this  drill,  unlike  the  IngersoU,  Art.  27,  the  piston  rotates  while  making 


Fig. 


the  downward  stroke.    Tlie  piston-rod,  o,  is  made  lighter  tlian  in 

ottier  drills.    This  gives  a  greater  surface  under  the  piston  for  the  pressure 
of  the  steam  on  the  up  stroke,  and,  consequently,  greater  lifting  power.     This 
is  useful  when  the  drill  sticks  in  the  hole. 
The  tripod  legs  are  of  bar  iron.    Their  length  is  adjustable. 


MACHINE   ROCK-DRILLS.  681 

Art.  36.  The  Pierce  hand  rock-drill  is  a  percussion  drill 
worked  by  a  crank  which  turns  a  disc  about  2  ft  iu  diam.  The  disc  has  a  semi-circular 
slot,  in  which  works  ^le  arm  which  raises  the  drill-rod.  This  arm, in  rising, compresses 
a  coil-spring,  which,  on  the  down  stroke,  drives  the  drill  against  the  rock.  An  iron 
ball,  weighing  30  lbs  or  more,  is  furnished  with  each  machine.  This  ball  may  be 
screwed  to  the  top  of  the  drill-rod,  for  giving  greater  force  to  the  blows  of  the  drill. 
The  ball  may  be  used  without  the  spring,  by  disengaging  the  latter. 

The  drill  makes  about  40  strokes  of  10  or  12  ins  per  minute;  and  bores  holes  from 
^  to  2]/^  ins  diam.  It  can  be  arranged  to  drill  to  depths  of  30  ft  and  over.  For 
sharpening  the  bits,  it  has  an  emery  wheel  attached,  which  is  turned  by  the  crank. 
The  latter,  at  such  times,  is  thrown  out  of  gear  with  the  disc. 

The  drill  is  mounted  on  a  rectangular  two-legged  frame,  about  5  ft  high 
by  2  ft  wide,  made  of  iron  tubes.  To  the  top  of  this  frame  a  third  leg  is  attached, 
by  adjusting  which  the  angle  of  the  drill-rod  with  the  vert  may  be  changed.  Like 
other  percussion  drills  worked  by  hand-power,  this  one  ceases  to  work  to  advantage 
when  said  angle  exceeds  about  45°. 

Art.  37.  Channeling  consists  in  making  long,  deep,  and  nai-row  cuts  in 
the  rock.  In  this  way  large  blocks  can  be  gotten  out  without  blasting  and  the  con- 
sequent danger  of  fracture.  This  is  ordinarily  done  by  boring  a  row  of  holes  about 
an  inch  apart  in  the  clear,  and  then  breaking  down  the  intermediate  spaces  by 
means  of  a  blunt  tool,  called  a  broach.  This  is  called  broach  channeling-. 
For  this  purpose  a  steam  drilling  machine  is  mounted  iipon  a  hor  bar  resting  upon 
two  pairs  of  legs.  The  hor  bar  is  placed  over  the  intended  row  of  holes,  and  the 
drill  is  slid  along  upon  it  from  one  hole  to  the  next.  In  using  the  broach,  the  rotat- 
ing apparatus  is  thrown  out  of  gear,  so  that  the  edge  of  the  broach  maintains  its 
position  in  line  with  the  row  of  holes. 

Art.  38.  The  Sannders  patent  channeling*  machine,  of  the 
Ingersoll  C!o,  consists  of  a  rock-drilling  machine,  having,  in  place  of  the  usual  drill- 
ing-bit, a  gang  of  tools  consisting  of  a  number  of  chisels,  clamped  together  side  by 
side,  and  thus  forming  a  cutting  tool  about  7  ins  long  by  %  inch  wide.  This  to(i 
has  as  many  cutting-edges  (each  as  long  as  the  tool  is  wide)  as  there  are  chisels. 
The  machine  is  supported  upon  a  carriage,  moving  on  a  track  parallel  with  the 
channel  to  be  cut.  The  tool  is  of  course  not  rotated ;  but  the  rifle-bar,  A,  Fig  4,  is 
employed  to  move  the  carriage  along  the  track  about  an  inch  after  each  blow.  The 
carriage  remains  stationary  while  a  blow  is  being  struck.  Under  favorable  circum- 
stances this  machine  has  cnt  from  80  to  100  sq  ft  of  channel  per  day  of  ten 
hours.  Its  i¥eight,  including  carriage,  is  about  5000  lbs. 

A  valve  is  provided,  by  which,  if  desired,  the  steam  may  be  shut  off  from 
the  piston  on  the  down  stroke,  so  that  said  stroke  may  be  made  with  only  the  iveight 
of  the  piston,  rod,  and  drill. 

Art.  39.  The  Ingersoll  Co  have  a  special  appliance,  designed  by  Mr.  W.  L. 
Saunders,  C  E,.  for  drilling*  and  blasting  rocks  under  urater,  even 
when  they  are  covered  by  a  considerable  depth  of  mud. 

Art.  40.  Air  compressors  for  rock-drills,  as  made  and  used  in  this  coun- 
try, are  mostly  hor,  direct-acting  engines.  That  is,  the  axes  of  the  steam-  and  air- 
cylinders  are  hor;  and  the  piston-rod  passes  directly  from  the  steam-cylinder  into 
the  air-cylinder.  A  fly-wheel  is  attached,  by  a  crank  and  connecting-rod,  to  the 
piston-rod.  Sometimes  the  steam-engine  is  separate  from  the  compressor,  and  the 
power  is  conveyed  to  the  latter  by  belts  or  gearing ;  or  water-power  may  be  used  in 
the  same  way.  The  air  is  forced  into  a  receiver,  which  is  generally  a  plate^iron 
cylinder,  3  or  4  ft  in  diam,  and  5  to  12  ft  long. 

If  the  air-  or  pumping-cylinder  of  the  compressor  is  so  arranged  as  to  take  in  air 
on  one  stroke  only,  and  force  it  out  into  the  receiver  upon  the  return  stroke,  it  ii 
"single-acting."  If,  at  each  stroke,  it  both  takes  in  and  forces  out  air,  it  i« 
*' double-acting."  If  the  compressor  has  only  one  air-cylinder,  it  is  **sin« 
gle."  If  it  has  two,  and  thus  practically  consists  of  two  single  compressors,  it  ift 
**  duplex." 

The  valves  may  be  either  "  poppet"  valves,  held  in  place  by  springs,  an(J 
operated  by  the  pressure  of  the  air  itself;  or  slide  valves,  operated  by  eccentrics 
and  rods,  as  in  steam-engines. 

The  compression  of  the  air  develops  heat.  This  is  removed  either  by  causing 
cold  water  to  circulate  through  the  air-piston,  and  through  jackets  surrounding,th« 
air-cylinder;  or  by  injecting  it  into  the  air-cylinder  iu  the  form  of  spray.  Or  both 
methods  may  be  used  together. 


682 


MACHINE   ROCK-DRILLS. 


Art.  41.  The  following  partial  list  of  Clayton  compressors,  compiled  from 
data  given  by  the  makers,  shows  the  dimensions  and  performance  of 
each.    We  give  also  a  list  of  their  receivers.  » 

CliAYTON  DOUBIiE-ACTING  AIR-COMPRESSORS.  Partial  List. 


Buplex  Direct-acting*  Compressors. 

Diam  of  steam -cylinders ins. 

"      air  "  ins. 

Length  of  stroke ins. 

Number  of  revolutions  per  minute 

Cub  ft  of  free  air  compressed  per  minute Actual. 

Approximate  wt  of  compressor lbs. 

Approx  number  of  rock-drills-   with  3-inch  cyls  sup- 
plied with  air  at  60  to  80  Ibd  per  dq  mch 

Single  Direct-acting*  Compressors. 

Diam  of  steam-cylinder ins. 

'*     air  "         ins. 

Length  of  stroke ins. 

Number  of  revolutions  per  minute 

Cub  ft  of  free  air  compressed  per  minute Actual, 

Approx  wt  of  compressor 

Approx  number  of  rock-drills    with  3-inch  cyls  sup- 
plied with  air  at  60  to  80  lbs  per  sq  inch 


Number,  designating  th 

of  the  ma«hiue. 

1 

2K 

4 

8 

10 

14 

8 

10 

14 

12 

13 

15 

fl20 

100 

100 

\  to 

to 

to 

(140 

130 

120 

136 

210 

438 

3000 

7000 

15000 

2 

4 

8 

8 

10 

14 

8 

10 

14 

12 

13 

15 

(120 

100 

100 

\  to 

to 

to 

(140 

130 

120 

68 

105 

219 

1650 

3850 

8250 

1 

2 

4 

18 

18 
24 

80 
to 
90 
900 
2500O 

18 


18 
18 
24 
80 
to 
90 
450 
13760 


*  The  price  of  a  compressor  alone,  to  be  worked  by  a  separate  steam-engine  or  water-power,  is  of 
course  less  than  that  of  the  above  compressor  and  engine  combined. 


Air-Receivers;   vertical  and  liorizontal. 

Diameter 
Inches. 

Length,        Approximate 
yeet.           weight,  lbs. 

Diameter, 
Inches. 

Length, 
Feet. 

Approximate 
weight,  as. 

33 

30 
36 

40 

f^                     700 

7  [            890 

8  1         1560 
6         (         1600 

1 

40 
40 
40 
40 

8 
10 
11 
12 

1675 
1900 
2000 
2100 

The  Air-Receivers  have  brass-face  pressure-gauge,  glass  water-gauge,  safety-valvei, 
blow-off  valve,  try-cocks,  flanges  and  connections  to  automatic  feed  on  compressor. 


TRACTION. 


683 


TKAOTION. 


Traction  on  common  roads,  and  canals ;  or  the  power  read  to  draw 

vehicles  aud  boats  along  them.    In  connection  with  this  subject  read  the  preceding  and  the  following 

The  following  table  shows  tolerable  approximations  to  the  force  in  fts  per  ton,  reqd  to  draw  a,  8t««B 
•oach  and  passengers,  up  ascents  on  the  Holyhead  turnpike  road  in  England,  (a  fine  road,)  by  horses  ; 
fts  ascertained  by  means  of  a  dynamometer.  The  entire  weight  was  1 J^  tons ;  but  in  the  table,  the 
results  are  given  per  single  ton.     From  the  nature  of  such  cases,  no  great  accuracy  is  attainable. 


Proportional 

Ascent  in  Ft. 

At  4  Miles 

At  6  Miles 

At  8  Miles 

At  10  Miles 

Ascent. 

per  Mile. 

per  Hour. 

per  Hour. 

per  Hour. 

per  Hour. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

lin    ro}4 

340.7 

210 

^  216 

225 

240 

1  "     20 

264. 

196 

•202 

212 

229 

1  "     26 

203.1 

155 

160 

166 

175 

1  "     30 

176. 

137 

142 

147 

154 

1  "     40 

132. 

114 

120 

124 

130 

1  "     64 

82.5 

109 

115 

120 

126 

1  "  118 

44.7 

102 

107 

113 

120 

1  "  138 

38.3 

99 

103 

109 

117 

1  "  156 

33.9 

98 

101 

106 

112 

1  "  245 

21.6 

93 

96 

101 

107 

1  "  600 

8.8 

81 

85 

91 

96 

Level. 

0. 

76 

60 

85 

91 

The  following  results,  most  of  them  with  the  same  instrument,  are  also  in  B)s  per  ton ;  with  a  four- 
wheeled  wagon,  at  a  slow  pace,  on  a  level;  and  the  roads  in  fair  condition. 

On  a  cubical  block  pavement 32  P>s  per  ton....... ....to  50. 

•'      McAdam  road,  of  small  broken  stone 62"     "     ''    probably  to  75. 

"      gravelroad 140"      "     " 

"      Telford  road,  of  small  stone  on  a  paving  of  spawls    46"     "    "  "         "75. 

"      broken  stone,  on  a  bed  of  cement  concrete 46"     "     "  "         "75. 

"      common  earth  roads 200  to  300.   On  a  plank  road  30,  to  50  fl)s. 

Tbe  tractive  power  of  a  borse  diministies  as  bis  speed  in- 
creases; and  perhaps,  within  certain  limits,  say  from  %  to  four  miles  per  hour, 
nearly  in  inverse  proportion  to  it.  Thus,  the  average  traction  of  a  horse,  on  a  level,  and  actually 
pulling  for  10  hours  in  the  day,  may  be  assumed  approximately  as  follows : 

Miles  per  hour.        Lbs.  Traction.                  Miles  per  hour.        Lbs.  Traction. 
H 333.33  214 111.11 

1     250.  2^ 100. 

IH 200.  2% 90.91 

IH K..  166.66  3     83.33 

1% 142.86  SJ^ 71.43 

2     125.  4     62.50 

If  he  works  for  a  smaller  number  of  hours,  his  traction  may  Increase  as  the  hours  diminish  ;  down 
to  about  5  hours  per  day  and  for  speeds  of  about  from  1^  to  3  miles  per  hour.  Thus,  for  5  hours  per 
day  his  traction  at  2^  "miles  per  hour  will  be  200  lbs,  &c.  When  ascending  a  hill,  his  power  dimin. 
ishes  so  rapidly,  from  having  partially  to  raise  his  own  weight,  (which  averages  about  1000  to  1100 
lbs,)  that  up  a  slope  of  5  to  1,  he  can  barely  struggle  along  without  any  load.    On  such  an  ascent, 

he  must  exert  a  force  equal  to  439  fts  per  ton,  or  of  196  fl)s  for 
the  1000  fts  of  his  own  weight.  Assuming  that  on  a  level  piece  of  good  turnpike,  he  would  when  haul- 
ing a  cart  and  load,  together  weighing  1  ton,  have  to  exert  a  traction  of  60  Kis ;  then  on  ascending  a 
hill  of  40  inclination,  (or  1  in  14.3  ;  or  369i^  ft  per  mile,)  he  would  have  to  exert  156  ttis,  against  the 
gravity  of  the  1  ton  :  and  67  lbs,  against  that  of  his  own  weight ;  or  223  Bbs  in  all.  He  may,  for  a  few 
mins,  "exert  without  injury,  about  twice  his  regular  traction.  This  calculation  shows  that  up  a  hill 
of  40,  an  average  horse  is  fully  tasked  in  drawing  a  total  load  of  one  ton  ;  and  should,  therefore,  be 
allowed,  in  such  a  case,  to  choose  his  own  gait ;  and  to  rest  at  short  intervals.  A  fair  load  for  a  single 
horse  with  a  cart,  at  a  variable  walking  pace,  workiug  10  hours  per  day,  on  a  common  undulating 
road  in  good  order,  is  about  half  a  ton,  in  addition  to  the  cart,  which  will  be  about  half  a  ton  more. 
With  two  horses  to  this  same  cart,  the  load  alone  may  be  about  13^  tons. 

Rem.     Since  the   action  of  gravity  is   the   same  on   good  roads   and  bad   ones,   it  follows   that 

ascents  become  more  objectionable  the  better  the  road  is. 

Thus,  on  an  ascent  of  2°,  or  184.4  ft  per  mile,  gravity  alone  requires  a  traction  of  78  Bbs  per  ton; 


684 


TRACTION. 


which  is  about  10  times  that  on  a  level  railroad  at  6  miles  an  hour ;  but  only  about  equal  to  that  on  a 
level  common  turnpike  road,  at  the  same  speed.  Therefore,  (to  speak  somewhat  at  random,)  it  would 
require  10  locomotives  instead  of  1 ;  but  only  2  horses  instead  of  1.  A  grade  of  1  in  35  ;  or  150  ft  to  a 
mile;  or  1°  38',  is  about  the  steepest  that  permits  horses  to  be  driven  down  a  hard  smooth  road,  in  a 
fast  trot,  without  danger.  It  should,  therefore,  not  be  exceeded  except  when  absolutely  necessary, 
especially  on  turnpikes.  ^ 

On  canals  and  otiier  waters,  the  liquid  ia  the  resisting  medium  that 

takes  the  place  of  friction  on  level  roads.  But  unlike  friction,  its  resistance  varies  as  the  squares  of 
the  vels  ;  at    least  from  the  vel  of  2  ft  per  sec,  or  1.364  miles  per  hour ;  tc 

that  of  n}4  ft  per  sec,  or  7.84  m  per  h.  As  the  speed  falls  below  13^  m  per  h,  the  resistance  varies  less 
and  less  rapidly ;  and  this  is  the  case  whether  the  moved  body  floats  partly  above  the  surface ;  or  is 
entirely  immersed.  In  towing  along  stagnant  canals,  &c,  the  vel  is  usually  from  1  to  2^-2  m  per  h; 
for  freight  most  frequently  from  1}^  to  2.  Less  force  is  required  to  tow  a  boat  at  say  2  m  per  h,  where 
there  is  no  current,  than  at  say  l>i  m  per  h,  against  a  current  of  %  m  per  h,  because  in  the  last  case 
the  boat  has  to  be  lifted  up  the  very  gradual  inclined  plane  or  slope  which  produces  the  current. 
The  force  required"  to  tow  a  boat  along  a  canal  depends  greatly  upon  the  comparative  transverse 
sectional  areas  of  the  channel,  and  of  the  immersed  portion  of  the  boat.  When  the  width  of  a  canal 
at  water-line  is  at  least  4  times  that  of  the  boat ;  and  the  area  of  its  transverse  section  as  great  as  at  least 
614  times  that  of  the  immersed  transverse  section  of  the  boat,  the  towing  at  usual  canal  vels  will  b« 
about  as  easy  as  in  wider  and  deeper  water.  VTith  less  dimensions,  it  becomes  more  difficult.  (D'Au- 
buisson.)  Much  also  depends  on  the  shape  of  the  bow  and  other  parts  of  the  boat ;  and  on  the  propor- 
tion of  its  length  to  its  breadth  and  depth.  Hence  it  is  seen  that  the  mere  weight  of  the  load  is  by  no 
means  so  controlling  an  element  as  it  is  on  land.  The  whole  subject,  however,  is  too  intricate  to  be 
treated  of  here.  Morin  states  that  naval  constructors  estimate  the  resistance  to  sailing  and  steam 
vessels  at  sea,  at  but  from  about  .5  to  .7  of  a  lb  for  every  sq  ft  of  immersed  transverse  section,  when 
the  vel  is  3  ft  per  sec,  or  2.046  miles  per  hour.     It  is  far  greater  on  canals. 

On  the  Schnylkill  Navigation  of  Pennsylvania,  of  mixed  canal 

and  slack  water,  for  108  miles,  the  regular  load  for  3  horses  or  mules,  is  a  boat  of  very  full  build ;  and  na 
keel;  100  ft  long,  173-^  ft  beam;  and  8  ft  depth  of  hold  ;  drawing  b}4  ft  when  loaded.*  Weight  of  boa^ 
about  65  tons;  load  175  tons  of  coal,  (2240  Bbs;)  total  weight  240  tons,  or  80  tons  per  horse  or  mule. 
On  the  down  trip  with  the  loaded  boats,  for  4  days,  the  animals  are  at  work,  actually  towing,  (except 
at  the  locks,)  for  18  hours  out  of  the  24  j  thus  exceeding  by  far  the  limits  of  time  usually  allowed  for 
continuous  effort.  ^ 

On  the  canal  sections,  (which  have  60  ft  water-line ;  and  6  ft  depth,)  the  speed  is\%  miles  per  hour ; 
and  on  the  deep  wide  pools,  2  miles. 

On  the  up  trip  with  the  empty  65-ton  boats,  the  average  speed  is  about  2J^  miles  per  hour.  The 
empty  boats  draw  16  to  18  ins  water ;  and  frequently  keep  on  without  stopping  to  rest  day  or  night 
through  the  entire  distance  of  108  miles.  The  animals  generally  have  2  or  3  days'  rest  at  each  end  of 
the  trip ;  but  are  materially  deteriorated  at  the  end  of  the  boating  season. 

If  our  preceding  assumption  of  143  2)s  traction  of  a  horse  at  1%  miles  per  hour,  is  correct,  the 

traction  of  the  loaded  boats  on  the  canal  sections  is  • =1.83  fts  per  ton. 

80  tons 

The  intelligent  engineer  and  superintendent  of  the  Sch  Nav,  James  F  Smith,  gives  as  the  results 
of  his  own  extensive  observation,  that  one  of  these  large  boats  loaded  (240  tons  in  all)  may,  without 
distressing  the  animals,  be  drawn  along  the  canal  sections,  for  10  hours  per  day,  as  follows :  By  one 
average  horse  or  mule,  at  the  rate  of  1  mile ;  by  two  animals,  at  \%  miles ;  and  by  three,  at  1%  miles 
per  hour.  When  four  animals  are  used  the  gain  of  time  is  very  trifling.  At  a  time  of  rivalry  among 
the  boatmen,  one  of  them  used  8  horses  ;  but  with  these  could  not  exceed  2)4  miles  per  hour  in  the 
canal  portions.  Two  or  more  horses  together  cannot  for  hours  pull  as  much  as  when  working  sepa- 
rately. 

If  our  preceding  short  table  of  the  traction  of  a  horse  at  diff  vels  for  10  hours  is  correct,  then  the 
traction  of  the  above  loaded  coal  boats  (240  tons)  on  the  canal  sections  of  the  navigation,  is  as  follows : 
The  last  column  shows  the  traction  in  H)s  per  sq  ft  of  area  of  immersed  transverse  section  where  largest; 
viz,  about  95  sq  ft. 

Horses.             Miles  per  Hour.                                                Lbs.  per  Ton.            Lbs.  per  Sq  Ft. 
1 1 f}§ 1.04 2.63 


•IJ^ If  4 1-39 3.50 


,1^ m- 


3  on  pools 2    f  :^-§ 1.56 3.95 

'  8 23^ |J§ 3.33 8.42 

3up-trip 21^ 3_0J) ^gj J2.50 

liacbine  Canal,  Canada,  120  ft  wide  at  water-line;  80  ft  at  bottom ;  depth 

on  mitre  sills  9  ft ;  6  horses  tow  loaded  schooners  with  ease. 

Before  the  enlargement  of  the  Erie  Canal,t  its  dimensions  were  40  ft  water-line  ;  28  ft  bottom ; 
4  ft  depth  of  water.  The  average  weight  of  the  boats  was  about  30  tons.  With  75  tons  of  load,  or  105 
tons  total,  they  were  towed  by  2  horses,  at  the  rate  of  about  2  miles  per  hour  ;  which  by  our  table  gives 
a  traction  of  nearly  2.4  lbs  per  ton.  The  boats  were  about  80  ft  long  ;  14  ft  beam  ;  full  'i%  ft  draught 
loaded ;  hence  the  traction  by  our  table  would  be  about  5.7  lbs  per  sq  ft  of  immersed  transverse  section. 

*  Cost  of  boats,  1884,  (Schuylkill  Canal)  about  $1800,    Annual  repairs  about 

$85.     Boats  last  16  to  20  years.     Length,  102  ftj  beam,  17J4  ft;   draft,  13^  to  53^  ft;  capacity,  180 
tons ;  weight,  about  58  tons  ;  speed,  with  3  mules,  \%  miles  per  hour. 

t  Length  363  miles ;  cost  $19680  per  mile.  The  enlarged  canal  has  70  ft;  42  ft ;  and  7  ft  of  water ; 
and  cost  $90800  per  mile  for  the  enlargement  only.  The  cost  of  the  several  canals  in  Pennsylvania 
has  ranged  between  $23000  and  $50000  per  mile. 


ANIMAL   POWER.  685 


While,  for  82-ton  loaded  boats  on  a  smaller  canal,  (the  boats  nearly  touching  bottom,)  the  traction  at 
\%  miles,  would  be  3J^  fts  per  ton ;  or  about  twice  as  great  as  the  above  1.78  fts.  Ii  also  would  be  5.7 
tts  per  sq  ft  of  immersed  section. 


AUIMAL  FOWEB. 

Art.  1.     So  far  as  regards  horses,  this  subject  has  been  partially  considered 

tinder  the  preceding  head.  Traction.  All  estimates  on  this  subject  must  to  a  certain  extent  be  vague, 
owing  to  the  difiF  strengths  and  speeds  of  animals  of  the  Same  kind ;  as  well  as  to  the  extent  of  their 
training  to  any  particular  kind  of  work.  Authorities  on  the  subject  differ  widely  ;  and  sometimes 
express  themselves  in  a  loose  manner  that  throws  doubt  on  their  meaning.  We  believe,  however, 
that  the  following  will  be  found  to  be  as  close  approximation  to  practical  averages  as  the  nature  of 
the  case  admits  of  with  our  present  imperfect  knowledge.  We  suppose  a  good  average  trained  horse, 
weighing  not  less  than  about  J^  a  ton,  well  fed  and  treated.  Such  a  one,  when  actually  walking  for 
10  hours  a  day,  at  the  rate  of  23^  miles  per  hour,  on  a  good  level  road,  such  as  the  tow-path  of  a  canal, 

or  a  circular  horsepath,*  Can  exert  a  continuous  pull,  draug'bt,  poller, 
or  traction,  of  100  lbs. 

Now,  2}4  miles  per  hour,  is  220  ft  per  min,  or  3%  ft  per  sec ;  and  since  10  hours  contain  600  min, 
his  day's  work  of  actual  hauling  on  a  level,  at  that  speed,  amounts  to 
min  ft  lbs 

600  X  220  X  100  =  13  200000  ft-fts  per  day. 
Or,  22000  ft-fts  per  min.  or  366%  ft-ttis  per  sect  Which  means  that  he  exerts  force  enough  during  the 
day  to  lift  13  200  000  Jbs  1  foot  high ;  or  1  320  000  lbs  10  feet  high ;  or  132  000  lbs  100  ft  high,  &c.  He  may 
exert  this  force  either  in  traction  (hauling)  or  in  lifting  loads.  If  he  has  to  raise  a  small  load  to  a 
great  height,  the  machinery  through  which  he  does  it  must  be  so  geared  as  to  gain  speed,  at  the  loss 
(commonly  but  improperly'  so  expressed)  of  power.  Whether  he  lifts  the  great  weight  through  a 
small  height,  or  the  small  weight  through  a  great  height,  he  exerts  precisely  the  same  amount  of 
force  or  power. 

Experience  shows  that  within  the  limits  of  5  and  10  hours  per  day,  (the  speed  remaining  the  ■ane,') 

tlie  draft  of  a  liorse  may  be  increased  in  about  the  same  pro- 
portion as  the  time  is  diminished  ;  so  that  when  working  from  5  to  lO  hours 
per  day,  it  will  be  about  as  shown  in  the  following  table.  Hence,  the  total  amount  of  13  200  000  ft-fti 
per  day  may  be  accomplished,  whether  the  horse  is  at  work  5,  6,  or  8,  &c,  hours  per  day.+  This,  of 
course,  supposes  him  to  be  actually  lifting  or  hauling  aU  the  time;  and  makes  no  allowance  for  stop- 
pages for  any  purpose. 

Table  of  draft  of  a  horse,  at  2}4  miles  per  honr,  on  a  level. 

Hours  per  day.  Lbs.  Hours  per  day.  Lbs. 

10 100  7  142^ 

9  lllj  6 166^ 

8 125  5  200 

Ilxperience  also  shows  that  at  speeds  between  %  and  4 
miles  an  hour,  his  force  or  draug-ht  will  be  inversely  in  pro* 
portion  to  his  speed.  Thus,  at  2  miles  an  hour,  for  10  hours  of  the  day,  his 
draught  will  be 

miles      miles  Jbs  Ybs 

2    :    2J4    :  :    100    :     125  draught. 
Atl}4  miles,  it  would  be  166%  B)s ;  at  3  miles, -83 J^  fts ;  and  at  4  miles,  62J^  lbs  ;  as  per  table  in 
Traction, 

Therefore,  in  this  case  also,  the  entire  amount  of  his  day's  work  remains  the  same  ;  §  and  within 

*  To  enable  a  horse  to  work  with  ease  in  a  circular  horse- walk,  its  diaro 

•hould  not  be  less  than  25  ft;  30  or  35  would  be  still  better. 

t  A  nominal  horse-power  is  33000  ft-Ibs  per  minute;  this  being  the  rate 
assumed  by  Boulton  and  Watt  in  selling  their  engines ;  so  that  purchasers  wishing  to  substitute 
steam  for  horses,  should  not  be  disappointed.  Their  assumption  can  be  carried  out  by  a  very  strong 
horse  day  after  day  for  8  or  10  hours ;  but  as  the  engine  can  work  day  and  ni>?ht  for  months  without 
stopping,  which  a  "horse  cannot,  it  is  plain  that  a  one-horse  engine  can  do  much  more  work  than  any 
one  such  horse.  Hence  many  object  to  the  term  horse-power  as  applied  to  engines  ;  but  since  every- 
body understands  its  plain  meaning,  and  such  a  term  is  convenient,  it  is  not  in  fact  objectionable. 
Boulton  and  Watt  meant  that  a  one-horse  engine  would  at  any  moment  perform  the  work  of  a  very 
strong  horse.    An  average  horse  will  do  but  22000  ft-lbs  per  min. 

t  It  is  plain  that  although  the  day's  labor  will  be  the  same,  that  of  an  hour,  or  of  a  min,  will  Tary 
with  the  number  of  hours  taken  as  a  day's  work.  It  must  be  remembered  that  a  working  day  of  a 
given  number  of  hours,  by  no  means  implies,  in  every  case,  that  number  of  hours. of  actual  work; 
but  includes  intermissions  and  rests. 

g  This  remark  about  speed  will  not  apply  to  loads  towed 
throu@:h  the  water.     Thus,  if  his  draught  at  2  miles  an  hour  be  125  fts ;  and 

at  4  miles,  62^  lbs  ;  he  will  on  land  draw  loads  in  these  proportions  ;  but  in  hauling' a  boat  throvgh 
the  water  at  the  greater  speed,  he  haS  to  encounter  the  increased  resistanceof  the  it^a^er  itself ;  which 
resistance  at  4  miles  is  much  more  than  twice  as  great  as  at  2  miles ;  probablv  4  times  as  great. 
Therefore,  at  4  miles  on  a  canal,  his  draught  of  623^  fits  would  not  suffice  for  a  load  half  as  great  as 
be  could  tow  with  his  draft  of  125  B)8  at  2  miles. 


686 


ANIMAL   POWER. 


all  the  foregoing  Iitnits  of  hours  and  speed,  may  be  practically  taken  to  be  about  13  200000  ft-S>8  per 
day ;  or  22000  ft-fts  per  min  of  a  day  of  10  hours.  But  it  does  not  follow  that  the  horse  can  alwayf 
in  practice  actually  lift  loads  at  that  rate ;  because  generally  a  part  of  his  power  is  expended  in 
overcoming  the  friction  of  the  machinery  which  he  puts  in  motion  ;  and  moreover,  the  nature  of  th« 
work  may  require  him  to  stop  frequeutly  ;  so  that  in  a  working  day  of  8  or  10  hours,  the  horse  may 
not  actually  be  at  work  more  than  5,  6,  or  7  hours. 

As  a  rough  approximation,  to  allow  for  the  waste  of  force  in  overcoming  the  friction  of  hoisting 
machinery,  and  the  weight  of  the  hoisting  chains,  buckets,  &c,  we  may  say  that  tlie  lisel'lll 

or  paying  daily  net  work  of  a  horse,  in  boisting-  by  a  com- 
mon g^in,  is  about  10  000000  ft-fts.  That  is,  he  will  raise  equivalent  to  10000000  fts  net  of 
water,  or  ore,  <fec,  1  foot.  The  load  which  he  can  raise  at  once,  including  chains,  bucket,  and  ao 
allowance  for  friction,  will  be  as  much  greater  than  his  own  direct  force,  as  the  diam  of  the  horse- 
walk  is  greater  than  that  of  the  winding  drum;  and  it  will  move  that  much  slower  than  he  does. 
His  own  direct  force  will  vary  according  to  the  number  of  hours  per  day  that  he  may  be  required  to 
work,  as  in  the  foregoing  table.  With  these  data,  the  size  of  the  buckets  can  be  decided  on ;  and  of 
these  there  should  be  at  least  two,  so  that  the  empty  one  at  the  bottom  may  be  filled  while  the  full  one 
at  top  is  being  emptied ;  so  as  to  save  time.     The  same  when  the  work  is  done  by  men. 

Art.  2.  A  practised  laborer  lianlin^  along:  a  level  road,  by 
a  rope  over  his  shoulders;  or  in  a  circular  path,  pushing  before  him  a 

hor  lever,  at  a  speed  of  from  1  ^  to  3  miles  per  hour,  exerts  about  )i  part  as  much  force  as  a  horse; 
•r  2  200  000  ft-Ibs  per  day ;  or  3666%  ft-fts  per  min  of  a  day  of  10  hours  of  actual  hauling  or  pushing. 

But  laborers  frequently  have  to  work  under  circumstances  less  advantageous  for  the»  exertion  of 
their  force  than  when  hauling  or  pushing  in  the  manner  just  alluded  to  ;  and  in  such  cases  they  cannot 
do  as  much  per  day.  Thus  in  turning  a  winch  or  crank  like  that  of  a  grindstone,  or  of  a  crane,  the 
continual  bending  of  the  body,  and  motion  of  the  arms,  is  more  fatiguing.  The  Size  Of  a 
IVinch  should  not  exceed  18  ins,  or  the  rad  of  a  circle  of  3  ft  diam ;  and  against 
it  a  laborer  can  exert  a  force  of  about  16  fts,  at  a  vel  of  2}^  ft  per  sec,  or  150  ft  per  min,  making  very 
nearly  16  turns  per  min ;  for  8  hours  per  day.  To  these  8  hours  an  addition  must  be  made  of  about 
H  part,  for  short  rests.  Or  if  a  working  day  is  taken  at  8,  or  10,  &c,  hours,  ^  part  must  generally  be 
taken  from  it  for  such  rests.  On  the  foregoing  data  an  hour's  work  of  60  min  of  actual  hoisting 
would  be 

ft)3  ft  min 

16  X  150  X  60  =  144000  ft-fts ; 
or,  deducting  4-  part  for  rests,  115200  ft-fts  per  hour  of  time,  including  rests.    In  practice,  however, 
a  further  deduction  must  be  made  for  the  fric  of  the  machine,  and  for  the  wt  of  the  hoisting  chains; 
and  in  case  of  raising  water,  stone,  ore,  &c,  from  pits,  for  the  wt  of  the  buckets  also.     As  a  rough 
average  we  may  assume  that  these  will  leave  but  100  000  ft-lbs  of  paying,  or  useful  work  per  hour; 

that  is,  that  a  man  at  a  winch  w^ill  actually  lift  equivalent  to 
100000  lbs  of  water,  ore,  dec,  1  foot  hig-h  per  hour's  time,  in- 
cluding rests.  This  is  equal  to  1666%  ft-fts  per  min  of  a  day  of  10  hours,  including  rests. 
Therefore,  in  a  day  of  10  working  hours  he  would  raise  1  000000  fts  net,  1  foot  high ;  Or  jUSt    1 

part  of  what  a  horse  would  do  with  a  gin  in  the  same  time.  We  have 
before  seen  that  in  hauling  along  a  level  road,  he  can  at  a  slow  pace  perform  about  %  of  the  daily 
duty  of  a  horse.  He  may  also  work  the  winch  with  greater  force,  say  up  to  30  or  even  40  fts ;  but 
he  will  do  it  at  a  proportionately  slower  rate ;  thus,  accomplishing  only  the  same  daily  duty. 
With  a  gin,  like  those  for  horses,  but  lighter,  with  2  or  more  buckets,  a  prac- 
tised laborer  will  in  a  working  day  of  10  hours,  raise  from  1  200000  to  1 400000  ft-fts  net  of  water,  ore, 
&c.  With  a  shallow  well  or  pit,  more  time  is  lost  in  emptying  buckets  than  in  a  deep  one ;  but  the 
deep  one  will  require  a.greater  wtof  rope.  To  save  time  in  all  such  operations  on  a  large  scale,  there 
should  be  at  least  two  buckets ;  the  empty  one  to  be  filled  while  thj  full  one  is  being  emptied.  It  is 
also  best  to  employ  2  or  more  men  to  hoist  at  the  same  time,  by  winches,  at  both  ends  of  the  axis; 
and  the  men  will  work  with  more  ease  if  the  winches  are  at  right  angles  to  each  other.  Each  winch 
handle  may  be  long  enough  for  2  or  3  men.  An  extra  man  should  be  emploved  to  empty  the  buckets. 
He  may  take  turns  with  the  hoisters.     The  same  remarks  apply  in  some  of  the  following  cases. 

On  a  treadwheel  a  practised  laborer  will  do  about  40  per  cent  more  dailj 
duty  than  at  a  winch  ;  or  in  a  working  day  *  of  10  hours,  including  rests,  he  will  do  about  1 400000  ft- 
lbs.  And  he  can  do  this  whether  he  works  at  the  outer  circumf  of  the  wheel,  stepping  upon  foot- 
boards, or  tread-boards,  on  a  level  with  its  axis  ;  or  walks  inside  of  it  near  its  bottom.  In  both  cases 
he  acts  by  his  wt,  usually  about  130  to  140  lbs ;  and  not  by  the  muscular  strength  of  his  arms.  When 
at  the  level  of  the  axis,  his  wt  acts  more  directly  than  when  he  walks  on  the  bottom  of  the  wheel; 
but  in  the  flr^t  case  he  has  to  perform  a  slow  and  fatiguing  duty  resembling  that  of  walking  up  a 
continuous  flight  of  steps;  while  in  the  second  he  has  as  it  were  merely  to  ascend  a  very  slightly  in- 
clined plane;  which  he  can  do  much  more  rapidly  for  hours,  with  comparatively  little  fatigue:' and 
this  rapidity  compensates  for  the  less  direct  action  of  his  wt.  Therefore,  in  either  case,  as  experience 
has  shown,  he  accomplishes  about  the  same  amount  of  daily  duty.  Treadwheels  may  be  from  5  to  26 
ft  in  diam,  according  to  the  nature  of  the  work.  They  are  generally  worked  by  several  men  at  once, 
and  may  at  times  be  advantageously  used  in  pile-driving,  as  well  as  in  hoisting  water,  stone,  &c. 

By  a  good  common  pump,  properly  proportioned,  a  practised  laborei 

will  in  a  day  of  10  working  hours,  raise  about  1 000000  ft-fts  of  water,  net.t 

Bailing  with  a  light  bucket  or  scoop,  he  can  accomplish  about 

200000  ft-lbs  net  of  water.  By  a  buCket  and  SWape,  (a  long  lever  rocking  vertically^ 
and  weighted  at  one  end  so  as  to  balance  the  full  bucket  hung  from  the  other ;  often  seen  at  country 

*The  working  day  must  be  understood  to  include  necessary' rests,  and  such  intermissions  as  the 
nature  of  the  work  demands  ;  but  does  not  include  time  lost  at  meals.  A  tcorking  day  of  10  hours 
may,  therefore,  have  but  8,  7,  or  6,  «fec  hours  of  actual  labor.  This  will  be  understood  when  we  here- 
after speak  of  a  working  day,  or  simply  a  day. 

t  Desagulier's  dstimates  of  daily  work  of  men  aod  horses  exceed  the  abore,  but  are  entirely  too  great 


ANIMAL  POWER.  687 

wells,)  600000  to 800  000.   In  the  last  he  has  only  to  pall  down  the  empty  bucket,  and  thereby  raise  th» 

counterweight.   By  2  biickets  at  the  ends  of  a  rope  suspended  over 

a  pulley,  500000  to  eOOOOO.    Here  he  works  the  buckets  by  pulling  the  rope  by  hand. 

By  a  tympau,  or  tynipauum,'^  worked  by  a  ireadwheel,  about  1 200000 
to  1400  000. 

By  a  Persian  wheel.f  a  ebain-pump,  a  cbain  of  buckets^  or 
an  Arcbiinedes  screw,  all  worked  by  a  treadwheel,  from  800  000  to  1000000 
ft-B)8.  Of  these  four,  the  tirst  three  lose  useful  effect  by  either  spilling,  leaking,  or  the  necessity  for 
raising  the  water  to  a  level  somewhat  higher  than  that  at  which  it  is  discharged. 

When  any  of  the  five  foregoing  machines  are  worked  by  men  at  winches,  the  result  will  be  about 
J^  less  than^by  treadwheels.  They  are  all  frequently  worked  also  by  either  steam, water,  or  horse-power. 

By  walking:  backward  and  forward,  on  a  lever  wbicb  rocks 
on  its  center,  a  man  may,  according  to  Robison's  Mech  Philosophy,  perform  a 
much  gi-eater  duty  than  by  any  of  the  preceding  modes.  He  states  that  a  young  man  weighing  135 
fts,  and  loaded  with  30  fts  in  addition,  worked  in  this  manner  for  10  hours  a  day  without  fatigue; 
and  raised  9}4  cubic  feet  of  water,  lli4  ft  high  per  min.  This  is  equal  to  3  984  000  ft-fts  per  day  of  18 
hours ;  or  66i0  ft-H»s  per  min  ;  or  nearly  -^  of  the  net  daily  work  of  a  horse  in  a  gin. 

A  laborer  standinpr  still,  can  barely  sustain  for  a  few  min,  a  load  of  100 

fiis,  by  a  rope  over  his  shoulder,  and  thence  passing  off  hor  over  a  pulley.  And  scarcely  as  much, 
when'(facing  the  load  and  pulley)  he  holds  the  end  of  a  hor  rope  with  his  hands  before  him.  He  can- 
Bot  push  hor  with  his  hands  at  the  height  of  his  shoulders,  with  more  than  about  30  lbs  force. 

"Weisbach  states  from  his  own  observation,  that  4  practised  men  raised  a  dolly  (a  wooden  beetle 
or  rammer,  of  wood;  with  4  hor  projecting  round  bars  for  handles)  weighing  120  ft>s,  4  ft  high,  at  th9 
rate  of  34  times  per  min,  for  4}^  min  ;  and  then  rested  for  43^  min  ;  and  so  on  alternately  througl) 
the  10  hours  of  their  working  day.  Therefore,  5  of  these  hours  were  lost  in  rests;  and  the  duty  per. 
formed  by  each  man  during  the  other  5  hours,  or  300  mins,  was 

120X4X34X300^^^^^^^^^^ 

In  tbe  old  mode  of  driving;  piles,  where  the  ram  of  400  to  1200  Tb» 

suspended  from  a  pulley,  was  raised  by  10  to  40  men  pulling  at  separate  cords,  from  35  to  40  fts  of  th« 
ram  were  allotted  to  each  man,  to  be  lifted  from  12  to  18  times  per  min,  to  a  height  of  3%  to  4%  feet 
each  time,  for  about  3  min  at  a  spell,  and  then  3  min  rest.  It  was  very  laborious ;  and  the  gangs  had 
to  be  changed  about  /lourly,  after  performing  but  J^  an  hour's  actual  labor. 

Hauling-  by  borses.  See  Traction.  When  working  all  day,  say  10  working 
hours,  the  average  rate  at  which  a  horse  walks  while  hauling  a  full  load,  and  while  returning  with 
the  empty  vehicle,  is  about  2  to  2J4  miles  per  hour;  but  to  allow  for  stoppages  to  rest,  &c,  it  is  safest 
to  take  it  at  but  about  1.8  miles  per  hour,  or  160  ft  per  min.  The  time  lost  on  each  trip,  in  loading 
and  unloading,  may  usually  be  taken  at  about  15  min.  Therefore,  to  find  the  number  of  loads  that  can 
be  hauled  to  any  given  dist  in  a  day,  first  find  the  time  in  min  reqd  in  hauling  one  load,  and  return- 
ing empty.  Thus :  div  twice  the  dist  in  ft  to  which  the  load  is  to  be  hauled  ;  or  in  other  words,  div 
the  length  in  ft,  of  the  round  trip,  by  160  ft.  The  quot  is  the  number  of  min  that  the  horse  is  in  mo- 
tion during  each  round  trip.  To  this  quot  add  15  min  lost  each  trip  while  loading  and  unloading  ;  the 
sum  is  the  total  time  in  min  occupied  by  each  round  trip.  Div  the  number  of  min  in  a  working  day 
(600  min  in  a  day  of  10  working  hours)  by  this  number  of  min  reqd  for  each  trip ;  the  quot  will  be  the 
number  of  trips,  or  of  loads  hauled  per  day. 

Ex.   How  many  loads  will  a  horse  haul  to  a  dist  of  960  ft.  in  a  day  of  10  working  hours,  or  600  min? 

Here,  960  X  2  =  1920  ft  of  round  trip  at  each  load.     And  -  -~  =  12  min,  occupied  in  walking.    And 

,      Jr\,      600  min  in  10  hours 

12-4-15  in  loading,  &c)=:27  mm  reqd  for  each  load.     Finally,  -—   =  ^ — : cr22.2.  or 

^  s.        '  .  H  .T      27  min  per  trip 

say  22  trips ,  or  loads  hauled  per  day. 

Table  of  number  of  loads  hauled  per  day  of  10  working 
hours.  The  first  col  is  the  distance  to  which  the  load  is  actually  hauled  ;  or  half 
the  length  of  the  round  trip.  The  cost  of  hauling  per  load,  is  supposed  to  be  for  one-horse  carts  ;  the 
driver  doing  the  loading  and  unloading ;  rating  the  expense  of  horse,  cart,  and  driver  at  $2  per  day. 

*  The  tympan  revolves  on  a  hor  shaft;  and  is  a  kind  of  large  wheel,  the  spokes,  arms,  or  radii  of 
which  are  gutters,,  troughs,  or  pipes,  which  at  their  outer  ends  terminate  in  scoops,  which  dip  into 
the  water.  As  the  water  is  gradually  raised,  it  flows  along  the  arms  of  the  wheel  to  its  axis,  where 
It  is  dischd.  The  scoop  wheel  is  a  modification  of  it.  It  is  an  admirable  machine  for  raising  large 
quantities  of  water  to  moderate  heights.  We  cannot  go  into  any  detail  respecting  this  and  other 
hydraulic  machines. 

t  A  kind  of  large  wheel  with  buckets  or  pots  at  the  ends  of  its  radiating  arms  ;  revolves  on  a  hor 
axis;  discharges  at  top.  The  buckets  are  attached  loosely,  so  as  to  hang  vert,  and  thus  avoid  spill- 
ing until  they  arrive  at  the  proper  point,  where  they  come  into  contact  with  a  contrivance  for  tilting 
and  emptying  them.  The  noria  is  similar,  except  that  the  buckets  are  firmly  held  in  place,  and  thus 
spill  much  water.     It  is  therefore  inferior  to  the  Persian  wheel. 

t  An  endless  revolvinp;  vert  chain  of  buckets.  D'Aubuisgon  and  some  others  erroneously  call  this 
the  Doria.    It  is  an  effective  machine. 


688 


ANIMAIv    POWER. 


Dist. 

No.  of 

Cost  per 

Dist. 

No.  of 

Cost  per 

Dist. 

No.  of 

Cost  per 

Feet. 

Loads. 

Load. 

Feet. 

Loads. 

Load. 

Miles. 

Loads. 

Load. 

Cts. 

Cts. 

Cts. 

50 

38 

5.26 

1500 

18 

11.11 

1 

7 

28.57 

100 

37 

5.il 

2000 

15 

13.3:J 

iH 

6 

33.33 

200 

34 

5.88 

2500 

13 

15.39 

i}i 

5 

40.00 

300 

32 

6.25 

3000 

11 

18.18 

2 

4 

50.00 

400 

30 

6.67 

3500 

10 

20.00 

3 

3 

66.67 

600 

27 

7.41 

4000 

9 

22.22 

4 

2 

100.00 

1000 

22 

9.09 

5000 

7 

28.57 

9 

1 

200.00 

If  the  loading  and  unloading  is  such  as  cannot  be  done  by  the  driver  alone;  but  requires  the  help 
of  cranes,  or  other  machinery,  an  addition  of  from  10  to  50  cts  per  load  may  become  necessary.  Haul- 
ing can  generally  be  more  cheaply  done  by  using  2  or  3  horses,  and  one  driver,  to  a  vehicle.  The  neat 
load  per  horse,  in  addition  to  the  vehicle,  will  usually  be  from  J^  to  1  ton,  depending  on  the  condition, 
and  grades  of  the  road.     From  13  to  15  cub  ft  of  solid  stone ;  or  from  23  to  27  cub  feet  of  broken  stone, 

make  1  ton.    In  estimating^  for  hauling-  roug-b  quarry  stone  for 

drains,  culverts,  <feC,  bear  in  mind  that  each  cub  yard  of  common  scabbled  rubble 
masonry,  requires  the  hauling  of  about  1.2  cub  yds  of  the  stone' as  usually  piled  up  for  sale  in  the 
quarry  ;  or  about  %  of  a  cub  yd  of  the  original  rock  in  place.     A  CUb  yd  Of  SOlid  StOne, 

i¥hen  broken  into  pieces,  usually  occupies  about  1.9  cub  yds 

perfectly  loose  ;  or  about  1%  when  piled  up.  A  strong  cart  for  stone  hauling,  will  weigh 
about  %  ton  ;  or  1500  lbs  ;  and  will  hold  stone  enough  for  a  perch  of  rubble  masonry  ;  or  say  1,2  pers 
of  the  rough  stone  in  piles.     The  average  weight  of  a  good  working  horse  is  about  3^  a  ton. 

Morin  gives  tlie  following  results  from  careful  experiments  made  bj 
him  for  the  French  Government.  The  draft  of  the  same  wheeled  vehicle  on  a  road,  may  in  practice 
be  considered  to  be, 

1st.    On  hard  turnpikes,  and  pavements;  in  proportion  to  th^ 

loads  ;  inversely  as  the  diams  of  the  wheels  ;  and  nearly  independent  of  the  width  of  tire.  It  increases 
to  uncertain  extents  with  the  inequalities  of  the  road  ;  the  stiffness  (want  of  spring)  of  the  vehicle ; 
and  the  speed ;  (considerably  less  than  as  the  square  roots  of  the  last.) 

2d.  On  soft  roads,  the  draft  is  less  witti  Tvide  tires  tban 
ivitli  narrower  ones ;  and  for  forming  purposes  lie  recommends  a  width  cd 
4  ins.    "With  speeds  from  a  walk  to  a  fast  trot,  the  draft  does  Bot  vary  sensibly. 


TRUSSES.  689 


TEUSSES. 

INTRODUCTION. 

General  Principles. 

1.  Truss  Design  a  Specialty.  The  design,  construction  and  erection 
of  trusses  have  become  a  specialty,  to  which  persons  confine  themselves  more 
or  less  exclusively,  and  thus  attain  a  degree  of  expertness  beyond  the  reach 
of  the  general  engineer.*  The  latter,  however,  should  have  a  knowledge 
of  the  subject,  sufficient  at  least  to  enable  him  to  form  a  well-grounded  opin- 
ion of  the  general  merits  of  a  design  and  to  guard  him  against  the  adoption 
of  one  involving  serious  imperfections.  In  a  volume  like  this  we  can  discuss 
only  general  principles. 

2.  The  Truss  Principle.  Theoretically,  a  truss  consists  of  a  number 
of  straight  bars,  joined,  near  their  ends,  by  perfectly  flexible  joints,  loaded 
only  at  these  joints,  and  so  arranged  that  all  its  internal  stresses  are  sus- 
tained by  its  members,  and  only  the  vertical  f  pressures,  due  to  the  weights 
of  the  truss  and  its  load,  are  transmitted  to  the  abutments. 

3.  Distinction  between  Beams  and  Trusses.  When  a  solid  beam 
(Fig.  2,  ^  7,  Transverse  Strength)  bends,  under  its  own  weight  or  under  that 
of  its  load,  all  the  fibers  above  the  neutral  axis  are  compressed,  while  all 
those  below  are  extended ;  and  the  resulting  change  of  length,  in  each  fiber, 
is  proportional  to  the  distance  of  the  fiber  from  the  neutral  axis;  but,  in  a 
truss,  the  loads  (including  the  weight  of  the  truss  itself)  are  theoretically 
regarded  as  divided  into  portions  which  are  concentrated  at  the  joints  be- 
tween the  members  and  which  act  through  the  cens  of  grav  of  their  cross- 
sections.  So  placed,  the  stresses  caused  by -them  could  not  act  transversely 
of  the  members,  as  in  a  beam,  causing  so-called  secondary  stresses,  but  must 
act  longitudinally  or  axially  of  the  members,  and  must  be  uniformly  distrib- 
uted over  their  entire  cross-sectional  areas.  This  is  the  distinguishing 
feature  of  all  trusses. 

4.  In  such  a  truss  the  material  would  be  used  most  economically,  and  the 
stresses  in  each  piece  and  in  each  part  of  such  piece  could  be  readily  and 
accurately  determined. 

5.  In  the  truss  of  a  well-designed  bridge  or  roof,  this  ideal  condition  is 
approximated  by  using,  for  the  principal  members,  straight  and  rather 
slender  pieces,  and  by  so  distributing  the  extraneous  load  that  it  shall  be 
applied  only  at  the  joints  between  the  members,  thus  subjecting  them 
chiefly  to  forces  acting  at  their  ends  and  in  the  directions  of  their  lengths. 
In  pin-connected  trusses  (see  ^  175)  the  joints  are  practically  flexible. 

Most  of  the  trusses  in  common  use  consist  of  two  long  members, 
usually  horizontal  (but  see  t  49),  called  chords,  extending  throughout 
the  span  and  connected  by  "web  members,  which  are  sometirnes  all  in- 
clined, and  sometimes  alternately  vertical  and  inclined.  Inclined  web 
members  are  called  diagonals. 

6.  Ties  and  Struts.  A  member  sustaining  tension  is  called  a  rod  or 
tie.  One  sustaining  compression  is  called  a  strut  or  post.  One  capable  of 
sustaining  both  tension  and  compression  is  called  a  tie-strut  or  a  strut-tie. 

7.  The  dimensions  of  a  truss  are  usually  measured  along  the  center  lines 
of  its  members;  and,  in  pin-connected  trusses,  the  pins  are  placed  at  the 
intersections  of  these  lines.  Hence,  the  measurements  are  usually  made 
from  "center  to  center  of  pins." 

8.  In  a  plate  girder,  the  flanges  are  usually  regarded  as  performing 
the  function  of  the  chords  of  a  truss,  and  the  web  as  performing  that  of  the 
web  members  of  a  truss. 

*  Railroad  companies  and  municipal  corporations  frequently  prepare  their 
own  bridge  specifications;  but  the  general  proportions,  number  of  panels, 
etc.,  are  often  left  to  the  judgment  of  the  bidders. 

t  We  here  suppose  the  truss  to  be  loaded  vertically.  If  the  load  is  other- 
wise applied,  as  in  the  case  of  the  wind  pressure  upon  a  horizontal  bracing 
truss,  the  pressure  on  the  supports  may  be  horizontal,  or  otherwise  inclined 
to  the  vertical,  but  all  the  internal  stresses  are  still  sustained  by  the  truss 
members. 

44 


690  TRUSSES. 

Lioading. 

9.  Dead  and  Live  Load.  In  bridges,  we  distinguish  between  the 
"dead"  and  the  "live"  load;  the  dead  load  comprising  the  weight  of  the 
permanent  structure — i.  e.,  of  the  bridge  itself,  with  its  trusses,  bracing  and 
floor  system ;  while  the  live  load  comprises  any  temporary  and  extraneous 
loads,  such  as  engines,  cars,  horses,  vehicles,  foot  passengers,  etc.,  which 
may  come  upon  the  bridge, 

10.  The  dead  load  is  usually  distributed  uniformly  along  the  span,  but 
the  loaded  chord  (that  carrying  the  roadway)  of  course  usually  receives  a 
greater  share  of.it  than  the  unloaded  chord.  The  live  load  comes  only  upon 
the  loaded  chord.  In  determining  stresses,  it  is  usual  to  consider  the  weight 
of  live  load  and  of  floor  system  as  being  on  the  loaded  chord,  and  the  rest  of 
the  dead  load  as  divided  equally  between  the  two  chords.  It  sometimes  hap- 
pens, however,  that  both  the  upper  and  the  lower  chords  carry  roadways. 
They  must  then,  of  course,  both  be  treated  as  "loaded,"  though  not  neces- 
sarily equally  loaded;  for  one  may  carry  a  railway  while  the  other  carries 
only  a  highway. 

Unsymmetrical   Loading.     Counterbracing. 

11.  Unsymmetrical  Loading.  In  Figs.  2  to  10,  the  loads  are  sup- 
posed to  be  placed  symmetrically. 

13.  If  this  could  be  the  case  in  practice,  the  compression  members  would 
never  be  called  upon  to  resist  tension,  or  the  tension  members  to  resist 
compression ;  and  the  trusses  in  Figs.  2  to  10  would  suffice  (supposing  each 
member  to  have  sufficient  strength),  even  though  the  compression  members 
were  incapable  of  resisting  tension  and  vice  versa.  Thus,  the  tension  mem- 
bers might  be  flexible  chains,  and  the  compression  members  might  be  posts, 
merely  abutting  against  supports  at  their  ends. 


p  a 


Fig.  1. 

.13.  But  in  a  truss.  Fig.  1  (a),  with  a  flexible  tie  in  the  panel,  p,  as  shown, 
the  load  W,  unsymmetrically  placed,  would  cause  failure,  as  indicated. 

14.  Counterbracing.  To  prevent  this,  those  members  which,  under 
moving  loads,  may  be  subjected  alternately  to  both  tension  and  compression, 
may  be  so  constructed  as  to  be  able  to  resist  both  kinds  of  stress.  That  is  to 
say,  the  tension  members  may  be  so  stiffened  as  to  be  capable  of  acting  as 
posts,  and  the  ends  of  the  compression  members  so  connected  to  the  chords 
that  those  members  can  also  act  as  ties.  This  is  the  expedient  usually  em- 
ployed in  trusses  without  vertical  web  members. 

15.  Counters.  In  trusses  with  rectangular  panels,  the  distortion,  Fig. 
1  (a),  caused  by  unsymmetrical  loading,  is  usually  prevented  by  the  intro- 
duction of  additional  members  called  counterbraces,  or  counters,  in  distinc- 
tion from  the  "main"  members,  which  last  are  designed  to  resist  the  norrnal 
stresses  due  to  uniformly  or  symmetrically  distributed  loads.  Thus,  in  Fig. 
]  {b)  the  unsymmetrical  load,  W,  tends  to  convert  the  rectangle,  p,  into  a 
rhomboid,  by  lengthening  its  diagonal,  W  d;  and  this  may  be  prevented  by 
the  introduction  of  an  oblique  tension  member  (counter)  in  the  line  of  that 
diagonal,  as  shown  by  dotted  line.  For  a  similar  reason,  such  a  counter  is 
inserted  also  in  the  corresponding  panel,  x  d. 

16.  Triangles.  It  will  be  noticed  that  the  introduction  of  counters 
reduces  the  truss  to  a  framework  made  up  exclusively  of  triangles. 

17.  It  might  at  first  sight  appear  that  the  several  parts  of  a  bridge  truss 
must  be  most  strained  when  covered  from  end  to  end  with  its  maximum 
load ;  but  this  is  true  only  of  the  chord  and  of  the  main  diagonals  and  verti- 
cals near  the  ends  of  the  truss.  The  other  web  members  may  be  more 
strained  by  a  -part  of  the  load,  placed  unsymmetrically  on  the  truss ;  so  that, 
although  correctly  proportioned  for  a  full  load,  they  may  be  too  weak  for  a 


BRACING.  691 

partial  one.  If  all  be  made  as  strong  as  the  end  ones,  they  will,  it  is  true,  be 
safe  for  a  passing  load ;  but  this  would  require  an  expense  of  material  that 
would  be  justified  only  in  the  case  of  moderate  spans,  especially  of  wood,  in 
which  the  additional  trouble  and  expense  of  getting  out  and  fitting  together 
pieces  of  many  different  sizes  may  more  than  counterbalance  the  saving  in 
material. 

18.  In  large  bridges,  where  the  live  load  is  small,  relatively  to  the  dead 
load,  but  little  counterbracing  is  needed,  and  that  at  and  near  the  center 
only ;  whereas,  in  a  very  light  bridge,  the  counters  should  extend  from  the 
center,  where  they  are  most  strained,  to  near  the  ends,  where  the  strain  upon 
them  is  least. 

Cross-bracing. 

19.  Bracing  between  Trusses.  Advantage  is  taken  of  the  proximity 
of  the  two  or  more  trusses  of  a  bridge,  standing  side  by  side,  to  connect  them 
by  cross-bracing,  thus  giving  to  the  entire  structure  far  greater  lateral  stabil- 
ity than  would  be  possible  in  the  single  trusses. 

20.  Thus,  lateral  bracing,  Fig.  39,  consists  of  horizontal  trusses  placed 
between  the  two  upper  chords  of  the  main  trusses,  or  between  the  two  lower 
chords,  or  both ;  the  chords  of  the  main  trusses  acting  also  as  the  chords  of 
the  lateral  trusses.  The  lateral  bracing  prevents  lateral  deflection  of  the 
chords. 

21.  Sway  bracing,  Fig.  64  (c)  (called  also  diagonal,  cross,  vibration  and 
wind  bracing),  consists  of  short  trusses  (usually  vertical)  crossing  the  bridge 
transversely  and  thus  connecting  the  two  trusses.  The  sway  bracing  has 
its  own  chords,  but  uses  parts  of  the  posts  of  the  main  truss  as  its  end  posts. 

22.  Portal  bracing.  Fig.  54  (a),  consists  of  sway  bracing  (usually  in  an 
inclined  plane)  joining  the  tops  of  the  end  posts  in  trusses  of  sufficient  depth 
to  permit  its  use.  The  portal  bracing,  with  the  end  posts,  forms  a  portal 
through  which  trains,  etc.,  enter  the  bridge. 

Types  of  Trusses. 

23.  The  simplest  form  of  truss  consists  of  a  single  triangle,  Figs. 
2  (a)  and  (6).  In  Fig.  (a)  the  load  produces  compression  in  the  rafters, 
tension  in  the  chord  or  tie  rod,*  and  compression  (  =  the  tension  in  the 
chord)  between  the  heads  of  the  rafters ;  in  Fig.  (6)  vice  versa. 

24.  The  truss  shown  in  Fig.  2  (a)  is  in  common  use  for  roofs  of  small  span, 
as  in  dwellings.  In  practice,  it  is  of  course  loaded  along  the  rafters,  and  not 
only  at  the  apex  as  in  Fig.  (a)  ;  but,  in  calculating  the  stresses  in  truss  mem- 
bers, we  commonly  first  assume  that  the  loads  are  concentrated  at  the 
intersections  of  the  members.  The  effect  of  their  actual  distribution  along 
the  members  is  then  determined  separately,  treating  the  members  as  beams. 


^Q  ^'') 


25.  In  Fig.  3  (a)  (called  a  King  truss),  the  vertical  tie  (improperly  called 
a  King  post),  and  in  Fig.  3  (b)  the  vertical  post,  simply  carries  the  weight  of 
the  load  to  the  apex,  i,  where  it  produces  the  same  effect  as  in  Figs.  2  (a) 
and  (6). 

26.  Hence,  neglecting  the  weights  of  the  vertical  tie  and  other  members, 
the  stresses,  caused  by  a  given  load,  W,  in  the  diagonals,  and  in  the  horizon- 
tal tie,  Fig.  3  (a),  are  the  same  (not  only  in  character,  but  also  in  amount) 
as  those  produced  by  an  equal  load,  W,  in  Fig.  2  (a).  Similarly,  those  m 
Fig.  3  (6)  correspond  with  those  in  Fig,  2  (6) . 

*  In  Figs.  2  to  12,  and  14  to  17,  double  or  heavy  lines  indicate  posts  or 
struts,  and  light  lines  indicate  ties. 


692  TRUSSES. 

27.  Figs.  4,  5,  and  6,  giving  modifications  of  the  simple  forms  shown 
in  Figs.  2  and  3,  illustrate  in  principle  most  of  the  bridge  trusses  in  com- 
mon use  for  spans  up  to  300,  400  or  even  500  feet.  See  Figs.  7  to  10,  ITf  35, 
etc. 

28.  In  Figs.  4,  5,  and  6,  there  is  an  upper  chord,  in  compression,  and  a 
lower  chord,  in  tension;  the  shorter  chord  sustaining  the  compression  be- 
tween the  heads  of  the  rafters,  Figs.  2  (a)  and  3  (a),  or  the  horizontal  tension 
between  the  feet  of  the  diagonals.  Figs.  2  (6)  and  3  (6).  Figs.  4  (a)  and  5  (a) 
are  modifications  of  Figs.  2  (a)  and  3  (a) ;  Figs.  4  (6)  and  5  (6)  of  Figs.  2  (6> 
and  3  (6) :   Fig.  6  (a)  of  Fig.  2  (a) ,  and  Fig.  6  (6)  of  Fig.  2  (6> 

(6) 

Fig.  4. 

29.  Figs.  4  (a)  and  4  (6)  may  be  regarded  as  showing  Figs.  3  (a)  and  3  (6) 
respectively,  with  the  vertical  member,  as  well  as  the  load,  split  in  two,  and 
the  two  parts  separated  by  horizontal  straining  pieces.  If  the  loads  are 
placed  symmetrically,  so  that  the  horizontal  pressures.  Fig.  4  (a),  or  tensions. 
Fig.  4  (6),  on  the  two  ends  of  the  shorter  chord,  are  equal,  the  two  diagonal 
counters  in  the  center  are  unnecessary. 


30.  Howe  and  Pratt  Systems.  In  Fig.  5  (a)  the  vertical  web 
members  ..re  in  tension,  and  the  diagonals  are  in  compression,  embodying 
the  "Howe"  principle,  used  in  bridges  with  wooden  diagonals;  while  in  Fig. 
5  (b)  the  verticals  are  in  compression,  and  the  diagonals  in  tension,  embody- 
ing tht  "Pratt"  principle,  used  in  bridges  with  metal  diagonals.  In  such 
bridges  long  compression  members  are  objectionable. 

W 

„ O 

(a)  \.£1_^X 

Fig.  6. 

31.  Warren  op  Triangular  Trusses.  In  Fig.  6,  illustrating  the 
"Warren"  or  "triangular"  truss,  the  web  members  are  all  diagonal,  and  are 
alternately  in  tension  and  in  compression.  They  divide  the  truss  profile 
into  isosceles  triangles. 

32.  Through,  Deck  and  Pony  Spans.  Figs.  4  (a),  5  (a)  and  6  (a), 
with  the  roadway  on  the  lower  chords,  are  called  "through"  spans,  and  Figs. 
4  (6),  5  (6)  and  6  (6),  with  the  roadway  on  the  upper  chord,  are  called  "deck" 
spans.  The  deck  span  permits  the  use  of  sway  bracing  (see  If  21)  between, 
and  throughout  the  depth  of,  the  two  or  more  trusses  forming  the  bridge, 
while  the  through  span  of  course  does  not ;  but  the  use  of  the  through  span 
is  often  required,  in  order  to  give  sufficient  head-room  for  boats,  floods, 
trains  on  crossing  roads,  etc.,  below  the  bridge.  A  truss,  loaded  on  the  lower 
chord,  but  too  shallow  for  lateral  bracing  (see  1  20)  between  the  upper 
chords,  is  called  a  "pony"  truss  (or  "pony  through"  truss). 

33.  Panels.  The  points  where  the  vertical  web  members  meet  the 
chords,  in  Figs.  4  and  5.  are  called  panel  points;  and  the  rectangular 
spaces,  an,nc,cd,  etc..  Fig.  5  (a),  between  the  verticals,  are  called  panels. 

34.  The  Warren  truss,  Fig.  6,  has  no  verticals,  as  essential  parts  of  it.  See 
H If  45  and  46.  Its  subdivisions  are  called  simply  triangles;  and  a  panel  is  a 
length  of  truss  equal  to  the  width  of  a  triangle.  A  panel  of  either  chord, 
however,  is  that  portion  of  it  between  two  panel  points. 


TYPES. 


693 


35..  Further  modifications  of  these  designs,  with  more  numerous  panels, 
are  shown  in  Figs.  7  to  10.  Figs.  7  (a)  and  8  (a),  with  verticals  in  tension, 
represent  the  Howe  truss  of  Figs.  4  (a)  and  5  (a),  while  Figs.  7  (b),  8  (6), 
9  (a)  and  9  (b),  with  diagonals  in  tension,  represent  the  Pratt  truss  of  Figs. 
4  (6)  and  5  (6).     Figs.  10  represent  the  Warren  truss  of  Fig.  6. 

36,  Fig.  8  (a)  represents  simply  Fig.  7  (a),  lowered  so  as  to  become  a  deck, 
instead  of  a  through  bridge ;  while  Fig.  8  (6)  represents  Fig.  7  (6)  converted 
from  a  deck  to  a  through  span  by  being  carried  on  vertical  end  posts. 

37.  In  Figs.  8,  the  vertical  end  posts,  and  the  horizontal  piece  at  each 
end  of  the  loaded  chord,  form  no  part  of  the  truss  proper.  The  latter 
simply  act  as  beams,  supporting  the  load  during  its  passage  from  the  abut- 
ment to  the  truss  and  vice  versa.  The  end  post  in  Fig.  (a)  supports  only 
one  end  of  this  beam,  while  that  in  Fig.  (6)  supports  half  the  truss. 


/I/I/MMMMY 


(«) 

Tlirong:li  Howe. 


(6) 


Beck  Pratt. 


Fig.  7. 


|/M/U^kKK|   ^^"^^^^ 


Beek  Howe. 


Fig.  8 


Tbrongb  Pratt. 


(a) 
Beck  Pratt. 


Fig.  9. 


Through  Pratt. 


m^:2' 


(a) 
Beek  Warren. 


'^  ^ 


Fig.  10. 


m 


(b) 
Through  Warren. 


38.  In  Figs.  8  the  middle  vertical  carries  no  part  of  the  load.  Theoret- 
ically it  serves  merely  to  prevent  deflection  of  the  two  unloaded  middle 
chord  panels  under  their  own  weight ;  but  in  practice  such  members  are  often 
inserted  for  the  purpose  of  obtaining  convenient  connections  for  lateral 
pieces,  such  as  floor  beams. 

39.  In  Figs.  9  (modifications  of  Fig.  7  (6))  and  in  Fig.  10  are  shown,  in 
principle,  the  most  common  forms  of  metal  bridge  truss,  used  as  deck  and 
as  through  spans  respectively. 

40.  In  Fig  9  (a),  as  in  Fig.  8,  the  vertical  end  posts  and  the  horizontal 
pieces  at  the  ends  of  the  loaded  chord  form  no  part  of  the  truss ;  and  in  Fig. 
9  (6),  as  in  Figs.  8,  the  middle  vertical  supports  only  the  unloaded  middle 
chord  panels. 


694 


TRUSSES. 


41.  Intersections.  In  deep  trusses,  two  or  more  sets  of  web  members 
are  sometimes  combined  in  one  truss,  with  one  pair  of  chords.  Thus  the 
two  simple  Pratt  trusses  shown  in  Figs.  11  (a)  and  (6)  combine  to  make  the 
"Whipple"  or  "double  intersection  Pratt"  truss,  Fig.  11  (c),  recently  in 
general  use. 


C«) 


(a) 


7NNXI   l/l/l^      ,,, 


Fig.  11. 


Fig.  12. 


42.  Similarly  the  two  simple  Warren  trusses,  in  Figs.  12  (a)  and  (6),  com* 
bine  to  form  the  double  intersection  Warren  of  Fig.  12  (c). 

43.  A  combination  of  four  systems  is  called  a  "quadruple  intersection" 
truss.     See  Fig.  59  (t). 

It      XI   V     t 

rry^  '^TN-^-mu^  T^^V  -r>^7  x-v?  XN/^ 


Fig 


44.  The  old  Towne  "lattice"  truss.  Fig.  13,  consisting  of  planks 
crossing  each  other  (usually  at  right  angles)  and  bolted  or  tree-nailed  to- 
gether at  their  intersections,  may  be  regarded  as  a  combination  of  several 
Warren  trusses. 


m~^ 


(a) 


Fig.  14.  (^^ 


45.  Sub-verticals.  In  deep  trusses,  where  the  horizontal  spread  of  the 
panels  is  considerable,  sub-verticals,  v.  Figs.  14  and  15,  are  often  used,  espe- 
cially in  Warren  trusses,  to  support  the  segments  of  the  loaded  chord.  See 
also  Figs.  59  (f),  (r),  and  (s). 


^      V  V         V         V  V         V  V         V     ^       ^     V  V  V     '     V  V  ^^w 


,,^^fmvmx\\ 


Fig.  15. 

In  the  "Baltimore"  truss,  Fig.  15  (6),  each  diagonal  is  braced,  at  its 
middle  point,  by  a  short  diagonal  strut  inclined  in  the  opposite  direction,  and 
a  sub-vertical  is  suspended  from  their  junction.  With  very  long  panels, 
sub-verticals  are  sometimes  used  for  the  panels  of  the  unloaded  chord  also. 
See  Fig.  15  (c). 


TYPES. 


695 


46.  Collision  strujts,  or  collision  posts,  S,  Figs.  59  (k),  (m),  (o), 
and  (t),  and  73  (a),  are  used  for  bracing  long  diagonal  end  posts  against  a 
blow  from  a  derailed  train. 


Fig;.  16. 

47.  Fink  and  Bollman  Trusses.  Figs.  16  show  two  obsolete  modi- 
fications of  Fig.  3  (6),  viz. :  the  Fink,  Fig.  16  (a),  and  the  Bollman,  Fig.  16  (6). 
The  large  bridge  over  the  Ohio  River  at  Louisville,  Ky.,  completed  1870,  is 
of  the  Fink  type.  The  Bollman  was  largely  used  on  the  Baltimore  and  Ohio 
Railroad  years  ago. 

48.  In  the  Fink  and  in  the  Bollman  truss  there  was  but  one  chord,  as 
shown.  This  chord  usually  carried  the  roadway.  Where  the  roadway  was 
placed  lower,  it  gave  the  truss  the  appearance  of  having  two  chords.  Under 
uniformly  distributed  loads,  in  the  Fink,  and  under  all  circumstances  in  the 
Bollman,  the  stress  in  this  chord  was  uniform  throughout.  In  the  Bollman 
(see  Fig.),  the  longitudinal  stresses  in  the  chord  were  all  applied  at  its  ends. 
Each  type  may  be  regarded  as  a  combination  of  several  suspension  trusses  like 
Fig.  3  (b) .  In  the  Bollman,  the  simple  trusses  were  all  of  the  same  span  and 
depth;  and  each  vertical  post,  except  the  central  one,  divided  its  simple 
truss  eccentrically.  The  Fink  principle  is  still  largely  used  in  metal  roof 
trusses.     See  Figs.  26. 


Fig.  17. 

49.  Curved  Chords,  Trusses  with  curved  or  "broken"  chords.  Fig.  17, 
are  frequently  used  for  long  spans.  The  members  themselves,  between  panel 
points,  are  always  straight.  In  the  bowstring,  Fig.  17,  the  panel  points 
of  the  upper  chord  lie  in  a  curve,  convex  upward.  In  the  crescent  truss, 
the  lower  chord  also  is  convex  upward.  The  bowstring  truss  has  the  ad- 
vantage, over  those  with  horizontal  upper  chords,  of  making  all  the  chord 
and  web  stresses  more  nearly  equal,  thus  simplifying  the  construction  and 
reducing  the  weight  of  the  trusses.  It  has  the  disadvantage  of  permitting 
no  overhead  bracing  near  the  ends  of  the  span.  If  the  curve  of  the  upper 
chord  is  made  parabolic,  the  dead  load  stress  is  uniform  throughout  the  lower 
chord,  and  in  each  vertical  (now  in  tension)  the  stress  is  equal  to  the  dead 
load  on  the  lower  chord.  The  diagonals  receive  no  dead  load  stress,  but 
are  called  into  action  only  by  eccentric  loads. 


Fig.  18. 

50.  The  Burr  truss.  Fig.  18,  at  one  time  much  used  for  wooden  bridges, 
was  a  combination  of  a  Howe  truss  and  an  arch. 


696 


TRUSSES. 


Camber.  < 

51.  Camber.  In  practice,  the  members  of  the  upper  and  lower  chords 
of  bridges  are  not  placed  perfectly  in  line,  but  so  that  the  chords  curve  slightly, 
with  the  convex  side  upward.  This  curve  is  called  the  camber.  Its  object 
is  to  prevent  the  truss  from  bending  down  below  a  horizontal  line  when 
heavily  loaded.     When  the  chords  are  cambered  (see  y  s  and  c  d.  Fig.  19), 


Fig.  1». 

they  become  approximately  concentric  arcs  of  two  large  circles,  of  which  the 
center  is  at  t;  and  the  upper  one  plainly  becomes  longer  than  the  lower. 
The  verticals,  instead  of  remaining  truly  vertical,  become  portions  of  radii  of 
the  arcs  mentioned ;  and,  although  their  lengths  remain  unchanged,  yet  their 
tops  are  farther  apart  than  their  feet;  and  this  renders  it  necessary  to 
lengthen  the  diagonals.     See  m  211-214. 

Cantilevers. 
53.  The    cantilever  principle  is  shown  in  Fig.  20,  where  A  and  B 


:j" 


m 


Fig.  20. 

represent  counterweights  or  anchorages.  Fig.  21  shows  the  Niagara  canti- 
lever bridge.  It  consists  of  two  cantilever  trusses,  ah,  a'  h',  connected  by 
an  ordinary  truss,  ba',  which  is  suspended,  by  a  vertical  link  at  each  end, 
from  the  ends  of  the  cantilever  trusses.  The  weight  of  the  truss  is  counter- 
balanced by  anchorages,  A  and  B,  or  by  weights,  or  both.  The  principal 
advantage  of  the  cantilever  is  that  it  may  be  built  outward  from  the 
piers  across  the  channel.  It  thus  greatly  facilitates  erection  in  cases  where 
false-work  cannot  well  be  used. 


Fig.  21. 

Movable  Bridges. 
53.  Movable    bridges,  including  draw,  swing  and  lift  bridges,  are   of 
three  general  classes ;  one  in  which  the  movable  part  slides  horizontally,  one  in 


^^^^^ 


Fig.  22. 

which  it  swings  horizontally,  and  one  in  which  it  swings  vertically.  Ordi- 
narily, the  movable  span  is  pivoted  near  the  middle,  and  swings  horizontally 
on  the  central"  swing"  or  "pivot"  pier,  as  in  Fig.  22.     In  such  cases  it  is 


SKEW   BRIDGES. 


697 


usually  mounted  on  a  central  pivot,  or  on  a  nest  of  rollers  or  wheels  running 
on  a  circular  track.  Such  a  bridge  must  be  so  designed  that,  when  it  is  swung 
open,  or  if  it  is  not  brought  to  a  bearing  at  the  ends  when  closed,  it  shall 
sustain  not  only  its  own  weight,  but  also  any  other  loads  that  may  come  upon 
it.  In  addition,  each  half  must  be  able  to  act  as  a  bridge  supported  at  both 
ends,  with  all  possible  live  loads;  for,  as  an  unbalanced  live  load  comes  on 
either  end,  that  end  will  be  brought  to  a  bearing.  In  elaborate  bridges;  pro- 
vision is  made  for  raising  the  ends  of  the  draw  span,  when  closed,  thus 
bringing  both  ends  to  a  firm  bearing,  and  the  floor  flush  with  that  on  the 
adjacent  abutment  or  fixed  span.  This  raising  is  usually  made  sufficient 
to  relieve  the  middle  pier  of  only  a  portion  of  the  load.  The  bridge  then 
acts  like  a  "continuous"  girder  (see  Transverse  Strength,  It  78,  etc.)  sup- 
ported at  three  or  at  four  points,  depending  upon  ttie  arrangement  of  the 
bearing  on  the  pivot  pier. 


Fig.  23. 

54.  Drawbridges  in  which  the  movable  part  swings  vertically,  may  either 
revolve  about  a  pivot,  or  they  may  roll,  as  in  the  Scherzer  rolling  lift  bridge. 
Fig.  23. 

55.  Skew  bridges  are  used  where  a  channel,  road,  etc.,  is  crossed  ob- 
liquely, and  where  it  is  inconvenient  to  have  the  abutments  perpendicular  to 
the  trusses.  For  simplicity  in  making  floor  connections,  etc.,  the  truss  is 
usually  so  designed  as  to  bring  the  panel  points  opposite  each  other,  as  in 
Figs.  24  and  25.     Where  the  skew  is  but  slight,  this  necessitates  a  difference 


Ca) 


Plan 


".#S^ 


Elevation 

Fig.  24. 

in  inclination  between  the  two  end  posts,  as  in  Fig.  24,  involving  complica- 
tion in  the  connections  for  the  portal  bracing.  But  where  the  skew  is  greater, 
it  may  be  possible  to  make  it  just  equal  to  one  or  more  even  panels,  adjusting 


Plan. 


Fig. 


the  panel  length  to  suit,  and  thus  leaving  each  truss  symmetrical,  as  in  Fig. 
25.  In  each  figure,  those  members  which  belong  only  to  the  farther  truss 
(the  upper  one  in  the  plan)  are  shown  by  dotted  lines. 


698 


TRUSSES. 


Roof  Trusses. 
56.  Roof  trusses  are  made  in  a  great  variety  of  forms.  Those  shown  in 
Figs.  26  are  common.  In  Fig.  26  (a),  part  of  load,  at  d,  compresses  the  rafter 
from  d  to  a,  while  the  remainder  compresses  the  strut,  d  h,  and  pulls  the 
rod  h  i  and  the  part-chord  h  a.  Similarly,  part  of  c  passes  through  c  a  to 
o,  and  the  remainder  through  c  k  d  h  i  to  the  apex  i.  Thus  each  load 
is  eventually  carried  by  the  members,  part  to  the  apex  and  part  along  a 
rafter  to  an  abutment.  It  will  be  seen  that  the  greatest  stresses  in  the  rafters 
and  in  the  chord  occur  near  the  ends.*     Sometimes  the  members  shown 


(a)    a 


Fig.  26. 

vertical  in  Fig.  (a)  are  inclined,  or  the  lower  chord  is  "broken,"  being  usually 
convex  upward.  Roof  trusses  are  often  composed,  as  in  Figs,  (b)  and  (c),  of 
two  Fink  trusses,  inclined,  and  leaning  against  each  other,  their  feet  being 
held  in  position  by  a  tie,  m  n,  and  the  rafters  forming  the  upper  chords  of  the 
Fink  trusses. 

STRESSES  EV  TRUSS  MEMBERS. 
General  Principles. 

57.  Conditions  of  Equilibrium.  In  trusses,  as  in  beams,  it  is  neces- 
sary and  sufl&cient,  for  equilibrium,  that  the  internal  stresses,  and  their 
moments,  shall  balance  the  external  forces  and  their  moments.  The  exter- 
nal forces  (viz.,  the  loads  and  the  end  reactions)  and  the  resulting  moments 
and  shears,  are  discussed  under  Statics,  Tfl  285,  etc.  We  here  discuss  the 
determination  of  the  internal  stresses.  For  the  fundamental  distinction 
between  beams  and  trusses,  see  Trusses,  1  3. 

58.  In  general,  the  stresses  in  the  members  are  found  by  means  of  the 
principles  of  moments  (Statics,  tt  301,  etc.),  and  of  Shears  (Statics,  ^"H  325, 
etc.),  making  use  of  the  force  parallelogram  (Statics,  ^H  35,  etc.)  or  force 
triangle  (Statics,  It  46,  etc.),  the  force  and  cord  polygons  (Statics,  HH  72, 
etc.,  86,  etc.)  and  the  influence  diagram  (Statics,  tt  339,  etc.). 

59.  A  very  convenient  method,  and  one  in  common  use,  is  that  described 
more  fully  in  tH  67,  etc.,  below,  where  the  truss  is  considered  as  being  cut 
through  by  a  section.  We  then  seek  to  ascertain  what  stresses,  in  the  mem- 
bers so  cut,  would  be  required  to  preserve  equilibrium. 

60.  Before  the  stresses  can  be  calculated,  and  the  truss  proportioned  to 
those  strains,  its  weight  must  be  known ;  for  this  constitutes  a  load,  and  there- 
fore affects  the  stresses.  But,  on  the  other  hand,  we  cannot  learn  its  weight 
until  we  know  the  sizes  of  its  different  members.  In  this  dilemma  we  must 
assume  for  it  an  approximate  weight,  based  upon  our  knowledge  of  some- 
what similar  trusses  already  built.  This  becomes  the  more  necessary  as  the 
truss  increases  in  size,  so  that  its  own  weight  becomes  greater  in  proportion 
to  that  of  the  load. 


*  If  the  diagonals  were  parallel,  their  stresses,  and  those  in  the  verticals, 
would  be  greatest  at  the  center  of  the  span,  and  least  at  the  abutments. 


STRESSES. 


699 


61.  To  distinguish  between  ties  and  struts;  from  the  point,  o. 
Figs.  27,  where  the  force  is  applied,  draw  o  c  to  represent  the  applied  force, 
in  the  direction  in  which  that  force  tends  to  move  the  point,  o;  and  upon  o  c 
as  a  diagonal  construct  the  force  parallelogram,  a  b.  Through  o  draw  i  i  paral- 
lel to  the  other  diagonal  a  b.  Then,  if  a  piece  be  on  the  same  side  of  i  i  with 
o  c,  it  is  a  strut ;  while,  if  it  be  on  the  opposite  side,  it  is  a  tie. 


Fig.  27, 


62.  Ties  and  struts  may  often  (as  in  Fig.  27)  be  readily  distinguished  by 
inspection,  by  imagining  the  piece  to  be  flexible,  like  a  rope  or  chain.  If  it  is 
seen  that  it  would  then  resist  the  force  acting  upon  it,  the  member  is  a  tie; 
if  not,  it  is  a  strut.  Or,  suppose  that  the  piece  is  not  secured  at  its  ends. 
If,  then,  it  is  seen  that  it  would  resist  the  force  acting  upon  it,  the  member  is 
a  strut;  if  not,  it  is  a  tie. 

63.  Or  we  may  proceed  as  follows:  In  Fig.  28  (a),  representing  joint  a, 
we  begin  with  the  known  net  vertical  reaction,  R  * ;  and  find  the  unknown 
stresses  in  the  chord  and  in  the  end  post  by  means  of  the  force  triangle, 
making  their  arrows  follow  the  known  direction  of  R.  Transferring  these 
arrows  to  the  respective  truss  members,  Fig.  (d),  we  find  that  the  chord 
pulls  away  from  a,  and  is  therefore  a  tie ;  while  the  end  post  pushes  toward  a, 
and  is  therefore  a  strut. 


(c) 


Fig.  28. 


64.  In  Fig.  (6),  representing  joint  6,  we  draw  P  upward  to  represent  the 
pressure  of  the  end  post  toward  b;  and  the  other  two  sides  of  the  force  tri- 
angle give  the  pressure  in  the  chord  member,  Q,  and  the  tension  in  the  tie,  T. 

65.  In  Fig.  (c),  representing  joint  c,  we  know  T,  M,  and  the  load,  W,  and 
we  obtain  the  tension,  N,  and  pressure,  S,  in  the  corresponding  members. 

*  Inasmuch  as  half  of  each  end  panel  rests  directly  upon  a  support,  and 
thus  adds  nothing  to  the  stresses  in  the  members,  we  must,  in  determining 
those  stresses,  use  only  the 

net   reaction  —  reaction  —  half  panel  load. 


700 


TRUSSES. 


66.  Tensile  stresses,  because  they  tend  to  elongate  a  member,  are  con- 
ventionally regarded  as  positive,  and  designated  by  -f,  while  compressive 
stresses  are  regarded  as  negative,  and  designated  by  — . 

Method    by    Sections. 

67.  Let  Fig.  29  (a)  represent  a  roof  truss,  with  three  equal  loads,  W,  of 
2  tons  each,  applied  at  a,  c  and  6,  respectively,  and  let  it  be  required  to  find 
the  stresses  produced,  by  those  loads  alone,  in  the  members  a  c  and  a  d. 
Suppose  the  portion  shown  in  Fig.  29  (6)  to  be  separated  from  the  rest  of  the 
truss,  as  shown,  by  cutting  through  the  members  a  c  and  a  d.  The  lower 
portions  of  those  members,  shown  in  (6),  are,  however,  supposed  to  be  held 
m  their  original  positions  by  the  stresses  Sc  and  Sd,  exerted  in  these  mem- 
bers themselves.  Taking  moments  about  the  right  support,  h,  Fig.  29  (a)- 
we  have,  for  the  upward  reaction  of  the  left  abutment,  a, 

3W 
R-    2-. 

68.  We  have,  then;  at  a,  Fig.  29  (6),  four  forces,  as  follows:  two  known 
forces,  viz.:  W,  vertically  downward,  »  2  tons,  and  R,  vertically  upward, 

=  ~2~?  aiid  two  unknown  forces.   So  and   Sd.     Now  So  makes  a  known 

angle,  A,  and  Sd  a  known  angle,  B,  with  the  vertical.  The  vertical  forces, 
W  and  R,  have,  of  course,  no  horizontal  resolutes  (see  Statics,  ^^  54,  etc.); 
and  their  vertical  resolutes  are  the  forces  themselves. 


Fig.  29. 

69.  The  horizontal  resolutes  of  the  inclined  forces.  So  and  Sd,  are,  re- 
spectively: Sc.sinA,  and  Sd.sinB;  and  their  vertical  resolutes  are: 
Sc.cosin  A,  and  Sd.cosin  B. 

70.  We  see,  by  inspection,  that  the  stress,  Sc,  in  the  rafter,  a  c,  is  com- 
pression, and  that  the  stress,  Sd,  in  the  lower  member,  is  tension ;  but,  for 
convenience,  we  may  at  first  assume,  in  advance,  that  all  of  the  unknown 
stresses  are  tensions  or  +.  Then  those  which  finally  appear  as  +  are  known 
to  b.e  tensions,  and  vice  versa.  Their  horizontal  resolutes,  in  this  case,  are 
therefore  both  taken,  for  the  present,  as  being  right-handed,  or  positive;  and 
their  vertical  components  upward  or  positive  also.  It  will  be  remembered 
(see  ^  66)  that  we  regard  tensions  as  positive,  and  compressions  as  negative. 

71.  Now,  in  order  that  the  four  forces  at  o,  viz.:  W  =  2  tons,  downward. 
3  W 

R  =  — 2"»  upward,  So  and  Sd,  may  be  in  equilibrium,  it  is  necessary: 

(1)  that  the  sum  of  their  horizontal  resolutes  be  zero,  or 
Sc-sin  A    -I-    Sd.sin  B    =    Qs; 

(2)  that  the  sum  of  the  verti-cal  resolutes  be  zero,  or 
R  —   W    +  Sc.cosin  A     h    Sd-cosin  B    =    0. 

Thus,  let  A  =  45°,    sin  A  =  0.707;     cosin  A  -  0.707. 
B  =  75°,    sin  B  =  0.966;     cosin  B  «=  0.259. 

Then  0.707  So  +  0.966  Sd  =  0  ; 

R  —  W  +  0.707  Sc  +  0.259  Sd  =  0; 


Sc  = 


—  0.966  Sd 


0.259   Sd  —  R  +  W 


0.707  0.707 

0.966  Sd  — -  0.259  Sd  =  0.707  Sd  =  R  —  W. 


CHORD  STRESSES. 


701 


72.  Again,  in  Fig.  30,  with  section  mv,  stress  in  ed  ■■  Wi 

\y  j^        —  9 

15  =*  —  9.     With  section  vy*  stress  in  c/  —  -  — 


-R  =  6  — 
With  sec- 


tion ux,  stress  in  gd  — 


Wi-f  Wo 


cos  9  cos  d' 

It  will  be  seen  that  these 


cos  0  cos  0 

forces,  all  acting  downward  on  the  part  truss  to  the  left  of  the  section, 
give  tension  in  ed,  and  compression  in  ef  and  gd. 

With  section  uz  we  cut  two  web  members,  gd,  and  gc;  but  the  stress  in  gd 

has  already  been  found    =  ^,  the  vertical  component  of  which  is  =  3. 

Hence,  stress  in  e-c  =  Wi  +  W2  +  W3  +  3  -—  R  =  6. 

73.  It  is,  however,  evident  from  inspection  that  the  middle  vertical  bears 
simply  the  middle  load,  W3  =  6 ;  for,  cutting  the  truss  by  a  curved  section, 
as  at  c,  and  examining  the  small  portion  thus  cut  out,  we  see  that  we  have 
but  two  vertical  forces — viz.,  the  central  load,  W3,  and  the  stress  in  the 
vertical  member;  and,  for  equilibrium,  these  two  must  be  equal. 


Fig;.  30. 


Chord  Stresses,  Moments. 

74.  For  the  chord  stresses.  Fig.  30,  let  P  =  panel  length  —  10  ft.     Then 
the  bending  moment  at  the  panel  point,  d,  is 

M  =  2  R  P  —  WiP 

=  15  X  20  —  6  X  10 

=  300  —  60  =  240. 
Cutting  the  truss  by  section  uv,  we  find  that,  of  the  three  members  cut, 
only  the  upper  chord  member,  eg,  has  a  moment  about  d.     Call  its  stress  S. 
Its  leverage  is  the  depth,  D,  of  the  truss,  =  12;  and,  for  equilibrium,  S  D  =» 

M.     Hence,  S  =  w  =  -j^   =  20. 

75.  Similarly,  taking  moments  about  e,  we  find  the  stress,  in  the  lower 

240 


chord  member,  /  d,  cut  by  the  section,  u  v,  to  be  -r^ 


20,  or  the  same  as 

the  stress  in  the  upper  chord  panel  cut  by  the  same  section.  Inspection 
shows  the  correctness  of  this  result ;  for  the  diagonal  strut,  e  f,  evidently 
delivers  to  the  upper  chord  panel,  e  g,  a,  compressive  stress  or  "chord  incre- 
ment" (see  H  77)  ==  the  tensile  stress  which  it  delivers  to  the  lower  chord 
panel,  /  d. 

76.  If  the  chord  members  are  inclined,  their  lever  arms  must  of  course  be 
measured  perpendicularly  to  them;  and  we  can  no  longer  use  the  vertical 
depth  of  the  truss  as  the  lever  arm. 

77.  Chord  Increments.  Fig.  30.  Each  diag  delivers  a  comp  stress  to 
the  upper  chord,  and,  in  trusses  with  parallel  chords,  an  equal  tensile  stress 
to  the  lower  chord.  Find  the  shear,  or  vertical  component,  Vi;  V2,  etc.,  of 
the  stress  in  each  diagonal,  beginning  with  the  end  post.  Then  the  "chord 
increments,"  h\,  fa,  etc.,  or  the  stresses  in  the  chord  members,  af  and  me, 
id  and  eg,  etc.,  due  to  the  several  diagonals  separately,  are 

hi  =  Vi  tan  0 

ho  =  V2  tan  B 

hz  =  V3  tan  e 
and,   for  the   total  stress  in   each   chord    member,    we   have,    Hi    =    M; 
Ht  —  ^1  -1-  A2 ;   H3  =*  Ai  +  ^2  +  ^3 ;   and  so  on. 


702 


TRUSSES, 


Shear. 

78.  Any  portion  of  a  shear  diagram  applies  to  all  those  members  through 
which  its  shear  travels,  up  to  the  panel  point  where  the  shear  undergoes  a 
change.  Thus,  in  Fig.  31  the  shear  diagram  on  the  right  of  the  Fig.  includes 
the  vertical,  tu;  that  on  the  left  includes  the  diagonal,  mo;  and  that  between 
the  loads  includes  the  diagonal,  t  n,  and  the  vertical,  m  n. 


A\XX/A 

•Ma               ^ 
a' 

_ Ife' 

Fi^.  31. 

79.  Shear  Influence  Diagram.  See  Statics,  ^I  325,  etc.  In  a 
trusB,  Fig.  32,  the  ordinates,  c'  g,  etc.,  to  the  line  a"  h'  (constructed  as  in  Fig. 
156,  Statics,  *||  349),  give  the  left  end  reactions;  and  those,  c'  h,  etc.,  to  the 
line  a'b",  give  the  right  end  reactions,  for  any  position  of  the  load ;  and  the 
resulting  shears  for  a  load,  W  (not  shown),  at  any  panel  point;  but  the 
shears  in  a  panel,  cd,  for  a  load,  W,  between  the  panel  points,  are  modified  by 
the  action  of  the  stringers  in  distributing  the  load  between  said  adjacent  panel 
points,  as  indicated  by  the  influence  lines,  qh,  etc.,  for  the  several  panels. 
Thus,  with  W  at  c  and  at  d,  respectively,  the  shear,  in  the  panel  cd,  is  repre- 
sented respectively  by  c'h  (negative)  and  by  d'q  (positive) ;  and,  as  the  load 
passes  from  c  to  d,  the  shear  in  the  panel  changes  from  c'h  to  d'g. 


Fig.  32. 

But  when  W  is    placed  at  any  point,  e,  between  c  and  d,  its  load  is . 
distributed  between  the  panel  points,  c  and  d,  by  the  stringer,  acting  as  a 
beam. 

80.  Thus,  drawing,  for  this  beam,  c  d  (as  for  the  whole  beam  in  Statics, 
Fig.  156),  the  panel  influence  lines  fd'  and  c's,  we  see  that,  as  W  moves  from 
d  into  the  panel  cd,  as  to  e,  the  truss  reaction  at  a  is  thereby  slightly  increased 
from  d'g  to  e'k  ;  but  at  the  same  time  a  portion  of  W,  represented  by  e't,  is 


WEB  STRESSES. 


703 


carried  by  the  stringer  to  c,  where  it  diminishes  the  shear  e'k  (due  to  the  truss 
reaction  at  a),  leaving  tk  as  the  value  of  the  shear  in  the  panel.  As  we  place 
W  successively  at  other  points,  farther  from  d,  and  approaching  o,  the  load 
carried  by  the  stringer  to  c,  and  represented  by  the  ordinates  from  c'd'  to 
fd',  continue  to  increase  faster  than  does  the  left  end  truss  reaction,  R, 
represented  by  the  ordinates  from  a'h'  to  a"h' ;  and  the  resulting  shears  in  the 
panel  are  represented  by  the  ordinates  from  id'  to  jq.  At  o,  the  part  load, 
o'j,  carried  to  c,  is  =  the  left  truss  reaction,  and  the  shear  in  the  panel  is  zero. 
With  the  load  between  o  and  c,  the  part  loads  carried  to  c,  and  represented  by 
the  ordinates  from  o'  c',  to  //,  are  greater  than  the  corresponding  left  end 
truss  reactions;  and  the  result  is  a  negative  shear  in  the  panel,  indicated  by 
the  ordinates  from  j  g  to  jf.  It  will  be  noticed  that  the  resulting  shears 
throughout,  both  positive  and  negative,  are  indicated  by  the  ordinates  from 
c'd'  to  hq* 

Reversing  the  process,  a  similar  argument  may  be  applied  to  the  panel 
influence  line  c's,  beginning  with  the  load  at  c,  with  negative  shear  in  panel 
=  c'hf  and  supposing  it  moved  across  the  panel  to  ciT  where  positive  shear 
in  the  panel  becomes  =  d'q. 
^  81.  In  the  case  of  a  uniform  load,  extending  on  to  the  span  from  the 
right  support,  6,  the  point  o  is  the  position  of  head  of  load  for  maximum  posi- 
tive shear  in  the  panel,  cd;  for,  in  the  case  of  a  uniform  load,  the  shear,  with 
head  of  load  at  e,  is  represented  by  the  area  (sum  of  all  the  ordinates) 
e'mqh'e^'y  and  manifestly  this  area  increases  as  the  head  of  the  load  ap- 
proaches o;  but  when  it  reaches  o,  the  area  above  a'h'  can  increase  no  further, 
and  when  it  passes  o,  the  negative  shears,  represented  by  the  ordinates  from 
o'c'  to  o'h,  begin  to  reduce  the  resultant  positive  shear. 

83.  Having  found,  by  any  method,  the  maximum  shear,  d'q,  due  to  a 
concentrated  load  at  d,  for  the  diagonal,  d  n,  Fig.  32,  and  the  reverse  maxi- 
mum shear,  c'  h,  due  to  the  same  load  at  c,  we  may  draw  an  influence 
line,  h  q,  which  gives,  as  before,  the  point,  o,  of  position  of  head  of  uniform 
load  for  maximum  stress  in  the  diagonal,  d  n,  from  which  (as  above)  we  find 
the  corresponding  position  of  the  head  of  a  series  of  concentrated  loads. 

In  practice,  the  influence  line  for  shear  is  of  value  chiefly  in  thus 
finding  the  position  of  load  producing  maximum  stress,  and  the  resulting 
stresses,  in  trusses  with  curved  chords,  such  as  Fig.  17.  In  such  a  truss, 
owing  to  the  inclination  of  the  members  of  the  upper  chord,  those  members 
take  some  of  the  shears  in  their  respective  panels,  and  the  stress  in  the  diag- 
onal is  therefore  less  than  the  shear  in  the  panel. 

Graphic  Determination  of  Dead  Load  Stresses. 

83.  Construct  first  a  diagram  of  the  truss,  as  in  Fig.  33  (a),  lettering  the 
spaces  between  the  members,  and  those  between  the  arrows  representing 
the  dead  loads.  Call  the  end  post,  1-3,  between  A  and  B,  "  AB,"  the  stress  in 
it  "ab,"  the  load  at  2,  *'cd,"  etc.,  using  capital  letters  for  panels  and  truss 
members,  and  small  letters  for  loads  and  stresses.  Adopt  a  suitable  scale  of 
forces,  and  construct  the  diagram.  Fig.  33  (6),  as  follows: 


0 


Co) 


c 
d 

V 

'        / 

6 

e     (ft) 

.     / 

\    ., 

^y            ^y 

\\ 

h 

/\            A 

7\ 

l9 
a 

'  \ 

r  ^ 

\ 

Fig:.  33. 


84.  Consider  first  the  point  1,  Fig.  33  (a).  There  are  here  three  forces 
in  equilibrium,  viz.,  ac,  ah  and  he.  Find  the  net  end  reaction,  R  =  ac,  and 
lay  it  off  upward  (since  it  acts  upward  on  1)  from  any  convenient  point,  a,  to 
c.  Fig.  33  (6).     From  a  draw  an  indefinite  line  ah  parallel  to  AB  and  from  e 

*  Since  fh,  j'o',  qd'  and  ke'  are  parallel,  me'  =  kt,  and  c'h  —  gf» 


704  TRUSSES. 

draw  cb  parallel  to  BC,  obtaining  the  force  triangle  ach  of  the  point  1.     The 
lengths  of  cb  and  ba  then  give  the  stresses  by  scale. 

85.  In  Fig*  33  (6),  the  arrow  on  ac  indicates  the  upward  direction  of  that 
force.  Following  around  the  triangle,  we  affix  arrows  (in  the  same  direction) 
to  cb  and  ba.  Supposing  these  arrows  now  to  be  transferred  to  the  corre- 
sponding members  in  Fig.  33  (a)  we  see  that  b  c  pulls  from  the  point  1,  show- 
ing that  6  c  is  tensile,  or  +,  while  b  a  pushes  toward  1,  showing  that  6  a  is 
compressive,  or  — . 

86.  The  characters  of  the  stresses  may  be  found  more  quickly  as  follows: 
Draw  a  circle,  Fig.  33  (c),  and  place  on  it  arrows  pointing  around  in  the 
direction  (counter-clockwise  in  this  case)  followed  around  the  truss  in 
constructing  the  load  line.  See  ^  92,  below.  Then  consider  any  panel  point, 
Fig.  33  (a),  and  follow  the  letters  in  the  spaces  around  that  point  in  the  direc- 
tion of  the  arrows  on  the  circle.  Note  the  order  of  the  letters,  and  follow 
the  corresponding  equilibrium  polygon,  Fig.  33  (6),  around  in  the  same  direc- 
tion. This  will  give  the  directions  in  which  the  forces  respectively  act  on  that 
point. 

87.  Thus,  consider  the  panel  point  2.  Following  around  2  in  the  direc- 
tion of  the  circle,  we  read  B,  C,  D,  E.  Turning  now  to  Fig.  33  (6),  and 
reading  b,  c,  d,  e,  we  find  that  on  be  we  go  from  right  to  left  (or  opposite 
to  the  direction  indicated  by  the  arrow  drawn  for  point  1) ;  hence  be  acts 
to  the  left  on  2,  and  BC  is  therefore  in  tension,  and  its  stress  be  is  +. 


1 5  O    1 

H 

o 

I     ^    \ 

/^ 

\  J 

7i/ 

/m 

/l2 

V  ra) 

.All    C    > 

U  i>  > 

I    -K    > 

\   X    1 

I    s   > 

\     T    ^V 

o 


(o) 
Fig.  33  (repeated) 


c 
d 

^          '/ 

h 

[e     (b) 

.     / 

\    ., 

\/           v 

\J 

,^ 

-7^\                         /'  \ 

7\^ 

ig 

a 

'     \, 

/    I 

t 

\ 

u 

88.  Given  now  the  stress,  be,  in  BC,  construct,  on  be,  the  force  polygon 
bcdefoT  the  four  forces  acting  on  the  point  2.  Thus,  from  c  lay  off  cd  down- 
ward, to  represent  the  dead  load  on  the  lower  chord  at  2.  Since  be  acts  as  a 
pull  from  the  left  on  2,  and  since  the  forces  must  follow  each  other  around  the 
polygon,  cd  must  evidently  be  drawn  downward  from  c  and  not  from  b. 
From  d  draw  an  indefinite  line  parallel  to  DE,  and  from  6  another,  parallel 
to  BE.  They  will  intersect  at  some  point,  as  e,  and  eb  and  de  will  then  repre- 
sent the  stresses  in  BE  and  DE. 

89.  Inspection  would  show  that  be  =  cd,  since  cd  is  the  only  force  acting 
on  2  with  a  vertical  component,  and  that  be  =  de  ;  but  the  construction  of 
the  force  polygon  bede  is  necessary  for  the  completion  of  the  diagram. 

90.  Having  now  found  the  stresses  in  DE,  BE,  and  AB,  and  knowing  the 
panel  load  ( =  ^  a)  at  the  point  3,  construct  the  polygon  g  ab  e  f  g.  This 
gives  e  f  and  /  g,  and  from  these  the  process  may  be  continued  and  the  dia- 
gram completed. 

91.  It  will  be  noticed  that,  in  some  cases,  a  point  oh  the  diagram,  Fig. 
33  (6),  is  given  more  than  one  letter.  Ordinarily  this  is  simply  a  coincidence, 
arising  from  overlapping  of  the  force  polygons.  In  some  cases,  however,  the 
coincidence  of  the  letters  shows  that  the  stress  in  the  member  is  zero. 

93.  In  practice  it  is  usual  to  construct  first  the  entire  load  line  ct,  thus: 
draw  first  the  net  reaction,  ac,  upward;  then,  following  around  the  truss 
counter-clockwise,  draw  all  the  other  exterior  (dead  load)  forces  in  their 
proper  order,  thus  cd,  dk,  kl.  Is,  st,  tv,  vp,  po,  oh,  hg,  ga.  The  stress  diagram 
may  then  be  constructed,  as  before. 


LIVE   LOADS. 
Live   Loads. 


705 


93.  It  might  at  first  be  supposed  that  each  member  of  the  truss  would 
receive  its  maximum  stress  when  the  train  completely  covered  the  bridge; 
but  this  is  true  only  of  the  chord  members.  In  the  truss  shown  in  Fig. 
33  (a)  each  web  member  receives  its  maximum  stress  when  the  greatest 
possible  shear  occurs  in  a  section  cutting  that  member. 


Fi^.  34. 


94.  In  Fig.  34,  the  main  diagonals  to  the  left,  and  the  counters  to  the  rigM 
of  the  center,  C,  are  shown.  Any  one  of  these  members  receives  its  maxi- 
mum stress  from  a  uniformly  distributed  load  when  the  load  extends  from  it 
to  the  right  support  b,  with  head  of  load  at  a  point,  o,  Fig.  32  (a),  found 
as  in  ^  81 ;  and  vice  verso  for  the  diagonals  inclined  in  the  opposite  direction. 
Each  vertical  receives  its  maximum  stress  when  the  load  extends  from  the 
farther  support  to  a  point,  o  (see  *[  81),  in  the  panel  beyond  the  vertical. 
This  statement  must  be  slightly  modified  when  the  concentrated  wheel 
loads  are  considered.     See  tif  97,  etc. 


^/IMXlXIXIZiV 


w 


Fig.  35. 


95.  Assumed  Uniform  Live  Load.  As  a  crude  approximation,  the 
engine  and  train  are  sometimes  considered  as  a  uniform  load  crossing  the 
bridge.  Fig.  35 ;  but  this  method,  ignoring,  as  it  does,  the  great  concentration 
of  weight  in  naodern  locomotives,  is  apt  to  be  either  unsafe  or  wasteful  of 
material.  This  assumption  is  proper  in  connection  with  wind  pressure  on 
train.     See  t  121. 


Fig.  36. 


96.  Concentrated  Excess  Loads.  Again,  to  provide  for  the  locomo- 
tive loads,  one  or  more  concentrated  excess  loads.  Fig,  36,  are  sometimes 
employed.  The  stresses  due  to  these  loads  may  be  computed  separately,  and 
added  to  the  stresses  produced  by  the  uniform  live  loads.  To  produce  the 
maximum  chord  stresses,  the  excess  loads  should  be  in  the  middle  of  the  span, 
and  the  train  load  should  cover  the  entire  bridge.  This  method  is  fairly 
approximate,  and  engineers  are  divided  as  to  whether  this  method  of  con- 
centrated e>acess  loads  should  be  used,  or  that  of  the  actual  or  "typical" 
locomotive  wheel  loads  as  explained  below, 

97.  Standard  or  Typical  **  Wheel "  Loadings.  In  the  method  of 
wheel  loads,  the  actual  stresses,  produced  by  the  heaviest  engines  likely  to 
cross  the  bridge,  are  considered.  Even  in  engines  of  nearly  the  same  weight, 
the  loads  may  be  differently  spaced,  and  spaced  at  intervals  of  odd  fractions 
of  an  inch,  rendering  computation  very  laborious.  For  this  reason,  and  in 
order  to  provide  for  the  use  of  heavier  engines  in  the  future,  it  is  customary  to 
consider  an  imaginary  or  "typical"  engine,  with  loads  and  spacing  given  in 
round  numbers,  the  stresses  from  which  shall  at  least  be  equal  to  those  pro- 
duced by  the  heaviest  engines  likely  to  be  used  during  the  life  of  the  bridge. 

The  live  loads  are  ordinarily  taken  as  consisting  of  two  typical  locomotives 
with  their  tenders,  followed  by  a  uniform  train  load.     See  Digests  of  Speci- 
fications. 
45 


706 


TRUSSES. 


98.  The  following  is  an  example  of  the  computation  of  live  load  stresses  by 
the  method  of  locomotive  wheel  loads : 

Fig,  37  (b)  represents  the  loads  on  one  rail,  corresponding  to  Cooper's 
Staridard,*  Class  E  40,  which  consists  of  two  coupled  consolidation  loco- 
motives, followed  by  a  train  considered  as  equivalent  to  a  uniform  load  of 
4000  lbs.  per  linear  foot.  In  the  diagram.  Fig.  37  (a),  all  loads  are  figured 
in  thousands  of  pounds,  moments  in  millions  of  foot-pounds,  and  distances 
in  feet. 

400 


(a) 


^ 

y 

^ 

- 

y^" 

/ 

/ 

I 

—[ 

/ 

/ 

r- 

/ 

/ 

p. 

r- 

^/' 

y 

^ 

r 

T~ 

^^, 

^^ 

X 

J 

^c^ 

.^' 

^^ 

^ 

i 

0       2 

0  a3t 

)        4 

0 

■0 

60 

[7 

0      \ 

0        9 

0      1 

0 

no      120     130      140     150        " 

g-\  1         ZHAO  07      tf    y        )U      iJ  1-i  in  1A       in  fb    1/1S    , 

(&)    40  OOOO    no  oo  <n  OOOO    qq  oo  \ 


10     20  20  20 '20      1313    1313     10     20  20  20  20      1313   1313       2  per  fOOt 

Loads  in  thousands  of  pounds 
Fig.  37. 

99.  Live  Load  Web  Stresses.  The  maximum  live  load  stresses 
will  occur  in  the  web  members  of  any  panel  of  the  truss  in  Fig.  34  or  38, 
when  the  live  load  produces  the  maximum  shear  in  that  panel.     It  can  be 

W 
shown  that  this  will  occur  when  P    =   »  where    P    =    the  live  load 

n 
on  the  panel  cut  by  the  section ;   W  =  the  total  live  load  on  the  truss, 
and  n    =    the  number  of  panels  in  the  truss.     This  equation  is  called  the 
criterion  for  maximum  shear. 

100.  The  following  table  is  based  upon  this  relation.  The  second  col- 
umn is  obtained  by  adding  successive  wheel  loads  to  P.  In  this  case, 
W  =  6  P,  since  our  truss  has  6  panels.  Let  any  wheel  be  at  a  panel 
point.  Then,  by  moving  the  wheel  a  little  to  the  left  or  right,  it  will  be 
included  in  or  excluded  from  P.  Hence  P  and  W  have  each  a  minimum 
and  a  maximum  value  for  each  wheel  at  the  panel  point. 

Value,  P,  of  load  on  Corresponding  value  of 

panel  to  left  of 
given  point. 
0  to  10,000 
10,000  to  30,000 
30,000  to  50,000 
50,000  to  70,000 
70,000  to  90,000 

101.  The  correct  position  of  live  load,  for  maximum  t^hear  in  any  panel, 
is  found  by  successive  trials.     When  the  correct  position  is  found,  the 


No.  of  wheel  at  any 
given  panel 
point. 
1 
2 
3 
4 
5 


W  for  maximum 
shear  in  panel. 

0  to    60,000 

60,000  to  180,000 

180,000  to  300,000 

300,000  to  420,000 

420,000  to  540,000 


*  "Transactions  Am.  Soc.  Civ.  Engrs,,"  vol.  xlii,  No.  858,  Dec,  1899, 
p.  227.     See  Digests  of  Specifications. 


LIVE   WEB   STRESSES. 


707 


moment  about  the  right  support  is  computed,  and  from  this  the  shear  is 
obtained.       For  example,  see  below. 

102.  These  operations  may  be  performed  by  computation,  with  or  with- 
out the  aid  of  graphic  methods.  As  the  method  of  computation  alone  is 
rather  tedious,  particularly  when  the  form  of  the  truss  is  complicated  by 
curved  chords  or  sub-panels,  and  as  the  graphic  method  is  abundantly 
accurate  for  all  practical  purposes,  and  has  the  advantage  of  direct  appeal 
to  the  eye,  only  the  latter  is  given  herewith. 

103.  The  "wheel  diagram,"  Fig.  37  (a),*  gives  (1)  a  stepped  "load 
line"  or  "shear  diagram,"  and  (2)  a  curved  "moment  diagram"  or 
"  equilibrium  polygon."     See  Statics,  ^^  359,  etc.  ~ 

104.  The  load  line  gives  the  total  live  load  to  the  left  of,  and  including, 
any  point. 

105.  The  moment  line  gives,  at  any  point,  the  (left-handed)  live  load 
moment,  about  that  point,  of  all  loads  to  the  left  of  and  including  that 
point.     Thus,  to  the  left  of  and  including  wheel  No.  5  we  have 


Wheel. 
1 
2 
3 
4 
5 


Load. 
10,000 
20,000 
20,000 
20,000 
20,000 


Distance  from 

wheel  5. 

23 

15 

10 

5 

0 


Moment  about  5 
in  ft. -lbs. 
230,000 
300,000 
200,000 
100,000 
0 


830,000 


Total,     90,000 

and  the  ordinates,  ab  to  the  load  curve,   and  ac  to  the  moment  curve, 
under  wheel  5,  measure  90.0  and  0.830  respectively. 

106.  Fig.  38  represents  the  truss,  to  the  same  scale  as  Fig.  37.    We  may 
call  this  a  "  truss  diagram."  f 


B 

C 

J) 

E 

F 

\ 

\ 

\ 

\ 
\ 
\ 
\ 

\ 

\ 

\ 

\ 
\ 
\ 
\ 
s 
\ 
\ 
,  \ 

a  b  c  d  e  f  g 

Fig.  38. 

107.  Example.  To  compute  the  maximum  shear  in  the  panel  be.  Fig. 
38,  first  find  that  position  of  the  load  which  will  produce  that  maximum 
shear.  As  a  guess,  place  the  truss  diagram.  Fig.  38,t  with  its  point  c  under 
wheel  2,  Fig.  37.  Examining  the  load  diagram,  over  the  right  end,  g,  of 
the  span,  we  see  that  we  now  have  a  total  load,  W,  of  284,000  lbs.  on  the 
span;  and  the  load  diagram,  over  wheel  2  (placed  at  point  c)  shows  (see 
also  table,  •!  100)  that  the  load,  P,  on  the  panel,  b  c,  is  now  somewhere 
between  10,000  and  30,000  lbs.;  but,  for  maximum  shear  in  the  panel,  be, 
the  load,  P,  on  that  panel  must  be  (see  Hlf  99  and  100)  =  W  -i-  n  = 
284,000  -f-  6  >  30,000  lbs.  Hence,  P  must  be  increased  by  moving  the 
train  diagram,  Fig.  37,  to  the  left  (or,  which  is  the  same  thing,  by  moving 
the  truss  diagram.  Fig.  38,  to  the  right)  until  wheel  3  is  over  c.  We  now 
have  W  =  292,000  lbs.;  P  =  anywhere  between  30,000  and  50,000;  and 
required  value  of  P,  for  maximum  shear,  =  W  -r-  n  =  292,000  -v-  6  = 
48,667  lbs.     Hence,  the  conditions  are  satisfied,  and  panel  be  receives  its 

*  Method  published  by  Ward  Baldwin,  "Engineering  News,"  vol.  xxii, 
Sept.  28,  1889,  p.  295.     See  also  letter,  "  Eng.  News,"  Dec.  28,  1889,  p.  615. 

t  For  the  following  discussion  it  will  be  found  convenient  to  make  a 
copy  of  Fig.  38,  or  simply  of  the  lower  chord,  on  a  separate  piece  of  paper 
which  may  be  applied,  in  dififerent  positions,  to  Fig.  37. 


708 


TRUSSES. 


maximum  shear,  when  wheel  3  is  at  c.  The  moment  diagram  shows,  ver- 
tically over  g,  the  live  load  moment  about  the  right  support  =  17,516,000 
ft. -lbs.;  and  the  moment  at  c  =  233,000  ft. -lbs. 

108.  Let  M  =  the  (left-handed)  live  load  moment  at  the  right  abutment, 
due  to  all  the  loads  on  the  span. 
L  =  the  span. 
1  m  =  the  (left-handed)  live  load  moment  at  the  panel  point  on 

the  right  of  the  panel  in  question. 
I  =  the  length  of  the  panel. 
V  =  the  shear  in  the  panel. 
Then  V  =     M    _  ^  ^  JT^SiaOOO  _  230000  ^   ^^^  ^  ^^  . 
Li  L  150  Zo 


400 


« 

S 


1 


(«) 


^ 

1 — 40 

^ 

4-30 

J 

/ 

v 

J 

— 

/ 

/ 

r- 

J 

/ 

/ 

r 

- 

r-" 

/ 

^ 

/ 

^ 

l" 

^ 

^ 

^ 

^ 

\-<r 

^C^ 

i^ 

■^ 

r^ 

^ 

/ 

0 

\~ 

0  a:ic 

A 

0 

0 

0 

} 

0 

^ 

0 

0 

^ 

)U 

1 

0     i. 

'i)    /. 

0    u 

0    u 

6     ^ 

(b) 


^    .     U    5        6    7     8    9       W     11  i2  t3  1A      15  16  17  IS    , 

^  nooo   po  no  zo  nnnn  on  qn 


i31. 


S  per  foot 


10    20  20  20  20     i313   1313     10    20  20  20^0      13 13   i3 13 

Loads  in  thousands  of  pounds 

Fig.  37  (repeated). 

109.  The  maximum  live  load  shears  in  the  other  panels,  similarly  com- 
puted, are  as  follows,  the  load  being,  in  each  case,  so  placed  as  to  give 
said  maximum  shear: 


Panel,  No. 

Mom.  M 

Mom.  m 

Shear, 

Stress, 

and  Posi- 

at Rt.  End 

at  Rt.  End 

Pounds. 

Pounds. 

tion  of 

of  Truss. 

of  Panel. 

_-          M           7W 

V 

Wheel. 

Ft.-lbs. 

Ft.-lt)s. 

^^l""T- 

S  = Li 

cos  ^t 

a6  4  at  6 

27.176,000 

480,000 

162.000 

aB. 

—217,200 

6c  3  at  c 

17,516.000 

230.000 

107,600 

Be 

+  144,200 

c<i  3  at  d 

10,816,000 

230,000 

63.000 

Cc 
Cd 

—63.000 
+84,500 

de  2  at  e 

4,936,000 

80,000 

29,700 

T}d 

-29,700 
+  39.800 

c/  2  at  / 

1,743,000 

80,000 

8.400 

E/ 

+  11,250 

the  reaction  of  the  left  support,  a,  and  -y-  =  so  much 


of  the  panel  load  as  goes  to  the  left  end  of  the  panel, 
t  ^  =  angle  between  diagonal  and  vertical. 


LIVE   CHORD   STRESSES. 


709 


110.  The  live  load  stress  in  the  hip  suspender,  B6,  is  due  entirely 
to  loads  upon  the  two  lower-chord  panels,  a  b  and  b  c.  Thus,  with  wheel 
4  at  b,  panel  length  =  a6  =  6c  =  25  ft.,  we  have: 

On  a  6  On  6  c 


Wheel.  Load,  W. 

Dist,  d 
from  a 

Stress 

on  B6  = 

.  wd-i- 25. 

Dist,  d, 
Wheel.  Load,  W.   from  c. 

Stress 

onB6  = 

wd  -^  25. 

1  10,000 

2  20,000 

3  20,000 

7 
15 
20 

2,800 
12,000 
16,000 

5  20,000         20 

6  13,000         11 

7  13,000           6 

16,000 
5,720 
3,120 

Total,  50,000 

30,800 

Total;  46,000 

24,840 

Total  load  on 
Wheel  4 

ab  = 
bc  = 

50,000 
46,000 
20,000 

Stress  in  B  6  from  ab           = 
."       "     "       "     wheel  4  = 

30,800 
24,840 
20,000 

Total  load  on  ac  -  116,000  '  Total,  75,640 

111.  For  any  given  set  of  loads  on  a  c,  the  maximum  stress  in  B  6  oc- 
curs when  the  load  on  a c  is  equally  divided  between  ab  and  b  c;  and  this 
ordinarily  occurs  while  some  wheel  (to  be  found  by  trial)  is  passing  b. 
Thus,  with  wheel  4  just  to  the  right  of  b,  we  have,  on  ab,  wheels  1,  2  and 
3,  ==  50,000;  and,  on  be,  wheels  4,  5,  6  and  7,  =  66,000  ibs.;  but,  with 
wheel  4  just  to  the  left  of  6,  we  have,  on  ab,  wheels  1,  2,  3  and  4,  = 
70,000;  and,  on  6  c  (neglecting  wheel  8,  which  now  enters  6  c),  wheels  5,  6 
and  7,  =  46,000  lbs.  Hence,  while  wheel  4  is  passing  6,  there  is  an  in- 
stant when  the  loads  on  a  6  and  on  6  c  are  equal,  and  at  that  instant  the 
stress  in  B6  reaches  its  maximum  (75,640  lbs.,  see  ^  110)  for  the  given 
set  of  loads. 

113.  Live  Load  Chord  Stresses.  The  criterion,  for  position  of  load 
for  maximum  bending  moment  in  any  section,  and  hence  for  maximum 

W         L  w 

stress  in  the  chord  members  at  that  section,  is =>  -y,  or  I    =    ^--^^rr ; 

w  I  W  ' 

where  W  =  total  load  on  the  truss,  w  ==  load  to  the  left  of  the  section,  L  = 
span  of  bridge,  and  I  =  length  of  segment  to  the  left  of  the  section. 

113.  To  find  the  position  of  load  for  maximum  moment  in  any  panel,  by 
means  of  the  moment  diagram,  Fig.  37^  place  a  wheel,  say  wheel  2,  at  the 
panel  point  at  the  right  of  the  given  panel.  From  the  intersection  on  the  load 
line  (usually  coinciding  with  the  x-axis)  vertically  over  the  left  support,  lay  a 
ruler  or  stretch  a  thread  to  the  intersection  of  the  load  line  with  a  vertical 
from  the  right  support.  If  the  line  so  constructed  recrosses  the  load  line  at 
a  point  vertically  over  the  section  in  question,  the  position  is  a  correct  one ; 
if  not,  it  is  incorrect.  To  facilitate  this  work,  it  is  well  to  use  a  truss  diagram. 
Fig.  38,  drawn  on  a  sheet  of  tracing  paper,  with  the  verticals  carefully  ex- 
tended from  the  panel  points  as  far  up  as  the  load  or  moment  lines  are  likely 
to  extend. 

114.  It  will  often  be  found  that  more  than  one  position  satisfies  the  cri- 
terion, and  that  some  one  of  these  may  give  greater  moments  than  the  others. 
Hence  it  is  well  to  look  for  all  possible  positions.  When  these  are  found,  de- 
termine the  moments,  thus:  On  the  moment  curve  find  the  two  points 
corresponding  (vertically)  with  the  left  and  the  right  support  respectively, 
and  join  these  points  by  a  straight  line.  When  the  head  of  train  has  not 
reached  the  left  support,  the  point  corresponding  with  the  left  support  is  in 
the  X-axis,  produced. 

115.  The  required  moment  is  measured  by  the  vertical  ordinate  distance 
along  the  section,  between  the  moment  curve  and  the  straight  line  just  con- 
structed. The  stress  in  the  chord  members  affected  is  equal  to  the  moment 
divided  by  the  depth  or  the  truss.  Using  these  methods,  the  following  re- 
sults are  obtained : 


Section. 

Wheel. 

Moment,  ft  .-lbs. 

Stress,  lbs. 

Members. 

B6 
Cc 
Cc 
Dd 
Bd 

4 
7 
8 
11 
12 

4,049,333 
6,211,667 
6,207.667 
7,044,000 
7,056,500 

144,600 
221,800 
Not  max. 
Not  max. 
252,000. 

ab  =  bc 
cd  =  — BO 

—CD  =  -DE 

710 


TRUSSES. 


Wind   Loads. 

116.  A  complete  bridge  is  subjected  not  only  to  vertical  loads,  due  to  dead 
load,  to  live  load,  and  to  impact  caused  by  inequalities  in  track  and  in  rolling 
stock,  but  also  to  horizontal  loads.  These  horizontal  loads  are  due  to  the 
transverse  action  of  wind,  or  of  centrifugal  forces  produced  by  the  train  in 
passing  around  a  curve  on  the  bridge,  and  to  the  longitudinal  traction  or 
"drag"  caused  by  stopping  or  starting  a  train  on  the  bridge.  Hence  it  is 
necessary  to  supply  horizontal  bracing,  which,  with  the  two  upper  chords 
or  the  two  lower  chords  of  the  two  vertical  trusses,  form  horizontal  trusses, 
known  as  the  upper  and  lower  lateral  systems,  Figs.  39  (a)  and  39  (6),  and 
sway  and  portal  bracing,  tH  21  and  22. 

117.  The  wind  is  considered  as  blowing  at  right  angles  to  the  bridge. 

118.  The  wind  produces  several  effects,  and  these  must  be  ascertained 
separately,  and  their  joint  effect  then  determined.    Among  these  effects  are: 

(1)  Direct  stresses  in  both  the  upper  and  lower  lateral  systems,  by  pressing 
wlirectly  upon  the  chords;  acting  horizontally  as  a  uniformly  distributed  load. 

(2)  Additional  direct  stresses  on  the  lateral  system  of  the  loaded  chord 
when  a  train  is  on  the  bridge,  owing  to  pressure  of  wind  against  the  train. 

(3)  An  overturning  moment  upon  the  bridge  as  a  whole,  thus  increasing 
the  dead  and  live  load  stresses  in  the  leeward  and  diminishing  those  in  the 
windward  vertical  truss. 

(4)  A  similar  overturning  effect  upon  the  train  and  its  wheels,  which  sinii- 
larly  modifies  their  pressures  upon  the  floor  beams  and  thus  the  stresses  in 
the  main  trusses. 


Fi^.  39. 


119.  The  wind  load,  acting  directly  upon  the  bridge,  is  assumed  to  be 
equally  divided  between  the  upper  and  lower  chords,  and  between  the  wind- 
ward and  leeward  trusses. 

120.  (1)  The  direct  wind  stresses  in  the  lateral  bracing,  due  to  the  pres- 
sure of  the  wind  on  the  truss,  are  found  as  are  the  stresses  in  the  main  trusses, 
due  to  dead  load;  the  horizontal  transverse  struts  of  the  lateral  bracing 
corresponding  to  the  verticals  of  the  main  trusses, 

131.  (2)  Direct  stresses  in  lateral  system  of  loaded  chords,  Fig.  39  (6), 
due  to  wind  on  train. 

Examining  any  panel,  as  c  d,  let 

w   =«  wind  pressure,  in  lbs.  per  lineal  foot  of  train ; 
■■  panel  length,  in  feet; 
wind  pressure,  in  lbs.,  per  panel  fully  occupied  by  train; 
=  number  of  panels  in  span  (  =  6  in  this  case) ; 
■  n  p  =  span,  in  feet ; 

■■  number  of  panels  from  left  support,  a,  to  and  including  the  panel, 
c  d,  under  consideration; 
distance,  a  d,  in  feet; 

length,  in  feet,  of  that  portion  of  the  panel,  c  d,  which  is  occu- 
pied by  train; 
(n  —  m)  v  -\-x  =  that  portion  of  the  span  which  is  occupied 

by  train,  in  feet; 
wind  pressure  on  train  for  a  pressure  of  1  lb.  per  lineal  foot ; 
truss  wind  reaction,*  at  a,  =  wt^  -^2  I 


V 
w  p 


I 


m  p   = 


t 


-=    panel  wind  reaction,*  at  c,  =  w  x^  -i-  2  p  \ 

>=   wind  shear  *  in  panel,  c  d,  =R  —  r  =  w  ^  -i-  2  1  —  w  x^  -i-2  p. 


*  See  foot-note  (t),  1  2. 


WIND    STRESSES. 


711 


The  horizontal  truss  reaction,*  at  a,  due  to  a  concentrated  horizontal 
pressure,  =  1,  acting  at  any  distance,  y  (not  shown),  from  d,  is  = 
In  —  m)  p  +  y 


;  and  the  horizontal  panel  reaction,  at  c,  due  to  the  same 
The  maximum  wind  shear,*  in  the  panel,  c  d,  due  to  wind 


n  p 

pressure,  is  = 

on  train,  occur's  (see  Hlf  79  to  81)  when  the  head  of  the  train  reaches  that 
point,  o,  at  which,  if  the  concentrated  load  be  placed,  these  two  reactions 
y  __   (n  —  m)  p  +  y 


V 


will  be  equal,  or  - 
,  P 

(n  —  m)  p 

X  '=  y  -^ 


n  p 


With  head  of  train  at  o,  we  have 


n  - 


Under  any  conditions,  the  wind  shear,*  S,  in  the  panel,  is  =  R  —  r,  where 


R  = 


tf?[(n  —  m)  p  +  xY 


jind   r  =  --. 


2  n  p  '   •^  2  p* 

Substituting  here  the  value  of  x,  just  found,  for  maximum  shear,  we  ob- 
tain, as  the  maximum  value  of  the  wind  shear  *  in  the  panel, 


w  p 


■  (n  — m)2. 


"^        2(n— 1) 

123.  (3)  Stresses  in  main  truss  members,  due  to  overturning  moment  of* 
wind  on  truss. 

Overturning  moment  =  (wind  panel  load  at  top  chord)  X  (number  of 
panel  points  in  span)  X  height  of  truss. 


Vertical  reaction  at  one  support  == 


1  overturning  moment 

2  width  between  trusses' 


Since  the  upper  lateral  system  carries  all  wind  loads  to  the  ends  of  the 
bridge,  the  end  posts  and  the  chords  (which  take  the  horizontal  components 
of  the  end  post  thrusts)  are  the  only  main  truss  members  affected. 


r" 
i 

I 

I 


(a) 
Fig.  40. 

133.  (4)  Stress  in  main  truss  members,  due  to  overturning  moment  of  wind 
on  train,  Fig.  40.  Let  h  =  height  from  center  of  gravity  of  lateral  system 
cf  loaded  chord  to  center  of  pressure  of  wind  on  train,  p  =  wind  pressure  per 
lineal  foot  of  train,  w  =  width  between  centers  of  gravity  of  trusses,  m  — 
overturning  moment,  per  lineal  foot  of  train,  v  =  added  vertical  load  on 

leeward  truss,  per  lineal  foot  of  truss.     Then  m  =  h  p,  and  r  =  — . 

w 

Impact,  Etc. 

134.  The  effects  of  impact,  due  to  inequalities  of  the  track;  those  of 
"drag,"  due  to  the  starting  and  stopping  of  trains;  and  those  of  centrifugal 
force  of  the  train  on  curves,  are  not  susceptible  of  rigorous  calculation, 
and  engineers  differ  in  their  requirements  respecting  provision  for  them. 
See  Digests  of  Specifications. 


*  See  foot-note  (t),  If  2. 


712 


TRUSSES. 


Determination  of  Maximum  and  Minimum  Stresses. 

135.  Where  specifications  make  allowable  unit  stresses  depend  upon  the 
relation  between  the  maximum  and  minimum  stresses  in  any  given  member, 
both  must  be  computed. 

In  computing  the  maximum  and  the  minimum  stress  in  any  member, 
bear  in  mind  that  a  condition  which,  of  itself,  would  have  a  certain  effect 
upon  the  stress,  may  bring  with  it  other  conditions  which  produce  a  greater 
effect  of  the  opposite  kind.  Thus,  although  the  action  of  wind  on  train 
would,  of  itself,  reduce  the  stresses  in  certain  members,  this  action  can  take 
place  only  with  train  on  bridge,  and  the  vertical  action  of  the  train  load 
would  ordinarily  increase  those  stresses  more  than  the  wind  action  would 
diminish  them. 

In  computing  minimum  stresses,  although  the  live  load  is  usually  to  be 
neglected,  we  must  of  course  not  neglect  the  dead  load,  which  is  always 
present. 

Curves    on    Bridgei|. 

136.  Wlien  tlie  tracli  on  a  bridge  is  curved,  it  is  usually  so  laid  that 
the  center  line  of  the  bridge  bisects  the  middle  ordinate,  m,  of  the  curve. 
See  Fig.  40  (a).  The  center  of  gravity  of  a  panel  load,  P,  at  the  center  of  the 
span  (supposing  it  to  stand  over  the  center  of  the  track)  is  thus  thrown  out  a 
distance  =  ^  m  from  the-  center  line  of  the  bridge,  or  a  distance  =  i  b  + 
i  m  from  the  inner  truss,  where  b  =  width  of  bridge  between  centers  of 
trusses.  Taking  moments  about  the  center  of  the  inner  truss,  we  have, 
therefore,  for  the  load,  W,  on  the  outer  truss,  due  to  P, 


W 


p  i  fe  +  i  ^   ^   p  ^  +  ^ 


b  26    • 

It  is  customary  (see  Digests  of  Specifications)  to  proportion  the  outer 
truss  on  the  safe  assumption  that  its  share  of  the  live  load,  at  each  panel 
point,  is  determined  by  the  formula  just  given,  and  to  design  the  inner  truss 
like  the  outer  one. 


« 

n       tt 

/ 

A 

\ 

\ 

^ 

X 

/ 

A 

m 

a                7 

% 

h 

^ 

a 

I 

n 

Fj^.  40  (a)  and  (b). 


Fig.  31  (repeated). 


Counterbracing. 

127.  In  a  truss  of  any  ordinary  form  (like  that  in  Fig,  31),  under  the  action 
of  a  uniformly  distributed  dead  or  live  load,  or  of  a  live  load  distributed 
symmetrically  as  regards  the  center  of  the  span,  the  shears  in  each  panel  on 
the  left  of  the  center  of  the  span  are  positive,  while  those  in  the  panels  on  the 
right  are  negative;  and  the  stresses  throughout  the  truss  are  such  that  the 
ties  sustain  tension,  and  the  struts,  compression;  the  tendency,  in  each 
panel,  being  to  elongate  that  diagonal  occupied  by  a  tie,  and  to  shorten  that 
diagonal  occupied  by  a  strut. 

But  the  tendency  of  an  eccentric  load,  such  as  those  shown  in  Fig.  31, 
is  to  reverse  the  shears  in  the  panels  between  it  and  the  center;  and.  if  this 
effort,  relatively  to  the  other  forces,  is  of  sufficient  magnitude  to  reverse  the 


EOOF   TRUSSES.  713 

final  shear  in  any  panel,  the  tendency  will  be  to  shorten  the  diagonal  occupied 
by  a  tie,  see  Fig.  1  (a),  and  to  lengthen  any  diagonal  occupied  by  a  strut. 

As  explained  in  11^  14  and  15,  this  condition  is  met,  in  the  Warren 
or  triangular  truss,  by  making  each  web  member  capable  of  resisting  both 
tension  and  compression ;  and,  in  trusses  with  both  vertical  and  diagonal 
web  members,  by  inserting  counters. 

In  a  drawbridge  or  swing  bridge,  not  only  the  web  stresses,  but  also 
the  chord  stresses,  are  reversed  when  the  draw  is  opened  or  closed. 

To  provide  against  possible  further  increase  in  live  loads,  over  those 
now  in  use,  specifications  sometimes  require  that,  wherever  the  live  and  dead 
load  stresses  are  of  opposite  character,  only  70  per  cent,  of  the  dead  load 
stress  shall  be  considered  as  effective  in  counteracting  the  live  load  stress. 
For  other  methods  of  making  similar  provision,  see  Digest  of  Specifications 
for  Steel  Railroad  Bridges. 

Roof  Trusses. 

128.  In  roof  trusses,  the  dead  load,  i.  e.,  the  weight  of  the  truss  itself 
and  that  of  the  purlins,  roof  covering,  etc.,  and  the  snow  load,  are  usually 
taken  as  uniformly  distributed.  In  many  cases  the  sum  of  the  dead  and 
snow  loads  is  divided  equally  between  the  two  supports.  In  other  words, 
the  end  reactions  are  equal. 

129.  The  weights  of  steel  trusses,  in  pounds  per  square  foot  of 
building  space  covered,  may  be  taken,  for  preliminary  estimate,  at  (0.05  to 
0.08)  X  span  in  feet,  according  to  design  and  loading.  Those  of  wooden 
trusses,  with  wooden,  iron  or  steel  tension  members,  may  be  taken  at  from 
one-tenth  to  one-fifth  less. 

If  it  is  found  that  the  weight  of  a  truss,  as  designed,  considerably  exceeds 
the  weight  assumed  for  it  in  advance,  it  should  be  redesigned,  assuming  a  new 
weight  slightly  greater  than  that  obtained  from  the  design. 

130.  The  weights  of  purlins,  of  steel  or  wood,  may  be  taken  at  from 
2  to  3  lbs.  per  square  foot  of  building  space  covered. 

131.  The  weight  of  roof  covering  may  be  taken  approximately  as 
follows : 

Corrugated  iron 2  to  3  lbs.  per  sq.  ft.  of  roof  surface. 

Slate, 7to9   " 

Shingles,  on  laths,    2  to  3    "  "  "  " 

If  on  boards,  add, 3  **  "  "  " 

If  plastered  below  the  rafters,  add,         6  "  "  "  " 

132.  The  snow  load,  in  States  north  of  lat.  35**,  may  be  taken  as  vary- 
ing (chiefly  with  latitude)  from  10  to  30  lbs.  per  sq.  ft.  of  horizontal  projec- 
tion of  roof  surface. 

133.  The  purlins,  stringers,  etc.,  should  be  so  arranged  as  to  carry 
the  weight  of  roof  covering  and  of  snow  directly  to  the  panel  points,  and  thus 
avoid  transverse  stresses  in  the  rafters. 

134.  Each  truss  bears,  besides  its  own  weight,  half  the  weight  of  roof  and 
snow  between  the  two  trusses  (or  truss  and  wall),  adjacent  to  it,  and  each 
panel  point  bears  half  the  load  between  two  panel  points  (or  panel  point  and 
end  support)  adjacent  to  it. 


I 

■      n                           J 

^1       .              r 

_L 

J_ 

m                                    1 

Fig.  41. 

Thus,  Fig.  41,  truss  TT  carries  a  weight  ==  that  on  the  surface  between 
the  two  dotted  lines,  DD  and  EE ;  and  panel  point  p  carries  a  weight  =  that 
on  the  rectangle  mn. 

135.  The  wind  is  regarded  as  blowing  horizontally  upon  one  side  of  the 
roof  and  as  exerting  a  uniformly  distributed  normal  pressure  upon  that  side. 
In  the  following  table  of  assumed  normal  pressures  against  sloping  sur- 
faces, under  horizontal  wind  pressures  of  40  lbs.  per  sq.  ft.,  the  values  in  the 
last  column  are  based  upon  Hutton's  experiments.  Here  a  is  the  angle 
between  the  sloping  roof  surface  and  a  horizontal  plane. 


714 


TRUSSES. 


Assumed  normal  wind  pressure,  P,  in  tbs.  per  sq.  ft.  Horizontal 

wind  pressure  ==  40  tbs.  per  sq.  ft.     a  =  angle  between  roof  surface  and 
horizontal  plane. 

P  P 


a. 

sin  8. 

40.sintt. 

Hutton. 

a. 

sin  a. 

40  .sin  a. 

Hutton. 

5° 

0.087 

3.5 

5.1 

35° 

0.574 

22.9 

30.1 

10° 

0.174 

7.0 

9.6 

40° 

0.643 

25.7 

33.3 

15° 

0.259 

10.4 

14.2 

45° 

0.707 

28.3 

36.0 

20° 

0.342 

13.7 

18.4 

50° 

0.766 

30.6 

38.1 

25° 

0.423 

16.9 

22.6 

55° 

0.819 

32.8 

39.4 

30° 

0.500 

20.0 

26.5 

60° 

0.866 

34.6 

40.0 

136.  The  directions  and  amounts  of  the  end  reactions  and  of  the  stresses 
in  the  members,  due  to  wind,  depend  upon  whether  one  or  both  supports 
are  fixed.  If  both  ends  are  fixed,  their  reactions  are  parallel  to  the  normal 
wind  pressure — i.  e.,  they  are  at  right  angles  to  that  side  of  the  roof  upon 
which  the  wind  is  blowing;  but,  if  one  end  is  free  to  slide  longitudinally  of  the 
truss,  its  reaction  is  taken  as  vertical  and  that  of  the  other  is  more  nearly 
horizontal  than  the  normal  wind  pressure.  When  one  end  is  free,  the  stresses 
must  be  determined  for  wind  blowing  on  the  fixed  side  (in  which  case  it  tends 
to  flatten  the  roof)  and  also  for  wind  blowing  on  the  free  side,  in  which  case 
its  horizontal  component  tends  to  shorten  the  tie-rod  and  to  raise  the  apex. 
The  stresses  in  the  members  of  roof  trusses  are  conveniently  found  by  means 
of  the  method  by  sections,  im  67,  etc.,  or  graphically,  as  below. 

137.  Fig.  42  (a)  illustrates  the  graphic  treatment  of  wind  stresses  for  Fig. 
42  (6),  under  the  three  conditions  named,  viz. : — case  1,  with  both  ends  fixed; 
case  2,  wind  blowing  against  the  fixed  side ;  and  case  3,  wind  blowing 
against  the  free  side.* 

In  Fig.  42  (a),  the  segments  ab,  be,  cd  and  de  represent  the  normal  wind 
pressures  at  the  panel  points  AB,  BC,  CD  and  DE  respectively,  and  ae 
therefore  represents  the  total  normal  wind  pressure  on  the  roof,  all  being 
exerted  against  the  left  side.* 

138.  In  case  1  (both  ends  fixed)  the  segments  fa  and  ef  of  the  solid  line  ea 
represent  the  left  and  right  reactions  respectively. 

139.  In  case  2  (wind  blowing  against  fixed  side)  the  reactions  are  repre- 
sented by  the  dash  line  ef'a;  and  in  case  3  (wind  blowing  against  free  side) 
by  the  dash-and-dot  line  ef^^a. 

140.  The  segments  /7  and  ff"  represent  the  horizontal  components  of  the 
right  and  left  wind  reactions  respectively  in  case  1 ;  and  f'f  that  of  the  wind 
reaction  of  the  fixed  end  in  case  2  or  case  3,  or  that  of  the  total  wind  reaction. 

141.  Having  found,  by  moments,  the  end  reactions  ef  and  fa  for  case  1; 
where,  Fig.  42  (6), 

,  A  s 

ef  =  ae  ^ ; 

Span 

the  vertical  reactions,  e/'  and  a/",  for  cases  2  and  3  respectively,  are  found  by 
dropping  perpendiculars  from  e  and  from  a,  Fig.  42  (a),  upon  gf  produced. 
The  reactions  of  the  fixed  ends  are  then  given  by  the  closing  line,  /'a,  in  case 
2,  and  by  ef  in  case  3. 

*  To  avoid  the  necessity  of  showing  two  skeleton  figures  and  two  diagrams, 
we  have  supposed  the  wind  to  blow  always  in  one  direction  (viz.,  against  the 
left  side)  and  first  one  end  of  the  truss  and  then  the  other  end  to  be  fixed. 
In  practice,  of  course,  the  reverse  of  this  is  the  case;  i.  e.,  one  end  or  the 
other  of  the  truss  (if  not  both)  is  fixed,  and  remains  so;  and  the  wind  may 
blow  against  either  side.  The  figure  and  diagram  will,  however,  answer  for 
this  latter  condition  also.  Thus,  if  the  wind  blow  against  the  left  side,  as 
shown,  and  if,  as  in  case  2,  that  side  is  fixed,  then  the  diagram,  using  the 
broken  lines,  e  f  a,  gives  the  stresses  in  the  members,  as  they  are  lettered. 
But  now  (the  left  end  remaining  fixed)  suppose  the  wind  to  blow  against  the 
free  side;  i.  e.,  from  the  right.  We  may  nevertheless  suppose  the  right  end 
fixed,  and  the  wind  blowing  against  the  left  side,  as  in  Fig.  (6),  and  find  the 
stresses  in  the  members  from  Fig.  (a)  as  it  stands,  us^ng  the  dash-and-dot 
diagram,  e  f  a;  but  we  must  then  remember  that  the  stresses  thus  found  for 
BG,  GF,  etc.,  on  the  left  of  the  truss,  Fig.  (6),  really  apply  to  the  correspond- 
ing members,  QE,  QF,  etc.,  on  the  right,  and  vice  versa 


EOOF  TRUSSES. 


715 


143.  The  stresses  in  the  web  members,  GH,  MN,  etc.,  Fig.  (b),  and  those 
in  the  several  members,  BG,  EM,  etc.,  of  the  rafters,  are  given  by  the  corre- 
sponding lines,  gh,  mn,  bg,  em,  etc.,  in  the  diagram.  Fig.  (a). 

143.  In  the  leeward  rafter,  in  this  case,  the  stress  in  the  three  segments, 
ME,  PE,  QE,  is  uniform  throughout,  and  moreover  it  is  the  same  in  each  of 
the  three  cases,  being  =  me  =  pe  =  qe. 

In  the  four  web  members,  LM,  MN,  NP,  PQ,  to  the  leeward  of  the  center, 
the  stress,  in  this  case,  is  zero,  being  represented  by  the  point,  Imnpq,  Fig. 
(a). 

144.  The  stresses  in  the  several  segments,  GF,  JF,  LF,  NF,  and  QF,  of  the 
horizontal  tie  rod,  Fig.  (6),  are  represented,  in  Fig.  (a), 

in  case  1  (both  ends  fixed)  by  gf,  jf.  If,  nf  and  qf; 
in  case  2  (wind  against  fixed  side)  by  gf,  jf,  etc. ; 
in  case  3  (wind  against  free  side)   by  gf,  jf,  etc. 
In  each  of  the  three  cases  there'  is  uniform  tension  in  the  three  leeward 


i^*X 


Fig.  42. 


segments,    LF,  NF   and   QF,  of   the   horizontal   tie    rod.     This   uniform 
tension  is 

in  case  1  (both  ends  fixed)  =  If    =  mf    =-nf; 

in  case  2  (wind  against  fixed  side)  =  If  —  mf^  =  nf; 

in  case  3  (wind  against  free  side)         =  If  =  mf  =  nf. 

145.  It  is  thus  seen  that,  in  oiir  Fig.,  with  horizontal  tie  rod,  the  dififer- 
ence  in  the  manner  of  supporting  the  ends  affects  only  the  horizontal  stresses 
in  the  members  of  that  rod,  and,  through  them,  the  manner  in  which  the 
horizontal  component  /'/''  of  the  wind  stress  is  distributed  between  the  two 
supports. 

146.  If  the  lower  chord  were  not  straight,  however,  the  stresses  in  the 
rafter  and  web  members  would  be  affected  by  the  difference  between  the 
three  cases. 

147.  The  final  or  resultant  stress,  in  any  member,  is  the  algebraic  sum  of 
the  dead,  snow  and  wind  loads  for  that  member.  In  some  cases,  the  wind 
toad  may  diminish  or  even  reverse  the  stresses  due  to  dead  and  snow  loads. 


716 


TRUSSES. 


148.  In  timber  roof  trusses  of  short  span,  Figs.  43  to  47,  for  roofs  of  dwell- 
ings and  other  small  buildings,  we  may,  with  sufficient  accuracy,  make  a 
liberal  assurnption  for  load,  to  include  wind  pressure.  In  discussing  these 
figures,  we  investigate  the  stresses  by  means  of  the  force  parallelogram. 
For  dimensions  of  such  trusses,  see  If  266. 

149.  In  the  wooden  roof  truss,  Fig.  43,  uniformly  loaded  along  each 
rafter,  let  H  I  =  the  weight  of  one  rafter  and  its  load.  Then  E  I  =  the  hori- 
zontal pressure  of  the  head  and  of  the  foot  of  that  rafter  (the  latter  being  the 
tension  in  the  chord),  and  H  E  =  the  inclined  pressure  at  its  foot. 


V     h 


y    P 


Fiff.  43. 


150.  In  Fig.  44,  make  G  R  =  HI.  Then  G  L  is  the  transverse  pressure 
of  the  load  against  the  rafter  as  a  beam,  and  L  R  is  a  longitudinal  pressure 
along  the  rafter,  forming  a  part  of  the  total  longitudinal  pressure. 

151.  If  G  R  were  concentrated  at  G,  L  R  would  be  uniform  from  G  to  a, 
and  would  not  be  exerted  above  G ;  but,  as  G  R  represents  a  load  uniformly 
distributed  along  the  rafter,  from  top  to  foot,  the  pressure  represented  by 
L  R  increases  uniformly,  from  nothing,  at  the  top,  o,  to  L  R,  at  the  foot,  a. 

153.  Of  the  transverse  pressure,  G  L,  one  half,  =  o  p,  is  sustained  at  the 
top,  o,  of  the  rafter,  and  the  other  half,  =  a  q,  at  the  foot,  a.  At  the  top, 
o  p  is  resolved  into  the  horizontal  pressure,  o  6  =  E  I,  against  the  head  of 
the  other  rafter,  and  a  uniform  thrust,  o  z,  along  o  a. 


^*^z>v. 

F? i 

X 

p          \^ 

/ 

l\^\l^ 

/ 

^ 

a 

^^ 

-     4 

^^ 

Fi 

s.  44.                t 

^/ 

153.  It  is  immaterial  whether  we  thus  resolve  o  p  directly  into  o  h  and  o  z 
(as  though  the  head  of  the  rafter  rested  against  a  vertical  wall  at  o),  or  whether 
we  first  resolve  it  between  the  two  rafters,  into  o  c  and  o  r.  For  in  the  latter 
case  we  must  add  to  o  c  a  thrust  ( =  or  =  cz^  produced  in  o  a  by  the  trans- 
verse pressure  (similar  to  o  p)  of  the  head  of  the  other  rafter ;  and  the  sum  of 
these  two  (o  c  and  o  r)  is  =  o  z. 

154.  The  total  longitudinal  thrust  in  the  rafter  increases  uniformly  from 
0  z,  Sit  the  top,  to  o  z  4-  L  R  =  a  A;,  at  the  foot,  where  it  combines  with  a  q 
(  =  half  the  transverse  load)  to  form  av  =  HE. 

155.  Tension  in  chord  =  l'Ei  =  tv  =  sk  —  n  q.  Vertical  pressure  on 
support  =  til  =  at  =  as-{-an  =  a8-\-st. 


ROOF   TRUSSES. 


717 


156.  In  Fig.  45,  having  found,  as  for  Fig.  43,  the  stresses,  etc.,  due  to 
the  rafters  and  their  loads,  remember  that  the  king  rod,  o  n,  supports  its 
own  weight  plus  the  portion  y  y  oi  the  chord  =  ^  the  chord.  Making  o  t  = 
this  combined  weight,  we  have  o  w  =  o  d  =  an  additional  pressure,  uniform 
throughout  each  rafter,  and  cm  =  cd  =  a,n  additional  tension  on  the  chord. 


Fig.  45. 

157.  In  Fig.  46,  assuming,  for  safety,  that  the  rafter,  ;  6,  is  divided  at  its 
center,  U,  make eo  =  the  weight  oizr  and  its  load  (zr  =  half  the  rafter) . 

158.  Then  e  i  =  an  additional  pressure  on  JJ  h,  e  k  =  pressure  on  U  c; 
« 1  =  s  A;  =  additional  tension  on-  half  chord,  c  b,  and  e  o  =  2  e  s  =  load  of 
and  on  z  r,  =  additional  tension  on  king  post  due  to  both  struts.  Then 
make  a  g  =  eo  +  weight  of  king  rod  +  weight  of  two  struts  +  weight  of  and 
on  y  y,  and  proceed  as  in  Fig.  45. 


Fig.  46. 


159.  Each  strut  will  thus.bear  half  of  the  weight  of  and  on  zr,  or  x  u,  only 
when,  as  in  Fig.  46,  the  inclination  of  the  strut  is  the  same  as  that  of  the 
rafter.  If  the  strut  is  steeper  than  the  rafter,  it  will  bear  more  than  half; 
but  if  it  is  less  steep  than  the  rafter,  it  will  bear  less  than  half ;  the  remainder 
being  in  every  case  borne  by  the  rafter. 

160.  The  introduction  of  the  struts  converts  each  rafter,  considered  as  a 
beam,  into  two  beams  of  shorter  span  and  bearing  less  loads. 


Fig.  47. 


161.  In  the  queen  truss,  Fig.  47,  make  og  =  total  tension  in  queen  rod  + 
half  weight  of  and  on  the  "straining  beam,"  z  w.  Longitudinal  pressure  on 
■3  tt;  =  tension  in  chord,  6  a,  =  I  E  +  o  c. 


718 


TRUSSES. 


Deflections. 
162.  The  total  deflection  of  a  truss  *  comprises  (1)  the  elastic  or 
temporary  deflection,  due  to  the  stretchf  of  its  several  members  under  the 
loading  applied  to  the  truss,  and  (2)  the  non-elastic  or  permanent  deflec- 
tion, due  to  looseness  of  its  joints.  In  good  construction  the  latter  is 
relatively  negligible  in  moderate  spans. 

The  total  elastic  deflection,  D,  of  a  truss,  at  any  point,  c,  is  made  up  of 
partial  elastic  deflections,  d,  d,  etc.,  at  c,  each  due  to  the  stretch,  A:,t  in 
some  member. 

Let  it  be  required  to  find  the  deflection  at  a  panel  point,  c  (usually  the  cen- 
ter of  a  span  or  the  end  of  the  arm  of  a  swing  bridge  or  other  cantilever) ; 
and,  for  any  load  or  system  of  loads,  let 
D    =   the  total  elastic  deflection  at  c; 
.d   =   the  partial  elastic  deflection,  at  c,  due  to  the  stretch,  k,-\  in  any 

member ; 
p   =   the  unit  stress  in  that  member ; 
P    =   the  total  stress  in  that  member; 
I   =   the  length  of  that  member; 

the  stretch  t  in  that  member  =  -^ ; 


k   = 
W   = 


that  load  which,  applied  at  c,  would  produce  the  stress,  P,  in  that 

member : 
P 


W 

E    =   the  modulus  of  elasticity  of  the  material,  =  p  ■ 


Pj 
k' 


163.  Equivalence   of  Work.     In  Fig.  48,  let  any  load,  W,  be  applied 
at  any  point,  c,  of  a  truss  or  bar.     Then,  for  a  small  deflection,  J  such  as  may 


*  See  "The  Application  of  the  Principle  of  Virtual  Velocities  to  the  Deter- 
mination of  the  Deflection  and  Stresses  of  Frames,"  by  Geo.  F.  Swain, 
Jour,  of  the  Franklin  Institute,  vol.  lxxxv,  1883;  "Trusses  with  Super- 
fluous Members,"  by  Wm.  Cain,  Van  Nostrand's  Magazine,  vol.  xxvii, 
No.  4,  October,  1882;  "The  Graphical  Solution  of  the  Distortion  of  a  Framed 
Structure,"  by  David  Molitor,  Jour.  Ass'n  Eng'ng  Societies,  vol.  xiii. 
No.  6,  June,  1894;  and  "The  Theory  and  Practice  of  Modern  Framed  Struc- 
tures," by  J.  B.  Johnson,  C.  W.  Bryan  and  F.  E.  Turneaure,  New  York, 
John  Wiley  &  Sons. 

t  For  brevity  we  here  use  the  word  "stretch"  to  signify  any  change  of 
length,  including  the  shortening  due  to  compression,  as  well  as  the  elonga- 
tion due  to  tension. 

t  For  the  sake  of  clearness,  the  stretches  and  deflections,  in  our  Figs.,  are 

p 
exaggerated  beyond  the  limit  within  which  the  ratio,  ^,,  would  remain  even 

W 
approximately  constant. 


DEFLECTIONS.  719 

be  permitted  in  trusses,  the  external  work,  W  c?,*  of  a  partial  deflection,  d, 
due  to  the  stretch,  /:,  in  any  member,  is  practically  =  the  internal  work, 
P  k,  of  overcoming  the  resisting  stress,  P,  in  that  member,  through  the  dis- 
tance, k;  or 

W  d  =  P  A;: 

Hence, 

d  =  ^k  =  uk;     or     ^=    p. 

In  words,  the  stretch,  k,  in  any  one  member,  is,  to  the  resulting  partial 
deflection,  d,  at  c,  inversely  as  is  any  stress,  P,  in  that  member,  to  a  Toad,  W, 
which,  if  applied  at  c,  would  cause  that  stress. 

Thus,  in  Fig.  48  (a),  where  k  is  in  the  same  direction  as  D,t  P  =  W,  and 
T>=k. 

In  Figs.  (6)  and  (c),J  suppose  the  strut  incompressible.     Then  D  is  due 

p 
solely  to  the  elongation  of  the  tie,  and  D  =  ^  k  =  u  k. 

p 

In  Fig.  (c),  ^  is  greater,  and  (for  a  given  stretch,  k,  in  the  tie)  D  is  there- 
fore greater,  than  in  Fig.  (6). 

164.  Deflection  Independent   of  Nature   of  Cause   of  Stretch. 

Now  it  is  evident  that  the  deflection,  d,  at  c,  depends  solely  upon  the  amount 
and  character  of  the  stretch,  k,  in  the  member,  and  is  independent  of  the 
nature  of  the  cause  of  that  stretch.  That  is  to  say,  any  change,  k  (however 
caused),  in  the  length  of  the  member,  necessarily  contributes  its  fixed  quota, 

p 
d  =  ^-r  k,  to  the  total  deflection,  D,  at  c.     In  other  words,  since  d  and  k  are 

W 
mere  distances,  and  since  u  is  simply  a  ratio,  the  relation  between  d  and  k  is 
a  purely  geometrical  one,  and  is  therefore  not  confined  to  deformations  pro- 
duced by  applied  loads,  but  is  applicable  also  to  those  produced  by  changes 
of  temperature,  to  intentional  lengthening  or  shortening  of  members,  or  to 
any  other  cause. 

Hence,  if  a  member  be  in  any  way  lengthened  or  shortened,  by  a  length,  k, 
p 
a  corresponding  change,  d,  =  i^.  .  A:  =  w  Ar,  takes  place  in  the  deflection  at  c. 
W 

For  instance,  if  we  place  any  system  of  loads  upon  a  truss,  and,  by  the 
principles  of  statics,  determine  the  resulting  total  stress,  P,  and  unit  stress 
p,  in  any  member;  we  have,  for  the  partial  deflection,  at  c,  due  to  the  stretch, 
k,  in  that  member,  under  the  given  system  of  loads, 

d    =   u  k;  (For  u,  see  H  165.) 

and,  since  k   =  -^^, 

p 

165.  To   obtain  the  ratio,   u,  =  ;^,  for  each  member,  we  suppose  a 

concentrated  load  applied  at  c;  and,  by  the  principles  of  statics,  find  the 
resulting  total  stresses,  P,  P,  etc.,  in  the  several  members.  If  the  supposed 
load,  at  c,  be  taken  =  unity,  the  stresses,  P,  P,  etc.,  so  found,  are  the  desired 
ratios,  u,  u,  etc. 


*  Strictly  speaking,  with  a  load  increasing  gradually  from  0  to  W,  and 
with  resulting  stress  increasing  gradually  from  0  to  P,  we  should  deal  with 
the  mean  load,  =  i^  W,  and  with  the  mean  stress,  =  ^  P,  in  each  member; 
but  it  will  be  seen  that  this  would  not  affect  the  equations  derived. 

t  Where,  as  in  Figs,  (a),  (b)  and  (c),  only  one  member  is  supposed  to 
change  its  length,  D  =  d. 


t  See  note  (|)  on  preceding  page. 


720 


TRUSSES. 


166.  Summation  of  Deflections.  The  total  deflection,  D,  at  c, 
under  the  given  system  of  loads,  being  =  the  sum  of  the  partial  deflections, 
d,  d,  due  (but  not  necessarily  equal)  respectively  to  the  stretches,  k,  k,  in 
the  several  members,  we  have 


2d 


,  p  u  I 
'     E    • 


Thus,  in  Figs,  (d)  and  (e),  we  assume  the  tie  extensible  and  the  strut  com- 


In  Fig.  (d),  W 
Hence  D  =  2 


(c),D  =  2mA; 


Pi  +  Pa;  and  WD  = 

P  P 

=  ^  '^1+  ^  'h  =  Ulh  +  U2h' 


Flk+F2k  =  (Fi-\-  PJ  k. 

W 
k  '=  ;^  .  k  =  k.     In  Fig. 


167.  Positive  and  Negative  Stretches.  In  some  cases  it  naay 
happen  that  the  change  of  length  of  a  member  diminishes,  instead  of  in- 
creasing, the  total  deflection  at  the  point,  c,  in  question,  and  must  therefore 


be  taken  as  negative  in  summing  up  the  values  oiu  k 


V  ul  ^ 


but  when  c  is 


the  middle  point  of  a  span,  or  the  end  of  a  cantilever,  all  the  changes  in  length 
of  the  members  ordinarily  contribute  to  the  deflection,  and  must  therefore 
be  taken  as  positive. 

Theoretically,  the  formula,  D  =  2  ^-^^,  applies  also  to  the  deflections 

of  arches,  dams  and  other  structures  composed  of  blocks;  but,  owing  to 
the  uncertainty  of  the  values  of  E,  and  to  the  relative  inaccuracy  of  finish 
in  masonry  work,  it  is  of  but  little  practical  utility  in  such  cases. 

168.  Redundant  or  Statically  Indeterminate  Members.  Trusses 

frequently  contain  members  whose  stresses  cannot  be  found  by  the  principles 
of  statics.  Thus,  in  Fig.  11  (c),  the  two  diagonal  tension  members  meeting 
at  the  top  of  either  end  post  are  said  to  be  redundant,  or  statically  indeter- 
minate, because  the  principles  of  statics  do  not  enable  us  to  determine  what 
proportion  of  the  total  load  goes  to  the  supports  through  each  of  the  two 
systems,  Figs.  11  (a)  and  (6),  composing  Fig.  11  (c).  But  the  deflection 
formula,  just  given,  enables  us  to  determine  the  stresses  in  such  members; 
for,  by  means  of  it,  we  may  find,  separately,  the  deflection  in  each  of  the  two 
systems,  Figs.  11  (a)  and  (6) ;  and  the  part  load,  transmitted  to  the  supports 
through  each  of  these  two  systems,  is  inversely  proportional  to  their  deflec- 
tions. 

BRIDGE    DETAILS    AND    CONSTRUCTION. 

General  Principles. 

169.  In  general,  a  truss  bridge  consists  essentially  of  two  or  more  verti- 
cal trusses,  AB,  CD,  Fig.  49,  placed  side  by  side,  and  connected  by  the 
floor  system,  which,  in  turn,  they  support;  and  bracing  (forming  a 
**  lateral  system")  is  supplied  between  opposite  chords,  where  practica- 
ble, in  order  to  maintain  the  trusses  parallel. 


H 

J> 


Fig.  49. 


170.  The  floor  system  consists  ordinarily  of  floor  beams  and  string- 
ers. The  floor  beams,  AC,  EF,  etc..  Fig.  49,  are  placed  transversely  to  the 
bridge,  and  are  attached  to  the  trusses  at  opposite  panel  points.  Connected 
with  these  and  perpendicular  to  them  or  parallel  to  the  trusses,  are  the 
stringers,  GH,  IJ,  etc.  In  railroad  bridges,  there  are  usually  two  or  more 
stringers  placed  side  by  side  and  running  the  length  of  the  bridge,  to  support 
the  ties.  In  city  highway  bridges,  these  stringers  are  usually  spaced  at  smaller 
intervals,  and  support  buckled  plates  or  other  form  of  flooring,  on  which  the 


DESIGN.  721 

paving  is  laid.  For  country  highway  bridges,  the  stringers  are  frequently 
of  wood,  placed  quite  near  together,  and  the  planks  of  the  floor  are  nailed 
or  spiiced  directly  to  them. 

Solid  floors  (see  Pencoyd  floor  sections)  add  to  the  rigidity  and  per- 
manence of  a  bridge,  and  give  increased  protection  to  traffic  below,  against 
injury  from  falling  bodies  or  in  case  of  derailment.  Their  shallowness  is  an 
advantage  where  head-room  is  an  object. 

171.  Any  load,  then,  is  carried  first  from  the  ties  or  floor,  etc.,  to 
the  stringers,  then  by  the  stringers  to  the  floor  beams,  and  finally  by  the 
floor  beams  to  the  panel  points  of  the  bridge,  where  it  is  carried  through  the 
trusses  to  the  supports. 

173.  Pedestals,  shoes  or  bed  plates,  Fig.  62,  bolted  to  the  piers,  support 
the  ends  of  the  trusses.  When  the  bridge  is  of  long  span,  so  that  the  expan- 
sion and  contraction  due  to  heat  and  cold  are  considerable,  expansion 
bearings,  Figs.  60,  62,  must  be  provided  at  one  end.  See  ^^  205,  etc. 
For  cross-bracing,  see  Tf^  19,  etc. 

General    Character    of   Design. 

173.  Flexible  and  Rigid  Tension  Members.  Adjustable 
Counters.  Until  recently,  eye-bars  have  generally  been  used  for  the 
tension  members  of  trusses.  These  are  long  flat  bars,  liable  to  yield  laterally 
under  compression,  and  furnished,  at  their  ends,  with  eyes  or  openmgs, 
through  which  pass  pins  connecting  them  with  the  other  members  of  the 
bridge ;  but  rigid  built  members,  capable  of  sustaining  some  compression,  as 
well  as  tension,  are  now  much  used  for  tension  members.  Counters  were 
usually  made  in  two  lengths,  and  were  adjustable,  the  two  lengths  being 
connected  by  turnbuckles ;  but  these  rendered  it  possible  to  bring  undue  and 
dangerous  stresses  in  the  panels,  and  they  are  now  giving  place  to  counters 
made  each  in  one  length. 

174.  Compression  members  are  ordinarily  "built  up"  of  angles  and 
plates,  or  of  channels  and  plates  with  latticing,  in  hollow  shapes,  bringing 
most  of  the  material  as  far  as  possible  from  the  neutral  axes  of  the  cross- 
section  and  thus  increasing  its  resisting  moment. 

175.  Pin  and  Riveted  Connections.  The  web  members  are  con- 
nected with  the  chords  either  by  pins  or  by  rivets.  In  the  former  case  the 
truss  is  said  to  be  pin-connected;  in  the  latter  case,  riveted.*  Until  re- 
cently, pin-connected  trusses  have  been  typical  of  American  practice;  but 
the  Americans  are  now  largely  using  riveted  trusses,  for  spans  up  to  from 
150  to  175  ft.,  while  the  Europeans  are  in  some  cases  using  pins.  The  prin- 
cipal advantage  claimed  for  the  riveted  joint  is  that  it  makes  a  stiffer  bridge 
and  one  that  will  not  rattle,  and  that  a  riveted  truss,  computed  as  if  pin-con- 
nected, will  have  an  additional  margin  of  safety  on  account  of  added  stiff- 
ness. In  the  pin-connected  bridge,  on  the  other  hand,  the  stresses  can  be 
much  more  accurately  determined,  and  deflection  may  take  place  without 
producing  twisting  or  bending  stresses  in  the  connections  themselves. 

176.  Tendency  to  Greater  Rigidity.  There  is  a  growing  tendency 
to  use  stifi'er  bracing,  to  design  at  least  all  short  braces  for  compression,  and 
to  make  even  the  longer  tension  members  of  channels  or  angles,  forming  a 
rigid  member.  Unless  pin-connected  eye-bars  are  of  exactly  equal  length, 
some  of  them  will  receive  more  than  their  share  of  the  total  stress. 

177.  Floor-beam  Connections.  In  the  United  States,  floor-beam 
connections  were  formerly  made  by  hanging  the  floor  beams  from  the  pins 
by  means  of  hangers ;  but  now,  where  possible,  the  ends  of  the  floor  beams 
are  riveted  directly  to  the  inner  sides  of  the  posts. 

178.  In  tension  members,  rivets  are  so  arranged  as  to  reduce  the  net 
effective  section  as  little  as  possible. 

179.  Compression  members  are  so  designed  as  to  place  most  of  the  material 
as  far  as  possible  from  their  neutral  axes,  and  they  are  sometimes  strength- 
ened by  auxiliary  ties  or  posts  supporting  them  at  their  middle  points,  'in 
cases  where  the  resulting  saving  in  material  for  the  member  will  be  consid- 
erably greater  than  the  expenditure  of  material  in  the  auxiliary  member. 

*  Riveted  trusses  are  unfortunately  called,  also,  "lattice  girders,"  "lattice 
trusses,"  "riveted  lattice  girders"  and  "riveted  lattice  trusses."     The  term 
"  lattice  "  is  often  applied  to  shallow  trusses  with  numerous  panels. 
46 


722 


TRUSSES. 


180.  So  far  as  possible,  compression  members  are  made  equally  strong 
against  bending  about  either  of  the  two  principal  axes,  AB  and  XY,  Fig.  52, 
of  their  cross-sections. 

181.  Where  the  same  mernber  occurs  many  times  in  a  bridge,  and  where, 
therefore,  an  excess  of  material  in  the  design  of  such  member  would  involve 
a  large  total  'vaste,  the  computation  of  the  member  is  repeated  many  times 
until  the  most  economical  section  is  found. 

183.  In  metal  trusses  the  shorter  members  are  usually  made  to  withstand 
compression,  and  the  longer  ones  tension,  this  being  more  economical  of 
material.  Thus,  the  Pratt  truss,  with  diagonal  tension  members,  is  used 
for  steel  bridges,  v/hile  the  Howe  truss  is  now  built  only  with  wooden  diago- 
nals. 

Tension  Members. 

^or.    X  ».  <•  ..  maximum  tensile  stress 

183.  In   eye-bars,  area  of  cross-section  =    — n n r- : —  . 

allowable  unit  tension 

184.  The  dimensions  of  the  heads  of  eye-bars  are  usually  determined  by 
the  manufacturers,  and  are  so  designed  as  to  give  ample  excess  %f  strength 
at  the  pin-holes;  so  that,  if  tested  to  destruction,  fully  two-thirds  of  the 
number  of  bars  tested  shall  break  in  the  body  of  the  bar,  this  being  usually 
required  by  specifications.  It  is  important  that  the  proportions  of  eye-bar 
heads  should  be  such  as  to  ensure  thorough  working  of  the  metal  in  the 
upsetting  process. 

^     /      I      M/    M/    \l/    \l/ 


O      JO      20      30      40      50      GO      70     80      90     lOO   ISO  X20 
Fig.  50. 

185.  Fig.  50  shows,  to  two  different  scales,  the  "packing"  (arrangement 
of  pins  and  eye-bars)  in  the  left  half  of  the  lower  chord  of  a  150  ft.  through 
(skew)  span  built  by  the  Phoenix  Bridge  Co.  in  1900  for  the  Philadelphia  and 
Reading  Railway  Co.  near  Reynolds,  Pa. 

186.  Built  Sections.  Hip  vertical  hangers,  non-adjustable  counters 
and  their  corresponding  mains,  are  usually  built  up  of  rolled  steel  shapes. 
A  section  in  common  use,  shown  in  Fig.  51,  consists  of  four  angles,  connected, 
at  intervals,  by  small  narrow  flat  bars,  riveted  to  the  angles  and  running 
across  zigzag  from  one  to  the  other.  When  single,  as  in  Fig.  51,  this  is  called 
"lacing" ;  when  double,  as  in  Fig.  52  (6),  "latticing."  The  shaded  area  of  the 
angles,  Fig.  51,  minus  that  of  the^rivet  holes,  is  taken  as  the  effective  section, 

187.  Minimum  Sections,  Specifications  (see  Digests)  usually  require 
the  use  of  some  minimum  section.  Thus,  in  a  counter  in  which  the  stress  is 
58,000  lbs.,  3.5  square  inches  of  cross-section  would  suffice ;  but  specifications 
frequently  forbid  the  use,  in  such  sections,  of  any  angle  smaller  than  3^  X  3^ 
X  I,  which  gives  9.20  square  inches  gross;  or,  deducting  one  rivet  hole  from 
each  angle,  7.68  square  inches  net  section. 

Compression  Members. 

188.  The  computation  of  a  compression  member  consists  of  a  series  of 
approximations ;  for  the  unit  stress  depends  upon  the  radius  of  gyration,  the 
radius  of  gyration  on  the  area  of  section  and  disposal, of  material  with  regard 
to  the  axes,  and  this,  in  turn,  on  the  unit  stress.  See  Pillars,  under  Strength 
of  Materials. 

189.  Fig.  52  (a)  shows  a  form  often  used  for  posts,  and  consisting  of  two 
channels,  placed  with  their  backs  outward  and  riveted  together  by  lacing. 
In  Fig.  52  (6)  the  channels  are  placed  with  their  backs  inward.  For  econ- 
omy, the  channels  should  be  so  spaced  as  to  make  the  radius  of  gyration  the 


COMPRESSION   MEMBERS. 


723 


same  about  either  axis,  A-B  or  X-Y.     The  radius  of  gyration  is  given  in 
the'  hand-books  of  mfrs  of  structural  shapes.     See  pp.  892,  etc. 

190.  The  upper  chord  section  is  frequently  built  up  of  two  channels 
and  a  plate,  or  in  some  such  form  as  shown  in  Fig.  53,  consisting  of  two  verti- 
cal plates  or  "webs,"  a  horizontal  top  plate  or  "cover,"  four  "angles,"  and 
flat  pieces  or  bars  on  each  side  of  the  bottom.  Lattice  bracing,  or  lacing,  is 
provided  along  the  bottom,  except  at  panel  points,  where  it  is  omitted  in 
order  that  the  post  and  the  ties  may  enter  the  chord  from  below.  In  pin- 
connected  trusses  the  axis  of  the  pin  lies  in  the  line  AB. 

191.  The  interior  width,  w,  depends  chiefly  on  the  space  required  by  the 
post  and  ties  which  meet  at  the  panel  point,  and  also  upon  the  height  of  the 
inside  rivet  heads.  Usually,  for  convenience  of  construction,  the  greatest 
width,  w,  required  is  kept  constant  throughout  the  upper  chord.  The  height, 
H,  depends  chiefly  on  the  size  of  eye-bar  head,  and  is  kept  constant.  The 
thicknesses  of  the  web  plates,  and  sometimes  also  those  of  the  bars  and 
angles,  are  varied,  along  the  chord,  in  order  to  provide,  at  each  point,  suffi- 
cient area  to  withstand  the  stress. 


'7'/^r\ j-^Tyyy 


■/ 

Fig.  51 


{a)  (b) 

Fig.  52. 


192.  The  end  post  is  to  be  considered  not  only  as  a  column,  but  also  as  a 
beam  subject  to  shear,  on  account  of  the  wind  blowing  against  the  top  of  the 
side  of  the  truss.  The  design  of  this  built-up  form  is  much  the  same  in  princi- 
ple as  that  given  above  for  a  post.  Certain  sections  are  tried,  and  then 
changed  if  necessary. 

193.  The  end  post  must  be  safe,  not  only  against  bending  about  the 
axis,  A-B,  Fig.  53,  under  compression,  but  also  against  bending  about  the 
other  axis,  X-Y,  under  the  combined  effect  of  compression  and  the  bending 
moment,  due  to  the  wind  blowing  against  the  top  of  the  truss.  See  Fig.  54  (a) . 


Fig. 


194.  The  portal  bracing,  above,  and  the  floor  beam,  below,  are  assumed 
to  prevent  bending  of  the  parts  of  the  posts  adjacent  to  them ;  and  we  may 
consider  either  half  of  either  post  (  =  ^  Z)  as  a  vertical  cantilever,  fixed  at 
one  end,  and  loaded,  at  the  other  end  (which  is  at  the  middle  of  the  post) 
with  a  load  =  wind  pressure  on  half  the  upper  portion  of  one  truss. 


724 


TRUSSES. 


195.  The  maximum  stresses,  due  to  compression,  of  course  occur  about 
the  middle  of  the  post,  while  those  due  to  the  wind  occur  near  the  ends. 
Hence  it  would  be  unreasonable  to  require  the  post  to  resist  all  of  both 
effects  simultaneously  throughout  it§  length;  and  specifications  therefore 
usually  allow  the  unit  stress,  due  to  dead,  live,  impact  and  wind  loading  com- 
bined, to  be  increased  to  21,000  lbs.  persq.  in.,  properly  reduced  by  formula 
foi  compression. 

196.  The  formula  for  the  strength  of  long  compression  members  is  as  fol- 
lows, where 

P    =   average  permissible  unit  load  on  column,  in  lbs.  per  sq.  inch; 
I   =   length  of  column ; 
r   =   radius  of  gyration  of  its  cross-section : 


P  = 


21,000 


1  + 


9000  r2 


197.  The  formula  for  extreme  fiber  stress  due  to  combined  compression 
And  bending,  is 

P    ^     Mb  T 


I  — 


PJ2 

Ec 


Where 


P    = 
A   = 

Mb  = 

T   = 

I   = 

I  = 

E   = 


longitudinal  compressive  force ; 

area  of  section ; 

bending  moment  due  to  transverse  load ; 

distance  from  neutral  axis  to  extreme  fibers; 

moment  of  inertia ; 

length  of  beam ; 

modulus  of  elasticity; 

coefficient.     See  Transverse  Strength,  T[  103. 


Joints. 
198.  Pin  Plates.  Where  a  pin  passes  through  one  or  more  shapes  of 
some  member,  it  often  happens  that  the  combined  surfaces  of  the  truss  mem- 
bers alone,  in  contact  with  the  pin,  are  insufficient  to  transmit,  by  bearing,  all 
of  the  stresses  to  be  delivered  to  the  member.  There  is  then  danger  of 
crushing  the  material  which  presses  against  the  pin.  To  obviate  this,  other 
shapes,  usually  flats  and  called  pin  plates  or  reinforcing  plates,  are  riveted 
to  the  member;  giving,  in  all,  sufficient  bearing  surface  for  the  pin.  See 
Fig.  55 ;  where  the  letters  denote : 

AA,  angles,  W,  web,  F,   filler, 

C,  cover,  P,   pin,  O,  outside  pin  plate, 

B,  bar,  J,  jaw,  T,   batten  plates. 


^^O^ 

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o    o    o    o    o 

o    o    o    o    o/^ 

w 

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O     O     O     O    0 

ooo  ooo 

o    o   o    o   o 

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ooo    \'^ ) 

O     O        '^J^^j 

\ao    o    o 

o    o   o    o    o 

o    o    o/ 

B' 


Fig.  55. 

199.  In  Fig.  56,  the  two  channels  form  the  whole  member  (except  the  lat- 
ticing, which  cannot  be  included  to  resist  compression)  and  the  pin  passes 
through  both  channels.  In  the  case  of  a  built-up  chord  section,  or  of  an 
end  post.  Fig.  53,  however,  the  webs   form  only  a  part  of  the  section; 


JOINTS   AND   PINS. 


725 


while  the  cover,  the  angles  and  the  flats  can  receive  no  stress  directly  from 
the  pin,  but  naust  receive  it  indirectly  from  the  web  and  from  the  pin  plates 
attached  to  it. 

300.  Where  a  pin  plate  is  placed  on  each  side  of  the  web,  the  outside  one 
must,  according  to  most  specifications,  cover  the  angles ;  and  there  must,  in 
addition,  be  a  "filler"  between  it  and  the  web. 

201.  Engineers  differ  as  to  the  manner  in  which  the  stresses  are  actually 
transferred  through  the  several  parts  of  a  pin  connection.  We  may  assume 
that  the  stresses  in  the  pin  plates  are  delivered  almost  directly  to  the  shapes 
of  the  member.  Thus,  the  outer  reinforcing  plate  probably  delivers  most  or 
all  of  its  stress  to  the  angles,  and  little  or  none  to  the  web. 

203.  In  each  angle,  those  rivets  which  pass  through  the  inner  pin  plate 
must  transmit,  by  means  of  thoir  bearing  against  the  angle,  the  sum  of  the 
stresses  which  they  take  by  shear  from  inside  and  from  outside.  In  other 
words,  these  rivets  are  in  double  shear. 

Pins. 

303.  The  pin  must  be  designed  to  resist  bending  stresses  from  the  mem- 
bers through  which  it  passes.  It  is  also  subject  to  shear,  but  this  is  seldom 
a  critical  point. 

304.  The  pin  requiring  the  greatest  cross-section  is  usually  either  the  one 
at  the  middle  of  the  span  and  in  the  lower  chord,  where  the  chord  stresses 
are  greatest,  or  the  one  at  the  joint  between  the  end  post  and  the  top  chord ; 
but,  as  the  pins  are  relatively  small  members,  all  the  other  pins  are,  for  the 
sake  of  uniformity,  usually  made  of  the  same  size  with  it. 


Fi^.  56. 


foundation 
Fig.  57. 


Expansion   Bearings. 

305.  Expansion  bearings  usually  consist  of  a  nest  of  carefully  turned 
rollers  placed  between  two  planed  surfaces,  shown  in  principle  in  Fig.  57. 

306.  The  rollers  are  steel  cylinders,  from  3  to  6  ins.  diam.;  and  1  to  4 
ft.  long;  planed  smooth.  From  4  to  8  or  more  of  these  are  connected  to- 
gether by  a  frame,  and  one  such  frame  is  placed  under  at  least  one  end  of  the 
truss.  The  rollers  rest  upon  a  strong  planed  cast  bed-plate;  bolted  to  the 
masonry  below.  Under  the  end  of  the  truss  is  a  similar  plate  by  which  it 
rests  on  the  rollers.  Since  a  truss  of  even  200  ft.  span  will  scarcely  change  its 
length  as  much  as  3  ins.  by  extremes  of  temperature,  the  play  of  the  rollers  is 
but  small.  They  are  kept  in  line  by  flanges  cast  along  the  side  of  the  bed- 
plate. Flanges  should  also  project  downward  from  the  upper  bed-plate, 
so  as  completely  to  protect  the  rollers  from  dust,  rain,  etc. 

307.  The  total  displacement,  allowed  for  the  free  end  of  the  truss,  is 
usually  specified  (see  Digests) ;  otherwise  it  may  be  taken  as 

(T  —  t)  span 
145,000     ' 


D  = 


where  T  and  t  =  the  max.  and  min.  temps,  respectively,  in  degrees  F.  The 
min.  temp,  to  be  expected  may  be  obtained  from  Weather  Bureau  records 
of  temps,  in  the  shade,  but  the  max.  should  be  taken  20°  or  30°  higher  than 
that  of  the  Weather  Bureau;  because,  in  bright  sunshine,  the  bridge  will 
1  )Ccome  much  hotter  than  the  air. 

308.  Rockers.     In  order  to  restrict  the  length  of  the  bearing,  where  the 
displacement  is  moderate,  rockers.  Fig.  62,  are  often  used  instead  of  rollers. 

309.  For  other  regulations  and  suggestions  regarding  design  of  roller 
bearings,  see  Digests  of  Specifications,  and  Figs.  60  and  61. 


e 


726  TRUSSES. 

Loads,  Etc. 

210.  Loads,  Clearance,  etc.,  for  Highway  Bridges.  See  also 
Digests  of  Specifications  for  Bridges. 

Weights  of  crowds.  At  the  Chelsea  bridge,  London,  picked  men, 
packed  upon  the  platform  of  a  weigh-bridge,  gave  a  load  of  only  84  lbs.  per 
sq.  ft.  At  Buckingham  Palace,  men,  wedged  as  closely  as  possible  upon  a 
space  20  ft.  in  diameter,  the  last  man  lowered  from  above,  among  the  others, 
save  120  lbs.  per  sq.  ft.  But  modern  experimenters  have  easily  obtained 
i.oadsof  from  140  to  150  lbs.  per  sq.  ft.  With  picked  men,  averaging  163.2 
lbs.  each,  all  facing  one  way,  carefully  packed,  and  confined  within  an 
enclosure  6  ft.  square  (0.9  sq.  ft.  per  man),  Prof.  L.  J.  Johnson,  at  Harvard 
University,  obtained  a  maximum  of  181.3  lbs.  per  sq.  ft.*  See  also  pages 
755,  etc. 

Where  the  enclosure  of  the  space  is  such  that  portions  of  the  persons, 
standing  against  the  enclosure,  may  project  beyond  it,  the  load,  per  unit  of 
space,  is  of  course  increased ;  and,  with  small  areas,  this  increase  may  be 
relatively  important. 

Camber. 

311.  Amount  of  Camber.  If  we  divide  the  span  in  feet,  by  50,  the 
quot.  will  ordinarily  be  a  sufficient  camber,  in  inches.  This  amounts  to  1  in 
600.  The  camber  to  be  used  is,  however,  usually  stipulated  in  the  specifica- 
tions. See  Digests.  A  well-built  bridge,  of  good  design,  should  not,  under 
its  greatest  load,  deflect  more  than  about  one  inch  for  each  100  feet  of  its 
span,  or  1  in  1200.     Indeed,  the  deflection  is  frequently  much  less  than  this. 

312.  The  excess  of  length  of  the  upper  chord  over  that  of  the  lower  one, 
given  the  span,  the  depth  of  truss  and  the  camber,  will  be  = 

8  X  depth  X  camber  ^ ^        gj, 

span  *  ' 

This  rule  applies  closely  with  any  camber  not  exceeding  0.02  of 
the  span. 

313.  Length  of  diagonal  c  h,  Fig.  58,  c  6   =    f^  a  c-  -{•  a  h^ ; 
where  a  c  ==  depth  of  truss,  and  ah  =  en  -\ ^ .  -km       mo 

214.  Sometimes  the  elongation  or  shortening,  produced  by 

the  loading,  is  computed  for  each  member,  and  the  length  of  each  member 
affected  is  correspondingly  changed.     See  Deflections,  H  162,  etc. 

Examples. 

215.  Figs.  59  (a)  to  (w),  to  a  uniform  scale  of  1  inch  =  60  feet,  serve  to 
indicate  current  practice  respecting  the  types  selected  for  different  spans, 
the  relation  between  span,  panel  length  and  depth,  the  spacing  of  stiffeners 
in  plate  girders,  the  arrangement  of  chord  and  web  members,  the  use  of  rigid 
and  flexible  members,  counters  and  turnbuckles,  in  trusses,  and,  approxi- 
mately, the  dimensions  of  rigid  members  and  of  gusset  plates,  as  seen  in 
elevation. 

216.  In  each  case  the  left  half  of  the  span  is  shown,  and  the  center  line  of 
the  span  is  indicated  by  a  dot-and-dash  line.  Through  spans  and  deck  spans 
are  distinguished  by  the  elevation  of  the  roadway,  as  approximate!:,'  indi- 
cated at  the  left  support. 

217.  In  Figs,  h  to  g,  representing  trusses,  rigid  members  are  indicated  by 
double  lines,  flexible  members  in  verticals  and  main  diagonals  by  single 
lines,  and  counters  by  dotted  lines.  In  pin  spans,  \o  avoid  confusion,  the 
rigid  members  are  shown  cut  off  near  the  pins. 

218.  Figs,  (a)  to  (o)  represent  standard  designs,  from  25  to  200  ft.  span,  by 
Mr.  Ralph  Modjeski,  C.E.,  for  the  Northern  Pacific  Railway  Co.;  Fig.  (p) 
a  250  ft.  railroad  span  designed  by  the  Pencoyd  Works  of  the  American 
Bridge  Co. ;  Fig.  iq)  a  308  ft.  railroad  span  by  the  Phcanix  Bridge  Co. ;  Figs, 
(r)  to  (0  designs  for  riveted  trusses  by  the  Elmira  Works  of  the  American 
Bridge  Co, ;  and  Fig.  (u)  a  riveted  railroad  bridge,  of  102  ft.  span,  designed 
by  the  Pencoyd  Works. 

*  Journal.  Ass'n  Engng  Socs,  Jan.,  1905. 


!7f 


EXAMPLES. 


727 


219.  Fig.  (a)  represents  a  beam  girder;  Figs.  (6)  to  (g)  plate  girders;  and 
Figs,  (h)  to  (q)  riveted  and  pin  trusses;  Figs,  (h)  and  (i)  being  riveted,  and 
Figs,  (j)  to  (q)  pin. 


(6)  #^  ^"-"^     c^|lllllll  \ie)  (g)  ^ 


-ti 


Ftg.  59  (a  to  9). 


728 


TRUSSES. 


320.  Fig.  (r)  represents  a  128  ft.  span  for  the  New  York  Central  and  Hud- 
son River  R.  R.,  and  Figs,  (s)  and  (t)  spans  of  143  ft.  and  160  ft.  respectively 
for  the  Delaware,  Lackawanna  and  Western  R.  R.  Figs,  (r)  and  (s)  are  modi- 
fications of  the  Baltimore  truss,  Fig.  15  (b),  and  Fig.(0  is  a  quadruplex Warren 
truss.     Fig.  (s)  is  a  skew  pan.     Fig.  (u)  is  a  "pony"  span. 


xxxyxyyyy 


Plan  of  Floor  and 
JBoUom  Bracing 

Fig.  59  («). 


Fig.  59  (t*). 


231.  Details.  Figs.  60  to  65  show  a  few  details  of  trusses  and  of  plate 
girders. 

222.  Figs.  60  and  61  show  left  end  connections  of  two  through  truss 
bridges  (with  roller  bearings)  designed  by  the  Pencoyd  Works;  Fig.  61 
representing  the  250  ft.  through  pin  span  shown  m  Fig.  59  (p),  and  Fig.  60 
a  124  riveted  through  span,  showing  the  portal  bracing. 


EXAMPLES. 


729 


730 


TRUSSES. 


Base  of  Bail 


Fig.  63. 


Scale  for  Figs,  62,  63, 
64. 

223.  Fig.  62  shows  the 
left  end  bearing  (with 
rockers)  and  a  floor  beam 
connection  for  the  stand- 
ard 200  ft.  deck  pin  span 
of  the  Northern  Pacific 
Railway,  Fig.  59  (n) ;  and 
Fig.  63  shows  a  floor  beam 
connection  of  the  160  ft. 
through  pin  span  of  the 
same  railway,  Fig.  59  (m). 


WEIGHTS. 


.1.  ^\f*  ■^^^^•t?'^  ^°^,  ^^  represent  respectively  a  50  ft.  deck  plate  girde 
the  N.  Pac.  Ry.  and  an  85  ft.  through  plate  girder  by  the  Pencoyd  Wo 


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Spans,  in  feet. 
Fig.  66. 

*  See  paper  by  Ralph  Modjeski,  C.E.,  in  "Journal  of  the  Western  Socie 
of  Engineers,"  Chicago,  Feb.,  1901,  vol.  vi.  No.  1. 


732  TRUSSES. 

weights  of  steel  railroad  bridges  designed  for  two  locomotives,  of 
146  tons  each,  and  a  uniform  train  load  of  4000  lbs.  per  foot  of  track.  The 
weights  include  the  two  beams,  girders  or  trusses  of  one  single-track  span, 
with  their  bracing,  metal  floor  system  and  end  bearings.  For  wooden  floor 
system,  add  400  lbs.  per  lineal  foot.  For  pin-connected  spans  (130  to  200 
ft.),  the  three  dash  diagrams  show,  respectively,  the  weights  of  two  trusses 
alone,  of  two  trusses  and  bracing,  and  of  two  trusses,  bracing  and  metal 
floor.     The  solid  curve  includes  weights  of  end  bearings. 

For  weights  of  combination  (wood  and  iron)  railroad  bridges,  see  ^  249. 

Highway  bridges  differ  so  widely,  as  to  service  and  design,  that  it  is 
scarcely  practicable  to  give  here  uaef  ul  data  as  to  their  weights. 

List  of  Large  Bridges. 

Each  bridge  here  given  is  believed  to  be  (1902)  the  largest  of  its  type  in 
the  world. 

Type                     Spanning  At  Span,  ft.    Built 

Truss, Ohio  River  Louisville  553       1893 

Swing,    Missouri  River  Omaha  520        1894 

Suspension, East  River 

("Brooklyn")  New  York  1595        1883 

Suspension, East  River 

("New^O  New  York  1600               * 

Arch  (metal) Niagara  River  Niagara  Falls  840       1898 

Arch  (stone), . .  • .  .Petruff  Valley  Luxembourg  277                * 

Cantilever Firth  of  Forth  Queensferry  1700        1890 

The  highest  viaduct  is  the  Gokteik  Viaduct,  in  Burmah,  with  a  maximum 

height  of  320  ft.,  and  a  total  length  (composed  of  short  spans)  of  2260  ft., 
buUt  in  1901. 

Timber  Trusses. 

226*  Timber  is  now  becoming  so  expensive,  except  in  unsettled  regions, 
and  the  labor  of  designing  3o  cheap,  that  it  is  no  longer  found  to  be  good 
practice  to  use  unnecessarily  heavy  timbers,  simply  for  the  sake  of  being  "on 
the  safe  side"  and  avoiding  computations.  Hence,  in  important  bridges, 
every  part  of  each  member  under  stress  is  usually  computed.  On  the  other 
hand,  the  strength  of  wood  is  so  uncertain  an  element  that,  when  in  doubt, 
it  is  best  to  adopt  that  assumption  which  will  require  the  larger  section;  and 
ample  factors  of  safety  should  be  used. 

227.  Compression  members  are  designed  as  columns  (see  Pillars, 
under  Strength  of  Materials)  *  and,  if  subjected  to  transverse  stresses  as  well, 
these  also  should  be  carefully  taken  into  account.  All  holes  and  other  reduc- 
tions of  section  must  of  course  be  deducted  from  the  gross  section. 

228.  In  the  tension  members  also,  all  reductions  of  section  must  be 
considered ;  but  iron  or  steel  rods  are  now  generally  used,  in  place  of  wood, 
in  tension  members. 

229.  In  addition,  care  should  be  taken  that  the  timber  can  withstand 
any  crushing  or  shearing  stresses  that  may  come  upon  it  or  be  set  up  in 
it.  Thus,  the  ends  of  posts  should  be  investigated,  to  see  that  they  are  safe 
against  crushing.  Where  a  post  meets  another  member  at  an  inclined  angle 
and  is  to  be  notched  into  it,  it  is  economy  to  compute  the  depth  of  notch 
required ;  as,  the  deeper  the  notch,  the  greater  the  gross  section  required  for 
the  notched  member.  Where  bolts  are  fastened  to  timbers  by  nuts,  washers 
should  invariably  be  placed  under  the  nuts,  and  the  size  of  washer,  necessary 
to  prevent  crushing  the  wood,  computed. 

230.  Where  the  wood  is  subjected  to  shearing,  as  where  a  bolt,  passed 
through  a  timber,  transmits  stress  by  the  bearing  of  its  side  against  the  inside 
of  the  hole,  or  where  there  is  a  step  or  table  which  may  be  sheared  off  by  the 
pressure  of  another  piece  against  it,  it  should  always  be  seen  that  there  is 
sufficient  surface  along  the  grain  of  the  wood  to  take  the  shear,  and  some 
allowance  should  be  made  for  the  possibility  that  the  grain  may  run  out  to 
the  surface  or  to  some  hole  before  all  the  stress  can  be  transferred. 


*  Under  construction,  1902. 


TIMBER    TRUSSES. 


733 


231.  Cross-section  of  tipper  Chord.  Since  it  would  be  inconve- 
nient, in  practical  construction,  to  change  the  section  of  a  timber  upper  chord 
at  different  points,  it  is  designed  throughout  to  withstand  the  maximum 
stress  occurring   between  any  two  panel  points.     Assume  width  of  chord 

member.     Findr^  =  (least  radius  of  gyration)^  *=>        19  Find  allowable 

unit  stress  according  to  column  formula  given  in  specifications  or  adopted, 
using  the  given  maximum  stress.  Find  area  required  for  this  unit  stress. 
Find  the  resulting  depth,  which  for  a  horizontal  or  inclined  member  is  pref- 
erably somewhat  greater  than  the  width,  to  allow  for  bending  moment  due 
to  its  own  weight.  If  this  does  not  give  a  good  commercial  size,  it  may  be 
well  to  revise,  in  order  to  obtain  a  better  section. 

333.  Struts  are  preferably  made  as  wide  as  the  upper  chord.  Each  strut 
must  be  designed  separately.  Obtain  r^,  allowable  unit  stress,  etc.,  as  for  the 
upper  chord.  For  economy,  the  struts  should  average  nearly  square,  even 
though  it  should  be  necessary  to  alter  the  section  of  the  upper  chord  in  order 
to  prevent  wide  deviation  from  a  square  section. 

233.  The  vertical  ties  (of  iron)  may  now  be  designed.     Area  of  cross- 
maximum   stress  T>      X  •.;r.     .  rt        ,.  mr    ^r.^  tTM 

section  =   -Tj TT n — i — —  •     But  see  Mmimum  Section,  1  187.      The 

allowable  unit  stress 
size  of  a  nut  is  usually  fixed  by  the  diameter  of  the  rod,  but  the  washers 
should  be  so  designed  as  not  to  crush  the  wood. 

334.  The  bearings  or  indentations,  required  in  the  upper  and  lower 
chords  to  hold  the  inclined  members  in  place,  may  now  be  computed.  The 
component  (in  the  strut)  perpendicular  to  the  face  or  faces  against  which  it 
presses,  is  computed,  and  the  necessary  depth  obtained,  assuming  the  width 
of  the  lower  chord  the  same  as  that.of  the  upper  chord  and  the  struts. 

335.  The  section  of  the  lower  chord  may  now  be  decided  upon,  since 
the  reduction  of  section,  due  to  indentations,  is  known. 

.  -       .  ,.  maximum  stress 

Area  of  net  cross-section  =  —7; r-; : — : • 

allowable  unit  stress 


Figr.  67. 


336.  End  Joint.  Fig.  67.  Many  different  designs  for  end  joints  have 
been  made,  proposed  and  discussed.  The  ends  of  the  straps  should  enter 
notches  in  the  lower  chord,  to  such  a  depth  that  the  total  stress,  taken  by  the 
end  fibers  of  the  sides  of  the  notches,  is  equal  to  the  stress  that  the  ends  of 
the  straps  can  resist  by  bending.  This  depth  can  be  found  by  successive 
trials,  or  by  means  of  two  algebraic  equations,  in  which  the  maximum  allow- 
able pressure  and  the  depth  of  notch  are  the  two  unknown  quantities. 

Determine  the  shearing  surface  required  to  transmit  this  stress  to  the  body 
of  the  lower  chord.  This  will  also  determine  the  space  between  the  notches 
and  the  end  of  the  lower  chord.     Compute  the  stress  (if  any)  that  remains  to 


734  TRUSSES. 

be  transferred,  and  design  the  long  inclined  bolts  and  their  washers  accord- 
ingly. Compute  also  the  compressive  area  and  depth  of  vertical  face  of 
lower  end  of  upper  chord,  required  to  transmit  the  horizontal  component  of 
its  thrust.  See  also  that  the  lower  bearing  presents  sufficient  surface  to 
resist  the  vertical  component.  The  keys,  between  the  bolster  and  the  lower 
chord,  must  be  designed  to  carry  the  horizontal  component  of  the  wind.  For 
safety,  friction  between  the  two  parts  should  be  neglected. 

237.  Figs.  68  show  joints  adapted  to  most  of  the  cases  that  occur  in 
practice  with  wooden  beams,  etc.  They  need  but  little  explanation.  Fig, 
(a)  is  a  good  mode  of  splicing  a  post ;  in  doing  which  the  line  o  o  should  never 
be  inclined  or  sloped,  but  be  made  parallel  to  the  axis  of  the  post ;  otherwise,' 
in  case  of  shrinkage,  or  of  great  pressure,  the  parts  on  each  side  of  it  tend  to 
slide  along  each  other,  and  thus  bring  a  great  strain  upon  the  bolts.  When 
greater  strength  is  required,  iron  hoops  may  be  used,  as  at  b,  h  and  /,  instead 
of  bolts.  Fig.  (6)  shows  a  post  spliced  by  4  fishing  pieces;  which  may  be 
fastened  either  by  bolts,  as  in  the  upper  part ;  or  by  hoops,  as  in  the  lower. 
The  hoops  may  be  tightened  by  flanges  and  screws  as  at  s;  or  thin  iron 
wedges  may  be  driven  between  them  and  the  timbers,  if  necessary.  Fig.  C 
shows  a  good,  strong  arrangement  for  uniting  a  straining-beam  k,  or  rafter  I, 
and  a  queen-post  u;  by  letting  k  and  I  abut  against  each  other,  and  confining 
them  between  a  double  queen-post  t  t;  n  n  are  two  blocks  through  which 
the  bolts  pass.  A  similar  arrangement  is  equally  good  for  uniting  the  tie- 
beam  w,  with  the  foot  v,  of  the  queens ;  with  theaddition  of  a  strap,  as  in  the 
figure.  Fig.  (e)  is  a  method  of  framing  one  beam  into  another,  at  right  angles 
to  it.  An  iron  stirrup,  as  at  /,  may  be  used  for  the  same  purpose ;  and  is 
stronger.  Figs,  g  h,  ij  are  built  beams.  When  a  beam  or  girder  of  great 
depth  is  required,  if  we  obtain  it  by  merely  laying  one  beam  flat  upon  an- 
other, we  secure  only  as  much  strength  as  the  two  beams  would  have  if  sep- 
arate. But  if  we  prevent  them  from  sliding  on  one  another,  by  inserting 
transverse  blocks  or  keys,  as  at  g;  or  by  indenting  them  into  one  another,  as 
at  i  j;  and  then  bolt  or  strap  them  firmly  together  to  create  friction ;  we  ob- 
tain nearly  the  strength  of  a  solid  beam  of  the  total  depth ;  which  strength  is 
as  the  square  of  the  depth.  See  Horizontal  Shear,  ^  51,  under  Transverse 
Strength. 

238.  The  strength  of  a  built  beam  is  increased  by  increasing  its  depth  at 
its  center,  where  it  is  most  strained ;  as  in  the  upper  chords  of  a  bridge.  This 
may  be  done  by  adding  the  triangular  strip  y  y  between  the  two  beams. 

239.  A  piece  of  plate-iron  may  be  placed  at  the  joints  of  timbers  when 
there  is  a  great  pressure ;  which  is  thus  more  equally  distributed  over  the 
entire  area  of  the  joint;  or  cast  iron  may  be  used. 

240.  Frequently  a  simple  strap  will  not  suffice,  when  it  is  necessary  to  draw 
the  two  timbers  very  tightly  together.  In  such  cases,  one  end  of  each  strap 
may,  as  at  x,  terminate  as  a  screw ;  and  after  passing  through  a  cross-bar  Z, 
all  may  be  tightened  up  by  a  nut  at  x.  Or  the  principle  of  the  double  key, 
shown  at  K,  may  be  applied.  Sometimes,  as  at  A,  the  hole  for  the  bolt  is 
first  bored ;  then  a  hole  is  cut  in  one  side  of  the  timber,  and  reaching  to  the 
bolt-hole,  large  enough  to  allow  the  screw  nut  to  be  inserted.  This  being 
done,  the  hole  is  refilled  by  a  wooden  plug,  which  holds  the  nut  in  place. 
Then  the  screw-bolt  is  inserted,  passing  through  the  nut.  By  turning  the 
screw  the  timbers  may  then  be  tightened  together. 

241.  When  the  ends  of  beams,  joists,  etc.,  are  inserted  into  walls  in  the 
usual  square  manner,  there  is  danger  that,  in  case  of  being  burnt  in  two,  they 
may,  in  falling,  overturn  the  wall.  This  may  be  avoided  by  cutting  the  ends 
into  the  shape  shown  at  m. 

242.  When  a  strap  o,  Fig.  R,  has  to  bear  a  strain  so  great  as  to  endanger 
its  crushing  the  timber  p,  on  which  it  rests,  a  casting  like  v  may  be  used  under 
it.  The  strap  will  pass  around  the  back  r  of  the  casting.  The  small  projec- 
tions in  the  bottom,  being  notched  into  the  timber,  will  prevent  the  casting 
from  sliding  under  the  oblique  strain  of  the  strap.  The  same  may  be  used 
for  oblique  bolts,  and  below  a  timber  as  well  as  above  it.  When  below,  it 
may  become  necessary  to  bolt  or  spike  the  casting  to  the  under  side  of  the 
timber.  When  the  pull  on  a  strap  is  at  right  angles  to  the  timber,  if  there 
is  much  strain,  a  piece  of  plate-iron,  instead  of  a  casting,  may  be  inserted 
between  the  strap  and  the  timber,  to  prevent  the  latter  from  being  crushed 
or  crippled ;  see  I  and  I. 


TIMBER   DETAILS. 


735 


f-^ i— K \     ^ '^ 


F=F=F^ 


Fig.  68. 


736 


TRUSSES. 


243.  Lower  Chord  Splice.  Owing  to  the  length  of  the  lower  chord, 
it  may  be  necessary  to  splice  it.  as  in  Fig.  69,  where  the  splice  is  a  tabled  fish- 
plate joint.  The  number  of  tables  to  be  used  is  largely  a  matter  of  trial.  The 
use  of  too  many  tables  involves  too  much  carpentry,  and  consequent  uncer- 
tainty as  to  distribution  of  pressures,  while  the  use  of  too  few  requires  deep 
notches,  which  may  too  greatly  reduce  the  section.  These  tables  must  be 
designed  to  resist  bearing  against  their  ends,  and  to  resist  being  sheared  off. 
Bolts  should  be  passed  through,  especially  at  the  ends  of  the  fish-plates,  to 
prevent  them  from  bending  outward ;  and  the  washers  should  be  so  designed 
as  to  transmit  safely  to  the  wood  all  the  stress  that  can  come  on  the  bolt. 


rtni. 


-^ 


Fig.  69. 


'W 


244.  Fig.  70  shows  a  lower  chord  splice  used  in  connection  with  the  stand- 
ard combination  (timber  and  iron)  Howe  truss  bridges  of  the  Chicago,  Mil- 
waukee &  St.  Paul  Railway  See  Hlf  249-251,  Figs.  73.  Four  of  the  clamp- 
blocks  shown  are  required  for  each  joint,  a  block  being  placed  upon  each  side 
of  each  stick  to  be  spliced.  The  two  opposite  blocks  forming  a  pair  are  held 
together,  and  against  the  stick,  by  four  through  bolts;  and  the  cylindrical 
lugs,  cast  on  the  surface  of  each  block,  enter  corresponding  holes,  bored  in 
the  face  of  the  stick.  The  two  blocks  on  the  same  side  of  a  stick  are  held 
together  by  the  hooked  clamp-bar.  The  clamp-key  is  driven  between  the 
left  hook  and  the  beveled  key-seat  on  the  left  block. 


!0      O      O      O 


Fig.  70. 

245.  Figs.  71  illustrate  a  small  wooden  Howe  truss  bridge.     The  top 

and  bottom  chords  are  each  made  up  of  three  or  more  parallel  timbers  c  c  c, 
placed  a  small  distance  apart,  to  let  the  vertical  tie-rods  r  r  pass  between 
them.  The  main  braces,  o  o,  are  in  pairs  or  in  threes.  The  pieces  compos- 
ing them  abut,  at  top  and  at  bottom,  against  triangular  angle  blocks,  s; 
which,  if  of  hard  wood,  are  solid;  and,  if  of  cast  iron,  hollow,  Fig.  {d),  and 
strengthened  by  inner  ribs.  These  blocks  extend  across  the  three  or  more 
chord  pieces.  Against  their  centers  abut  also  the  counterbraces  e.  These 
are  single  pieces  in  small  bridges ;  or  in  pairs,  in  large  ones ;  and  pass  between 
the  pieces  which  compose  a  main  brace.     Where  the  wooden  braces  and 


TIMBER  TRUSSES. 


737 


counters  cross,  they  are  bolted  together.  In  Fig.  (c?),  the  dotted  lines  show 
the  strengthening  ribs;  and  the  lug,  x,  keeps  the  block  in  place.  The  vertical 
tie-rods  r  r,  of  iron,  are  in  pairs,  threes,  or  fours,  etc.,  according  to  size  of 
bridge ;  with  a  screw  and  nut  at  each  end.  The  heads  and  feet  of  the  braces 
and  counters  abut  square  against  the  angle  blocks;  and  are  often  kept  in 
place  only  by  tightening  the  screws  of  the  vertical  ties.  The  end  posts,  p 
and  d,  the  end  ties,  i  c  and  b  y,  and  the  horizontal  extensions,  g  i  and  w  b,  of 
the  upper  chord,  form  no  part  of  the  truss  proper. 


246.  In  large  spans,  to  prevent  the  pressure  at  the  ends  of  the  diago- 
nals from  crushing  the  chords,  the  angle  blocks  are  sometimes  cast  with  deep 
projecting  flanges  under  their  bases.  These  flanges,  passing  between  the 
pieces  which  compose  a  chord,  extend  to  the  opposite  face  of  the  chord. 
There  the  flanges  bear  upon  broad  washers  at  the  ends  of  the  vertical  rods. 


^ 


c 

-II- 


^S 


Fig.  72. 

247.  A  common  form  of  lateral  bracing.  Fig.  72,  resembles  a  Howe 
truss  laid  flat  on  its  side.  In  it  the  diagonals  of  the  cross  are  struts  of  timber; 
and  the  pieces  r  r  are  round  rods.  One  of  the  struts  is  whole,  with  the  excep- 
tion of  a  slight  mortise  on  each  vertical  side,  at  its  center,  for  receiving  tenons 
cut  on  the  inner  ends  of  the  two  pieces  which  compose  the  other  diagonal. 
At  the  sides  of  the  chords,  the  ends  of  the  diagonals  rest  upon  a  ledge  (shown 
by  the  dotted  line  i  i),  about  li  inches  wide,  cast  at  the  bottom  of  the  cast- 
iron  angle  block.  The  tie-rod  r  r,  passing  through  the  chords  of  both  trusses, 
being  tightened  by  means  of  the  nut  s,  holds  the  diagonals  firmly  in  place ; 
and,  in  case  of  their  shrinking  a  little  in  time,  they  can  be  again  tightened  up 
by  the  same  means. 

The  cast  angle  block  is  as  deep  as  a  brace ;  its  thickness  need  not  exceed 
half  an  inch,  in  a  large  bridge.     It  has  holes  for  the  passage  of  the  rod  r  r. 

47 


738 


TRUSSES. 


248.  Dimensions  for  each  of  two  trusses  of  a  single  track  Howe 
bridge  for  highway  or  suburban  electric  railway,  with  cars  weighing  20  to 
25  tons  each.  Timber  not  to  be  strained  more  than  800  lbs,  per  sq.  inch; 
nor  iron  more  than  5  tons  per  sq.  inch.  Iron  supposed  to  be  of  rather  supe- 
rior quality,  requiring  from  25  to  27  tons  (60,480  lbs.)  per  sq.  inch  to  break  it. 
The  rods  to  be  upset  at  their  screw-ends.  To  each  of  the  two  sides  of  each 
lower  chord  is  supposed  to  be  added,  and  firmly  connected,  a  piece  at  least 
half  as  thick  as  one  of  the  chord  pieces,  and  as  long  as  three  panels,  at  th« 
center  of  the  span. 


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End  Rod. 

§ 

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Brace. 

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3 

3>^ 

3 

IX 

349.  Standard  "combination"  (wood  and  iron)  Howe  truss  rail- 
road bridges,  Chicago,  Milwaukee  &  St.  Paul  Railway,  1891-1902. 

The  bridges  are  designed  for  a  train  load  of  4000  lbs.  per  lineal  foot. 
Wind  stresses  are  computed  for  a  pressure  of  500  lbs.  per  lineal  foot  of 
train.  Compressive  stresses  in  braces,  max.  505,  min.  92  lbs.  per  sq.  inch. 
Tensile  stresses;  in  main  truss  rods,  max.  12,450,  min.  8500  lbs.  per  sq. 
inch  of  net  area  at  root  of  thread.  Lateral  rods,  max.  15,000  lbs.  per  sq. 
inch  of  net  section. 

Timber.  Cross-ties  and  guard  rails,  white  oak ;  top  chord  packing  blocks, 
white  pine;  all  other  timber,  white  or  Norway  pine  or  Douglas  fir. 

Combination   Bridges,   Chicago,  Milwaukee  &  St.  Paul  Railway: 


Total  length, 
feet,* 

Span,  feet,*. , 

Panels,    ..... 

Upper  chord, . 

Lower  chord,  t 

Main     braces, 
M: 
At  center,  t 
At  ends, .  . 

Counters,  C: 
At  center,  t 
At  ends, .  .  , 

Vertical  rods 
At  center,  t 

At  ends, . 

Weight,  lbs.,§ 


82 

93 

103 

114 

125 

136 

77 

88 

99 

110 

121 

132 

7 

8 

9 

10 

11 

12 

4,  7x10 

4,  7x10 

4,  8x10 

4,  8x10 

4,  8x10 

4,  8x10 

4,  7x14 

4,  7x14 

4,  8x14 

4,  8x14 

4,  8x15 

4,  8x15 

2,  10x10 

2,    8x10 

2,  10x10 

2,  10x10 

2, 10x10 

2,  10x10 

2,  10x12 

2, 12x12 

2,  12x12 

2,  12x14 

2, 12x14 

2,  14x14 

2,  8x10 

1,  10x12 

1,  8x10 

1,  8x10 

1,  8x10 

1,  10x12 

1,8x8 

1,    8x8 

1,8x8 

1,8x8 

1,8x8 

1,    8x8 

3.  li 

3,   H 

3,    If 

3,  U 

3,    If 

3,    U 

3,  2i 

3,    2\ 

3,   2i 

(3,  2i 
1  2,  2 

3,   2i 
2,   2 

3,    2f 
2,   2 

130,300 

155,500 

182,000 

209,400 

233,100 

259,900 

147 

143 

13 
4,  8x12 
4,  8x15 


2,  10x10 
2,  14x14 

1,  8x10 
1,8x8 

3,    If 
3,   2i 
2,  2 
286,500 


*  Lengths  and  spans  are  given  to  the  nearest  foot.  The  panel  length  ia 
10  ft.  Hi  ins.  Each  span  is  longer,  by  one  panel,  than  the  next  preceding. 
Depth,  in  all  cases,  25  feet  in  clear  of  chords.  Width,  14  ft.  6  ins.  in  clear 
between  chords. 

t  For  lower  chord  splice,  see  Fig.  70,  -1[  244. 

X  Owing  to  the  fact  that  the  spans  have  alternately  odd  and  even  numbers 
of  panels,  there  is  some  ambiguity  as  to  the  dimensions  of  web  members  at 
center  of  span. 

§  Weight  includes  the  two  trusses,  bracing  and  floor  system  for  one  single- 


COMBINATION  TRUSSES. 


739 


Vpper  cliord, 

and 

lateral  bracing 


Stringers  b"  X  12" 


(a) 


J  i:iJ ;  W!  I 


11 


lWWWW-^4fl-fK^#'HHm-tt- 


2   Wall  Plates  12"xl2' 
Half  end  elevatio^v 

Loiver  chord, 

lateral  bracing 

and 

floor  system 


Figr.  73 


track  span.     Of  the  total  weight  of  the  bridge,  the  trusses  have  about  o3 
per  cent.,  lateral  systems  26,  floor  system  18,  and  wall  plates  3  per  cent. 


740 


TRUSSES. 


250.  In  all  spans,  lateral  bracing,  6X6  ins.  at  center,  8X8  ins.  at  ends; 
of  span;  lateral  rods,  li  ins.  at  center,  1^  to  1|^  ins.  at  ends;  collision  struts, 
S  (one  at  each  end  of  each  truss),  6  X  14  ins,;  transverse  portal  brace,  B, 
between  ends  of  upper  chords  (one  at  each  end  of  bridge),  10  X  12 ins.; 
diagonal  portal  braces,  D  (two  at  each  end  of  bridge),  6X8  ins. 

The  floor  beams  are  10  X  16  ins.,  20  ft.  long,  14  ft.  clear  span,  and  2 
ft.  6  ins.  apart  center  to  center.  The  stringers  are  5  ins.  wide,  12  ins.  deep, 
placed  as  shown  in  the  end  view.  Fig.  (6).  Ties,  8  ins.  wide,  6  ins.  deep,  1  ft. 
apart,  center  to  center.  Guard  rail,  8  ins.  wide,  5  ins.  deep.  Under  each 
end  of  the  lower  chord  are  two  timber  wall  plates,  12  X  12  Ins.,  20  ft.  long. 

251.  Figs.  73  show  the  99  ft.  span.  Fig.  (a)  is  a  side  elevation  of  half 
the  span  with  top  and  bottom  lateral  bracing  and  floor  system ;  Fig.  (6)  a 
half  end  elevation,  showing  portal  bracing;  Fig.  (c)  a  panel  point  con- 
nection (the  same  for  upper  and  for  lower  chord) ;  Fig.  (d)  a  cast-iron 
angle  block  for  same ;  and  Fig.  (e)  a  cast-iron  angle  block  for  lateral  bracing. 

Metal    Roof    Trusses. 

252.  Among  the  types  commonly  used  for  metal  roof  trusses  are  the  trian- 
gular truss,  Fig.  26,  and  the  arch  truss ;  the  triangular  truss  being  used  for 
short  spans,  and  the  arch  truss  for  long  spans.  See  ^*S[  255,  etc.,  and 
Fig.  75. 

253.  In  roof  trusses  of  short  span,  carrying  light  loads,  the  minimum 
sections  prescribed  in  bridge  specifications  will  often  suffice  for  all  the  mem- 
bers. The  connections  are  made  by  means  of  rivets  and  flat  plates,  some- 
what as  shown  in  Fig.  74. 


Fig.  74. 


254.  In  designing  such  trusses,  no  matter  how  light  the  stresses,  care 
should  be  taken  to  avoid  eccentric  loadings,  which  are  apt  to  occur  where  the 
members  are  not  repeated  symmetrically  on  both  sides  of  the  flat  plates. 

255.  Train-shed  Roof,  Broad  Street  Station,  Pennsylvania 
Railroad,  Philadelphia.  Figs.  75  (a)  to  (/).  Built  1893  by  Pencoyd 
Bridge  and  Construction  Co.,  erected  by  Railroad  Co.     ""  ~" 


108'  6";  length,  589' 2i''. 
shown  in  Fig.  (c). 


Span,  300'  8";  rise, 
Twenty  trusses,  arranged  in  10  pairs,  3  pairs 


256.  Each  truss  consists  of  two  arched  rafters,  A  B  and  B  C,  and  a  hori- 
zontal chord,  A  C,  with  three  pin  joints.  A,  B  and  C,  Fig,  (a). 

Each  rafter  is  composed  of  two  chords,  14  radial  braces  and  26  diagonals. 
For  the  sake  of  appearance,  the  chords  are  extended  across  the  top  panels, 
occupied  by  the  two  triangular  apex  members,  and  are  there  connected  by 
a  sliding  joint, 

257.  The  horizontal  chord  AC,  Fig.  (a),  lies  below  the  floor  of  the 
train-shed,  and  is  suspended,  at  intervals,  from  girders  which  support  that 
floor  and  which  are  carried  by  iron  columns  in  the  lower  story. 

258.  End  bearings.  One  foot  of  each  truss  rests  upon  a  fixed  shoe,  bolted 
down  to  the  pier;  the  other  on  a  nest  of  11  steel  rollers. 

259.  At  each  end  of  the  roof,  a  horizontal  wind  truss,  WW,  Fig.  (a),  is 
suspended  from  the  rafters,  its  ends  resting  upon  brackets  riveted  to  their 
iower  chords. 


METAL    ROOF    TRUSSES. 


741 


360.  Between  these  horizontal  wind  trusses  and  the  rafters,  and  just  be- 
hind the  glass  curtain  closing  each  end  of  the  roof,  are  placed  vertical 
wind  trusses  (not  shown),  with  horizontal  and  diagonal  bracing,  in  planes 
normal  to  the  main  trusses.  These  vertical  trusses  are  hung  from  the 
lower  panel  points  of  the  rafters,  Fig.  (a).  They  resist  the  wind  pressure  on 
the  curtain,  and  transmit  it,  through  the  horizontal  wind  trusses,  to  the 
lower  chords  of  the  rafters. 

261.  Two  rafters,  AB  and  BC,  Fig.  (a),  of  one  truss,  together  weigh 
138,000  lbs.;  chord  (two  lines  of  eye-bars)  16,000,  total  154,000  lbs.  One 
pair  of  trusses,  308,000.  Framework  of  entire  train-shed,  including  trusses, 
purlins,  rafters  and  wind  bracing,  about  7,000,000  fbs. 


Saunch  Elevation^ 

(b) 


Scale  for  Figs,  lto6 


lOO 
Fig.  75. 


200 


300 
Feet 


263.  Assumed  extraneous  loads,  in  lbs.  per  sq.  ft.  of  ground  plan :  rafters 
9.5;  laterals  and  purlins,  7.5;  ventilator  and  skylight  framing,  9.5;  covering 
and  skylights,  15;  snow,  17;  wind,  35. 

263.  The  traveler.  Figs,  (a)  and  (/),  was  of  timber,  and  was  so  designed  as 
to  permit  the  use  of  the  old  train-sheds  while  the  new  roof  was  being  erected. 

264.  The  top  section,  from  apex  to  joint  9,  Figs,  (a)  and  (c),  was  riveted 
up  at  mill  and  placed  in  position  as  a  whole,  as  was  also  the  foot  section  from 
joint  27  to  the  top  of  the  triangular  foot.  The  remainder  was  riveted  on  the 
traveler.  To  erect  one  pair  of  trusses,  with  the  purlins,  etc.,  connecting  it 
with  the  pair  last  erected,  required  about  10  days.  Shifting  the  traveler 
through  a  distance  about  equal  to  its  own  length,  to  receive  the  next  pair  of 
trusses,  required  about  3  minutes,  and  was  done  by  means  of  the  engines  used 
for  hoisting. 


742  TRUSSES. 

265.  Dimensions  of  Arched  Roofs  of  Large  Span. 


Arches, 

Roof. 

Span. 

Rise. 

Length. 

Area 
Covered. 

Train-sheds. 

Pennsylvania        Railroad, 
Philadelphia.          Broad 
Street  Station, 

ft. 

300 
252 

259 

199 

240 
209 

362 
368 

ins. 

8 
8 

0 

0 

7 

9 
0 

ft.  ins. 

108  6 
90  0 

88  3 

94  0 

107  0 

78  8 

149  0 
206  0 

ft.  ins. 

589  2 
652  6 

506  8 

652  0 

706  0 
836  0 

1,380  0 
1,268  0 

eq.  ft. 
177,150 

Pennsylvania        Railroad, 
Jersey  City 

164,900 

Phila.  &  Reading  Railroad, 
Philadelphia .   * '  Reading 
Terminal/'  Market  St., .  . 

New  York  Central  &  H.  R. 
R.R.,  New  York.  Grand 
Central  Station, 

131,250 
129,856 

Midland  Railway,  London. 

St.  Pancras  Station, 

Cologne,  Germany.  Nave, . 

Exposition  Buildings. 

Machinery     Hall,      Paris, 
1889.     Nave, 

169,400 
175,200 

500,600 
466,600 

Manufactures  and  Liberal 
Arts  Building,  Chicago, 
1893 

Timber  Roof  Trusses. 

266.  Dimensions  for  small  timber  roof  trusses,  Figs.  43  to  47, 
^•[i  148,  etc.,  of  white  pine.  Span  =  4  X  rise.  Combined  weight  of  trusses, 
roof  and  load,  including  snow  and  an  allowance  for  wind,  40  fbs.  per  square 
foot  of  roof  surface.  Trusses  12  feet  apart,  center  to  center.  Safety  factor 
=  3;  b  =  breadth;  d  =  depth.  For  Fig.  46,  struts  4.5  X  4.5  ins.  would 
suffice ;  but,  for  practical  reasons,  the  struts  are  generally  made  as  wide  as 
the  rafters.  For  Fig.  47,  the  straining  beam  is  12X12  ins.  For  Figs.  45, 
46  and  47,  two  sets  of  dimensions  are  given ;  the  finst  for  chord  unloaded;  the 
second  for  chord  loaded  with  100  ibs.  per  square  foot. 


Span 
Ft. 

30 
30 
40 
60 

Rise. 
Ft. 

Rafters. 
Ins. 

Chord. 

King  or 

Queen 

Rod. 

Ins. 

Diam. 

Fig. 

Timber. 
Ins. 

Iron. 
Ins. 
Diam. 

Chord. 

b. 

d. 

b. 

d. 

43 
45 
46 
47 

7.5 
7.5{ 

5 
6.5 

8.5 

6 

8      - 
10 
12 

10 
9 
11 
8 
10 
12 
14 

5 
6.5 

8.5 

6 

8 
10 
12 

10 
9 
11 
8 
15 
12 
12 

1 
1 

1.5 

1*.*5 

i* 

H 

1 

2 

unloaded 

unloaded 

loaded 

unloaded 

loaded 

unloaded 

loaded 

i 


TRANSPORTATION    AND    ERECTION.  743 


TRANSPORTATION  AND  ERECTION. 

367.  Girders  should  be  loaded  for  transportation  on  flat  cars,  with  web 
vertical,  and  with  bearings  at  the  points  distant  i  span  from  the  ends.  Where 
too  long  for  two  cars,  one  or  more  idle  spacing  cars  are  used  and  the  points 
of  support  are  pivoted. 

268.  Girders  may  be  erected  by  means  of  gin  poles,  derricks,  gallows, 
etc.,  or  they  may  be  skidded  from  the  cars  and  lowered  to  place  by  jacks  and 
blocking.  Gin  poles  should  have  at  least  four  guys,  with  tackles,  for  easy 
adjustment.  Hoisting  may  often  be  done  by  means  of  a  locomotive  running 
on  the  tracks  of  that  part  of  the  structure  which  is  already  completed. 
Ropes  are  used  at  about  1  their  ultimate  strength. 

369.  Viaducts  are  usually  built  from  above,  by  means  of  an  overhead 
projecting  traveler.  Sometimes  by  means  of  a  cableway ;  but  this  method  is 
slow.  In  some  cases  the  traveling  tower  is  on  the  ground,  and  reaches  to  the 
top  of  the  viaduct.  Or,  the  viaduct  may  be  erected  by  means  of  false-work, 
or  from  an  existing  structure. 

370.  Long  span  bridges  are  usually  built  upon  a  platform  of  false- 
work or  on  a  row  of  trestle  bents,  well  braced. 

371.  Erection.  Upon  the  false-works  the  lower  chords  are  first  laid, 
as  nearly  level  as  may  be.  The  upper  chords  are  then  raised  upon  temporary 
supports  which  foot  upon  the  one  that  carries  the  lower  chord.  The  upper 
chords  are  first  placed  a  few  inches  higher  than  their  intended  positions,  in 
order  that  the  web  members  may  readily  be  slipped  into  place.  When  the 
web  members  are  in  place,  the  upper  chords  are  gradually  lowered  until  all 
rests  upon  the  lower  chords.  The  screws  are  then  gradually  tightened,  to 
bring  all  the  surfaces  of  the  joints  into  their  proper  contact;  and  by  this 
operation  (the  upper  chord  members  having  the  necessary  excess  of  length), 
the  camber  is  formed,  and  the  lower  chords  are  lifted  clear  of  the  false- 
works ;  the  truss  now  resting  only  upon  its  permanent  supports. 

373.  False-work  is  ordinarily  constructed  of  hemlock  or  pine,  costing 
about  S20  per  1000  ft.  bpard  measure.  Allow  about  $15  per  1000  ft.  B.  M.  for 
framing,  etc.  $5  to  $15  per  1000  ft.  B.  M.  may  usually  be  obtained  for  old 
material  ("salvage")- 

The  naain  members  are  wsually  12  X  12  ins.,  and  the  diagonals  3  X  12. 
Bolts,  i  inch.  Owing  to  the  temporary  nature  of  false-work  and  to  the  sal- 
vage which  may  be  obtained  for  it  if  it  is  not  too  badly  cut  up,  it  is  advisable 
to  use  plenty  of  material  of  good  standard  sizes,  especially  in  longitudinal 
bracing,  which  may  be  placed  between  alternate  pairs  of  bents,  forming 
towers. 

373.  In  soft  bottoms,  the  false-work  may  rest  upon  piles,  to  which 
the  uprights  of  the  false-work  may  be  notched  and  bolted,  or  banded.  Not 
less  than  4  piles  per  bent  should  be  used.  As  many  as  24  have  been  used. 
Piles  should  be  braced  below  water-mark.  Bents  should  be  built  in  stories 
of  from  12  to  30  feet  each.  Connections  should  be  made  by  means  of  side 
pieces  or  fish-plates. 

374.  With  rock  bottom,  in  a  strong  current,  it  may  be  expedient  to 
sink  cribs  filled  with  stone,  as  a  foundation  for  the  false-work. 

375.  The  erection  of  cantilevers  and  suspension  bridges  requires 
much  time ;  but  their  use  is  often  necessitated  by  the  impossibility  of  erectii^g 
false-work. 

376.  Renewal  of  bridges  may  be  accomplished  by  displacement ;  either* 
by  protrusion,  where  the  new  span  is  skidded  longitudinally  along  the  track; 
by  transverse  displacement,  where  both  old  and  new  spans  are  placed  on 
tracks  running  normally  to  the  bridge;  by  vertical  displacement,  either  ris- 
ing or  descending;  or  by  pivotal  displacement,  the  old  span  swinging  out 
about  a  pivot,  and  the  new  span  swinging  into  place  about  another  pivot. 

377.  Cautions.  In  erection  and  renewal,  consider  dead  weight  of 
bridge,  effect  of  impact  of  current  of  stream,  impact  of  boats,  ice,  drift,  etc., 
especially  when  floods  are  to  be  apprehended,  and  strains  of  hoisting  tackle. 


744  TRUSSES. 

For  rigidity,  a  liberal  safety  factor  must  be  used.  Drift  may  pile  up  and 
form  a  dam.  Trestle  bents,  in  the  water,  increase  the  velocity  and  scour 
of  the  current,  and  may  thus  cause  undermining.  False-work  may  be  pro- 
tected against  drift  by  fender  piles.  Provide  against  eccentricities  of  wind 
stress.  Numerous  accidents  have  shown  the  expediency  of  guarding  the 
unfinished  truss  itself  against  high  winds.  All  lateral  and  other  wind 
bracing  should  be  in  place,  and  secured,  before  the  false-works  are  re- 
moved and  the  trusses  allowed  to  rest  upon  their  final  bearings. 

Avoid  dropping  tools,  etc.  Even  very  small  pieces,  falling  from  a  great 
height,  are  dangerous  to  life  and  even  to  the  bridge.  Hooks  in  tackles  are 
liable  to  break  or  to  pull  out.  Travelers  should  be  well  guyed  and  clamped, 
and  carefully  watched. 


TRUSS   SPECIFICATIONS.  745 

DIGESTS     OF     SPECIFICATIONS     FOR    BRIDGES 
AND  BUILDINGS. 

(1)  Steel  railroad,  highway  and  electric  railroad  bridges. 

(2)  Combination  (wood  and  steel)  railroad  bridges. 

(3)  Steel  roof  trusses,  framework  and  buildings. 

The  following  Digests  of  Specifications  for  Bridges  and  Buildings  are  in- 
tended primarily  to  give  a  general  view  of  the  essential  features  of  current 
practice  in  such  matters,  and  only  secondarily  to  indicate  the  practice  of 
any  particular  company. 

(1)  DIGEST  OF    SPECIFICATIONS  FOR    STEEL  RAILROAD 
AND    HIGHWAY  BRIDGES. 

List  of  Specifications  Used. 

A9.     American  Bridge  Company, 

General  Specifications  for  Steel  Railroad  Bridges,  1900. 
Aa,  American  Bridge  Company, 

General  Specifications  for  Steel  Highway  Bridges,  1901. 

B,  Baltimore  &  Ohio  Railroad  Company, 

General  Specifications  for  Railroad  and  Highway  Bridges,  Roofs, 
jlnd  Steel  Buildings,  1901. 

C,  Copper,  Theodore  — , 

General  Specifications  for  Steel  Railroad  Bridges  and  Viaducts, 
1901. 
Cc,   Cooper,  Theodore  — , 

General  Specifications  for  Steel  Highway  and  Electric  Railway 
Bridges  and  Viaducts,  1901. 

D,  Delaware,  Lackawanna  &  Western  R.  R.  Company, 

Specifications  for  Steel  Railroad  Bridges,  October,  1899;  revised 
to  July,  1900. 

E,  Erie  Railroad  Company, 

General  Specifications  for  Bridges,  1900. 
G,     General  practice. 
Oo,  Osborn  Engineering  Company, 

General  Specifications  for  Highway  Bridge  Superstructures,  1901. 
P,      Pennsylvania  Railroad  Company, 

Standard  Specifications  for  Steel  Bridges,  January  1,  1901. 
R,     Philadelphia   &  Reading  Railway  Company, 

Specifications  for  Steel  Bridges,  1898;  revised  February,  1901. 
Y,      New  York  Central   &  Hudson  River  R.  R.  Leased  and  Operated  Lines, 

General  Specifications  for  Steel  Bridges,  1900. 


I.  GENERAL  DESIGN. 

Limiting  Spans  for  Different  Types. 

Beams  and  Girders.       A  Aa  B            C  Cc  D  Y 

ft  ft  ft           ft  ft  ft  ft 
Rolled    beams,    solid 

floor,  etc., up  to  up  to  up  to  up  to  up  to  up  to  up  to 

20  40  20           20  40  20  25 

Plate  girders 20  to  25  to  20  to  20  to  20  to  20  to  25  to 

100  80  100         120  80  100  100 
Trttssfs 

Riveted  trusses,* 100  to  40  &  100  to  75  to  40 &  90  to  100  to 

140  over  120         150  over  160  200 

Pin  trusses, over  over  over  over  over  150  &  200  & 

140  140.  120  120  120  over  over 

Riveted  trusses,*  under  100  ft;  pin  trusses,  over  100  ft,  Oo. 

Depth  of  truss,  min,  =  one-eighth  of  span,  Oo. 

♦Unfortunately  called    "lattice  girders,"  A;    "lattice    trusses"    and 
**  riveted  lattice  girders,"  C;   "  riveted  lattice  trusses,"  D,  Y. 


B 

B* 

C 

c 

D 

D 

El 

El 

E2 

E2 

746  TRUSSES. 

A,  Aa,  Am  B  Co;    B,  B   &  O;    C,  Cc,  Cooper;    D,  D  L   &W;   E,  ErieJ 

Classification  of  Highway  and  Electric  Railroad  Bridges. 

Aa     Cc 

TAX*  City  bridges  having  buckle-plate  floors,  and  paving  on  concrete 
A    <  base. 

(A2*  City  bridges  having  plank  flooring. 

Suburban  or  interurban  bridges  for  heavy  electric  cars. 
Town  or  country  bridges  for  light  electric  cars  or  heavy  loads. 
Country  bridges  for  ordinary  highway  traffic . 
Bridges  for  heavy  electric  or  motor  cars  only. 
"    light 

Camber. 

Top  chord  panels  longer  than  lower  chord  panels  by  one-eighth  of  an  inch 
in  10  ft  =  1  in  960,  A,  B,  C,  E,  R,  Y. 

Highway  bridges,  three-sixteenths  of  an  inch  to  every  10  ft,  Cc. 

About  three-fourths  of  an  inch  in  100  ft  =  1  in  1600,  D. 

Sufficient  to  bring  joints  of  compression  chord  to  a  square  bearing  when 
truss  is  fully  loaded.  Each  member  built  longer  or  shorter  in  proportion  to 
the  stress  to  which  it  is  subject  under  a  full  dead  and  a  full  live  load,  so  that 
under  full  loading  it  will  have  its  normal  length,  Oo. 

Cross   Section  of  Bridge. 

Gage,  usually,  4  ft  8^  ins.    Distance,  cen  to  cen  of  tracks,  12  to  13  ft. 

Width  between  trusses  or  girders  in  deck  spans.  Pin  spans,  min, 
0.05  span.  Riveted  trusses  (D),  10  ft.  Plate  girders  (G),  5  to  7  ft.  Spans 
not  over  60  ft,  7  ft;  60  to  100  ft,  8  ft;  over  100  ft,  about  one-twelfth  of 
span,  D.     Plate  girders  (T),  over  60  ft,  in  proportion  to  height. 

Clearance  in  through  spans,  on  tangents,  G. 

a    3  to  3^  ft 

b  7    ft 

c     5  to  5^  ft  J  I 

d    4  to  6    ft  ff'U-b-^    T 

e  10  to  14  ft  '  ' 

f      1  to    5  ft  J-4> 

h  20  to  22  ft  i  '       /   / 

Minimum  Clearance  on  Curves.  ^^'^' 

Same  min  clearance  as  on  tangents,  A ;  Ditto  for  car  74  ft  long,  48  ft  cen 
to  cen  of  trucks,  10  ft  wide,  B ;  Ditto  for  car  75  ft  long,  54  ft  cen  to  cen  of 
trucks.  Additional  clearance  =  0.8  d  ins  on  each  side;  =  1.6  d  ins  between 
tracks ;  where  d  =  degree  of  curvature  =  central  angle  subtended  by  a  chord 
of  100  ft,  C;  increase  lateral  clearance  at  top  of  car  2.5  ins  for  each  inch  of 
superelevation  of  outer  rail,  C.  Cen  line  of  bridge  bisects  middle  ordinate 
and  is  parallel  to  chord,  Y. 

Highway  Bridges.  Headway,  14  ft,  Oo.  For  classes  A,  B,  C,  and 
E,  min  =  15  ft,  Aa,  Cc;  for  a  width  of  6  ft  over  each  track,  Aa.  For 
Class  D,  12.5  ft,  Aa,  Cc.  Horizontal  clearance.  Min  14  ins  greater 
than  width  of  roadway  between  wheel  guards,  Aa.  For  electric  cars,  6.5  ft 
from  cen  of  track,  Aa ;  7  ft,  Cc.  On  curves,  provide  for  a  car  of  45  ft  extreme 
length,  8  ft  wide,  20  ft  between  truck  centers,  Aa.  Width  between  centers 
of  trusses,  min  =  0.05  span,  G. 

Tension  Members. 

In  general,  hip  verticals  aiid  one  or  two  panels  of  lower  chord  at  each  end 
of  span  are  required  to  be  of  rigid  section,  so  as  to  resist  both  tension  and 
compression. 

Angles,  used  as  tension  members,  must  be  fastened  by  both  legs,  C,  D ; 
or  the  section  of  one  leg  only  will  be  considered  effective,  C. 

Adjustment. 

Avoid  adjustable  members.  A,  Aa,  D  (P  except  in  counters).     Avoid 

*  Cc,  Classes  A  and  B  shall  be  designed  to  carry,  at  any  future  time,  a 
double  track  electric  railway. 


T-r\i 


TRUSS   SPECIFICATIONS.  747 

G,  gen'l;  Oo,  Osb'n;  P,  Pa;  R,  R'd'g;   T,  NYC;  Aa,  Cc,  Oo,  H'way. 

adjustable  counters,  C,  Counter  rods  and  ties  in  pin  spans  adjustable,  Y; 
screw  ends  upset,  C,  E,  Y,  screw  threads  U.  S.  Standard,  C,  Cc,  D,  E ;  diam- 
eter, at  base  of  thread  greater  than  in  body  of  bar  by  one-sixteenth  of  an  inch, 
D;  about  10  per  cent,  Y;   17  per  cent,  Oo. 

Rods  with  welded  heads  must  be  of  wrought  iron,  Oo.  Loops  must  de- 
velop full  strength  of  bar,  Oo. 

Compression   Members. 

End  posts  and  upper  chords  have  2  webs,  a  cover  plate  on  top  flange, 
batten  or  tie  plate,  and  lacing  on  bottom  flange,  G 

Not  more  than  one  plate,  and  that  not  thicker  than  one-half  of  an  inch 
(in  highway  bridges  three-eighths  of  an  inch),  shall  in  general  be  used  as  a 
cover  plate,  C,  Cc.  Cover  plate  must  not  extend  more  than  4  ins  beyond 
outer  row  of  rivets,  D. 

Joints  between  sections  spliced  on  all  sides  with  at  least  2  rows  of 
closely  pitched  rivets  on  each  side  of  joint,  C.  Abutting  surfaces  faced  A 
(D,  P,  except  in  top  flanges  of  girders),  E,  R,  Y.  No  reliance  on  abutting 
surfaces,  E. 

Lattice  Bars. 

Width,  from  1.5  to  2.5  ins.  Thickness,  in  single  lattice,  one-fortieth, 
distance  between  rivets,  in  double  lattice,  one-sixtieth,  G.  If  over  seven- 
sixteenths  of  an  inch,  use  angles,  Oo.  Angle  with  axis  of  member;  in  single 
lattice  60°,  double  45°,  G. 

Pitch,  width  of  channel  +  9  ins,  C,  Cc ;  8  X  least  width  of  segment,  P,  R. 

Double  lattice  bars  riveted  together  at  their  intersections,  C,  D. 

Batten  Plates.  (Tie  Plates,  Stay  Plates.)  Min  length  generally  =  from 
0.75  to  1.5  X  its  own  width.  Min  width,  9  ins,  or  0.66  X  own  length,  or  = 
least  width  of  member,  Oo.  Min  thickness  =  three-eighths  of  an  inch  or  one- 
fiftieth  to  one-sixtieth  of  distance  between  centers  of  rivets,  G.  Rivet  spac- 
ing (D)  max  4  ins  cens. 

Pin  Joints. 

Eye  Bars.  Thickness,  min,  0.625  inch,  or  0.2  X  width  of  bar,  Oo. 
Heads  upset,  rolled  or  forged.  A,  R.  No  welds  allowed  A,  D,  except  (R)  to 
form  loops  of  laterals,  counters  or  sway  rods,  R.  Upset  and  die-forged,  Y. 
Heads  not  more  than  one-sixteenth  of  an  inch  thicker  than  body,  D,  P. 
Bars  annealed,  G  ;  before  boring,  D.  No  forge  work  after  boring,  R.  Bars 
to  be  placed  side  by  side  must  be  bored  at  the  same  temperature.  A,  Aa,  Y. 
Pins  must  pass  through  without  driving.  Eye  bars  working  together  must 
be  clamped  together,  and  bored  at  one  operation,  Oo. 

Distance  between  pin-holes  max  variation,  one  sixty-fourth  to  one  thirty- 
second  of  an  inch ;  or  one  sixty-fourth  of  an  inch  in  from  20  to  25  ft,  G. 

Built  Tension  Members.  Net  section  through  pin  hole  =  1.25  to  1.50 
X  net  section  through  body  of  member.  Net  section  back  of  pin  hole  = 
0.75  X  net  section  through  pin  hole,  or  =  0.80  to  1.00  X  net  section  through 
body  of  member,  G ;  proportion  for  double  shear  on  section  from  back  of  pin 
to  end  of  plate,  Oo ;  length  of  plate  back  of  pin,  min,  2.5  ins,  Oo.  Distance, 
back  of  eye  to  back  of  member,  greater  than  radius  of  pin,  Y. 

Pin  Holes.  Clearance  between  pin  and  hole,  from  one-fiftieth  to  one 
thirty-second  of  an  inch,  G. 

Pin  Plates  or  Reinforcing  Plates.  At  least  one  plate  on  each  side 
must  extend  not  less  than  6  ins  beyond  edge  of  batten  plate,  G. 

Pins.     Up  to  7  ins  diam.  rolled,  P;  over  7  ins,  forged,  C. 

Diameter,  min,  from  0.66  X  to  0.85  X  largest  dimension  of  any  of  its  eye 
bars,  G. 

Plate  Girders. 
Min  depth;  about  one-ninth  to  one-twelfth  of  span,  G. 
Proportions  of  Web  and  Flange.     Bending  moments  resisted  entirely 
by  the  flanges,  shear  resisted  entirely  by  the  web  plate,  C,  Cc,  R;  except 

when  the  web  is  made  in  one  length  or  is  fully  spliced  to  resist  the  bending 
stresses,  in  which  case  one-sixth  of  area  of  cross  section  of  web  plate  may  be 
considered  effective  as  flange  area,  Oo. 


748  TRUSSES. 

A,  Aa,  Am  B  Co;   B,  B   &  O;    C,  Cc,  Cooper;   D,  D  L   &  W;   E,  Erie* 

One-eighth  of  gross  area  of  web  included  in  flange,  A,  Aa,  B ;  if  length  = 
90  ft  or  over,  E ;  if.  length  is  less  than  50  ft,  only  the  cover  plate  and  the 
horizontal  legs  of  the  flange  angles  are  to  be  included  in  the  flange  area,  E ; 
no  part  of  web  included  in  flange,  C,  Cc,  D,  P,  R,  Y. 

Web.  Thickness,  min,  three-eighths  of  an  inch,  G ;  in  highway  bridges 
(Cc,  Oo),  five-sixteenths  of  an  inch. 

Total  shear,  acting  on  side  next  to  abutment,  to  be  taken  as  transferred 
into  flange  angles  within  distance  =  depth  of  girder,  A,  Aa,  B,  E,  Oo,  P. 

Web  Splices.  A  plate  on  each  side  of  web,  G;  at  least  three-eighths 
inch  thick.  A,  B ;  at  least  five-sixteenths  inch,  or  three-fourths  as  thick  as 
web,  and  wide  enough  for  2  rows  of  rivets  on  each  side  of  splice,  Oo. 

Stiffeners.  Generally  required  at  ends  and  at  points  of  concentrated 
load.  Intermediate  stiffeners  required  usually  when  unsupported  distance 
between  flange  angles  exceeds  50  to  60  X  web  thickness ;  when  shear  ex- 
ceeds 10,000  —  75  X  -T .— ,'^ —  ,  C :  in  highway  bridges  when  shear  exceeds 

thickness 
12,500  —  90  X  -^-^  ,  Cc ;  when  shear  exceeds  ^0.000  X  m  ^^^^^  ^  _ 
thickness  ^    ,         d- 

"^  3,000  t2 
web  thickness,  d  =  distance  between  flanges,  Oo. 

Spacing  usually  =  depth,  or  =  5  or  6  ft. 

Unit  stress^  max,  10,000  —  45  — ,  C ;  in  highway  bridges,  12,000  —  55  — , 
r  r 

where  1  =  length  of  stiffener,  r  =  its  least  radius  of  gyration,  Co. 
Dimensions  of  angles  usually  31  X  3  X  is  to  5  X  3^  X  f. 
Flanges. 
Unbraced  length  of  flange  (compression  flange,  C,  P)  max  =  12  X  width, 

B,  P,  B,  Y;  16  X  width.  A,  C,  Cc;  20  X  width,  D;  in  highway  bridges. 
=    20  X  width,  Aa,  =  25  X  width,  Oo. 

Comp.  flange  has  same  gross  area  as  tension  flange.  A,  Aa,  B,  C,  Cc,Oo,  P. 

Cover  plates  must  not  extend  more  than  5  ins,  or  8  X  thickness  of  first 
plate,  beyond  outer  line  of  rivets.  A,  Aa,  C,  Cc.  If  of  unequal  thickness,  the 
heaviest  plates  are  next  the  angles,  and  the  lightest  outside.  A,  Aa,  B,  C, 
Cc,  Y.  One  must  extend  full  length  of  girder,  B,  E.  Others  must  be  long 
enough  to  take  2  extra  rows  of  rivets  at  each  end,  C,  P. 

Bracing,  Riveting,  Bearings.  See  Bracing,  Riveted  Joints,  and  Bear- 
ings, below. 

Beam  Girders. 

Beams  in  groups  of  2,  3  or  4  for  each  rail,  10-inch  channel  separators,  about 
3  ft  apart,  riveted  to  webs,  B. 

Bracing. 

Composed  of  rigid  members,  riveted.  A,  Aa,  C,  Cc,  T;  members  inter- 
sect each  other,  and  other  members  to  which  they  are  connected,  on  common 
center  lines,  passing  through  all  centers  of  gravity.  Attachments  riveted 
symmetrically  in  all  directions,  Y. 

For  Beam  Girders.     Ten-inch  channel  strut  at  each  end;  with  2  or  3 
beams,  angle  bracing  between  girders ;  with  4  beams,  angle  struts  about  6  ft 
apart.     Connections  to  have  at  least  3  rivets,  B. 
Lateral. 

B,  in  through  spans.  Top  bracing;  portal  struts  at  ends;  intermediate 
struts  as  deep  as  the  chords ;  single  angles,  3^  X  3^^  X  I,  intersecting  in  each 
panel,  for  single  track;  double  angles,  latticed,  for  double  track. 

B,  in  through  spans,  bottom  bracing;  end  bottom  strut  and  intermediate 
angles,  riveted  to  each  other  and  to  stringers  at  each  intersection.  Not  less 
than  4  rivets  at  each  intersection  and  at  each  end  connection. 

B,  in  deck  bridges,  complete  upper  and  lower  systems  at  each  panel. 

Oo,  bottom  end  struts  in  all  spans,  whether  deck  or  through. 

Y,  top  and  bottom  lateral  bracing  in  all  deck  bridges  and  in  all  through 
bridges  having  sufficient  head  room.     Lower  lateral  bracing  in  all  through 


TRUSS  SPECIFICATIONS.  749 

G,  gen'l;  Oo,  Osb'n;  P,  Pa;   R,  RM'g;  Y,  N  Y  C;  Aa,  Cc,  Oo,  H'way. 

bridges.  Upper  system  in  all  stringers  framed  between  and  riveted  to  floor 
beams  where  length  of  stringers  exceeds  15  times  width  of  stringer  flange.  • 
Y,  in  deck  bridges  without  metal  floor  system,  upper  bracing  of  cross-struts 
at  each  panel  point,  composed  of  4  angles  latticed,  with  same  depth  as  upper 
chord;  stiff  diagonals  intersecting  in  each  panel  and  riveted  to  each  other 
at  each  intersection. 

For  Plate  Girders. 

Deck.  Between  upper  flanges,  angles  with  at  least  4  rivets  at  connections. 
Through ;  lower  lateral  bracing  of  angles,  intersecting  in  each  panel,  riveted 
to  each  other  and  to  stringers  at  each  intersection,  B. 

Lateral  bracing  angles  generally  of  same  size  as  those  in  stiffeners,  R. 

Cc,  in  highway  bridges,  a  buckle  plate  floor  may  be  considered  as  the  re- 
quired system  of  lateral  bracing  at  the  floor  level. 

Sway  (diagonal,  cross,  vibration  or  wind)  and  Portal. 

Proportioned  to  resist  unequal  loading  of  trusses  in  double  track  spans,  E, 
R ;  end  sway  bracing  to  transmit  all  horizontal  forces  to  abutment,  E ;  to 
carry  half  the  max  stress  increment  due  to  wind  &  centrif .  force,  A,  Aa,  B,  P. 

In  deck  spans,  at  each  panel  point,  A,  Aa,  D,  E,  P,  R,  Y. 

Overhead  bracing  in  through  spans  whose  depth  exceeds  25  ft,  C,  D,  P, 
Y;  in  highway  bridges,  20  ft,  Cc;  25  ft,  Oo. 

In  pony  trusses  and  through  plate  girders,  at  ends  and  at  each  floor  beam 
or  cross  strut,  A,  Aa,  R;  at  every  panel  point,  D. 

In  through  and  half  through  plate  girders,  at  each  floor  beam  and  at  each 
end,  or,  if  there  is  a  solid  floor,  not  over  8  ft  apart,  Y. 

In  deck  plate  girders,  rigid  cross  frames  at  ends  and  max  20  ft  apart,  Y; 
sway  frames  of  at  least  4  angles  at  ends  and  at  points  12  to  14  ft  apart,  B; 
through,  not  more  than  12  X  flange  width  along  top  flange,  B. 

Riveted  Joints. 

Rivet  Holes.     In  I  beams,  must  be  drilled,  B. 

May  be  punched;  in  steel  not  over  f  to  f  inch  thick,  G. 

Sub-punched  one-eighth  of  an  inch  smaller,  and  reamed  to  one-sixteenth 
of  an  inch  larger,  than  rivet,  in  steel  over  five-eighths  to  three-fourths  of  an 
inch  thick ;  in  connections  for  floor  beams  and  stringers  to  main  trusses  or 
girders,  E. 

No  drifting  allowed,  A,  B,  C,  D,  E,  R. 

No  interchange  of  pieces  after  reaming,  D,  P. 

Hole  larger  than  rivet  by  one-sixteenth  of  an  inch,  G. 

Die  larger  than  punch  by  max  one-sixteenth  of  an  inch,  G, 

Distance  from  edge  of  plate  to  center  of  rivet.  Min,  1.25  to  1.5  ins,  or  1.5 
to  2  diams  of  rivet.     Max,  4  to  5  ins,  or  8  X  thickness  of  plate,  G. 

Pitch      Min  =  3  X  diam  of  rivet,  general;  preferably  4  X  diam,  Oo. 

Max  pitch  in  line  of  stress,  5  to  6  ins,  or  16  X  thickness  of  thinnest  outside 
plate  connected;  normal  to  stress,  30  to  50  X  thickness  of  thinnest  outside 
plate  connected.  At  ends  of  compression  members  (or  of  built  members  in 
tension,  B)  ;  for  a  length  of  1.5  to  2  X  width  or  depth  of  member,  3.5  to  4  X 
diam  of  rivet,  G. 

In  plate  girders,  for  rivets  connecting  web  to  a  top  flange  supporting  the 
track,  max  =  3  ins,  R. 

Rivets.  Diam  generally  three-fourths  or  seven-eighths  inch.  Heads 
hemispherical,  G.     Height  of  head,  min  =  0.6  diam,  R. 

Driving.  Avoid  hand  riveting.  Machines,  direct-acting,  worked  by 
steam,  hydraulic  pressure  or  compressed  air,  capable  of  maintaining  applied 
pressure  after  upsetting,  G. 

Floor. 

Floor  Beams.  Depth,  min,  =  i  X  length,  Y.  In  railroad  bridges  and 
important  highway  bridges,  riveted  to  posts  of  trusses  or  to  webs  of  plate 
girders,  G.  Given  also  a  bearing  on  lower  flange  of  girder  or  on  a  bracket, 
G.     In  default  of  such  bearing,  increase  number  of  rivets  by  25  per  cent,  R. 

Hangers,  when  permitted,  not  adjustable,  C,  Cc.  Hangers  made  of  plates 
or  shapes,  Oo. 

Stringers*     Depth,  min,  =  i  X  length,  Y.     In  highway  bridges,  Classes 


750  TRUSSES. 

A,  Aa,  Am  B  Co;   B,  B   &   O;  C,  Cc,  Cooper;   D,  D  L   &  W;  E,  Erie; 

^1  and  A2,  of  steel;  Classes  B,  C,  and  E,  track  stringers  of  steel;  Class  D,  of 
wood  or  steel,  Cc.  In  railroad  bridges,  and  preferably  in  highway  bridges, 
riveted  to  webs  of  floor  beams  and  supported  by  their  flanges  or  by  brackets, 

B,  R.  Value  of  this  bearing  neglected  in  determining  number  of  rivets 
required,  R. 

Spacing,  cen  to  cen,  6  ft,  6  ins,  A,  B,  C,  D,  Y;  5  ft,  E;  double  track, 
through,  generally  6  ft,  R;  single  track,  8  ft,  R. 

Trough  Floors.  Troughs  rectangular,  built  of  plates  and  angles,  and 
riveted  to  main  girders  or  trusses  by  angles,  and,  when  practicable,  by  bracket 
angles  under  the  lower  horizontal  plates.  Gusset  plates  riveted  to  girders 
and  troughs  at  distances  of  not  over  8  ft,  Y. 

Bottom  filled  with  a  binder  composed  of  1  cu  ft  of  clean,  sharp  gravel, 
screened  to  one-fourth  of  an  inch,  to  1^  gallons  of  No.  4  asphalt  paving  com- 
position or  enough  to  fill  voids.  Gravel  first  heated  to  300°  F  and  the  whole 
mixed  at  that  temperature,  Y. 

Wooden  Floor.     Continued  over  abutments,  A,  B,  C,  R. 

Ties  or  Floor  Beams.     Long  leaf  yellow  pine  or  white  oak,  G. 

Width,  8  ins.  A,  B ;  9  ins.  R. 

Depth,  8  ins,  for  7  ft  span  of  tie,  to  14  ins  for  12  ft,  B ;  12  ins,  R ;  10  ins,  Y. 

Notched  down  one-half  of  an  inch;  max  1^  ins,  B. 

Spacing.  Usually  6  ins  clear;  16  ins  cen  to  cen,  R.  Every  3d,  4th,  or 
5th  tie  fastened  to  stringer  by  f  inch  bolt  or  lag  screw,  G. 

Wooden  Joists  in  Highway  Bridges.  Width,  min  3  ins  or  0.25  X 
depth;  spacing,  max,  2  to  2.5  ft.  Ends  of  joists  lap  past  each  other  at  bear- 
ings on  floor  beams,  with  0.5  inch  space  between  them  for  circulation  of  air. 

Wooden  Floor  Beams  for  Electric  Railroad  Bridges,  Classes  El 
and  E2  Min  6X6  ins,  spacing  max  6  ins,  notched  down  one-half  of  an  inch 
and  secured  to  girders  by  three-fourths-inch  bolts  not  more  than  6  ft  apart. 
From  center  of  span  toward  end,  so  notched  (Cc)  as  to  reduce  camber. 

Guard  Rails.  6X8  ins,  yellow  or  white  pine,  G.  Inner  face  not  less  than 
3  ft  3  ins  from  center  of  track.  A;  3  ft  7i  ins,  B;  5  ft  4  ins,  Y;  7  ft  li  ins 
apart,  clear,  R.  Notched  i  to  H  ins  over  ties,  G.  Fastened  to  every 
3d  or  4th  tie  (to  each  tie,  R)  and  at  splices  by  three-fourth-inch  bolt  or  lag 
screw,  G.     Splices  over  floor  timbers,  with  half-and-half  joints  of  6  ins  lap,  G. 

Wheel  Guards  and  Curbs  in  Highway  Bridges.  Wheel  guards 
6X4  ins,  blocked  up  from  floor  plank  by  blocks  2X6  ms,  12  ms  long,  not 
more  than  5  ft  apart,  bolted  to  stringers  through  blocking  pieces,  three- 
fourths-inch  bolts,  G.  In  electric  railroad  bridges  (Cc,  Class  E)  guard 
timbers  min  5X7  ins,  notched  1  inch  over  floor  timber  and  secured  by  three- 
fourths-inch  bolt  to  every  third  floor  timber  and  at  each  splice. 

Buckle  Plates.  Min  five-sixteenths  of  an  inch  thick  for  roadway,  one- 
fourth  of  an  inch  for  footwalk,  crown  2  ins,  for  widths  of  4  ft  under  roadway, 
5  ft  under  footwalks.  Preferably  in  continuous  sheets  of  panel  lengths. 
May  be  pressed  or  formed  without  heating. 

Bearings  on  Abutments  and  Piers. 

Permissible  load  on  masonry  foundations,  max,  pounds  per  sq  inch. 
400,  A,  Aa,  P;  300,  B;  250,  C,  Cc,  D,  E,  R;  dead  load,  500;  live  load, 
250,  Y. 

Bed  Plates.  Of  medium  steel,  C,  Cc.  Min  thickness,  three-fourths  to 
1  inch;  in  highway  bridges,  one-half  of  an  inch.  Max  fiber  stress  12,000  lbs 
per  sq  inch,  E. 

Where  ends  of  two  spans  rest  on  one  pier,  spans  are  tied  together,  or  have 
bed  plate,  three-eighths  to  three-fourths  of  an  inch  thick,  continuous  under 

Sheetlead,  one-eighth  to  one-fourth  of  an  inch  thick,  between  bed  plate 
and  masonry,  G.  <■  i     •  u 

Anchor  bolts,  1  to  1.25  ins  diam,  9  to  12  ins  in  masonry,  G;  fastened  with 
sulphur,  R;  with  cement,  C,  Cc,  Y, 

Pedestals.  Of  riveted  plates  and  angles,  C;  or  cast  steel,  Y.  Base 
plate  and  connecting  angles,  min  three-fourths  to  seven-eighths  of  an  inch 
thick,  B,  C,  Y.     2  rows  of  rivets  in  vertical  legs,  C,  Y. 


TRUSS   SPECIFICATIONS.  751 

G,  gen'l;  -Oo,  Osb'n;  P,  Pa;   R,  R'd'g;  Y,  NYC;  Aa,  Cc,  Oo,  H'way. 

Expansion  Bearings.  Provide  for  temperature  range  of  150^  F,  A,  C, 
E,  F,  R;  for  expansion  of  1  inch  in  100  ft,  D,  Y. 

One  end  sliding,  usually  in  spans  less  than  60  to  90  ft,  G. 

One  end  on  friction  rollers,  usually  in  longer  spans,  G ;  in  all  trusses,  Y. 

Hinged  bolster  at  each  end,  in  spans  from  80  to  100  ft,  G. 

Rollers  rest  on  bars  3X1  inch,  spaced  2  ins  and  riveted  to  bed  plate,  B» 
Free  ends  anchored  against  lifting  and  against  moving  sideways,  C.  Y. 

Rollers.  Of  machinery  steel,  C,  Cc.  Min  diam,  3  to  4  ins,  A,  B,  D,  E,  P, 
R;  3  ins  up  to  100  ft  span,  1  inch  for  each  additional  100  ft,  C,  Y.  Wax 
pressure  on  rollers,  in  lbs  per  linear  inch,  700  yd,  R;  1200  Vd,  A,  B,  P; 
300  d,  C,  D,  E ;  600  d,  Oo;  d  =  roller  diam,  ins.     Length,  ins,  =  900  yd,  Y. 

II.  MATERIAL. 

Rolled  and  Cast  Steel  and  Iron. 

Rolled  steel  in  superstructures  in  general. 

Cast  steel  in  bed  plates  in  special  cases,  in  machinery  of  movable  bridges. 

Rolled  iron  in  loop-welded  rods,  P;  in  laterals  and  unimportant  mem- 
bers, R. 

Cast  iron  in  bed  plates  in  special  cases  and  in  machinery  of  movable 
bridges. 

Rolled  Steel,  Grades. 

Soft.     In  general,  in  all  principal  parts. 

Medium.  In  pins,  friction  rollers,  lateral  bolts,  bearing  plates,  eye-bars, 
sliding  plates  and  bed  plates;  permissibly  (C)  in  compression  in  chords, 
posts,  and  pedestals. 

Rivet.     In  rivets. 

Machinery.     In  expansion  rollers,  C. 

Rolled  Steel.     Manufacture. 

(See  also  Digest  of  Specifications  of  Internat'l  Ass'n  for  Testing  Materials.) 

All  to  be  made  by  open-hearth  process. 

Slabs  for  rolling  plates  are  hammered  or  rolled  from  ingots  of  at  least  twice 
their  cross-section.  A,  B. 

Plates  up  to  36  ins  wide  rolled  in  universal  mill,  D,  R;  or  have  edges 
planed,  D. 

Rolled  Steel.     Manipulation. 

Annealing.  Eye-bars  heated  to  uniform  dark  red  and  allowed  to  cool 
slowly,  P ;  members  worked  at  blue  heat  are  heated  to  a  uniform  bright  red 
(not  exposed  to  direct  flame)  and  allowed  to  cool  slowly,  B. 

Steel  must  not  be  welded,  B.     No  reliance  upon  welded  steel,  C. 

No  work  put  upon  steel  at  or  near  blue  heat,  or  between  boiling-point  and 
point  of  ignition  of  hardwood  sawdust,  C. 

Rolled  Steel.     Shop  Work. 

Sheared  edges  of  steel  thicker  than  five-eighths  inch  shall  be  plaVied,  B. 

All  sheared  edges  (in  medium  steel,  D)  shall  be  planed  off  to  a  depth  of 
0.25  inch,  D,  Y;  except  web  plates  of  girders  over  36  ins  deep  when  covered 
by  flange  plates,  and  fillers  where  sheared  edges  are  not  seen,  D.  Grinding 
not  accepted  as  equivalent  of  planing,  except  for  lattice  bars,  Y. 

No  sharp  or  unfilleted  re-entrant  corners  permitted,  D,  Y.  Where  a 
plate,  angle  or  shape  has  been  cut  into,  the  fillet,  as  well  as  the  cut,  must  be 
finished  with  sharp  cutting  tool,  or  with  chisel  and  file,  so  that  no  sign  of  the 
punched  or  sheared  edge  remains,  D. 

Angles  or  bent  plates,  used  as  end  connections  on  girders,  floor  beams  or 
stringers,  must  be  accurately  fitted,  so  that  when  the  member  is  milled  to 
length  not  more  than  one-sixteenth  of  an  inch  will  be  taken  off  these  connec- 
tions at  their  roots,  D. 

Material  bent  by  punching  must  be  straightened  before  bolting  up,  R, 
Web  plates,  if  buckled,  hiust  be  cold-rolled  to  remove  the  buckles,  D. 

Spliced  chord  sections  must  be  assembled  and  strung  out  in  shop  in  lengths 
of  not  less  than  three  sections,  and,  after  being  drawn  up  into  contact  at 


752  TRUSSES. 

A,  Aa,  Am  B  Co;  B,  B  &  O;   C,  Cc,  Cooper;  D,  D  L  &  W;  E,  Eriej 

joints  and  lined  up  with  splice  plates  in  jdiace,  the  field  rivet  holes  shall  be 
reamed  to  a  tit  before  taking  apart,  and  tne  assembled  parts,  with  their  splice 
plates,  match-marked,  D. 

Riveted  members  must  have  all  parts  pinned  up  and  drawn  together  before 
riveting  up,  D.  * 

In  cases  of  skew  work,  or  of  complicated  connections,  or  of  a  large  number 
of  pieces  of  one  and  the  same  kind,  the  work  shall  be  set  up  and  fitted  to- 
gether in  the  shop,  sufficiently  to  insure  against  any  misfit,  D. 

Abutting  surfaces  at  ends  of  sections  of  compression  members,  and  ends 
of  members  to  be  framed  together,  are  usually  required  to  be  faced. 

Rolled  Steel.     Requirements. 

See  also  Digest  of  Specifications  of  International  Association  for  Testing 
Materials. 

TENSILE    TESTS. 

Specimens  of  Medium,  Soft  and  Rivet  Steel.     For  tests  of  full  size  eye-bars 
see  below. 
Ultimate  Strength,  u,  and  Elastic  Limit,  el,  in  thousands  of  lbs  per  sq 
inch.     Elongation,  s  (stretch),  and  reduction  of  area,  a,  in  percentages  oi 
original  dimensions.     Elongation  measured  in  a  length  of  8  ins. 


Medium  or 
"Pin"  Steel. 

Soft  or  "Bridge" 
Steel. 

Rivet  Steel. 

A,  Aa    . . 

B 

C,  Cc  ... 

D 

E    ... 

Oo    .. 

P  .... 

R    ... 

Y  .... 

u 
60-70 

60-68 
62-70 

60-70 
62-70 

60-68 

62-70 

el 
0.5  u 

0.5u 
0.5  u 

35 
33 

0.5  u 

0.6  u 

s 
22 

22 

22 

22 
17 

20 

25 

a 
40 

u 
52-62 
58-63 
54-62 
54-62 
56-64 
52-62 
52-62 

52-60 

56-64 

el 
0.5    u 

30 
0.5    u 
0.5    u 
0.58  u 

32 

28 

{"is"} 

0.6    u 

s 

25 
25 
25 
26 
27 
25 
25 

25 

26 

a 

45 
50 

50 

u 

48-58 
51-56 
50-58 
48-56 

50-60 
48-56 

48-56 

48-56 

el 
0.5  u 

27 
0.5  u 
0.5  u 

30 

28 

28 

s 

26 
26 
26 

28 

26 

28 

28 
28 

a 

56 
55 

Specimens  from  metal  over  five-eighths  of  an  inch  thick,  el  =  0.56  u,  Y. 

"  '*     eye-bars,  same  requirements  as  for  medium  steel,  D. 

"  "         u  =  63,  B. 

**  "  "         over  1.5  ins  thick,  deduct  from  el  1  for  each  one- 

eighth  of  an  inch ;  el,  min  =  20,  C. 

"  "  •*         and  pins  u  =  62-70,  1  =  0.6  u,   s  =  25,  a  = 

45,  Y. 

"  "     pins,  s  =  15,  C,  Cc. 

"  "         "     (medium,  soft  or  rivet)  s,  5  per  cent  less,  A. 

"    .  "        "     (soft)  s  =  20,  B. 

"  "        "    and  rollers,  s  =  10,  D. 

••  "     rollers  and  bearing  plates,  u  =  70-78,  s  =  22,  Y. 

BENDING    TESTS. 

In  medium  steel,  specimen  to  bend  through  an  angle  of  180°  around  a  bar 
of  diam  =  1  to  1^-  X  thickness  of  specimen,  without  showing  fracture  on 
outside  of  bend ;  in  soft  and  rivet  steel,  to  bend  flat  upon  itself. 

NICKING   TEST. 

When  nicked  and  bent  around  a  bar  of  diam  =  thickness  of  rod,  rivet 
steel  shall  show  a  gradual  break  and  a  fine,  silky,  homogeneous  fracture,  D. 

DRIFTING    TEST. 

Center  of  hole  as  in  ordinary  practice,  or  1.5  to  1.87  ins  or  2  diams  from 
edge  of  plate;   enlarge  to  1.25  to  1.50  diam,  G. 


TRUSS  SPECIFICATIONS.  753 

G,  gen'l;  Oo,  Osb'n;   P,Pa;  R,  R'd'g;  Y,  NYC;   Aa,  Cc,  Oo,  H'way. 

ANGLE    TEST. 

Angles  of  all  thicknesses  must  open  flat.  Angles  not  over  one-half  of  an 
inch  thick  must  bend  shut,  cold,  under  hammer  blows  without  sign  of  frac- 
ture, B. 

TEST    PIECES. 

Minimum  section,  usually  one-half  sq  inch.  Length,  min,  8  to  12  ins. 
Tests  are  usually  required  for  each  melt  or  blow. 

TESTS    OF    FULL    SIZE    EYE-BARS. 

Ultimate  lbs  per  Elastic  limit  lbs  per 

sq  inch                                  sq  inch                 Elongation  per  cent 
min                                        min  min 

A     Ao              f  5,000  less  than )  ,,^  u  ^ 

A,  Aa Ismail  specimen/ 10  between  necks 

B    55,000 0.5  ult i}nK'^^^''°^''u^? 

(10  between  necks  || 

C,  Cc 56,000 10  between  necks 

D    58,000 30.000 12  in  10  ft 

Oo  *  *. !  *. '. ".  *  *. '  *55,bb0  .*  .* .*  .*  * .' .' .'  .*  .* .'  .*  .*  .*  .* .' .'  * .'  .* .' .' .' .' .'  * .'  *12.5  in  15  ft 
P 48,000 27,000 14^  in  10  ft 

(58,000*1                    '     n  t^  ,iU  f  13  between  necks 

R    i  56,000  tj   "-^  "^^ (10  between  necks 

( 48.000  t 27,000 15  between  necks 

Y    58.000 33,000 10  in  20  ft 

In  general  not  over  4  per  cent  of  total  number  of  bars  in  bridge  will  be 
tested,  R ;  at  least  4  per  cent,  and  not  less  than  3  bars,  B. 

75  per  cent  of  fracture  must  be  silky,  the  remainder  fine  granular,  R. 
Break  in  head  shall  not  be  cause  for  rejection — 

(a)  if  bar  develops  10  per  cent  elongation  (12.5  per  cent  in  15  ft,  Oo)  and 
the  required  ultimate  strength  (ultimate  56,000,  C,  55,000,  Oo)  and  if  not 
more  than  one-third  of  all  the  bars  tested  break  in  the  head,  A,  C,  Oo. 

(b)  if  bar  stretches  14  per  cent  and  if  a  second  bar  breaks  in  body  and  the 
average  stretch  of  the  two  bars  is  not  less  than  16  per  cent,  P. 

Company  pays  for  bars  which  meet  requirements,  less  scrap  value,  G. 

TESTS  OF  COMPLETED  STRUCTURE. 

Specified  loads,  or  their  equivalent,  passed  over  structure  (in  railroad 
bridges  at  a  speed  of  not  over  60  miles  per  hour,  and  brought  to  a  stop  at  any 
point  by  means  of  air  or  other  brakes)  or  maximum  load  rested  upon  struc- 
ture for  12  hours.  After  test,  structure  must  return  to  its  original  position 
and  must  show  no  permanent  change  in  any  part,  C. 

COMPOSITION. 

Phosphorus,  max  percentage. 

In  acid  steel,  0.06  to  0.08;  in  basic  steel,  0.04  to  0.06;  in  castings  0.08. 

Sulphur,  max  percentage,  0.04  to  0.06. 

MAXIMUM    PERMISSIBLE    VARIATION    FROM     SPECIFIED    CROSS-SECTION 
OR    WEIGHT. 

2.5  per  cent,  G,  except  in  extra  wide  plates,  D,  Oo,  P. 

In  plates  over  40  ins  wide,  in  proportion  to  width,  up  to  5  per  cent  in  plates 
90  ins  or  wider,  D. 

1.5  per  cent ;  where  plates  36  ins  and  wider  form  40  per  cent  of  total,  2  per 
cent  in  excess,  Y. 

Long  plates,  ^  inch  out  of  line  in  20  ft,  f  inch  in  40  ft,  R. 

Shapes  or  plates,  3  per  cent  short  in  thickness;  plates  80  ins  wide,  5  per 
cent,  R. 

*  t  Medium  steel.  *  Bars  not  over  10  sq  ins.  f  20  sq  ins.  Proportional 
values  for  intermediate  areas,  t  Soft  steel.  3  In  bars  not  longer  than  20 
ft  between  necks.     ||  In  bars  longer  than  20  ft  between  necks.     1[  Max,  16. 

48 


754  TRUSSES. 

A,  Aa,  Am  B  Co;    B,  B    &  O;    C,  Cc,  Cooper;    D,  D  L    &  W;   E,  Erie? 

Steel  Castings. 
Manufacture.     Open  hearth,  A,  Aa,  D,  P,  Y;  acid,  Y;  annealed,  P, 
R,  Y.     Carbon,  per  cent,  0.25  to  0.40,  G. 
Phosphorus,  pei-  cent,  max,  0.08,  B,  Y. 

TENSILE    TESTS, 


C,  D,E,  P,  R.  . 


Size  of 

test  piece, 

ins. 


J^  square 

or 
i  round 

r  i  round 
<  about  6 
(.    long 


Ultimate 

Elastic 

Elonga- 
tion 
per  cent, 
in  2  ins 
min. 

Reduc- 

strength, 
lbs  per  sq 

limit, 
lbs  per  sq 

tion 
of  area 

inch,  min. 

inch,  min. 

per  cent. 

65,000 

33,000  1 

to 

or        [ 

10  to  15 

20,    P 

70,000 

0.5  ult  ) 

( 55,000) 

(a)-^    to   y 

20 

(65,000) 

(72,0001 

(b)'       to     [ 
(80,000) 

0.5  ult 

15 

25 

BENDING  TEST. 

Y  (a),  for  general  purposes,  bed  plates,  pedestals,  etc.,  to  bend  90°,  to  a 
radius  =  diameter  of  test  piece. 

Y  (b),  for  drawbridge  rollers,  etc. 

Rolled  Iron. 

Requirements  in  Osbom's  specification  for  highway  bridges,  Oo.  Made 
from  puddled  iron  or  rolled  from  fagots  or  piles  of  No.  1  wrought  iron  scrap, 
alone  or  with  muck  bar.  Tensile  strength,  min,  48,000  lbs  per  sq  inch  (50,- 
000,  R) ;  yield  point,  25,000  lbs  per  sq  inch  (26,000,  R) ;  elongation,  20  per 
cent  in  8  ins ;  in  sections  weighing  less  than  0.654  lb  per  lineal  ft,  15  per  cent. 
Specimens  cut  from  bar  as  rolled  must  .end  through  an  angle  of  180°  under  a 
succession  of  light  blows  ^  when  nicked  and  bent,  fracture  shall  be  generally 
fibrous  and  free  from  coarse  crystalline  spots ;  not  over  10  per  cent  of  the 
fractured  surface  "hall  be  granular ;  specimens  heated  bright  red  shall  bend 
through  an  angle  of  180°  under  a  succession  of  light  blows  not  delivered 
directly  on  the  bend;  must  not  show  red-shortness.  In  flat  and  square 
bars,  one-thirty-second  of  an  inch,  in  round  iron  0.01  inch,  variation  either 
way  in  size  will  be  allowed,  Oo. 

Cast    Iron. 

Tough  gray  iron,  A,  D,  E,  R;  unless  otherwise  specified,  A,  R. 

Transverse  strength.  Bar  1  inch  square,  12  ins  span,  to  bear  2,500  lbs, 
center  load.  Must  deflect  0.15  inch  before  rupture,  G.  Bar  1  inch  square, 
4,5  ft  span,  to  bear  500  lbs  center  load,  E,  R. 


Phosphor  Bronze. 

1  inch  cube,  under  compression,  elastic  limit,  20,000  lbs. 
lbs,  permanent  set  max  one-sixteenth  of  an  inch,  B. 


Under  100,000 


Timber. 

Sap  wood  not  allowed  in  more  than  10  per  cent  of  the  pieces  of  one  kind, 
and  no  piece  will  be  accepted  showing  sap  covering  more  than  0.25  X  the 
width  of  the  piece  on  any  face  at  any  point,  or  more  than  half  the  thickness 
of  any  plank  at  its  edge,  at  any  point,  Oo. 


TRUSS   SPECIFICATIONS.  755 

G,  gen'l;    Oo,  Osb'n;  P,  Pa;  R,  R'd'g;  Y,  N  Y  C;  Aa,  Cc,  Oo,  HVay. 

III.  LOADS. 

1.  Vertical  Loads. 

(Dead  and   Live  Loads  and  Impact.) 
Dead  Loads  in  Steam  Railroad  Bridges. 
Dead  load  =  weight  of  metal  -f  n  lbs  per  lineal  ft  of  track,  C,  D,  E,  R,  Y. 
n  =  400,  C,  D,  E;  n  =  500,    R;  n  =  620,  Y. 

Timber  taken  at  4.5  lbs  per  ft,  B.M.,  G.     Ballast,  110  lbs  per  cu  ft,  C. 
Rails,  splices,  and  joints  taken  at  100  lbs  per  lineal  ft  of  track,  A,  B,  C» 
Rails,  splices,  guard  rails,  etc.,  at  160  lbs  per  lineal  foot  of  track,  P. 
Two-thirds  of  dead  load  assumed  to  be  carried  by  loaded  chord,  Y;  in 
spans  less  than  300  ft,  B;  in  longer  spans  calculate  distribution,  B. 

Dead  Loads  in  Highway  and  Electric  Railroad  Bridges. 

Iron,  3.33  lbs  per  lineal  ft  of  bar  1  sq  inch  area,  Oo. 

Steel,  3.40  "     "        "      "    "    "     1  sq  inch  area,  Oo. 

Timber  per  ft  board  measure,  4,  Aa;  creosoted,  5,  Oo;  oak,  4.5,  Cc,  Oo| 
other  hard  woods,  4.5,  Cc;  yellow  pine,  4,  Oo;  spruce  and  white  pine,  3.5, 
Ce;  white  pine  and  cedar,  3,  Oo. 

Concrete,  etc.,  lbs  per  cu  ft,  130,  Aa;  stone  concrete,  125,  Oo;  cinder 
concrete,  100,  Oo.     Stone,  150,  Oo,  granite,  160,  Aa. 

Brick,  150,  Aa;  125,  Oo;  sand,  100,  Oo.     Asphalt,  130,  Aa;  90,  Oo. 

Rails,  fastenings,  splices  and  guard  timbers,  100  lbs  per  lin  ft  of  track,  Aa. 

Live  Loads  for  Steam  Railroad  Bridges. 
THEODORE  COOPER'S  STANDARD  LOADING.* 


,       d   d   d   d        ...*       ».      d    d   d  d        *    *      *    *        TT 

ja  qoqo  0  0  9  9  4  OQOQA^ILW' 


Fig.  1. 

TWO    CONSOLIDATION     LOCOMOTIVES,    WITH    THEIR    TENDERS    AND     TRAINS. 


Load  in 

lbs  on  one 

pair 

of  wheels 

Train 

for  each  track. 

load, 

Truck 

lbs  per 

(bogie) 

Driver 

Tender 

lin  ft. 

Class 

b 

d 

t 

U 

E  27 

13,500 

27,000 

17,550 

2,700 

E  30 

15,000 

30,000 

19,500 

3,000 

E  35 

17,500 

35,000 

22,750 

3,500 

E  40 

20,000 

40,000 

26,000 

4,000 

E  50 

25,000 

50,000 

32,500 

5,000 

A,  adopts  Cooper's  loading.     See  C,  above. 

B,  Cooper's  Class  E  50,  unless  otherwise  specified. 

D,  E,  P,  R,  Loads  and  spaces  differing  slightly  from  Cooper's. 
Y,  Cooper's  Class  E  40. 

*  In  Mr.  Coooer's  system  of  standard  loading,  the  No.  (27  to  50)  follow- 
ing the  letter  E  in  the  class  designation  gives  the  load,  d,  on  one  pair  of 

40 
drivers,  in  thousands  of  pounds.     In  each  class,  d^2b=  —  t   =   10  IT. 

Since  these  ratios  are  constant  for  all  classes,  the  stresses  due  to  any  class  are 
proportional  to  the  number  of  the  class.  The  cost  and  weight  of  metal,  in 
bridges  of  all  kinds,  built  under  C  specifications,  will  be,  in  each  class,  about 
10  per  cent  greater  than  in  the  class  next  lighter. 


756  TRUSSES. 

A,  Aa,  Am  B  Co;    B,  B   &  O;    C,  Cc,  Cooper;   D,  D  L   &  W;   E,  Erie; 

ALTERNATIVE    LOADINGS. 

Use  Fig.  1  or  the  alternative,  whichever  gives  the  greater  stresses. 


WW  WW 

^  Q  Q3 

Ft,  1^ 9 >}< — 7 — t — 7^^ 

Figs.  2  and  3. 


Load  on  one  pair  of  wheels. 
(d    =6ft;W  =  W  =  50,000  lbs,  Above  E  40,  60,000  lbs.  C. 
Fig.  2  Ad    =7ft;W  =  W  =  65,000  lbs,  D. 

U    =  7  ft;  W  =  W  =  60,000  lbs;  L  =  4,500,  Y. 
FiD-  q  /  ^1  =  9  f^ :  W  =  W  =  66,000  lbs        )     w 
■^ig-  ^-  (ds  ==  da  =  7  ft;  w  =  w  =  30,000  lbs/     ^• 

Add  30  per  cent  in  figuring  floor  beams,  stringers,  hangers,  suspenders, 
and  other  floor  connections.  Add  0  to  30  per  cent  for  spans  from  100  ft 
down  to  25  ft,  D. 

ON    CURVES. 

Distribution  of  live  load  between  the  two  trusses. 

W  =  P  — -  .- — ;  where  W  =  proportion  of  live  load  borne  by  the  outer 

truss;  P  =  live  load  at  panel  considered;  m    =  middle  ordinate  of  entire 
curve  on  span;  b  =  dist  betw  cens  of  trusses.    Make  both  trusses  alike,  B,Y. 

SPECIAL    LOADINGS. 

For  rivets  connecting  upper  flange  angles  with  web  in  deck  girders  carrying 
the  floor  directly  on  the  top  flanges,  and  in  deck  spans  with  wooden  floor 
beams,  when  distance  between  trusses  exceeds  6  ft,  50,000  lbs  on  one  pair  of 
drivers,  distributed  equally  oyer  three  ties  or  floor  beams,  P. 

For  floors,  the  load  on  a  single  pair  of  engine  wheels  distributed  over  4 
ties,  B;  over  3  ties,  C.  For  trough  floors,  60,000  lbs  on  one  pair  of  wheels, 
distributed  over  two  troughs,  Y. 

THREE-TRUSS    BRIDGES. 

In  double-track  deck  spans,  all  three  trusses  of  equal  strength,  C. 
In  plate  girder  bridges  of  more  than  one  track,  center  girder  figured  for 
0.75  X    the  live  load,  E. 

FUTURE    INCREASE    OF    LIVE    LOADING. 

Only  70  per  cent  (50  per  cent,  R)  of  the  dead  load  shall  be  considered 
effective  in  counteracting  live  load  stress.  A,  R.     Use  1.5  X  live  load,  E. 


"  That  the  heavier  of  these  engines  (see  *C,'  imder  'Standard  Loading,' 
above)  is  close  to  the  possible  maximum,  considering  the  limitations  of  the 
permissible  cross-section  of  existing  railroads  and  the  mechanical  details  of 
design  and  proportions,  is  not  improbable.  That  the  economical  tendency 
toward  heavier  and  heavier  engines  will  in  the  near  future  reach  the  heavier 
class  E  50  upon  the  most  important  roads  is  to  be  expected.  The  cars  will 
also  follow  the  same  tendency  for  many  kinds  of  traffic,  as  experience  justi- 
fies the  advance.  There  are  now  in  use  self-dumping  coal  cars  of  a  nominal 
capacity  of  100,000  pounds,  which  have,  on  four  axles,  a  total  load  of  146,000 
pounds  (10  per  cent  increase  over  nominal  capacity)  on  a  wheel  base,  for  two 
adjacent  cars,  of  17  ft,  2  ins.  These  cars  on  all  ordinary  bridges  produce 
strains  equivalent  to  those  of  E  33." — Theodore  Cooper. 

Members  subject  to  reversal  of  stress  must  be  so  designed  that  a  live  load 
n  per  cent  greater  than  that  specified  shall  not  increase  their  unit  stresses 
more  than  n  per  cent,     n  =  25,  C;  50,  B;  100,  P. 


TRUSS  SPECIFICATIONS.  757 

G,gen'l;   Oo,  Osb'n;  P,  Pa;  R,  RM'g;  T,  NYC;  Aa,  Cc,   Oo,  H'way, 

Live  Loads  for  Highway  and  Electric  Railroad  Bridges. 

Aa,  For  the  Floor  and  its  For  the  Trusses, 

(Am.  Bridge  Co.)  Supports. 

and  Uni-       Per  lin  ft  of  Per  sq  ft  of 

Cc,  Concentrated.       form      single  track,    remaining  floor. 

(Theo.  Cooper.)  (c).  (Proportionally  for  inter- 

Wagon       Car  mediate  spans.) 

(a),  (b) 

on         Per      Spans    Spans      Spans     Spans 
Class.*  each      sq  ft,     up  to     200  ft       up  to      200  ft 

track,       lbs      100  ft       and        100ft        and 
tons        tons  over  over 

lbs         lbs  lbs  lbs 

A 24  ..  100      1,800     1,200        100  80 

B   12     or    24  100       1,800     1,200  80  60 

C    12     or    18         100       1,200     1,000  80  60 

Up  to 
75  ft 

D 6.  ..  80         ..  ..  80  55 

El 24  ..        1,800     1,200 

E2 18  ..        1,200     1,000 

(a)  On  two  axles,  10  ft  cens  (and,  Aa,  5  ft  gage) ;  in  classes  A,  B,  and  C, 
assumed  to  occupy  a  width  of  12  ft  in  single  line  (or,  Cc,  22  ft  in  double 
line)  on  any  part  of  the  roadway. 

(b)  On  two  axles,  10  ft  centers. 

(c)  In  classes  A,  B,  and  C,  on  remainder  of  floor,  including  footwalks.  In 
class  D,  on  total  floor  surface. 

Oo.  Osborn  Engineering  Co.  Highway.  May  specify  any  combination 
of  the  following  loadings,  according  to  character  of  bridge  and  of  load. 

Uniform  loads,  lbs  per  sq  ft.  For  spans  up  to  150  ft,  100  on  roadway  and 
80  on  sidewalks,  or  80  on  both.     For  spans  over  150  ft,  80  or  60  on  both. 

A  steam  road  roller;  axles  11  ft  apart,  forward  roll  4  ft  face,  two  rear  rolls 
5  ft  cens  and  each  20  ins  face,  15,000  or  9,000  lbs  on  forward  roll  and  10,000 
Oi  6,000  lbs  on  each  rear  roll ; 

A  horse  roller,  12,000  lbs  on  roll,  5  ft  face; 

A.  wagon  load,  10,000  lbs  on  two  axles,  8  ft  apart,  5  ft  gage; 

Two  electric  cars  on  each  track;  Fig.  a. 

A  train  of  electric  cars  on  each  track;  Fig,  b, 

A  train  of  coal  cars  of  60,000  lbs  capacity;  Fig,  c. 

Fig.  a 

lhs.\  30,000 I       I  SOjOOO  \ 

_Q Q_ 


ft.  H-T  >l<      — 20 »K-7^-^ 


Fiq,  1) 

lh8.\ 50,000  or  80,000 |  ^^  |  >    50, OOO  or  80, OOP | 

.—Q Q _Q Q Q Q Q Q_ 

ft.      ^6  >i<       -le       -M  e  ^     12-    >l<  6->t* le ^k-e->ki« 


^^^'.^  92,000  I 


n    n 9    n        no- Q    O 

It.   Vg  >[<  10 ^5-^8—^5-^ lO- ^S-^rS 

♦Class  A,  city  bridges.  Class  D,  ordinary  country  highway. 

"      B,  suburban  or  interurban.         "     El,  heavy  electric  railway  only. 
"      C,  heavy  country  highway.         "    E2,  light  electric  railway  onlv. 


758  TRUSSES. 

A,  Aa,  Am  B  Co;   B,  B   &  O;    C,  Cc,  Cooper;   D,  D  L  &  W;   E,  Erie; 

FUTURE    INCREASE    OF    LIVE    LOADING.       HIGHWAY. 

In  electric  railroad  bridges,  Class  E,  only  70  per  cent  of  dead  load  stress  to 
be  considered  as  effective  in  counteracting  the  live  load  stress,  Aa.  For 
bridges  carrying  electric  or  motor  cars,  counters  so  proportioned  that  a 
future  increase  of  25  per  cent  in  the  specified  live  load  shall  not  increase  the 
unit  stress  more  than  25  per  cent,  Cc. 

Impact. 

I  =  S  .  ;  where  I  =  impact  stress  to  be  added  to  the  live  load  stress ; 

S  =  calculated  max  live  load  stress;  1  =  length  in  feet  of  loaded  distance 
which  produces  the  maximum  stress  in  the  member,  A. 

1=  S  (o.l  +  Y-y^Qol)'  ^^-  ^'  Bouscaren,  C.  E. 

In  Highway  Bridges;  I  =  25  per  cent  of  live  load  stresses  Aa;  X  =-  L^  -i- 
(L  +  D),  where  L  and  D  =  live  and  dead  load  stresses,  Oo« 

2,  Horizontal  Forces. 

(Drag,   Centrifugal  and  Wind.) 
(a)  Longitudinal. 

_  Drag.  In  bridges  for  steam  and  electric  railroads,  provide  for  a  longitu- 
dinal force,  at  the  rails,  =  0.2  of  the  max  live  load.  In  double  track. (Y), 
provide  for  trains  moving  either  way. 

(b)  Transverse. 

(1)   Centrifugal  Force. 

F  =  centrifugal  force;  W  =  weight  of  train  on  bridge;  d  =  degree  of 
curvature  =  central  angle  subtended  by  a  chord  of  100  ft;  v  =  velocity  in 
miles  per  hour;  c  =  a  coefficient. 

A,  F  =  c  d  W.     For  d  up  to  5°,  c  =  0.03.     Deduct  from  c  0.001  for  each 
degree  over  5°.     Train  on  each  track. 

B,  R,  F  =  0.02  of  the  live  load  for  each  deg  of  curvature.    B,  up  to  5°.    De- 

duct 0.001  for  each  degree  over  5°. 

C,  F,  computed  for  v  =  60  —  3  d  on  steam  railroads,  =  40  on  electric  rail- 

roads; force  acting  5  ft  above  base  of  rail. 

D,  V  =  60. 

E,  F  =  force  due  to  that  uniform  load  which  would  produce  the  max  speci- 
fied live  load  bending  moment  on  span;  v  =  60. 

Y,  F  =   W  v2  d  -^  85,666.     Up  to  d  =  4°,  v  =  60.     For  d  over  4°,  v  =- 
60  —  2d.      Max  train  load  on  each  track. 

(2)  Wind. 

(a)    ON    RAILROAD    BRIDGES. 

Wind  pressure,  in  lbs  per  sq  ft,  =  w;  in  lbs  per  lin-ft  =  W. 
w  =  either  30  lbs  per  sq  ft  on  exposed  surface  of  trusses  and  floor  and  on 
that  of  a  train  of  10  ft  average  height,  beginning  30  ins  above  rail  base; 
or  50  lbs  per  sq  ft  on  exposed  surface  of  trusses  and  floor;  whichever 
gives  the  greater  stresses,  A,  P. 
In  truss  spans  over  200  ft  &  in  plate  girders,  w  =  30  lbs  per  sq  ft  of  exposed 
surf  of  1  girder  and  floor,  4-  W  on  train  for  lower  chord,  as  below,  B. 
W  =  L  +  U.     L  =  pressure  in  lbs  per  lin  ft  on  loaded  chord,  U  on  un- 
loaded chord.      W  includes  both  wind  on  bridge  and  wind  on  train. 
Wind  on  bridge.     L  =  U  =  150,  B,  C,*  D,  R;    =  200,  E. 
L  ==  200,  on  double  track  300,  acting  8  ft  above  rail  top ;  "I  ^ 
U  ==  150,  on  double  track  225,  acting  at  cen  of  chord,         J      * 
Wind  on  train.     L  =  300,  B,  D,  R,t  Y;    =  450,  C,t  =  400.  E. 

*In  spans  over  300  ft,  add  to  U  10  lbs  to  each  additional  30  ft,  C 

t Acting  7.5  ft  above  rail,  R. 

JActing  6  ft  above  rail  base.     Includes  lateral  vibrations  of  trains. 


TRUSS  SPECIFICATIONS.  759 

G,  gen'l;   Oo,  Osb'n;  P,  Pa;  R,R*d»g;  Y,  N  Y  C;  Aa,  Cc,  Oo,  H'way. 

A.  Wind  stress,  Sw,  in  any  truss  member,  C  (main  truss  member,  D ; 
chord  or  end  post,  R),  need  be  considered  only  (1)  when  Sw  exceeds  30  per 
cent,  C  (25  per  cent,  D,  R),  of  max  stress,  S,  due  to  dead  and  live  loads. 
Then  increase  section  to  bring  Sw  within  limit,  C,  D,  R.  (2)  When  Sw, 
alone  or  in  combination  with  temperature  stress,  can  balance  or  reverse  S,  C. 

Anchorage.  In  determining  the  requisite  anchorage  for  the  loaded  struc- 
ture, the  train  is  assumed  to  weigh  800  lbs  per  lineal  foot,  A,  B,  C,  P;  600 
lbs  per  lineal  foot,  R. 

(b)    ON    HIGHWAY    AND    ELECTRIC    RAILROAD    BRIDGES. 

Either  30  lbs  per  sq  ft  on  the  exposed  surface  of  all  trusses  and  floor,  + 
150  lbs  (180,  Oo)  per  lineal  foot  of  a  train  covering  the  span;  or  50  lbs  per 
sq  ft  on  the  exposed  surface  of  all  trusses  and  floor;  whichever  gives  the 
greater  stresses,  Aa,  Oo. 

On  each  chord,  150  lbs  per  lin  ft,  of  span,  due  to  bridge,  and  on  the  loaded 
chord  150  lbs  per  lin  ft  of  span  additional  due  to  train.  For  spans  exceeding 
300  ft,  add  10  lbs  on  each  chord  for  each  additional  30  ft,  Cc. 

Wind  stresses  (in  truss  members,  Cc;  in  chords  and  end  posts,  Oo)  to  be 
provided  for  only  when  the  wind  stress  exceeds  25  per  cent  of  the  max  dead 
and  live  load  stresses  (of  the  sum  of  all  other  stresses,  Oo),  or  when  the  wind 
stress  (alone  or  in  combination  with  temperature  stress,  Cc)  can  (neutralize 
or,  Cc)  reverse  the  stress  in  the  member,  Cc,  Oo. 

IV.  STRESSES  AND  DIMENSIONS. 

Effective  Span  and  Depth. 

^  In  pin  spans,  span  and  depth  are  measured  between  centers  of  pins.  In 
riveted  trusses  the  span  is  measured  between  centers  of  end  bearings  and 
the  depth  between  centers  of  gravity  of  chord  sections.  In  plate  girders  the 
span  is  measured  between  centers  of  end  bearings,  and  the  depth  between 
centers  of  gravity  of  flange  areas  or  over  backs  of  flange  angles,  whichever  is 
the  less.  In  floor  beams  the  span  is  measured  between  centers  of  trusses, 
and  in  stringers  between  centers  of  floor  beams,  G. 

Limiting  Unit  Stresses. 
Tension. 
Net  section.  The  net  section  of  any  tension  member  or  flange  is  deter- 
mined by  a  plane  cutting  the  member  square  across  at  any  point.  The  great- 
est number  of  rivet  holes  which  can  be  cut  by  the  plane,  or  come  within  an 
inch  of  the  plane,  is  deducted  from  the  gross  section,  B.  The  rupture  of  a 
riveted  tension  member  is  considered  equally  probable,  either  through  a 
transverse  line  of  rivet  holes,  or  through  a  diagonal  line  of  rivet  holes  where 
the  net  section  does  not  exceed  by  30  per  cent  the  net  section  along  the 
transverse  line,  C,  Cc.  • 

In  deducting  rivet  holes  for  net  section,  their  diameter  is  taken  at  one- 
eighth  of  an  inch  greater  than  that  of  the  cold  rivet,  G ;  for  countersunk 
rivets  (Oo),  one-fourth  of  an  inch  greater. 

Maximum  permissible  tensile  stresses,  in  lbs  per  sq  inch. 

Medium      Soft 
Steel        Steel 

A,  Aa,  Under  vertical  forces  only  or  horizontal  forces  only  17,000     15,000 

Under  vertical  and   horizontal   forces  combined         21,000     19,000 

B,  For  "Bridge"  (soft)  and  Rivet  Steel,  same  as  Medium  Steel  under  A. 

D.     For  soft  steel :  For  dead  load ;  live  load. 

Eye-bars 14,000  9,000 

Built  sections 12,500  8,500 

Counters    8,500 

For  dead  and  live  load. 
Hip   suspenders,  floor  beam  hangers,  members  sub- 
ject to  sudden  loading 7,500 

Tension  flanges  of  plate  girders  and  rolled  beams 9,000 

Bracing 12,000 


760  TRUSSES. 

A,  Aa,  Am  B  Co;   B,  B   &  O;    C,  Cc,  Cooper;   D,  D  L   &  W;   E,  Erie; 

Main  members  of  trusses,  flanges  and  webs  of  girders  and  floor  beams 
for  double  track,  floors  and  girder  flanges  with  ballast  floor,  add  10 
per  cent 
For  medium  steel,  add  10  per  cent. 
_  /         min    stress  \ 

E.      8,000  (l+„-^--t~)- 

Oo.     (Highway  Bridges.)     Medium  steel,  22,000 ;  soft  steel,  20,000 ;  wrought 

iron,  18,000. 
P.      M  =  max  calculated  stress  in  member  ^ 

m  =  min  "  "       "         " 

Let  r    =  S  ;  let  k  -  l^^.     Then  M  (1  +  k)  shall  not  exceed  15,000. 
M  1   +  r 

Long  hip  verticals  must  have  25  per  cent  excess  strength;  short  floor 
beam  hangers  50  per  cent  excess,  P. 

Y.  Soft  steel.     Chords  and  web  members  of  trusses,  and  flanges  of 
plate  girders,  floor  beams  and  stringers. 

Dead  load  and  drag 16,000 

Live  load  and  centrifugal  force 8,000 

MAXIMUM    STRESSES    IN    TIMBER,    LBS    PER    SQ    INCH. 

Trans-  Bear- 
verse          End  ing     Shear 
For  Highway  Bridges,  Oo.            load-        bear-        Short  across  along 
ing            ing        column  *  fibre     fibre 

White  oak   1,400         1,300          1,000  550         300 

Long  leaf  pine 1,600          1,300          1,000  350         200 

White  pine 1,100            900            700  200         150 

Hemlock     950            850            650  200         100 

Extreme  fibre  stress,  in  floor  beams,  max,  yellow  pine  and  white  oak, 
1,200  lbs  per  sq  inch;  white  pine  and  spruce,  1,000,  Aa,  Cc. 

Compression. 

p  =  permissible  working  stress  in  compression  member,  in  lbs  per  sq  inch. 

f   =  generally  the  permissible  stress  in  tension  member,  in  lbs  per  sq  inch. 

a  =  a  coeflScient, 

1    =  length  of  piece,  in  ins,  between  cens  of  connection, 

r  =  least  radius  of  gyration  of  cross-section  of  member,  ins. 


Live  load 

f  a 

8,000  18,000 

to  to 

8,500  24,000 


f  a 

.  „    fin  medium  steel 17,000  11,000 

^^*  t  In  soft  steel 15.000  13,500 

B.  In  soft  steel 17,000  11,000 

C.  See  below. 

Dead  load 
f  a 

r  12,000  18,000 

to  to 

(12,500  24,000 

E.     f  =  8.000  (l  +  "^^"  stress \     ^  ^  36,000  with  both  ends  fixed;  a  => 
\  max  stress/ 

24,000  with  one  end  fixed;  a  =  18,000  with  both  ends  hinged. 
Oo.   (Highway  Bridges.)     f  =  22,000  for  medium  steel,  20,000  for  soft  steel, 

18,000  for  wrought  iron ;  a  as  in  E,  above. 
P.    f  =  15,000;  a  =  13,500. 

R.   f  =  6,500   (l  +  H^^^^f^^J^");  if  max  =  8,000;  a  =  40,000  with  flat 
\  max  stress  / 

ends;  a  =  20,000  with  pin  ends. 
When  one  end  is  pinned,  p  =  mean  of  values  derived  as  above. 
For  angle  iron  struts,  see  below. 

♦Length  not  over  12  X  least  side.     fMin  stress  =  dead  —  live  load  stress. 


TRUSS   SPECIFICATIONS.  761 

G,  gen'l;  Oo,  Osb'n;  P,  Pa;  R,  R'd'g;   T,  N  YC;  Aa,  Cc,  Oo,  H'way. 

Y.  Soft  steel  in  chords  and  web  members : 

f  a 

For  dead  load  and  drag 16,000  18,000 

For  live  load  and  centrifugal  force 8,000  18,000 

C,  Cc.  p  =  M  — c-^. 
r 
For  medium  steel  in  stationary  structures : 

Dead  load  Live  load 

Mo  M  c 

Chord    segments,  stiffeners.  20,000  90  10,000  45 

For  highway  bridges 24,000  110  12,000  55 

End  and  other  posts {  .J^'^^Z     {to  IS     {to     t]^     {tolg 

Forhighwaybrid.es {,„~     {,„  fg     {,^1'^     {,„« 

Lateral  struts,  rigid  bracing  ) 

for  railroad  and  highway  >      13,000  60  8,666  40 

bridges \ 

For  soft  steel,  deduct  15  per  cent;  for  movable  structures,  deduct  25  per 
cent. 
^,  Angle  iron  struts. 

With  flat  ends,  p  =  9,000  —  30  - ;  with  pin  ends,  p  -  9,000  —  34  -t.    In 
r  r 

lateral  and  cross  struts,  add  30  per  cent. 
Ijength  of  compression  members,  max,  =  40  to  45  diameters,  or  100  to 
120.  In  highway  bridges,  120  to  140  r,  Aa ;  100  to  120  r,  Cc ;  125  to  150  r,  Oo, 
where  r  =  least  radius  of  gyration. 

Unsupported  width  (distance  between  rivets)  of  plates  subject  to  com- 
pression, max  =  45  X  thickness,  Oo;  30  X  thickness,  C,  Cc,  D;  in  cover 
plates  of  top  chords  and  end  posts,  40  X  thickness,  C,  Cc,  D  j  or,  if  a  greater 
width  is  used,  effective  section  shall  be  taken  as  40  X  thickness,  C,  Cc. 
Distance  between  supports  in  line  of  stress,  max  =  16  X  thickness,  Oo. 

Timber  columns,  whose  length  exceeds  12  X  their  least  sides,  in  highway 
bridges,  Oo. 

Q 

Max  unit  stress  =  ^2 — 

^+  1,000  d2 
where  C  =  1,000  lbs  per  sq  inch  for  white  oak  and  long  leaf  pine,  700  for 
white  pine,  650  for  hemlock ;  1  =  length  of  column,  between  supports,  ins ; 
d  =  least  side,  ins,  Oo. 

Alternating  Stresses. 

Total  sectional  area  of  member  to  be  made  =  sum  of  areas  required  for 
both  stresses,  A,  B. 

Area  sufficient  to  resist  either  stress  plus  0.8  (0.6,  R;  1.0,  T)  X  the  lesser 
stress,  C,  Cc,  D,  R,  T. 

Permissible  working  stress,  in  lbs  per  sq  inch : 

a  =  8,000  ( 1  +  ,    ""'''  ''T"  °^  l"''^".^^^).  E. 

V  2  X  max  stress  of  greater  kmd/ 

M  =  max  calculated  stress  of  greater  kind,  >. 

m  =  max  calculated  stress  of  lesser  kind,  ) 

Let  r  =  z--.     Let  k  =  ., .     Then  M  (1  +  k)  shall  not  exceed  i 

M  2  —  r  V 

15,000  lbs  per  sq  in,  J 

IN    BRIDGES    FOR    HIGHWAYS    AND    FOR    ELECTRIC    RAILROADS. 

In  Classes  A,  B,  C,  and  D,  members  proportioned  for  that  stress  which  re- 
quires the  larger  section.  In  Classes  E  1  and  E  2,  make  sectional  area  =  sum 
of  areas  required  for  the  two  stresses,  Aa.  Members  designed  to  resist  either 
stress  and  given  25  per  cent  excess  of  strength  in  their  joints  and  connec- 
tions, Oo^ 


762  TRUSSES. 

A,  Aa,  Am  B  Co;   B,  B   &  O;    C,  Cc,  Cooper;   D,  D  L   &  W;   E,  Erie; 

Shear  and  Bearing  Stresses. 

Shear  In  web  plates,  max,  lbs  per  sq  inch.  10,000,  B ;  4,000,  E ;  5,000, 
B;  13,000,  P;  in  medium  steel,  10,000,  A,  Aa;  in  soft  steel,  9,000,  A,  Aa; 
across  grain,  6,000,  D;  with  grain,  5,000  (net  section),  D;  dead  load,  10,- 
000,  Y;  live  load,  5,000  (gross  section),  Y. 

Shear  and  Bearing  on  Rivets,  Bolts  and  Pins,  Maximum,  in  lbs 
per  sq  inch. 

Shear  Bearing 

Medium  Soft  Medium  Soft 

A,   Aa,   B 12,000 11,000 24,000 22,000 

C    9,000 9,000 15.000 15,000 

Cc        10,000 10,000 18,000 18,000 

Oo     10,000 10,000 22,000 20,000 

D,  R     7,500 7,500 12,000 12,000 

Y,  Shear  =  0.75  S  ;  bearing  =  1.50  S.  S  =  permissible  unit  stress  in  tension. 

In  field  riveting,  increase  number  of  rivets  25  per  cent,  A,  A  a,  B,  Oo,  P; 
if  machine  driven,  10  per  cent.  A,  Aa,  P;  in  stringers  and  floor  beams,  one- 
third,  P.-    Take  0.66  to  0.80  X  stress  as  above,  C,  Cc,  D,  R,  Y. 

In  floor  connections,  use  0.8  X  stresses  as  above,  C,  Cc;  add  20  per  cent 
to  number  of  rivets,  Y. 

In  wind  and  sway  bracing,  use  1 .25  to  1 .5  X  stresses  as  above,  C,  Cc^  D,  R. 

Rivets  with  countersunk  heads  taken  at  0.75  X  value  of  rivets  with  full 
heads,  P. 

Bearing,  on  phosphor  bronze  disks,  5,000  lbs  per  sq  inch,  B, 

Bending  Stresses. 

Stress  in  extreme  fibres,  under  bending  moments,  max,  lbs  per  sq 
inch. 

In  pins  and  bolts,  25,000,  B;  18,000,  C;  20,000,  Cc;  15,000,  D,R;  16,000, 
Y;  in  pins,  closely  packed,  25,000,  Oo;  in  medium  steel,  25,000;  in  soft 
steel,  22,000,  A,  Aa,  P.  Centers  of  bearings  of  strained  members  taken  as 
points  of  application  of  the  stresses.  A,  Aa,  R.  Applied  forces  considered 
as  uniformly  distributed  over  the  middle  half  of  the  bearing  of  each  member, 
C,  Cc.     Bending  calculated  t'cm  distances  between  centers  of  bearing,  Oo. 

In  rolled  beams  and  channels,  14,000,  P. 

In  wooden  floor  beams,  1,000,  A,  B,  C,  P. 

Compound  Stresses. 
Compound  (axial  and  bending),  maximum,  lbs  per  sq  inch. 
In  end  posts  of  through  spans,  dead  +  live  -f  wind  +  bending,  max  = 
15,000,  R. 

Proportion  the  member  to  resist  sum  of  direct  stress  plus  0.75  bending 

stress.  A,  Aa,  B,  P,  R.     Max  =  ' — ^ ,  where  I  =  length,  ins;  r  == 

^  '^  40,000  r2 
least  radius  of  gyration,  ins. 

If  pins  are  out  of  neutral  axis  of  section,  max  must  include  the  additional 
stress  due  to  the  eccentricity,  R. 

Bending  moment  at  panel  points  assumed  equal  and  opposite  to  that  at 
the  center.  A,  Aa.  If  fibre  stress  due  to  weight  of  member  alone  exceeds  10 
per  cent  of  the  allowed  unit  stress  on  such  member,  the  excess  must  be  con- 
sidered in  proportioning  the  areas,  C,  Cc,  R. 

Minimum  Dimensions. 
Minimum  thickness  of  plates,  in  railroad  bridges,  three-eighths  of  an 
inch  for  main  members,  five-sixteenths  of  an  inch  for  laterals;  in  highway 
and  electric  railroad  bridges,  five-sixteenths  to  one-fourth  of  an  inch.  Min 
diam  of  rod.  three-fourths  of  an  inch,  Oo.  Rods  and  bars,  min  .section,  1  sq 
inch,  D,  R;  counters  1.5  sq  ins,  D,  P.  Posts,  in  pin  spans,  min  width  10 
ins,  A.  In  posts  of  through  spans,  channels  min  10  ins,  B.  Angle,  min,  3.5 
X  3  X  five-sixteenths,  B. 


TRUSS  SPECIFICATIONS.  763 

G,  gen'l;  Oo,  Osb'n;  P,  Pa;  R,  R'd'g;  T,  N  Y  C;  Aa,  Cc,  Oo,  H'way. 

V.  PROTECTION. 

At  Shop.  After  removing  loose  scale  and  rust ;  1  coat  pure  boiled  linseed 
oil.  A,  Aa,  B,  D,  E,  P,  R;  raw  linseed  oil,  C,  Cc;  with  10  per  cent  in 
weight  of  lampblack,  D ;  standard  red  lead  paint,*  Y, 

Inaccessible  Parts.  2  coats  iron  ore  paint  in  pure  linseed  oil.  A,  Aa,  B, 
C,  Cc,  E,  R;  standard  red  lead  paint,*  Y;  1  coat,  D ;  1  heavy  coat  red  lead 
in  raw  linseed  oil,  P;  2  coats,  18  lbs  red  lead  in  1  gal  boiled  linseed  oil,  Oo. 

Finished  Surfaces.     Coated  with  white  lead  and  tallow.     General. 

Surfaces  in  Contact.  Painted  before  joining.  A,  Aa,  B,  C,  Cc,  R,  Y; 
with  2  heavy  coats  red  lead  in  raw  linseed  oil  on  each  surface,  Y. 

After  Erection.  2  additional  coats  of  paint  in  pure  linseed  oil,  A,  Aa, 
B,  C,  Cc;  2  coats  of  paint,  of  different  colors,  R;  2  heavy  coats  asphaltum 
varnish,  Y. 

At  least  48  hours  allowed  for  drying  of  each  coat,  T, 

Columns,  etc.,  for  5  ft  above  surface  of  street,  etc.,  2  heavy  coats  as- 
phaltum varnish;  under  sides  of  bridges,  rest  of  columns,  etc.,  2  heavy 
coats  standard  white  paint ;  *  ballast  side  of  trough  floors,  1  part  by  weight 
refined  Trinidad  asphalt  and  3  parts  straight  run  coal  tar  pitch  at  300°  F,  Y. 

Wherever  there  is  a  tendency  for  water  to  collect,  the  spaces  must  be  filled 
with  a  waterproof  material,  C,  Cc. 

First  coat  paint  ui  graphite  or  carbon  primer,  Oo. 

In  highway  bridges,  upper  surfaces  of  metal  floor  plates  thoroughly  coated 
with  asphalt,  Oo. 

VI.  ERECTION. 

The  Contractor  is  usually  required 

(1)  to  unload  materials  after  delivery,  to  furnish  falseworks  and  appli- 
ances, to  remove  the  old  bridge,  to  alter  existing  bridge  seats ; 

(2)  to  drill  and  set  anchor  bolts,  to  erect  and  adjust  the  superstructure, 
and  sometimes  to  furnish  and  place  the  wooden  floor  beams ; 

(3)  to  remove  falseworks  and  appliances ; 

(4)  to  keep  the  road  open  for  traffic  and  to  avoid  interference  with  any 
other  thoroughfare  by  land  or  water  and  interference  with  other  contractors ; 
to  furnish  and  pay  watchmen;  to  keep  material  clean  and  in  good  order; and 
to  assume  all  risks  of  damage  to  persons  or  property  by  reason  of  storms, 
floods  or  other  casualties ; 

(5)  to  furnish  pilot  nuts  for  the  protection  of  the  ends  of  pins  in  driving. 


(3)  DIGEST  OF  SPECIFICATION   FOR  COMBINATION    RAIL- 
ROAD BRIDGES. t. 

By  Baltimore  and  Ohio  Rcilroad  Co.,  1901. 

I.  GENERAL,  DESIGN. 

Type,  Howe. 

Rods  of  steel,  with  upset  ends ;  standard  nut  and  lock  nut  on  each  end. 
Cast  iron  joint  boxes  and  packing  spools. 

Steel  gib  plates  from  1.25  ins  thick  for  1.25  inch  rod,  to  1.75  ins  thick  for 
2.5  inch  rod. 

Splices  in  lower  chord  generally  of  steel  construction. 

II.  MATERIAL. 

Lumber,  Georgia  yellow  pine,  white  oak  or  white  pine. 

Rolled  steel.     Open  hearth.     Ultimate  strength  60,000  lbs  per  sq  Inch, 

*  Standard  red  lead  paint.     5  gals  contain  100  lbs  pure  red  lead,  4  gals  pure 
raw  linseed  oil,  one-half  pint  Japan,  free  from  benzine,  Y. 

Standard  white  paint.     5  gals  contain  42  lbs  pure  white  lead  in  oil,  21  lbs 
white  zinc  in  oil.  3  gals  pure  raw  linseed  oil,  Y. 

At  least  48  hours  between  coats,  and  between  final  shop  coat  and  load- 
ing, Y. 

t  To  be  used  only  for  temporary  purposes. 


764  TRUSSES. 

permissible  variation,  5,000  lbs;  elastic  limit,  30,000  lbs;  elongation  25  per 
cent  in  8  ins;  to  bend  180"  flat  upon  itself. 

III.  LOADS. 

Dead  Load. 
Timber  taken  at  4.5  lbs  per  foot  board  measure.     Track  100  lbs  per  lin  ft. 

Live  Load. 
Max  intended  load  +  25  per  cent,  to  provide  for  increase  and  impact. 

IV.  STRESSES  AND  DIMENSIONS. 

TJmiting  Unit  Stresses. 

Timber,  lbs  per  sq  inch,  max     Yellow  pine        White  pine        White  oak 

Bending  or  direct  tension 1,200  800  1,000 

Columns  under  17  diams  in  length    900  600  750 

Columns  over  17  diams  in  length . .  1,200-18  n  800-12  n  1,000-15  n 

where  n  =  length  -5-  least  thickness; 
n  max  =  40. 

Shearing,    along  grain 150  100  200 

Bearing,  in  direction  of  grain 1,500  1,000  1,250 

Bearing,  perpendicular  to  grain .  . .    350  200  ^  500 

In  columns  made  up  of    several  sticks  placed  side  by  side,  and   bolted 
together  at  intervals,  each  stick  treated  as  an  independent  column. 

Steel  rods,  max  unit  stress  =  12,000  lbs  per  sq  inch. 

Floor  beams  designed  to  carry  the  dead  load  and  the  heaviest  engines  in 
service  without  impact  allowance.     Reinforce  for  future  increase  of  loads. 

For  loadings  in  excess  of  that  used  in  designing,  reduce  speed  from  60  to 
15  miles  per  hour,  as  loads  increase  to  limit  of  25  per  cent  increase  of  load. 

V.  PROTECTION. 

Steel  rods,  gibs,  etc.,  1  coat  of  paint  in  shop;  2  after  erection. 
Wood,  at  joints  and  at  points  of  contact,  to  be  painted. 
Bolt  and  rod  holes  to  be  saturated  with  paint. 


(3)  DIGEST  OF  SPECIFICATION  FOR  ROOF  TRUSSES, 
STEEL.  FRAMEWORK  AND  BUILDINGS. 

By  Baltimore  and  Ohio  Railroad  Co.,  1901. 
I.  GENERAL   DESIGN. 

Made  principally  of  shapes.  No  adjustable  members,  except  in  lateral 
bracing.  Lateral  bracing  proportioned  for  a  full  wind  pressure  of  30  lbs  per 
sq  ft  of  exposed  surface,  acting  in  any  direction.  Tension  members  in  brac- 
ing must  in  all  cases  pull  directly  against  a  stiff  strut.  If  building  is  enclosed 
and  the  work  is  exposed  to  the  action  of  gases,  no  open  spaces  less  than  1 
inch  wide  left  between  members,  or  open  pockets  inaccessible  for  painting. 

II.  MATERIAL. 

Min  thickness,  0.25  inch.  When  subject  to  the  action  of  gases,  five-six- 
teenths inch  if  building  is  open;  0.375  inch  if  enclosed. 

m.  LOADS. 

Snow,  20  lbs  per  sq  ft  of  horizontal  projection  of  roof  surface.  Wind,  30 
lbs  per  sq  ft,  horizontal,  in  any  direction.     Min  total,  40  lbs  per  sq  ft. 

Covering.  For  roofs,  and  for  sides  unless  otherwise  ordered,  corrugated 
sheets,  No.  22  gage,  26  ins  wide;  corrugations,  2.5  ins;  3  ins  for  slope  of  1 
on  2 ;  6  ins  for  less  slope.     Purlins  hot  more  than  4  ft  apart  between  centers. 

IV.  STRESSES. 

Columns  sustaining  roof  are  considered  as  hinged  at  base,  unless  so  an- 
chored as  to  be  absolutely  fixed. 

Unit  stresses,  if  subject  to  no  moving  load  other  than  wind,  see  B,  in 
Digest  of  Specifications  for  Steel  Bridges,  and  Digest  (2)  of  B  &  O  R  R  Speci- 
fication for  Combination  Bridges.  Stresses  given  in  the  latter  to  be  in- 
creased 25  per  cent. 

V.  PROTECTION. 

Three  coats  of  paint.  If  exposed  to  gases,  use  bridge  paint  (see  B  in 
Steel  Bridge  Specifications) ;  if  not,  use  standard  building  paints. 


SUSPENSION   BRIDGES. 


765 


SUSPENSION  BEIDGES. 


Art.  1.    Table  of  data  required   for  calculating:  the  main 
cbains  or  cables  of  suspension  bridg-es.  Original. 


Tension  on  all 
the  Main 

Tension  at  the 

Deflection 

in  parts 
of  the 
Chord. 

Deflection 
in  Deci- 
mals of 

the  Chord. 

Length  of 
Main  Chains 
between  Sus- 
pension Piers, 
in  parts  of  the 
Chord. 

Chains  at 
either  Suspen- 
sion Pier,  in 
parts  of  the 
entire  Sus- 
pended Wt. 
of  the  Bridge, 
and  its  Load. 

Center  of  all 

the  Main 
Chains ;  in 
parts  of  the 
entire  Sus- 
pended Wt. 
of  the  Bridge, 
and  its  Load. 

Angle  of 
Direc- 
tion  of 

theChains 
at  the 
Piers. 

Natural 
Sineofthe 

Angle  of 
Direction 

of  the 
Chains,  at 
the  Piers. 

Natural 
Cosine  of 
the  Angle 

of  Direc- 
tion of  the 
Chains  at 
the  Piers. 

Deg.  Min. 

1-40 

.025 

1.002 

5.03 

5.00 

5    43 

.0995 

.9950 

1-35 

.0286 

1.002 

4.40 

4.37 

6    31 

.1135 

.9935 

1-30 

.0333 

1.003 

3.78 

3.75 

7    36 

.1322 

.991t 

1-25 

.04 

1.004 

3.16 

3.12 

9      6 

.1580 

.9874 

1-20 

.05 

1.006 

2.55 

2.51 

11    19 

.1961 

.9806 

1-19 

.0526 

1.007 

2.43 

2.38 

11    53 

.2060 

.9786 

1-18 

.0555 

1.008 

2.30 

2.25 

12    32 

.2169 

.9762 

1-17 

.0588 

1.009 

2.18 

2.12 

13    14 

.2290 

.9734 

1-16 

.0625 

1.010 

2.06 

2.00 

14      2 

.2425 

.9701 

1-15 

.0667 

1.012 

'.94 

1.87 

14    55 

.2573 

.9663 

1-14 

.0714 

1.013 

..82 

1.74 

15    57 

.2747 

.9615 

1-13 

.0769 

1.016 

1.70 

1.62 

17      6 

.2941 

.9558 

1-12 

.0833 

1.018 

1.57 

1.49 

18    33 

.3180 

.9480 

l-ll 

.0919 

1.022 

'        1.46 

1.37 

19    59 

.3418 

.9398 

1-10 

.1 

1.026 

1.35 

1.25 

21     48 

.3714 

.9285 

1-9 

.1111 

1.033 

1.23 

1.12 

23    58 

.4062 

.9138 

H 

.125 

1.041 

1.12 

1.00 

26    33 

.4471 

.8945 

1-7 

.1429 

1.053 

1.01 

.881 

29    45 

.4961 

.8726 

3-20 

.15 

1.058 

.972 

.833 

30    58 

.5145 

.8574 

% 

.1667 

1.070 

.901 

.750 

33    41 

.5547 

.8320 

1-5 

.2 

1.098 

.800 

.625 

38    40 

.6247 

.7808 

•225 

1.122 

.747 

.555 

42      0 

.6690 

.7433 

H 

.25 

1.149 

.707 

.500 

45    00 

.7071 

.7071 

.3 

.3 

1.205 

.651 

.417 

50    12 

.7682 

.6401 

H 

,  .3333 

1.247 

.625 

.375 

53      8 

.8000 

.600© 

.4 

.4 

1.332 

.589 

.312 

58      2 

.8483 

.5294 

9-20 

.45 

1.403 

.572 

.278 

60    57 

.8742 

.4855 

14 

.5 

1.480 

.559 

.250 

63    26 

.8944 

.4472 

These  calculations  are  based  on  the  assumption  that  the  curve  formed  by  the  main  chains  is  a 
parabola  ;  which  is  not  strictly  correct.  In  a  finished  bridge,  the  curve  is  between  a  parabola  and  a 
•atenary  ;  and  is  not  susceptible  of  a  rigorous  determination.  It  may  Save  SOUie  trOU* 
ble  in  making"  the  drawings  of  a  suspension  bridge,  to  remember  that  when  the 
«leflection  does  not  exceed  about  yW  of  the  span,  a  segment  of  a  circle  may  be  used  instead  of  the 
true  curve ;  inasmuch  as  the  two  then  coincide  very  closely  ;  and  the  more  so  as  the  deflection  be- 
comes  les»  than  -j^g-.  The  dimensions  taken  from  the  drawing  of  a  segment  will  answer  all  the  pur- 
poses of  estimating  the  quantities  of  materials. 

The  deflection  usiially  adopted  by  engineers  for  great  spans  is 

about  yL  to  yL  the  span.  As  much  as  J-j  is  generally  confined  to  small  spans.  The  bridge  wiU 
be  stronger,  or  will  require  less  area  of  cable,  if  the  deflection  is  greater  ;  but  it  then  undulates  more 
readily  ;  and  as  undulations  tend  to  destroy  the  bridge  by  loosening  the  joints,  and  by  increasing  the 
momentum,  they  must  be  specially  guarded  against  as  much  as  possible.  The  usnal  mode  of  doing 
this  is  by  trussing  the  hand-railing;  which  with  this  view  may  be  made  higher,  and  of  stouter  tim- 
bers than  would  otherwise  be  necessary.  In  large  spans,  indeed,  it  may  be  supplanted  by  regular 
bridge- trusses,  sufficiently  high  to  be  braced  together  overhead,  as  in  the  Niagara  Railroad  bridge, 
where  the  trusses  are  18  ft  high  ;  supporting  a  single-track  railroad  on  top ;  and  a  common  roadwaj 
of  19  ft  clear  width,  below.* 

*  The  writer  believes  himself  to  have  been  the  first  person  to  suggest  the  addition  of  very  deep 
trusses  braced  together  transversely,  for  large  suspension  bridges.  Early  in  1851,  he  designed  such 
a  bridge,  with  four  spans  of  1000  ft  each  ;  and  two  of  500;  with  wire  cables ;  and  trusses  20  ft  high. 
It  was  intended  for  crossing  the  Delaware  at  Market  Street.  Philada.  It  was  publicly  exhibited  for 
several  months  at  the  Franklin  Institute,  and  at  the  Merchants'  Exchange;  and  was  finally  stolen 
from  the  hall  of  the  latter.  Mr  Roebling's  Niagara  bridge,  of  800  ft  span,  with  trusses  18  ft  high,  wal 
not  commeneed  until  the  latter  part  of  1852 ;  or  about  18  months  after  mine  had  been  publicly  er 
bibited. 


766 


SUSPENSION   BRIDGES. 


Another  very  important  aid  is  found  in  deep  longitudinal  floor  timbers,  firmly  united  \<>iere  their 
ends  meet  each  other.  These  assist  by  distributing  among  several  suspender-rods,  and  by  that 
means  along  a  considerable  length  of  main  cable,  the  weight  of  heavy  passing  loads  ;  and  thus  pre- 
vent the  uudue  undulation  that  would  take  place  if  the  load  were  concentrated  upon  only  two  ovposite 
suspenders.  With  this  view,  the  wooden  stringers  under  the  rails  on  the  Niagara  bridge  are  made 
virtually  4  ft  deep.     The  same  principle  is  evidently  good  for  ordinary  trussed  bridges. 

Another  mode  of  relieving  the  main  cables  is  by  means  of  cable-stays ;  which  are  bars  of  iron,  or 
wire  ropes,  extending  like  cy.  Fig  1,  from  the  saddles  at  the  points  of  suspension  c,  d,  obliquely  down 
to  the  floor,  or  to  some  part  of  the  truss.  In  the  Niagara  bridge  are  (i4  such  stays,  of  wire  ropes  of 
1%  inch  diam;  the  longest  of  which  reach  more  than  quarter  way  across  the  span  from  each  tower. 
They  transfer  much  of  the  strain  of  the  wt  of  the  bridge  and  its  load  directly  to  the  saddles  »t  the  top 
of  the  towers;  thereby  relieving  every  part  of  the  main  cable,  and  diminishing  undulation.  They 
end  at  cand  d,  where  they  are  attached,  not  to  the  cables,  but  to  the  saddles.  They  of  course  do  not 
relieve  the  back  stays. 

The  g-reatest  dangler  arises  from  tlie  action  of  strong*  winds 
striking*  below  tlie  floor,  and  either  lifting  the  whole  platform,  and  letting 
it  fall  suddenly  ;  or  imparting  to  it  violent  wavelike  undulations.  The  bridge  of  1010  ft  span  across 
the  Ohio  at  Wheeling,  by  Charles  EUet,  Jr,  was  destroyed  in  this  manner.  It  is  said  to  have  undu- 
lated 20  ft  vertically  before  giving  way.  It  had  no  effective  guards  against  undulation  ;  for  although 
its  hand-railing  was  trussed,  it  was  too  low  and  slight  to  be  of  much  service  in  so  great  a  span. 
Many  other  bridges  have  been  either  destroyed  or  injured  in  the  same  way.  When  the  height  of  the 
roadway  above  the  water  admits  of  it,  the  precaution  may  be  adopted  of  tie-rods,  or  anchor  rods, 
under  the  floor  at  different  points  along  the  span,  and  carried  from  thenca,  inclining  downward,  to 
the  abutments,  to  which  they  should  be  very  strongly  confined.  In  the  Niagara  Railroad  bridge  56 
such  ties,  made  of  wire  ropes  1^  inch  diam,  extend  diagonally  from  the  bottom  of  the  bridge,  to  th» 
rocks  below.     They,  however,  detract  greatly  from  the  dignity  of  a  structure, 

Mr  Brunei,  in  so'rae  cases,  for  checking  undulations  from  violent  winds  striking  beneath  the  plat 
form,  used  also  inverted  or  up-curving  cables  under  the  floor.  Their  ends  were  strongly  confined  to 
the  abuts  several  ft  below  the  platform  j  and  the  cables  were  connected  at  intervals,  with  the  plat- 
form, so  as  to  hold  it  down. 

Art.  2.  The  angle  adg,  or  a  c  i,  Fig  1,  which  a  tang  dg  or  ci  to  the  curve  at 
either  point  of  suspension  c  or  d,  forma  with  the  hor  line  cd  or  chord,  is  called  tlie  angle  ©f 
direction  of  tbe  main  chains,  or  cables,  at  those  points.  Frequently  the  en<I>i 
cA,  and  dr,  of  the  chains,  called  the  bacllStayS,  are  carried  away  from  the  suspension  pier<J 
in  straight  lines ;  in  which  case  the  angles  Idr,  e  c  h,  formed  between  the  hor  line  e  I  and  the  chat » 
Itself,  become  the  angles  of  direction  of  the  backstays. 


Twice  the  deflection  a  b 


Sine  of  ang>le  of  direction  adg= ,__ 

■j/ (twice  the  detiection;2  -}-  (Half  the  chord)* 
Note  1.     The  direction  of  the  tang  d  g  or  c  i,  can  be  laid  down  on  a  drawing,  thus :  Continue  tke 
line  a  b,  making  it  twice  as  long  as  a  b  ;  then  lines  drawn  from  d  and  c  to  its  lower  end,  will  be  tangs 
to  the  parabolio  curve  at  the  points  of  suspension. 

Note  2.  If  the  chord  e  CT  be  not  hor,  as  sometimes  is  the  case,  the  angle 
must  be  measured  from  a  hor  line  drawn  for  the  purpose  at  each  point  of  suspension;  as  the  two 
angles  will  in  that  case  be  unequal,  the  piers  being  of  unequal  heights. 

Tension     on     all     the     main  Half  the  entire  suspended  weight  of  the  cle?- 

chains  or  cables,  tog*ether,  _ span  and  its  load 

at  either  one  of  the  piers,  ~" 
c  or  a,  Fig  1. 


Sine  of  angle  of  direction  a  d  g 


•\/ 1\^  o«„«\'7  I   /.>  r»„«\o        Half  the  entire  sus- 
or  =  y  (>^  SpanjM- (2  Defl)2  ^      pe„ded  weight  of 
2  Deflection  the  clear  span  and 

its  load. 


I^ension    on    all    the   main 
cliains  or  cables,  together, 
at   the   middle,   h,  of  the  = 
span,  Fig  1. 


or  = 


Half  the  entire  suspended        r.    •         «         i      * 

weight  of  the  clear  span  X  Cosine  of  angle  of 

audits  load  direction  adg 

Sine  of  angle  of  direction  adg 

Half  the  entire  suspended  weight  of  ^  Half  the 
tbe  clear  span  and  its  load  ^      span 

Twice  the  deflection 

The  diflFbetween  the  tensions  at  the  middle,  and  at  the  points  of  suspension,  is  so  trifling  with  the 
proportion  of  chord  and  deflection  commonly  adopted  in  practice,  viz,  from  about  ^^  to  ^^-,  that  it 
is  usually  neglected;  inasmuch  as  the  saving  in  the  weight  of  metal  would  be  fully  compensated  for 
by  the  increased  labor  of  manufacture  in  gradually  reducing  the  dimensions  of  the  chains  from  the 
points  of  suspension  toward  the  middle;  and  in  preparing  fittings  for  parts  of  many  different  sizes. 
The  reduction  has,  however,  been  made  in  some  large  bridges  with  wrought-iron  main  chains ;  buf 
i'A  non«  wi:h  wire  cables. 


SUSPElSrSION   BRIDGES. 


767 


Art.  2A.  As  it  is  sometimes  convenient  to  form  a  rough  idea  at  the  moment,  of 
the  size  of  cables  required  for  a  bridge,  we  suggest  the  following  rule  for  finding  approximately  tha 
area  in  sq  ins  of  solid  iron  in  the  wire  required  to  sustain,  with  a  safety  of  3,*  the  weight  of  the  bridge 
itself,  together  with  an  extraneous  load  of  1.205  tons  per  foot  run  of  span ;  which  corresponds  to  100 
tts  per  sq  ft  of  platform  of  27  ft  clear  available  width.  This  suffices  for  a  double  carriage-way.  and 
two  ff)Otways.  The  deflection  is  assumed  at  y^^  of  the  span ;  and  the  wire  to  have  an  ultimate 
strength  of  36  tons  per  solid  square  inch, 

For  spans  of  100  ft  or  more, 

RuLB.  Mult  the  span  in  feet,  by  the  square  root  of  the  span.  Divide  the  prod  by  100.  To  the 
quot  add  the  sq  rt  of  the  span.     Or,  as  a  formula, 

Area  of  solid  metal  of  all  span  X  sqrtof  span 


the  cables  ;  in  square  ins  ; 
for  spans  over  100  feet 


-\-  sqrt  of  span. 


For  spans  less  than  100  feet,  proportion  the  area  to  that  at  100  ft. 

If  a  defl  of  yq  is  adopted  instead  of  y^,  the  area  of  the  cabl«s  may  be  reduced  very  nearly  y  partr 

The  following"  table  is  drawn  up  from  this  rnle.    The  3d  col 

gives  the  united  areas  of  all  the  actual  wire  cables,  when  made  up,  including  voids.     (Original.) 


Span 

in 
Feet. 

Solid  Iron 

in  all   the 

Cables. 

Areas  of 
all  the 
Finished 
Cables. 

span 
Feet. 

Solid  Iron 
in  all  the 
Cables. 

Areas  of 
all  the 
Finished 
Cables. 

span 

Feet. 

Solid  Iron 
in  all  the 
Cables. 

Areas  of 
all  the 

Finished 
Cables. 

1000 
900 
800 
700 
600 
500 

Sq.  Ins. 
348 
800 
254 
212 
171 
134 

Sq.  Ins. 
446 
385 
326 
272 
219 
172 

400 

350 

300, 

250 

200 

175 

Sq.  Ins. 
100 
84 
69 
55 
42 
36.4 

Sq.  Ins. 

128 
108 

89 

71 

150 
125 
100 
75 
50 
25 

Sq.  Ins. 

30.6 
25.2 
20 
15 
10 
5 

Sq.  Ins. 
39.2 
32.3 
25.6 
19.2 
12.« 
6.4 

Having  the  areas  of  all  the  actual  cables,  we  can  readily  find  their  diam.     Thus,  suppose  with  a 

172 
span  of  500  ft,  we  intend  to  use  four  cables.    Then  the  area  of  each  of  them  will  be  -—  =  43  sq  ins< 

and  from  the  table  of  circles,  we  see  that  the  corresponding  diamjs  full  1%  ins. 

The  above  areas  are  supposed  to  allow  for  the  increased  wt  of  a  depth  of  truss,  and  other  additions 
necessary  to  secure  the  bridge  from  violent  winds,  and  from  undue  vibrations  from  passing  loads. 

When  these  considerations  are  neglected,  and  a  less  maximum  load  assumed,  the  following  descrip* 
tions  of  the  Wheeling  and  Freyburg  bridges  show  what  reductions  are  practicable.  Weight,  saffl* 
eisntly  providsd  for,  is  of  great  service  in  reducing  undulation. 

We  do  not  think  that,  diagonal  horizontal  bracing  should,  as  is  usual,  be  omitted  under  the  floor. 
It  may  readily  be  effected  by  iron  rods. 

All  the  cables  need  not  be  at  the  sides  of  the  bridge.  One  or  more  of  them  may  be  over  its  axis  5 
especially  in  a  wide  bridge.  One  wide  footpath  in  the  center  may  be  used,  instead  of  two  narrow 
ones  at  the  sides. 

The  platform  or  roadway  should  be  slightly  cambered,  or  curved  upward,  to  the  extent  say  of  about 

^i  Q-  of  the  span. 


»  The  writer  must  not  be  understood  to  advocate  a  safety  of  3  againsi  iiiO  lbs  per  sq  ft,  in  addition 
♦o  the  weight  of  the  bridge,  in  all  cases.  He  believes  that  limit  to  be  abou*  a  sufficient  one  for  a  pro- 
perly designed  wire  suspension  bridge  for  ordinary  travel ;  but  for  an  important  railroad  bridge,  he 
would  (according  to  position,  exposure,  &c)  adopt  a  safety  of  at  least  from  4  to  6  against  the  greatest 
possible  load,  added  to  the  wt  of  the  bridge.  A  train  of  cars  opposes  a  great  swrface  to  the  action  of 
side  winds :  and  trains  must  run  during  violent  storms,  as  well  as  during  calms  ;  but  a  large  opea 
Vidge  for  common  travel  is  not  likely  to  be  detsely  crowded  with  people  during  a  severe  storm. 


768 


SUSPENSION   BRIDGES. 


Art.  3.  Tension  on  the  back-stays,  c  /*  anrt  d  r,  Fig  1,  and 
strains  on  the  piers,  or  towers,  or  pillars.  If  the  angle  of  direction  cidg  and 
the  angle  Idr,  between  the  back-stays  and  the  horizontal,  are  equal  to  each  other,  the  tension  on  the 
back-stays  will  be  equal  to  that  on  the  main  cables  at  the  tops  of  the  piers ;  and  the  pressure  on  the 
piers  will  be  vertical ;  but  if  the  two  angles  are  unequal,  then  these  tensions  and  pressures  will  de- 
pend, to  a  very  important  extent,  upon  the  manner  in  which  the  chains  or  cables  are  fixed  to,  or  laid 
upon,  the  tops'  of  the  piers. 

Art.  4.     In  Figs 

2,  3  and  4,  the  piers 
dnm  are  supposed  to  be  im- 
movable ;  and  the  cables 
k  d  u,  passing  over  them, 
rest  immedia*^  ^  upon  hori- 
zontal rollers,  which  have  no 
other  motion  than  that  of  re- 
volving about  tfieir  horizon- 
tal axes  ;  the  frame  to  ivhich 
they  are  attached  being  bolt- 
ed to  the  top  of  the  pier.  On 
these  rollers  the  cables  slide, 
when  changes  of  loading 
or  of  temperature  produce 
changes  in  their  directions. 

In  this  case  the  ten- 
sion on  the  back- 
stays is  equal  to  that  on 
the  main  cable.  See  Funio- 
ular  Machine. 

To  find  the  direction  and 
amount  of  thepreSSUrO 

on  the  pier ;  from  d. 

Fig  2.  3  or  4,  lay  off  rf«  and 
dr,  each  equal,  by  scale,  to 
the  tension,  in  tons,  on  the 
mai.i  chain  at  d ;  and  from  » 
and  r  lay  itotf  to  t>.  In  other 
words,  draw  the  parallelo- 
gram dsvr,  and  its  diagonal 
rf  V.  Then  will  d  v  give  the 
direction  and  amount  of  the 
pressure  upon  the  pier. 

When,  as  in  Fig  2,  the 
angles  a  d  g  and  I  d  ti  are 
equal,  the  pressure  d  v  will 
be  vertical,  and  equal  to  the 
entire  weight  of  the  clear 
span  and  its  load. 

When,  as  in  Figs  3  and  4, 
the  angles  adg  and  Idu  are 
unequal,  the  pressure  d  v 
will  not  be  vertical,  but  will 
incline  from  d  toward  the 
smaller  angle. 

When,  as  in  Fig  Z,  adg  exceeds  Idu,  the  prassure  d  v  will  be  less  than  the  entire  weight  of  the 
dear  span  and  its  load. 

When,  as  in  Fig  A,  Idu  exceeds  adg,  the  pressure  d  v  will  be  greater  than  the  entire  weight  of  th< 
clear  span  and  its  load. 

If  we  suppose  symmetrical  piers,  d  n  to,  to  be  used  in  each  case,  tbe  base  m  n  of  that  in  Fig  2,  maj 
be  much  narrower  than  in  the  other  two  figs  ;  because,  the  direction  of  d  u  being  vertical,  the  pressan 
bas  no  tendency  to  overturn  the  pier.  In  Fig  2,  the  masonry  of  the  pier  should  be  laid  in  the  usual 
horizontal  courses,  in  order  that  its  bed  joints  may  be  at  right  angles  to  the  pressure  upon  them. 

But,  in  Figs  3  and  4,  if  the  bases  were  made  as  narrow  as  in  Fig  2,  the  lines  of  direction  dv,  of  the 
pressure,  would  fall  outside  of  them  ;  and  the  piers  would  consequently  be  in  danger  of  overturning. 
Also,  the  stones  of  the  masonry,  if  laid  in  horizontal  courses,  would  have  a  tendency  to  slide  on  each 
•ther.     To  prevent  this,  the  beds  should  be  at  right  angles  to  d  v. 

In  Fig  3.  the  obliquity  of  the.  pressure  would  tend  to  slide  the  base  of  the  pier  outward  as  shown 
by  the  arrow  ;  but  in  Fig  4,  inward.  This  tendency  is  produced  by  the  horizontal  component  of  the 
force  dv.  The  amount  of  this  may  be  found  thus,  in  either  fig:  From  d  downward  draw  a  vert  line 
as  in  Fig  4 ;  and  from  v  a  hor  one,  meeting  it  in  z,  then  v  z,  measured  by  the  same  scale  of  tons  as 
before,  will  give  this  horizontal  force,  and  d  z  will  give  the  vertical  component  of  the  pressure  d  v. 
The  effect  upon  the  pier,  of  the  one  pressure  d  r  is  precisely  the  same  as  would  be  produced  upon  it  by 
one  vertical  force  equal  to  dz  and  a  horizontal  one  equal  to  vz  acting  at  the  same  time,  as  explained 
under  Composition  and  Resolution  of  Forces. 

If,  in  either  fig,  we  draw  the  vert  lines  s^?  and  ro,  see  Fig  4,  thend  o,  measd  by  the  foregoing  scale, 
will  give  the  tons  of  horizontal  pull,  and  r  o  the  vertical  pressure,  produced  on  the  pier  by  the  back- 
stay ;  and  p  d  and  p  s  will,  in  like  manner,  give  the  corresponding  forces  produced  by  the  main  chain. 
If  we  add  together  ro  and  ps  they  will  be  found  to  be  equal  to  d  z,  and  if  we  subtract  d  o  from  p  d, 
their  difference  will  equal  v  z.  It  is  this  difference  only  that  tends  to  slide,  or  t-o  upset,  the  pier  ;  the 
other  portions  of  d  o  andp  d  neutralizing  each  other  in  that  respect. 
The  foregoing  strains  may  all  be  calculated,  thus: 

Horizontal  pnll  inward  by  the  main  chain  =  Tension  x  Cosineof  odsr 

'*  '*       Ontward   by  the  back-stay  =  Tension  X  Cosineof  «du. 

Vertical  pressnre  by  main  ehaiit  =  Tension  x  Sine  or  adg. 

*'  **  **    back-stay  =  Tension  X  sine  ofl  d «. 


SUSPENSION   BRIDGES. 


769 


Art*  5.  If  the  cables  pass  freely  over  a  loose  pin,  d.  Fig  4  A,  supported  by  a  link  L, 
kanging  from  the  Pxed  pin  z,  and  capable  of  moving  freely  about  both 
cf  its  pins;  the  tension  in  the  back-stay  will,  as  before,  be  equal  to     __. 
that  in  the  main  cable;  and  the  direction  and  amount  of  the  strain     Jb  l^t4  A 
on  the  piers  will  be  found  in  the  same  way  as  for  Figs  2,  3  and  4; 
namely :  lay  off  rf  s  and  d  r,  each  equal  to  the  tension,  and  draw  the 
parallelogram  davr.    Then  will  d  v  give  the  amount  and  direction  of 
the  strain  on  the  piers.  This  last  will,  of  course,  be  transmitted  through 
the  pins  and  the  link.   The  amount  of  tension  on  the  link  will  be  given 
by  the  length  of  d  u  ;  and  the  link  (being  free  to  move)  will  be  in  line 
"With  this  tension.   The  shearing  strain  on  each  pin  is  also  given  by  d  v. 

Art.  6.  But  if  the  ends  of  the  cable  and  back-stay. 
Pigs  4  B,  4  C  and  4  D,  at  the  top  of  the  pier,  be  made  fast  to  a  truck 
«r  wagon  which  is  supported  by  rollers  on  a  smooth  platform  on  top  of  the  pier,  the  axles  of  the 
rollers  being  fixed  in  the  truck  ;  then  the  strain  on  the  back-stay  will  not  be  the  same  as  that  on  the 
cable,  unless  the  angles  ad  g  and  Idu  are  equal,  as  in  Fig  4  B. 

li  adg  exceeds  I d' u,  as  in  Fig  4  C,  the  strain  on  the 
back-stay  will  be  less  than  that  on  the  cable,  and  vice  versa 
(Fig  4  D). 

But,  in  either  case,  if  the  top  of  the  pier  is  horizontal, 
as  is  usually  the  case,  the  horizontal  components  of  the 
strains  on  the  cables  and  on  the  back-stays,  will  be  equal, 
and  will  thus  counteract  each  other,  and  there  will  conse- 
quently be  no  horizontal  or  oblique  strain  on  the  pier. 
That  is,  the  strain  on  the  pier  will  be  vertical. 

To  find  the  amonnt  of  the  ten- 
sion on  the  back-stay,  and  of  the  pres- 
sure on  tlic  pier;  on  dg  in  either  Fig.  4 
B,  4  C  or  4  D,  lay  ott'  ds,  equal,  by  scale,  to  the  tension 
on  the  cable  at  d.  Draw  dv  perpendicular  to  the  surface 
m«  on  which  the  rollers  rest.  We  assume  that  mn  is 
horizontal,  as  is  generally,  hut  not  necessarily,  the  case  ; 
and  dv,  therefore,  vertical.  Draw  s  v  horizontal,  or  par- 
allel to  m  «.* 

Then  a  v  will  give  the  horizontal  pull  of  the  main  cable 
en  the  wagon,  and  dv  will  give  the  vertical  pressure  of 
the  wheel  d  on  the  tower  (to  which  that  of  the  wheel  d'  has 
yet  to  be  added).  From  d'  lay  off  d'  o  horizontal,  and  equal 
tosv;  and  draw  r  o  vertically.  Then  d'r  will  give  the 
amount  of  the  pull  on  the  back-stay ;  and  ro  will  give  the 
vertical  pressure  of  the  wheel  d'  on  the  pier;  which 
must  be  added  to  d  w  for  the  total  vertical  pressure. 

Or  the  various  strains  may  be  calculated,  thus: 

Horizontal     pull)       „.     ,,„,     ^     .      ^    . 

sv  ^rao  a't  the  U'SK'?J-,"p'r  =  ^'=d'd"'x'"'"-<"«'''- 
top  of  tile  pier       J 


Strain  il'  r  in  back- 
stay 
Pres  on  pier,  perp 


.  Horizontal  pull  «  «  or  d'o  at  top  of  j 
pier,  or  at  middle  of  span         " 


Cosine  of  I  d'  u. 


Tension  da 
on    main  X 
cable  at  d 


Sine  of  \ 
adg 


_  /Tension  d'  r 
'  von  back-stay 


^  Sineof\ 
^    Id'u) 


Art.  7.  When,  as  is  sometimes  the  case  in  light 
bridges,  the  piers  are  posts,  P  Fig  4  E,  of  wood  or  iron,  hinged  at 
the  bottom,  and  having  the  cables  and  back-stays  firmly  fixed  te 
their  tops ;  from  d  draw  d  s,  equal,  by  scale,  to  the  tension  on  the  main  cable  at  d ;  and  d  w  toward 
the  foot  of  the  post.  From  s  draw  s'r  parallel  to  the  back-stay,  and  meeting  d  m;  in  r.  Then  will 
s  r  give  the  strain  in  the  back-stay,  and  d  r  will  give  the  amount  and  direction  of  the  pressure  upon 
the  post. 

Art.  8.    As  in  the  Niagara  bridge,  the  cables  often  merely  rest  upon 

movable  trucks,  or  saddles,  T  Fig  4  F,  curved  on  top  to  avoid  sudden  bends  in  the  cables,  and  resting 
upon  loose  rollers  which  lie  upon  a  thick  horizontal  iron  plate  bolted  to  the  top  of  tbe  pier,  and  ar« 
free  to  move  horizontally.   In  such  cases  the  angles  ad  g  and  I  d'u  are  made  equal ;  so  that  the  pul38 

*  The  lines  a  I  and  a  v  must  be  drawn  parallel  to  the  surface  m  n  on  which  thetvagon  reata. -vhetber 

said  surface  be  horizontal  or  inclined. 

49 


770 


SUSPENSION   BRIDGES. 


d  s  and  d' r  are  equal,  as  are  also  their  horizontal  components  prf  and  d'o  •  and  the  pressures  on  tb» 
pier  are  vertical ;  and  if  changes  of  temperature  or  of  loading  produce  slight  changes  in  the  angles 
a  d  g  and  I  d'  xc,  the  truck  will  (by  reason  of  the  inequality  thus  brought  about  between  the  hori- 
zontal components)  move  far  enough  to  restore  the  equality  between  the  angles,  and  between  the 
horizontal  components,  and  consequently  the  pressure  upon  the  pier  will  at  all  times  be  vertical. 

Art.  9.    To  fiiui,  approximately,  tlie  leng^tli  of  a  main  chain 

C  5  d;  Pfg.  1  ;  having  the  span  c  d,  and  the  middle  dett  a  b.    See  preceding  table,  Art  1. 


Half  length  of  main  chain  =  "^1%  (detl2)  4-  (i^  chord>2. 

In  Menai  bridge  the  chord  cd  is  579.874  ft;  and  the  detl  is  43  ft. 

According  to  the  above  formula,  the  entire  length  is  588.3  feet.  By  actual  measurement  the  chain 
is  precisely  590  feet.    The  approximate  rule  below  gives  589.764  ft. 

Note.  The  lengths  obtained  by  this  rule  are  only  approximate,  because  the  calculation  is  base* 
npon  the  supposition  that  the  chains  form  a  parabolic  curve  :  whereas,  in  fact,  the  curve  of  a  finished 
bridge  is  neither  precisely  a  parabola,  nor  a  catenary,  but  intermediate  of  the  two. 

The  following  simple  rule  by  the  writer  is  quite  as  approximate  as  the  foregoing  tedious  on«, 
when,  as  is  generally  the  case,  the  defl  is  not  greater  than  ^^  of  the  chord,  or  span. 

Length  of  main  chain  when  defi  does  not  exceed  one-twelfth  of  the  span  =  chord  -j-  .23  defl. 

Art.  10.  To  find,  approximately,  tbe  length  of  the  vert 
snspendin^  rods  x  y,  «&:c.  Fig  1 ;   assuming^  the  curve  to 

be  a  parabola. 

Let  X,  Pig  1,  be  any  point  whatever  in  the  curve ;  and  let  a; «;  be  drawn  perp  to  the  chord  c  d ;  and 
st/perp  to  a6;  then  in  any  parabola,  as  a  c^  :  atv^  :  :  ah  z  b  f.  And  6 /thus  found,  added  to  6t, 
(which  is  supposed  to  be  already  known.,  being  the  length  decided  on  for  the  middle  suspending  rod,i 
gives  X  jT,  the  length  of  rod  reqd  at  the  point  x;  and  so  at  any  other  point. 

If  hf  thns  found  be  taken  from  the  middle  deflection  re  h. 

It  leaves  tr  a? ;  3.ud  thus  auy  deflection  w  x  of  the  main  chain  or  cable,  may  bo 
found  when  we  know  its  hor  dist,  aw,  from  the  center,  a,  of  the  span. 

In  the  foregoing  r«Ie,  the  floor  of  the  bridge  is  supposed  to  be  straight  ;  but  generally  it  is  raised 
toward  the  center;  and  in  that  case,  the  rods  must  first  be  calculated  as  if  the  floor  were  straight, 
and  the  requisite  deductions  be  made  afterward.  When  it  rises  in  two  straight  lines  meeting  in  the 
center,  the  method  of  doing  this  is  obvious.  When  an  arc  of  a  circle  is  used,  its  ordinates  may  be 
calculated  and  deducted  from  the  lengths  obtained  by  this  rule. 

Or,  having  drawn  the  curve  by  the  rule  for  drawing  a  parabola,  the  dimensions  can  "be  approx- 

imated to  by  a  scale.  The  adjustments  to  the  precise  lengths  must  be  made  during  the  actual  con- 
struction of  the  bridge,  by  means  of  ritats  on  their  lower  screw-ends.  The  rods  require,  therefore, 
only  to  be  made  long  enough  at  first. 

The  towers,  piers,  or  pillars,  firhieh  npliold  the  chains  or 
cables,  admit  of  an  enaless  variety  in  desig'n.  According  to  cir- 
cumstances, they  may  consist  each  of  a  single  vertical  piece  of  timber,  or  a  pillar  of  cast  or  wrought 
iron ;  or  of  two  or  more  such,  placed  obliquely,  either  with  or  without  connecting  pieces  ;  like  the 
bensS  of  a  trestle.  Or  they  may  be  mad*  (with  any  degree  of  or- 

namentation) of  cast-iron  plates ;  as  in  iron  house-fronts.  Or  they  may  be  of  masonry,  brick,  or 
concrete ;  or  of  any  of  these  combined. 

.  £ach  of  the  suspendingp-rods,  throtigh  which  the  floor  of  the  bridge  is 
■pheld  by  the  main  chains,  requires  merely  strength  sufficient  to  support  safely  the  greatest  load 
that  can  come  upon  the  interval  between  it  and  half-way  to  the  nearest  rod  on  each  side  of  it ;  in- 
cluding the  wt  of  the  platform,  &c,  along  the  same  interval. 

In  anchoring^  the  backstays  into  the  g'round,  it  is  necessary  to 

secure  for  them  a  sufficiently  safe  resistance  against  a  pull  equal  to  the  strain,         upon  the  backstay. 

As  to  the  anchorage  of  the  cables  Tselow  tlie  surfece  of  the  ground, 

natural  rock  of  firm  character  is  the  most  favorable  material  that  can  present  itself.  When  it  is  not 
present,  serious  expense  in  masonry  must  be  incurred  in  large  spans,  in  order* to  secure  the  necessary 
weight  to  resist  the  pull  of  the  cables.  Our  Figs  43^  give  ideas  of  the  modes  most  frequently  adopted. 
For  a  very  small  bridge,  such  as  a  short  foot-bridge,  for  instance,  the  backstays  may  simply  be  an- 
chored to  large  stones,  t.  Fig  A,  buried  to  a  sufficient  depth.  Or,  if  the  pull  is  too  great  for  so  simple 
a  precaution,  the  block  of  masonry,  mm,  may  be  added,  enclosing  the  backstay.  A  close  covering 
of  the  mortar  or  cement  of  the  masonry  has  a  protecting  effect  upon  the  iron. 

To  avoid  the  necessity  for  extending  the  backstays  to  so  great  a  dist  under  ground,  they  are  usually 
curved  near  where  they  descend  below  the  surface,  as  shown  at  B,  D,  and  E ;  so  as  sooner  to  reach 
the  reqd  depth.  This  curving,  however,  gives  rise  to  a  new  strain,  in  the  direction  shown  by  the 
arrows  in  Figs  B  and  D.  The  nature  of  this  strain,  and  the  mode  of  finding  its  amount,  (knowing 
the  pull  on  the  backstay,)  are  very  simple ;  and  fully  explained  under  the  head  of  Funicular  Ma- 
chine. The  masonry  must  be  disposed  with  reference  to  resisting  this  strain,  as  well  as 
that  of  the  direct  pull  of  the  backstay.  With  this  view,  the  blocks  of  stone  on  which  the  bend  rests 
should  be  laid  in  the  position  shown  in  Fig  D  ;  or  by  the  single  block  in  Fig  B.  Sometimes  the  bend 
is  made  over  a  cast-iron  chair  or  standard,  as  at  x.  Fig  F,  firmly  bolted  to  the  masonry. 

Fig  E  shows  the  arrangement  at  the  Niagara  railway  bridge  of  8213^  ft  span.     The  wire  backstays 

end  at  cc;  and  from  there  down  to  their  anchors,  they  consist  of  heavy  chains;  each  link  of  which 

Is  composed  of  (alternately)  7  or  8  parallel  bars  of  flat  iron,  with  eye  ends,  through  which  pass  bolts 

Each  of  the  7  bars  of  each  link  is  1.4  ins  thick,  by  7  ins  wide,  near  the 


SUSPENSION   BRIDGES. 


771 


lowest  part  of  the  chain ;  but  they  gradually  increase  from  thence  upward,  until  at  c,  c,  where  they 
unite  with  the  wire  cable,  the  sectional  area  of  each  li7ik  is  93  sq  ins.  These  chain  backstays  pass  in 
a  curve  through  the  massive  approach  walls.  (28  ft  high,)  and  descend  vertically  down  shafts  s,  s,  25 
ft  deep  in  the  solid  rock.  Here  they  pass  through  the  cast-iron  anchor-plates,  to  which  th«y  are  con- 
fined below  by  a  bolt  3%  ins  diam.  The  anchor-plates  are  6}^  feet  square,  and  2J^  ins  thick ;  except 
for  a  space  of  about  20  ins  by  26  ins,  at  the  center  where  the  chains  pass  through,  where  they  are  1 


foot  thick'.  Through  this  thick  part  is  a  separate  opening  for  each  bar  composing  the  lowest  link. 
From  this  part  also  radiate  to  the  outer  edges  of  the  lower  face  of  the  plate,  eight  ribs,  2}^  ins  thick. 
The  shafts  s,  s,  have  rough  sides,  as  they  were  blasted  ;  and  average  3  ft  by  7  ft  across  ;  except  at  the 
bottom,  where  they  are  8  ft  square.  They  are  completely  filled  with  cement  masonry,  with  dressed 
beds,  well  in  contact  with  the  sides  of  the  shafts;  and  thoroughly  grouted,  thus  tightly  enveloping 
the  chains  at  every  point;  as  does  also  the  masonry  of  the  approach  wall  ww;  which  extends  28  ft 
above  ground ;  and  is  6  ft  thick  at  top,  and  lOJ^  ft  thick  at  its  base  on  the  natural  rock. 

D,  Figs  43^,  shows  a  mode  that  may  be  used  in  most  cases,  for  bridges  of  any  span.  The  depth 
ftnd  the  area  of  transverse  section  of  the  shaft,  and  consequently  the  quantity  of  masonry  in  it,  will 
depend  chiefly  upon  whether  it  is  sunk  through  rock,  or  through  earth.  If  through  Arm  rock,  then 
If  its  sides  be  made  irregular,  and  the  masonry  made  to  fit  securely  into  the  irregularities,  much  re- 
liance may  be  placed  upon  it  to  assist  the  weight  of  the  masonry  in  resisting  the  pull  on  the  back- 
stays.    Earth  also  assists  materially  in  this  respect. 

F  is  the  arrangement  in  the  Chelsea  bridge  of  333  feet  span,  across  the  Thames,  at  London  ;  Thos. 
Page,  eng.  The  space  from  one  wall  6  6,  to  the  opposite  one,  is  45  feet;  and  is  built  up  solid  with 
brickwork  and  concrete ;  except  a  passage-way  4  ft  wide,  and  5  ft  high,  along  the  backstay ;  and  a 
small  chamber  behind  the  anchor-plates.    It  rests  chiefly  on  piles. 

The  arrangement  by  Mr  Brunei,  in  the  Charing  Cross  bridge,London,*lBvery  similar.  In  it  also 
the  entire  abutment  rests  on  piles  ;  and  is  40  ft  high,  30  ft  thick,  and  solid,  except  a  narrow  passage- 
way along  the  chains.    The  backstays  extend  into  it  60  ft.     Span  676  feet.     Defl  50  feet. 

G  is  intended  merely  as  a  generalhint,  which,  variously  modified,  may  find  its  application  in  the 
case  of  a  small  temporary,  or  even  permanent  bridge ;  for  the  number  of  pieces,  t,  t,  &c,  may  be  in- 
creased to  any  necessary  extent ;  and  they  may  be  made  of  iron  or  stone,  instead  of  wood. 

In  order  tliat  the  backstays  may  be  accessible,  they  are  fre- 
quently carried  through  openings  left  in  the  masonry  for  the  purpose^  Thus,  the  masses,  mm, 
of  masonry,  at  A  and  B,  Figs  4J^,  instead  of  being  made  solid,  may  consist  of  two  parallel  walls, 
between  which  the  backstay  may  pass ;  and  the  anchor-stones,  or  anchor-plates,  will  extend 
across  the  space  between  the  walls,  and  have  their  bearings  against  the  ends  of  the  walls.  In  D, 
E,  and  P,  the  cable  may  be  supposed  either  to  be  tightly  surrounded  by  the  masonry  and  grouted  to 
it,  or  else  to  be  surrounded  by  a  cylindrical  passage-way  like  a  culvert,  so  as  to  be  at  all  times  acces- 
sible. 

Soft  friable  stone  must  be  carefully  excluded  from  such  parts  of  the  anchorage  as  are  most 
directly  opposed  to  the  pull  of  the  backstays. 

If  blocks  of  stone  large  enough  for  securing  good  bond  are  not  procurable,  heavy  T-rails,  bars  of 
iron,  or  I-beams,  may  be  advantageously  introduced  for  that  purpose. 

The  masses  must  be  founded  at  such  a  depth  as  not  to  slide  by  the  yielding  of  the  earth  in  front 
of  them. 

For  safety.  It  is  well  to  disregard  the  effect  of  friction  in  diminishing  the  tension  on  the  backstay, 
and  to  regard  that  tension  as  continuing  uniform  throughout  the  backstay  to  its  end,  even  when  the 
backstay  is  curved  and  imbedded  in  the  masonry,  as  at  E,  Figs  434. 

The  side  parapets  should  be  high  and  stout,  so  as  to  act  as  stiffening 
trusses,  and  should  not  be  restricted  to  service  as  mere  hand-rails  or  guards.  As  a  rule  of  thumb 
their  depth  maybe  made  =  M  V  span,  provided  the  depth  be  not  less  than  that  required  for  a 
hand-rail.  The  parapets  should  be  stoutly  constructed,  with  special  attention  to  the  strength  of 
their  joints,  for  these  are  exposed,  by  the  undulations  and  lateral  motions  of  the  bridge,  to  violent 
deranging  forces  in  all  directions. 


*  Removed  to  Clifton,  England,  in  1863,  and  replaced  by  an  iron  truss  railway  and  foot  bridge. 


772 


EIVETS   AND   RIVETING. 


EIVETS  AND  EIVETING. 

Tbe  welgbts  in  the  following  table  of  course  include  the  head ;  but  the  lenstlis,  as  usual, 
are  taken  "  under  the  head ;  "  or  are  those  of  the  shanks  only.  In  practice,  discrepancies  of  5  or  6 
per  ct  in  wt  may  be  expected. 


Length 

Diameters  of  Bivet«  in  Inches. 

of  Shank. 
Ins. 

r.   1 

K 

% 

H 

Vs     1 

1 

IVs 

1'4 

Weixht  of  100  Rivets,  in  pounds. 

^ 

3.0 

8.5 



3.8 

9.9 

"17.3 



1 

4.6 

11.2 

19.4 

25.6 

38.9 



1^ 

5.4 

12.6 

21.5 

28.7 

43.1 

65.3 

"91.5 

123 

74 

6.2 

13.9 

23.7 

31.8 

47.3 

70.7 

98.4 

133 

6.9 

15.3 

25.8 

34.9 

51.4 

76.2 

105 

142 

2 

7.7 

16.6 

27.9 

37.9 

55.6 

81.6 

112 

150 

i 

8.5 

18.0 

30.0 

41.0 

59.8 

87.1 

119 

169 

9.2 

19.4 

32.2 

44.1 

64.0 

92.5 

126 

167 

10.0 

20.7 

34.3 

47.1 

68.1 

98.0 

133 

176 

8^ 

10.8 

22.1 

36.4 

50.2 

72.3 

103 

140 

184 

K 

11.5 

23.5 

38.6 

53.3 

76.5 

109 

147 

193 

12.3 

24.8 

40.7 

56.4 

80.7 

114 

154 

201 

T4: 

13.1 

26.2 

42.8 

59.4 

84.8 

120 

161 

210 

4 

13.8 

27.5 

45.0 

62.5 

89.0 

125 

167 

218 

^ 

14.6 

28.9 

47.1 

65.6 

93.2 

131 

174 

227 

15.4 

30.3 

49.2 

68.6 

97.4 

136 

181 

236 

% 

16.2 

31.6 

51.4 

71.7 

102 

142 

188 

244 

5 

16.9 

33.0 

53.5 

74.8 

106 

147 

195 

253 

¥ 

17.7 

34.4 

55.6 

77.8 

110 

153 

202 

261 

M 

18.4 

35.7 

57.7 

80.9 

114 

158 

209 

270 

19.2 

37.1 

59.9 

84.0 

118 

163 

216 

278 

6 

20.0 

38.5 

62.0 

87.0 

122 

169 

223 

287 

K 

21.5 

41.2 

66.3 

93,2 

131 

180 

236 

304 

7 

23.0 

43.9 

70.5 

99.3 

139 

191 

250 

321 

^ 

24.6 

46.6 

74.8 

106 

147 

202 

264 

338 

8 

26.1 

49.4 

79.0 

112 

156 

213 

278 

355 

9 

29.2 

54.8 

87.6 

124 

173 

234 

306 

389 

10 

32.2 

60.3 

96.1 

136 

189 

256 

333 

423 

11 

35.3 

65.7 

105 

148 

206 

278 

361 

457 

12 

38.4 

71.2 

113 

161 

223 

300 

388 

491 

The  diam  of  rivets  for  bridge  work  is  from  3^  to  1  inch ;  usually  %  to 
%;  and  for  plates  more  than  .5  inch  thick,  it  is  about  1.5  times  the  thickness; 
and  for  thinner  ones  about  twice ;  but  these  proportions  are  not  closely  adhered 
to.  The  eomiiion  form  of  rivets  as  sold  is  shown  at  R,  Figs  3,  a  head 
and  the  shank  in  one  piece ;  and  S  shows  the  same  when  after  being  heated 
white  hot  it  is  inserted  into  its  hole,  and  a  second  head  (conical)  formed  on  it  by 
rapid  hand-riveting  as  it  cools.  l¥hen  longer  than  about  6  ins  they 
are  cooled  near  the  middle  before  being  inserted,  lest  their  contraction  in  cooling 
should  split  off  their  heads.  The  hemispherical  heads  often  seen,  called  snap 
heads,  are  formed  by  a  machine.  The  two  heads  alone  require  about 
as  much  iron  as  3  diams  length  of  shank.  lieng^th  of  a  head  =  about  1 
diam  of  shank ;  and  its  ividth  about  2  diams  of  shank. 


Riveting  of  ISteani  and  \¥ater  Tig^ht  Joints. 

Joints  for  boilers  and  water-tight  cisterns  are  usually  proportioned  about 
as  per  the  following  table  by  Fairbairn ;  and  are  made  as  shown  either  by  Fig  1, 
or  Fig  2.    Fig  1  is  called  a  sing^le-riveted,  and  Fig  2  a  double-riveted 

lap-joint.     The  dist  a  a,  or  c  c,  is  the  lap. 

Mr  Fairbairn  considers  the  strength  of  the  single-riveted  lap-joint  to  be  about 
.56 ;  and  that  of  the  double-riveted,  about  .7  that  of  one  of  the  full  unholed 


RIVETS   AND    RIVETING. 


773 


plates,  when  both  joints  are  proportioned  as  in  his  following  table.    But  some 

later  experimenters  consider  about 
^  r~\    '^  a     ^        /^ .5  and  .6  as  nearer  the  correct  aver- 

age. Experiments  on  the  subject 
are  quite  conflicting;  and  it  is 
plain  that  no  one  set  of  propor- 
tions can  precisely  suit  all  the  dif- 
ferent qualities  of  plate  and  rivet 
iron.  With  fair  qualities  of  both, 
there  is  every  reason  to  rely  upon 
__.        .  XT'*       O  '^  ^°^  '^  ^^^  about  one-seventh 

Jr  IQ  1.  t*  IQ  ^.  part  less  than  Fairbairn's  assump- 

tion) as  safe  for  practice.    These 
proportions  include  friction  (Art  4),  without  which  they  would  be  about  A  and  .5. 


Fairbairn's  table  for  proportioning:  tlie  rivetingr  for  steam 
and  water-tj^bt  lap-joints. 


Thickness  of 
each  plate. 

Diameter  of 
rivets. 

Length  of shank 
before  driving. 

From  center  to 
center  of  rivets. 

Lap  in  single 
riveting. 

Lap  in  double 
riveting. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

3-lG        ' 

% 

Vs 

1^ 

1^ 

2  1 

^16 

Ps 

i4 

1| 

1^ 
1>I 

5}4 

A 

13-16 
15-16 
IK 

1% 

% 

2 

2% 

RiTeting:  of  iron  j^irders,  bridg^es,  <S:c. 


'N 


Kn 


g'  V    \y*e 


(t 


OOOiO      O 
OOOJOD    o 

oooLo    o 


Figs  3. 


^V7^S7~S?       N/     \/'    N/^ 


PP 


Art.  1.  Tbe  subject  of  riveting*  is  abstrnse,  and  involved  in 
much  uncertainty ;  and  experimental  results  are  very  discrepant.  We  here  pro- 
pose merely  to  confine  ourselves  to  what  is  considered  the  best  joint ;  and  for 
safety  we  shall  omit  friction;  see  Art  4.  In  girder  and  bridge  work  the  lap- 
joints  above  described  are  seldom  used.  Instead  of  them,  the  plates  p,  Figs  3,  to 
be  joined,  are  butted  up  square  against  each  other,  thus  forming  a  butt-joint, 
i  i.  Fig  D;  and  are  united  by  either  a  single  covering-plate,  cover,- 
wrapper,  fish-plate,  or  welt  e  e,  Fig  K ;  or  the  best  of  all  by  two  of  them, 
as  at  A,  or  0  0,  0  o,  Fig  B.  In  what  follows,  the  term  plate  never  includes  the 
covers.  The  single  coyer,  like  the  lap-joint,  allows  both  plates  and  cover  to  bend 
under  a  strong  pull,  somewhat  as  at  W,  thus  weakening  them  materially ;  whereas 
the  double  cover  o  o,  o  o,  Fig  B,  keeps  the  pull  directly  along  the  axis  of  the  plates, 
thus  avoiding  this  bending  tendency.  It  also  brings  the  rivets  into  double  shear, 
thus  doubling  their  strength.  When  there  is  but  one  cover,  it  should  be  at  least 
as  thick  as  a  plate ;  and  when  there  are  two,  experience  shows  that  each  had  bet- 
ter be  about  two-thirds  as  thick  as  a  plate,  although  theory  requires  each  to  bo 
but  half  as  thick  as  a  plate. 


774  RIVETS   AND    RIVETING. 

The  leiig^tti  vr  nr  of  covers  across  the  joint  is  equal  to  that  of  the  joint 
Butts  require  twice  as  many  rivets  as  laps,  because  in  the  lap  each 
rivet  passes  through  both  the  joined  plates;  and  in  the  butt  through  only  one. 

The  rivets  and  plate  on  one  side  only  (right  or  left)  of  the  joint- 
line  i  i  of  any  properly  proportioned  butt-joint  D,  represent  the  full  strength 
of  the  joint,  inasmuch  as  those  on  one  side  pull  in  one  direction,  against  those  on 
the  other  side,  which  pull  in  the  opposite  direction.  Therefore  in  designing  such 
joints  we  need  keep  in  mind  only  those  on  one  side,  as  is  done  in  what  follows. 
Thus  a  single,  double,  or  triple-riveted  butt-joint  D  implies  one,  two,  or  three 
rows  of  rivets  on  each  side  of  the  joint-line  i  i,  and  parallel  to  it.  In  a  propr 
erly  proportioned  lap  the  strength  is  as  all  the  rivets,  because  one-half  of  them 
do  not  pull  against  the  other  half,  but  one  end  of  every  rivet  pulls  in  one  direc- 
tion, and  its  other  end  in  the  opposite  direction. 

The  net  iron,  net  plate,  or  net  joint,  is  that  which  is  left  between 
the  rivet  holes,  and  outside  of  the  two  outer  ones,  all  on  a  straight  line  drawn 
through  the  centers  of  the  holes  of  one  row.  Its  width  and  area  are  called  the  net 
ones  of  the  joint.    That  between  other  rows  does  not  increase  the  strength. 

In  Figs  3,  N,  and  K,  the  rivets  are  in  sing-le  shear,  while  those  in  A  and  B 
are  in  double  shear. 

Art.  2.  Bridg-e-joints  are  not  required  to  be  steam  or  water- 
tig'ht  like  those  of  boilers  or  cisterns  ;  and,  therefore,  by  increasing  the  breadth 
of  the  overlap,  or  the  length  of  the  covers,  the  rivets  may  be  placed  in  several 
rows  behind  each  other,  as  the  3  rows  of  3  rivets  each  in  M  and  D,  instead  of  only 
one  row  of  9  rivets,  as  in  L.  By  this  means,  without  losing  any  of  the  strength  of 
the  9  rivets,  or  of  the  net  iron,  we  may  narrow  the  width  of  the  plate  to  an  ex- 
tent equal  to  the  combined  diams  (6  in  this  case)  of  the  holes  thus  dispensed  with 
in  the  one  row.  Moreover,  by  using  more  than  one  row  we  lessen  the  weakening 
effect  shown  at  W.  This  mode  of  placing  the  rivets  directly  behind  each  other  in 
several  rows,  as  at  M,  and  at  the  left-hand  half  of  Fig  D,  constitutes  Mr  Fair- 
bairn's  chain  riveting- ;  but  the  joint  will  be  somewhat  stronger  if  the  rivets 
are  placed  in  zig-zaging  order,  as  in  the  right-hand  half  of  Fig  D. 

The  dist  apart  of  the  rowrs  from  cen  to  cen  should  not  be  less 
than  2  diams.  It  is  questionable  to  what  extent  .this  increase  in  the  number  of 
rows  may  be  carried  without  an  appreciable  loss  of  strength  in  the  rivets  conse- 
quent upon  the  impossibility  of  quite  equalizing  the  strains  on  the  separate  rows. 
But  it  is  probable  that  if  we  do  not  exceed  2  or  3  rows  in  laps,  or  the  same  num- 
ber on  each  side  of  the  joint-line  in  butts,  we  may  in  practice  assume  that  each 
row,  and  each  rivet,  is  nearly  equally  strained. 

Rivet-holes  are  usually  of  about  one-sixteenth  inch  greater  diam  than  the 
original  rivet,  so  as  to  allow  the  hot  rivet  to  be  easily  inserted.  The  subsequent 
hammering  swells  the  diam  of  the  rivet  until  it  fills  the  hole.  We  may  either 
take  this  increased  diam  of  rivet  into  consideration,  as  we  have  done,  in  calcula- 
ting its  shearing  and  crippling  strength,  as  explained  farther  on,  or  with  reference 
to  increased  safety  we  may  omit  it.  Brilled  rivet-holes  are  said  to  be  better 
than  punched  ones,  as  the  drilling  does  not  injure  the  iron  around  them;  but  on 
the  other  hand  their  sharper  edges  are  said  to  shear  the  rivets  more  readily. 
Hence,  such  edges  are  sometimes  reamed  off.  Both  these  points  are,  however, 
disputed ;  and  both  modes  are  in  common  use. 

The  dist  from  the  edge  of  a  hole  to  the  end  of  a  plate  or  cover  should 
not  be  less  than  about  1.2  diams,  to  prevent  the  rivets  from  tearing  out  the  end 
of  the  plate ;  nor  nearer  the  side  edge  of  a  plate  than  half  the  clear  dist  between 
two  holes  as  given  by  the  Rule  in  Art  5.  The  first  is  rather  more  than  Fairbairn 
directs. 

Rivet  holes  weaken  the  net  iron  left  between  them,  not  only  by  the 
loss  of  the  part  cut  out,  but  either  by  disturbing  the  iron  around  them,  or  perhaps 
by  changing  the  shape  of  the  net  line  of  fracture,  which  may  not  then  resist 
tension  as  well  as  while  it  was  a  continuous  straight  line.  Some  deny  both  cause 
and  eflfect  entirely,  each  party  basing  its  opinion  on  experiments.  But  the  mass 
of  evidence  seems  to  the  writer  to  show  that  the  net  iron  loses  on  an  average 
about  one-seventh  of  the  strength  due  to  the  net  width.  With  a  view  to  safety, 
which  we  consider  to  be  of  paramount  importance,  we  shall  in  what  follows 
assume  (until  the  question  is  definitely  settled)  that  there  is  such  a  loss  of 
strength  in  the  net  iron. 

Riveted  joints  for  resisting  compression  should  depend,  not  as 
might  be  supposed  upon  their  butting  ends,  but  upon  either  the  shearing  or  the 
crippling  strength  of  the  rivets;  for  contraction  or  bad  work  may  throw  the 


RIVETS   AND    RIVETING. 


775 


Sressure  on  the  rivets.    Machine  riveting'  is  somewhat  stronger  than  that 
one  (as  is  assumed  in  our  examples)  by  hand.    Tlie  thickness  of  plates 

used  in  girders,  tubular  bridges,  &c,  is  usually  .25  to  .5  inch ;  with  thicker  ones 
up  to  1  inch  sparingly  in  large  ones.  A  packing  piece,  as  the  shaded  piece 
in  P,  is  one  inserted  between  two  plates  to  prevent  their  being  bent  or  drawn 
together  by  the  rivets. 

Art.  3.  A  riveted  Joint  may  yield  in  three  ways  after  being 
properly  proportioned,  namely,  by  the  shearing  of  its  rivets;  or  by  the  pulling 
apart  of  the  net  plate  between  the  rivet  holes ;  or  by  the  crippling  (a  kind  of 
compression,  mashing,  or  crumpling)  of  the  plates  by  the  rivets  when  the  two  are 
too  forcibly  pulled  against  each  other.  It  also  compresses  the  rivets  themselves 
transversely,  at  a  less  strain  than  the  shearing  one;  and  this  partial 
yielding  of  both  plates  and  rivets  allows  the  joint  to  stretch,  and  may  thus 
produce  injurious  unlooked-for  strains  in  other  parts  of  a  structure,  considerably 
before  there  is  any  danger  of  actual  fracture.  Or  in  steam  and  water  joints  it  may 
cause  leaks,  without  farther  inconvenience,  or  danger.  For  a  long  time  this 
crippling  had  entirely  escaped  notice,  and  it  was  supposed  that  the  only  important 
point  in  designing  a  riveted  joint  was  that  the  tensile  strength  of  the  net  plate, 
and  the  shearing  strength  of  the  rivets  should  be  equal  to  each  other. 

The  crippling  strength  of  a  joint  is  as  the  number  of  rivets,  in  a  lap, 
or  the  number  on  one  side  of  the  joint-line  in  a  butt  X  diam  X  thickness  of  joined 

Slate.  This  product  gives  the  crippled  area  of  the  joint.  We  shall  here  call  the 
iam  X  thickness  of  plate,  the  crippling  area  of  a  rivet.  If  there  are  2  or 
more  plates  (not  covers)  on  top  of  each  other  at  one  joint,  their  united  thickness 
is  used  for  finding  the  crippling  area.  The  ultimate  crlpjpling  unit, 
by  which  the  above  product  is  to  be  multiplied  for  the  actual  ultimate  crippling 
strength  of  the  joint,  may  be  safely  taken  at  about  60000  fibs,  or  26.8  tons,  per  sq 
inch. 

The  dlani  of  a  rivet  in  ins  to  resist  safely  a  given  single-shearing 
force  is  found  thus :  Mult  the  shearing  force  by  the  coef  of  safety,  that  is  by  the 
number,  3,  4,  or  6,  &c,  denoting  the  required  degree  of  safety.  Call  the  product  g. 
Mult  the  ultimate  shearing  strength  per  sq  inch  of  the  rivet-iron,  by  the  decimal 
.7854.  Call  the  product  6.  Divide  g  by  6.  Take  the  sq  rt  of  the  quotient.  The 
shearing  force  and  the  shearing  strength  must  both  be  in  either  Bbs  or  tons. 

Or  by  a  formula, 


Diam  iu  ins 


=v 


Shearing  force  X  coef  of  safety 
Ult  shearing  strength  per  sq  inch  X  .7854 


If  the  rivet  is  to  be  double-sheared,  first  mult  only  half  the  shearing 
force  by  the  coef  of  safety.    Then  proceed  as  before. 

Or,  near  enough  for  practice,  mult  the  diam  in  single  shear  by  the  decimal  .7. 

The  ultimate  shearing  unit  for  average  rivet-iron  maybe  taken  at 
about  45000  fts,  or  20.1  tons  per  sq  inch  of  circular  sheared  section. 


Table  of  ultimate  single  shearing:  strength  of  rivets. 

(market  sizes),  in  single  shear ;  at  45000  lbs  or  20.1  tons  per  sq  inch. 

This  table  is  not' to  be  used  when  as  in  our  "  Example,"  Art  5,  the 
crippling  strength  of  the  rivet  governs  the  strength  of  the  joint. 

If  the  rivet  is  in  double  shear  it  will  have  twice  the  strength  in  the 
table. 

For  the  diam  in  double  shear  to  equal  the  strength  in  the  table,  mult 
the  diam  in  the  table  by  the  decimal  .7 ;  near  enough  for  practice  ;  strictly,  .707. 


Diam. 
Ins. 

Diam. 
Ins. 

fts. 

Tons. 

Diam. 
Ins. 

Diam. 
Ins. 

fts. 

Tons. 

Diam. 
Ins. 

Diam. 
Ins. 

fts. 

Tons. 

^ 

.125 

552 

.246 

.562 

11183 

4.99 

1 

1.000 

35343 

15.8 

.187 

1242 

.554 

Vs 

.625 

13806 

6.16 

1.062 

39899 

17.8 

K 

.250 

2209 

.986 

.687 

16705 

7.46 

1^ 

1.125 

44731 

20.0 

.812 

3452 

1.54 

% 

.750 

19880 

8.88 

1.187 

49838 

22.2 

% 

.375 

4970 

2.22 

.812 

23332 

10.4 

1^4 

1.250 

55224 

24.6 

.437 

6765 

3.02 

% 

.875 

27060 

12.1 

1.312 

60885 

27.2 

y^ 

.500 

8836 

3  94 

.937 

31064 

13.9 

1% 

1.375 

66820 

29.8 

776 


RIVETS   AND    RIVETING. 


The  tensile  strengrth  of  a  properly  proportioned  joint  is 

equally  as  either  the  sectional  area  of  the  net  plate  (not  covers)  across  the  cen- 
ters of  only  one  row  of  rivets  |  or  as  the  shearing  or  the  crippling  (as  the  case 
may  be)  areas  of  all  the  rivets  m  a  lap,  or  of  all  the  rivets  on  one  side  of  the 
joint-iine  in  a  butt.  The  tensile  strength  of  fair  quality  of  plate  iron,  before  the 
rivet  holes  are  made,  averages  about  45000  lbs,  or  20.1  tons  per  sq  inch;  but  we 
shall  for  safety  assume,  as  stated  in  Art  2,  that  the  making  of  the  holes  reduces 
the  strength  of  the  net  iron  that  is  left  about  one-seventh  part,  or  to  38500  Bbs, 
or  17.2  tons  per  sq  inch. 

Rem.  £ven  tbis  is  considerably  too  ^reat  for  laps,  or  for  butts 
with  one  cover,  owing  to  the  weakening  of  the  iron  in  such  by  the  bending  shown 
at  W,  Figs  3.    But  we  are  not  speaking  of  such. 

Art.  4.  Tbe  friction  betireen  tbe  plates  in  a  lap,  or  between  the 
plates  and  the  covers  in  a  butt,  produced  by  their  being  pressed  tightly  together 
by  the  contraction  of  the  rivets  in  cooling,  adds  much  to  the  strength  of  a  joint 
while  new,  perhaps  as  much  as  1.5  to  3  tons  per  sq  inch  of  circ  section  of  all  the 
rivets  in  a  lap,  or  of  all  on  one  side  of  a  single-cover  butt ;  or  3  to  6  tons  of  all  on 
one  side  of  a  double-cover  butt.  In  quiet  structures,  this  friction  might  continue 
to  exist,  either  wholly  or  in  part,  for  an  indefinite  period ;  but  in  bridges,  &c,  sub- 
ject to  incessant  and  violent  jarring  and  tremor,  it  is  probably  soon  diminished, 
or  entirely  dissipated.  Hence  good  authorities  recommena  not  to  rely  on  it,  and 
it  is,  therefore,  omitted  in  what  follows. 

Art.  5.  We  now  give  rules  for  finding  the  number  of  rivets  required  for  a 
doable  cover  butt-joint  (the  only  Mnd  of  which  we  shall  treat),  ajad  their 
clear  or  net  distance  apart.  This  dist  +  one  diam  is  the  pitch  of  the  rivets,  oi' 
their  dist  from  center  to  center.  The  principle  of  the  rule  will  be  explained 
further  on,  at  Art  7. 

First,  select  a  diam  of  rivet  either  equal  to  or  greater  than  .85  times  the 
thickness  of  the  plate.  In  practice  they  are  generally  1.5  times  for  plates  )^  inch 
or  more  thick ;  and  2  for  thinner  than  }/^  in. 

Second,  mult  the  greatest  total  pull  in  pounds  that  can  come  upon  the  entire 
joint  by  the  coef  (3,  4,  or  6,  &c)  of  safety,  and  call  the  product  p. 

Third,  multiply  the  crippling  area  of  the  rivet  (that  is,  its  diam  X  the  thick- 
ness of  plate)  by  60000.  The  prod  is  the  ult  crippling  strength  of  a  rivet.  Call  it  m. 

Fourth,  divide 2?  by  m.  The  quotient  will  be  the  number  of  rivets  to  sustain 
the  given  pull  with  the  reqd  degree  of  safety. 

Then,  the  clear  distance  apart  will  be 

Number  of  rows  X  Diam  X  60000 
38500 

Fifth.  The  clear  dist  from  either  end  hole  of  a  row  to  the  side  edge  of  the  plate, 
ihould  be  not  lees  than  half  the  clear  dist  between  two  rivets  in  a  row. 

Fxample.  A  double-cover  butt-joint  in  .5  inch  thick  plate  is  to  bear  an  actual 
pull  of  33750  fts,  with  a  safety  of  4;  or  not  to  break  with  less  than  33750  X  4  = 
135000  fibs.     How  many  rivets  must  it  have ;  and  how  far  apart  must  tliey  be  ? 

First,  Here  .85  times  the  thickness  of  the  plate  is  .5  X  .85  =  .425  inch;  there* 
fore,  our  rivets  must  not  be  less  than  .425  inch  in  diam ;  but  we  will  take  .75  inch 
diam. 

Second,  The  greatest  pull  X  coef  of  safety  =  33750  X  *  =  135000  lbs  =  p. 

Third,  The  crippling  area  of  a  rivet  X  60000  =  .75  X  .5  X  60000  =  22500  ==  vu 

Fourth,  —  =  ■  =  6  rivets  required  on  each  side  of  the  joint-line. 

And  the  clear  space  or  net  width  between  them  will  be,  if  the  6  rlTCts 
are  in  one  ro^r : 

Diam  X  60000       45000       ,  ,^„«  . 

38500 =-  38500  =  ^'^^^^  ^°«- 

1  9188 
And  the  pitch   =»   net  space  +  diam  =«  1.1688  +  .75  =  1.9188  Ins,  =     '^    -  . 
Ba  2.56  diams.  **^ 

In  practice,  to  avoid  troublesome  decimals,  we  might  make  the  net  space  1.2  ins; 
and  the  pitch  1.95;  but  to  show  farther  on  the  working  of  the  rule,  we  adhere  to 
the  more  exact  ones. 

Fifth,  The  clear  dist  from  each  end  hole  to  the  side  edge  of  the  plate  is  half  of 
1.1688  =  .5844  ins. 

The  entire  -width  of  net  iron  is  equal  to  one  clear  space  X  number  of 
rivets  =  1.1688  X  6  =  7.0128  ins ;  and  the  entire  width  of  plate  is  equal  to  ona 
pitch  X  number  of  rivets,  =»  1.9188  X  6  «=  11.5128  ins. 


RIVETS  AND  RIVETING.  777 

The  area  of  cross  section  of  unholed  plate  Is  11.5128  X  -S  =  5.7564  sq  ins ;  its  ten« 
Bile  strength  before  the  boles  are  made  is  6.7564  X  45000  =  259038  fts, 

135000 
The  strength  of  our  joint,  omitting  friction,  is  therefore  n-Qn^  ==  '^^  °^  *^^*  of  tht 
original  unholed  plate.  2o9038 

If  tbe  6  rivets  are  In  2  rows  of  3  rivets  each,  the  clear  dist  be- 
tween tw^o  rivets  in  one  row  will  be  twice  as  great  as  before,  or  twice  1.1688 
=  2.3376  ins.  Pitcb  =  2.3376  +  .75  =  3.0876  ins  =  3.0876  -r-  .75  =  4.12  diams. 
Clear  dist  from  end  hole  to  side  edge  of  plate  =  half  of  2.3376  =  1.1688. 
Entire  width  of  net  iron  =  2.3376  X  3  =  7.0128  ins.  Entire  width 
of  plate  =  8.0876  X  3  =  9.2628  ins.  Area  of  cross  section  of  nnholed 
plate=9.2628x. 5^4.6314  sq  ins.  Ultimate  tensile  strength,  unholed 
=  4.6314  X  45000  =  208418  Bbs.    Ult  strength  of  riveted  joint,  omitting 

friction  =  =  .65  of  that  of  the  unholed  plate. 

20o-41o 

Thus  we  see  that  the  arrangement  with  two  rows  gives  the  same  strength  as  one 
row,  with  a  less  total  width  and  area  of  plate.    It  of  course  requires  longer  covers. 

If  the  6  rivets  are  in  3  rows  of  2  rivets  each,  the  area  of  cross 
section  of  the  unholed  plate  is  4.2565  sq  ins.    Its  tensile  strength, 

191542  lbs.    Strength  of  riveted  joint  =»  =  .7  of  that  of  the  unholed  plate. 

The  entire  width  of  net  iron  (7.0118  ins);  its  area  (7.0128  X  .5  =  3.5064  sq  ins); 
and  its  ultimate  tensile  strength  (3.5064  X  38500  =  135000  lbs),  are  the  same  in  each 
case.  The  last  is  the  required  breaking  strength  of  the  joint,  as  in  the  beginning 
of  our  example ;  and  is  equal  to  the  combined  crippling  strength  of  tbe  six  riyete. 

Art.  6.  The  distance  apart  of  the  rows,  from  center  to  center  of 
rivets,  should  not  be  less  than  two  diameters  of  a  rivet-hole. 

Rem.  1.  With  our  constants  for  tension,  shearing,  and  compression,  the 
rivets  will  not  yield  first  by  shearing  in  a  double-cover  butt  (and 
of  course  in  double  shear),  except  when  the  diam  is  either  equal  to  or  less  than 
.85  of  the  thickness  of  the  plate,  which  will  rarely  happen.  At  .85  the  crippling 
and  shearing  strength  of  a  rivet  are  equal  when  using  our  assumed  coeffs  of  crip- 
pling, shearing,  and  tension. 

Rem.  2.  Our  example  was  chosen  to  illustrate  the  rule.  It  will  rarely  hap- 
pen in  practice  that  the  rule  will  give  a  number  of  rivets  without  a  fraction  ;  or 
that  may  be  divided  by  2  and  by  3  without  a  remainder.  In  case  of  a  fraction,  it 
is  plainly  best  to  call  it  a  whole  rivet;  although  the  joint  thereby  becomes  a  trifle 
stronger  than  necessary.  Or  rivets  of  a  slightly  dilf  diam  may  be  used.  If  the 
number  of  rivets  comes  out  say  7  or  9,  we  may  make  2  rows  of  3  and  4,  or  of  4  and 
5,  &c.  Moreover,  the  width  of  the  plate  is  frequently  fixed  beforehand  by  some 
requirement  of  the  structure,  and  we  must  arrange  the  rivets  to  suit,  taking  care 
in  all  cases  to  maintain  the  calculated  area  of  net  iron  in  one  row,  <fec. 
Rem.  3.  We  have  (as  we  at  first  said  we  should  do)  confined  ourselves  to  the 
simple  butt-joint  with  2  covers,  and  with  the 
rivets  in  either  1,  or  in  2  or  more  parallel  rows 
on  each  side  of  the  joint-line ;  this  being  the 
strongest  and  the  one  in  most  cotnmon  use  in 
engineering  structures.  Necessity  at  times 
calls  for  less  simple  arrangements,  for  which 
we  cannot  aflbrd  space,  and  the  strength  of 
I  ^  which  is  not  so  readily  calculated.  These 
u  g  sometimes  yield  results  which  appear  strange 

to  the  uninitiated ;  thus,  this  lap-joint  breaks 
across  the  net  iron  of  one  plate,  along  either  c  c  or  o  o,  where  tlvere  is  most  ofit^  and 
where,  therefore,  it  might  be  supposed  to  be  the  strongest. 

Rem.  4.  The  folio w^ing  table  shows  approximately  the  comparative 
•trengths  of  the  common  forms  of  joints  when  properly  proportioned ;  varying 
with  quality  of  sheets,  and  of  rivets : 


With 
friction. 

The  original  unholed  plate 1.00 

Double-riveted  butt  with  two  covers 80 

Double-riveted  butt  with  one  cover 65 

Single-iiveted  butt  with  one  cover 50 

Double-riveted  lap 65 

Single-riveted  lap 50 


Without 

friction. 

1.00 

.64 

.52 

.40 

.62 

.40 


778 


RIVETS  AND   RIVETING. 


Rem.  5.  The  above  tabular  strengths  for  the  lap-joints  will  be  approx« 
imately  attained  hj  adopting  the  following  proportions,  according  as  the  joint  is 
double-  or  single-riveted. 


Calling  thickness  of  plate 

Then  make  diam  of  rivet 

"  "      breadth  of  lap 

"  "      pitch  from  cen  to  cen 

*'  "      dist  from  end  of  plate  to 

edge  of  holes 

"         "      dist  apart  of  rows  from 
cen  to  cen 


Double  riv.  zl^zas* 

In  thicknesses. 


1. 

1.67 
9.0 
7.0 

2.0 

3.33 


.6 
1.0 
5.4 
4.2 


2.0 


Single 

rir. 

In  thicknesses. 

In  diams. 

1. 

.6 

1.67 

1.0 

5.67 

3.4 

4.5 

2.7 

2.0 

1.2 

Rem.  6.    If  two  or  more  plates  on  top  of  each  other,  as  the 

four  in  A  B  or  M  H,  are  to  be  jointed  together  so  as  to  act  as  one  plate  of  the 
thickness  c  c,  the  diams  of  the  rivets,  and  the  thickness  of  the  covers  c  c,ee  will 
depend  upon  whether  the  junctions  of  the  plates  are  all  in  one  line  with  each 
other  as  at  c  c,  in  A  B,  or  whether  they  break  joint  with  each  other  as  at  0,  1,  2,  3 
inMH. 


It  is  plain  that  the  two  covers  c  c  by  means  of  their  connecting  rivets  convey 
from  A  to  B,  across  the  joint  c  c,  all  the  strength  that  partly  compensates  for  the 
severance  of  the  four  plates  at  that  joint ;  whereas  the  two  covers  e  e,  e  e,  and 
their  rivets  in  like  manner  convey  from  n  of  one  single  plate,  to  o  of  the  adjoining 
one,  across  the  joint  between  those  two  letters,  only  the  strength  that  partly  com- 
pensates for  the  severance  of  that  single  plate ;  and  so  with  the  joints  at  1,  2,  and 
3.  Therefore  the  covers  c  c,  and  their  rivets,  must  be  four  times  as  strong  as  those 
at  any  one  of  the  four  joints  0,  1,  2,  3.  The  first,  c  c,  are  to  be  regarded  as  joining 
two  solid  plates  A  and  B,  each  of  the  fourfold  thickness  c  c ;  and  the  others  as 
joining  two  of  the  single  thickness.  The  covers  c  c  will,  therefore,  each  be  about 
two-thirds  of  the  thickness  c  c ;  and  the  others  each  about  two-thirds  as  thick  as 
a  single  plate.  Thus,  suppose  each  of  the  4  plates  in  A  B  or  M  H  to  be  ^  inch  thick ; 
making  c  c  3  ins.  Then  each  cover,  c,  is  %  of  3  ins,  or  2  ins  thick  ;  or  the  two  covers, 
cc,  together  4  ins,  which  is  thus  the  effective  thickness  of  the  joint,  cc.  But  each 
cover,  e  e,  is  only  %  of  %  inch,  or  ^  inch  thick ;  and  the  effective  thickness  of  joint 
at  either  0,  1,  2,  or  3,  is  that  of  the  3  unbroken  plates  plus  that  of  the  2  covers,  or 
(3  X  ^  +  (2  X  3^)  =  314  ins. 

Art.  7.  Principle  of  the  Rule  in  Art  5.  With  our  constants  for 
shearing  (45000  lbs  per  square  inch)  and  for  crippling  (60000  lbs  per  square  inch),  and 
with  diameter  of  rivet  equal  to,  or  greater  than,  .85  times  the  thickness  of  the  plate, 
as  by  our  rule,  the  crippling  strength  of  a  double  cover  butt  joint  will  be  equal  to,  or 
less  than,  its  shearing  strength.  Therefore,  to  avoid  waste  of  material,  either  in  the 
plate  or  in  the  rivets,  we  must  make 

""^Ltsro^e^roVof"  ilet''  =  Crippling  .trength  of  aU  the  rivet,.    Or. 

wi^tifnf  V  Thickness  ^  Tension  _  Crippling  area  y.  Crippling  ».  Total  number 
plate  of  plate    ^     unit     ~    of  one  rivet    ^       unit      '^     of  rivets. 

Now,  by  Art  3,  the  crippling  area  of  a  rivet  is  =  diam  of  rivet  X  thickness  of 
plate.  We  take  the  crippling  unit  at  60000  ft>8 ;  and  the  tension  unit  at  38500  Tb& 
Therefore  (transposing)  we  must  make 

m  ^  1      X     .^xu  Diam  of  ^  Thickness  ^  cnnnA  vx  Total  number 
Total  net  width  „     rj^et     X    of  plate    X  ^^^^  X       of  rivets 

of  plate  -  ^ — 

^  Thickness  of  plate  X  38500 


RIVETS   AND    RIVETING.  779 

By  making  the  clear  distance  between  each  end  rivet  of  a  row  and  the  side 
edge  of  tlie  plate  =  half  the  clear  distance  between  two  rivets  in  a  row ;  and 
calling  the  sum  of  the  two  end  distances  one  space,  we  have 

Number  of  spaces  _  Number  of  rivets 
in  a  row  ~  in  a  row. 

So  that 
The  clear  distance  between  two  rivets  in  a  row, 

which  is 

_    Total  net  width  of  plate 

~~  Number  of  spaces  in  a  row 
is  also 

_    Total  net  width  of  plate 
Number  of  rivets  in  a  row 

Diam  of  ^  Thickness  ^  oc\t\(\(\  s^  Total  number 
rivet     ^    of  plate     ^  ^""""  ^       of  rivets 
•  ~         Thickness  ^  Qocnn  v  Number  of  rivets 

of  plate    ^  "^^^^^  ^         in  a  row. 
But 

Total  number  of  rivets  ^^      , 

=  Number  of  rows. 


Number  of  rivets  in  a  row 


Therefore,   omitting  "thickness  of  plate,"  common  to  both  numerator  and 
denominator,  we  have,  as  in  rule  in  Art  5, 

Clear  distance  _  Diam  of  rivet  X  60000  X  Number  of  rows 
apart  —  33500 

But  if  tlie  diameter  of  tbe  rivets  is  less  tban  0.85  times  tbe 
tbickness  of  tbe  plates,  the  shearing  strength  of  a  double-cover  butt  joint 
(with  our  assumed  constants  for  shearing  and  crippling)  is  less  than  its  crippling 
strength.     In  such  cases,  for  the  clear  distance  between  two  rivets  in  a  row,  say 

_  , .  Circular  area  of  a  rivet  X  Shearing  unit  ,     ^ 

Clear  distance  =       Thickness  of  plate  X  Tension  unit  ■      ><  ^ 

JBem.  1.  Butt  joints  in  double  sbear,  or  with  2  covers,  being  the 
only  ones  here  considered,  and  inasmuch  as  rivets  may  always  be  used  with  a  diam 
greater  than  .86  of  the  thickness  of  the  plate,  we  may  in  practice  always  use  the 
Rule  in  Art  5  for  such  joints;  and,  therefore,  we  gave  it  alone. 

Rem.  2.  Wben  usin^  tbese  rules  for  otber  kinds  of  Joint, 
such  as  laps,  or  butts  with  single  covers,  remember  that  the  rivets  in  such  are  in 
sing^le  shear;  and,  therafore,  we  can  use  Rule  in  Art  5  (for  crippling)  only  when 
the  diam  is  either  1.7  or  more  times  the  thickness  of  plate.  If  less,  use 
Rule  above  for  shearing:;  all  on  the  assumption  that  our  foregoing  coefs  of 
crippling  and  shearing  are  used. 

But  tbe  coef  for  tension  must  be  cbang-ed  for  each  kind  of  these 
other  joints,  to  allow  for  the  weakening  effects  of  the  bending  shown  at  W,  Figs 
3,  as  deduced  approximately  from  experiment.  The  writer  believes  that  the  fol- 
lowing tension  units  will  give  safe  approximate  results  without  friction.  For 
double-cover  butts,  double-riveted,  38500  lbs  per  sq  inch,  as  adopted  above. 
For  double-riveted  laps,  or  one-cover  butts,  28000.  For  sing:le-ri veted 
laps,  or  one-cover  butts,  24000.  But,  as  before  remarked,  no  great  certainty  is 
attainable  in  riveting. 

Rem.  3.  A  Joint  may  fail  by  crippling  without  the  facts  being 
known  or  even  suspected,  for  it  does  not  imply  that  anything  breaks,  but 
merely  that  the  joint  has  stretcbed ;  and  this  might  not  be  detected  even  on 
a  slighi  inspection  of  it.  Still  it  might,  and  probably  often  has  sufficed  to  endanger, 
and  even  destroy  both  bridges  and  roofs  by  generating  strains  where  none  were 
provided  for. 


780  RAILROAD   CURVES. 

RAIIiROAI)  CURVES. 

I>efiiiitioiis. 

A  circular  railroad  curve  ahcd.  Fig.  1,  is  an  arc  of  a  circle  joining  two  straight 
lines,  or  tang^ents,  e  t  and  i  z^  in  the  survey. 

Tlie  point  of  curve  is  the  beginning  a  of  the  curve,  or  th,at  end  of  it 
first  reached  by  the  survey  in  its  progress. 

Tlie  point  of  tang^ent  is  the  other  end  d  of  the  curve. 

Tlie  point  of  intersection  or  apex  is  the  point  i  where  the  two  tan- 
gents e  i  and  i  z  intersect. 

P.  C,  P.  T.  and  P.  I.  The  stakes  driven  at  the  point  of  curve,  point  of 
tangent,  and  point  of  intersection  are  marked  P.  C,  P.  T.  and  P.  I.  respectively, 
and  the  points  and  stakes  are  commonly  referred  to  by  those  letters.  The  point 
of  intersection,  however,  is  not  always  located. 

The  apex  distance  *  is  the  distance  aior  d  i  measured  along  a  tangent, 
from  either  end  a  or  d  of  the  curve,  to  the  apex  i  or  intersection  of  the  two 
tangents. 

t 


A  curve  may  be  located  by  setting  up  the  transit  at  the  point  (as  a)  where  the 
curve  is  to  join  either  tangent,  laying  off  equal  angles  iab,bac,  cad,  and  meas- 
uring off  the  equal  chords  (usually  100  feet>  ab,bc,  cd.f  Inasmuch  as  these 
equal  chords  are  usually  laid  oflf  with  the  full  length  of  a  chain  or  steel  tape,  we 
shall  call  them  cliains,  to  distinguish  them  from  other  chords,  such  as  a  c  or 
a  d,  etc.,  which  may  be  drawn  to  the  curve. 

Tlie  total  angle  of  the  curve  is  the  angle  ^?:  2;  between  the  two  tangents. 
It  is  equal  to  the  central  angle  aod  subtended  by  the  curve. 

The  degree  of  curvatnre  is  the  angle  aob,boc,  etc.  subtended  at  the 
center  by  a  chain.    It  is  equal  to  the  deflection  angle  J  m  6  c  formed  be- 

*The  apex  distance  is  often,  but  unfortunately,  called  the  "  tangent,"  and 
sometimes  the  "tangent  distance." 
t  But  see  Sub-chains. 
t Many  writers  call  iab,bac,  etc.  the  deflection  angle. 


BAILEOAD   CURVES. 


781 


tween  any  chain  be  and  the  extension  6m  of  an  adjoining  chain  ab,  or  to  the 
angle  tsn  formed  between  the  tangents  at  and  b  n  which  touch  the  curve  at  the 
two  ends  a  and  6  of  a  chain.  It  is  therefore  the  angle  through  which  the  direc- 
tion of  the  line  deflects  in  that  portion  ot  the  curve  subtended  by  one  chain. 
The  sharper  the  curve,  the  greater  the  deflection  angle  and  the  shorter  the 
ra<liiis  oa,  ob,  etc. 

A  one-degree,  two-degree,  three-degree,  etc.  curve  is  one  whose  deflection 
angle  or  degree  of  curvature  is  1"^,  2°,  3°,  etc. 

Tlie  tang^ential  ang-le  is  the  angle  iab,bac,  etc.  used  in  laying  off"  the 
curve.    It  is  equal  to  one-half  the  deflection  angle. 

The  deflection  distance.  Let  any  chain,  as  o6,  Fig.  2,  be  extended  to 
w,  and  bm  made  ^  ab.  Then  the  distance  m  c  from  m  to  the  end  c  of  the  next 
chain  6  c  is  called  the  deflection  distance  of  the  curve. 

Tlie  tangential  distance.  Let  any  chain,  as  6  c,  be  extended  to  v,  and 
let  cv  be  made  =  6  c.  Also  let  C2  =  ct;  be  laid  otf  upon  a  tangent  to  the  curve  at 
c.    Then  vz  is  called  the  tangential  distance  of  the  curve. 

By  means  of  the  deflection  and  tangential  distances  given  in  the  tables,  pp. 
784-786,  a  curve  may  be  located  without  a  transit  by  means  of  a  chain  (or  100 
feet  tape)  for  measuring  bm^cv,  etc.,  and  a  rod  or  tape,  etc.  for  measuring  m  c, 
vz,  etc. 

An  ordinate  is  any  line  drawn  from  a  chord  to  the  curve,  at  right  angles 
to  the  chord,  whether  the  chord  be  a  "  chain  "  or  not. 

The  middle  ordinate  is  the  ordinate  en,  Fig.  2,  at  the  middle  point  e 
of  a  chord  6  c. 


Sub-chains,  etc.  For  facility  of  explanation  we  have  hitherto  treated  of 
curves  as  being  composed  entirely  of  full  chains  ;  but  such  curves  seldom  occur 
in  practice.  Usually,  after  dividing  a  curve  into  as  many  full  chains  as  possi- 
ble, there  is  a  fraction  of  a'  chain,  or  .m&-chain,  left  over.  Besides,  the  chances 
are  that  a  curve  will  not  begin  or  end  at  a  full  100-feet  station  of  the  survey, 
but  at  a  point  between  two  such  stations,  as  in  Fig.  3;  and,  inasmuch  as  it  is 
desirable  to  carry,  throughout  the  curve,  the  same  numbering  of  the  stations  as 
we  have  on  the  tangents,  the  P.  C.  and  the  P.  T.  are  in  these  cases  treated  as 
fractional  stations. 

Thus,  in  Fig.  3,  the  P.  C,  at  a,  is  supposed  to  be  41  feet  beyond  station  122. 
"We  therefore  call  the  P.  C,  in  this  case,  station  122  -f  41,  and  station  123  thus 
falls  in  its  proper  place,  at  b,  100  feet  in  advance  of  station  122,  as  measured,  first 
along  the  tangent  a;  a  41  feet  to  the  P.  C,  and  thei5  along  the  sub-chain  a  6  of  59 
feet. 

Similarly,  the  P.  T.,  in  Fig.  3,  happens  at  a  point,  d,  on  the  tangent  dy,  20.8 
feet  back  from  station  125,  or  79.2  feet  (the  length  of  the  sub-chain  cd)  beyond 
station  124  or  c.    The  P.  T.  therefore  becomes  station  124  +  79.2. 


782 


RAILROAD   CURVES. 


Owing  to  the  occurrence  of  the  sign  -f-  (plus)  in  the  number  of  the  P.  C.  or 
P.  T.  of  a  curve  beginning  or  ending  with  a  sub-chain,  such  a  station  is  com- 
monly called  a  "plus." 


Figr.3 


The  snb-tang'eiitial  an^le.  Fig.  3,  is  the  angle  bat(=adb)  or  dch 
{=  dac)  subtended  by  a  sub-chain  ;  the  vertex,  a,  d  or  c,  of  the  angle  lying  in 
the  curve  itself.  We  shall  give  the  names  initial  and  final  sub-tangential 
angles  to  the  angles  hat  and  dc^  subtended  by  the  initial  and  final  sub-chains 
a  b  and  c d  respectively.  If  a ^  be  made  =  ab,  and  ch  =  cd,  the  chords  tb  and 
h  d  are  called  the  initial  and  final  snb-tang;ential  distances  respec 
tively. 

SSnb-deflection  angles.  Fig.  3.  The  initial  sub-deflection  angle  is 
the  angle  sbc  formed  between  the  first  full  chain  b  c  and  the  extension  6  s  of  the 
initial  sub-chain  a b.  The  final  sub-deflection  angle  is  the  angle  kcd  between 
the  final  sub-chain  cd  and  the  extension  ck  of  the  preceding  full  chain  b  c.  If 
6 5  be  made  =  6  c,  and  ck  =  cd,  the  chords  sc  and  kd  are  called  the  initial  and 
final  sab-deflection  distances  respectively. 

A  long:  chord  is  a  chord,  a c  or  ad,  Figs.  1  and  3,  subtending  two  or  more 
chains  or  sub-chains. 

A  simple  cnrve.  Figs.  1  to  3,  is  one  in  which  the  radius  remains  of  con- 
stant length  throughout  and  in  which  the  curvature  is  all  in  one  direction. 
Such  a  curve  is  therefore  an  arc  of  but  one  circle. 


A  componnd  curve,  as  ab  cd,  Fig.  4, is  one  in  which  the  curvature  is  ail 
in  one  direction,  but  which  consists  of  circular  arcs  described  with  two  or  more 


EAILEOAD   CUEVES. 


783 


diflerent  radii,  as  ob,  o'b,  o" e,  etc.    The  portions,  as  ab,bc,  cd,  described  with 
the  diflfereiit  radii,  are  called  the  brancbes  of  the  curve. 


Fig. 


A  reverse  curve,  a  be,  Fig.  5,  consists  of  two  curves  immediately  adjoin- 
ing one  another  (i.  e.,  without  any  straight  track  between  them)  and  curving  in 
opposite  directions.  The  radii  ob,  o'b,  of  the  two  portions,  or  branches,  of  a 
reverse  curve  may  be  of  equal  or  of  unequal  length,  and  the  total  angles,  aob^ 
bo' c,  of  the  two  branches,  may  be  equal  or  unequal. 


RAILROADS. 


Table  of  Radii,  Middle  Ordinates,  <&c,  of  CarTes. 

Contains  no  error  as  great  as  1  in  the  last  figure. 


Chord  100  feet. 


Ang.  of 
Deft. 

Bad. 
In  ft. 

Defl. 
Dlst. 
in  ft. 

Tang. 
Dist. 
In  ft. 

Mid. 
Ord. 

Ang.  of 
Defl. 

Had. 
in  ft. 

Defl. 
Dist. 
in  ft. 

Tang. 
Dist. 
in  ft. 

Mid. 
Ord. 

O   ' 

I 

343775 

.029 

.014 

.004 

0  ' 

1  36 

8581 

2.798 

1.396 

.349 

2 

171887 

.058 

.029 

.007 

38 

3508 

2.851 

1.425 

.356 

3 

114592 

.087 

.043 

.011 

40 

3438 

2.909 

1.454 

.364 

4 

85944 

.116 

.058 

.014 

42 

3370 

2.967 

1.483 

.371 

5 

68755 

.145 

.072 

.018 

44 

3306 

3.025 

1.512 

.378 

6 

67296 

.175 

.087 

.022 

46 

3243 

3.084 

1.542 

.385 

7 

49111 

.204 

.102 

.025 

48 

3183 

3.142 

1.571 

.393 

8 

4-2972 

.233 

.116 

.029 

50 

3125 

8.200 

1.600 

.400 

9 

38197 

.262 

.131 

.033 

52 

3070 

8.257 

1.629 

.407 

10 

34377 

.291 

.145 

.036 

54 

3016 

3.316 

1.658 

.414 

11 

31252 

.320 

.160 

.040 

56 

2964 

3.374 

1.687 

.422 

12 

28648 

.349 

.174 

.043 

58 

2913 

3.433 

1.716 

.429 

13 

26444 

.378 

.189 

.047 

3 

2865 

3.490 

1.745 

.436 

14 

24556 

.407 

.203 

.051 

2 

2818 

3.549 

2.774 

.443 

16 

22918 

.486 

.218 

.054 

4 

2773 

3.606 

1.803 

.451 

18 

21486 

.465 

.232 

.058 

6 

2729 

8.664 

1.832 

.458 

17 

20222 

.494 

.247 

.062 

8 

2686 

3.723 

1.861 

.465 

18 

19098 

.524 

.262 

.065 

10 

2645 

3.781 

1.890 

.473 

19 

18094 

.653 

.276 

.069 

12 

2605 

3.839 

1.919 

.480 

20 

17189 

.682 

.291 

.078 

14 

2566 

3.897 

1.948 

.487 

21 

16370 

.611 

.305 

.076 

16 

2528 

3.956 

1.978 

.495 

22 

15626 

.640 

.320 

.080 

18 

2491 

4.014 

2.007 

.502 

23 

14947 

.669 

.334 

.083 

20 

2456 

4.072 

2.036 

.609 

^ 

14324 

.698 

.349 

.087 

22 

2421 

4.131 

2.065 

.616 

•i6 

13751 

.727 

.863 

.091 

24 

2387 

4.189 

2.094 

.623 

26 

13222 

.756 

.378 

.095 

26 

2355 

4.246 

2.123 

.631 

27 

12732 

.785 

.392 

.098 

28 

2323 

4.305 

2.152 

.638 

28 

12278 

.814 

.407 

.102 

30 

2292 

4.363 

2.182 

.545 

29 

11854 

.844 

.422 

.105 

32 

2262 

4.421 

2.210 

.552 

30 

11459 

.873 

.436 

.109 

34 

2232 

4.480 

2.240 

.560 

31 

11090 

.902 

.451 

.113 

86 

2204 

4.537 

2.268 

.667 

32 

10743 

.931 

.465 

.116 

38 

2176 

4.596 

2.298 

.574 

33 

10417 

.960 

.480 

.120 

40 

2149 

4.654 

2.327 

.582 

34 

lOUl 

.989 

.494 

.123 

42 

2122 

4.713 

2.356 

.589 

35 

9822 

1.018 

.509 

.127 

44 

2096 

4.771 

2.385 

.696 

36 

0549 

1.047 

.523 

.131 

46 

2071 

4,829 

2.414 

.603 

37 

9291 

1.076 

.538 

.134 

48 

2046 

4.888 

2.444 

.611 

38 

9047 

1.105 

.552 

.138 

50 

2022 

4.946 

2.473 

.618 

39 

8815 

1.134 

.567 

.142 

52 

1999 

5.003 

2.501 

.625 

40 

8594 

1.164 

.582 

.145 

54 

1976 

5.061 

2.530 

.632 

41 

8385 

1.193 

.596 

.149 

56 

1953 

5.120 

2.560 

.640 

42 

8185 

1.222 

.611 

.153 

58 

1932 

5.176 

2.588 

.647 

43 

7995 

1.251 

.625 

.156 

8 

1910 

5.235 

2.618 

.654 

44 

7813 

1.280 

.640 

.160 

2 

1889 

5.294 

2.647 

.662 

45 

7639 

1.309 

.654 

.164 

4 

1869 

5.350 

2.675 

.669 

46 

7473 

1.338 

.669 

.167 

6 

1848 

5.411 

2.705 

.676 

47 

7314 

1.367 

.683 

.171 

8 

1829 

5.467 

2.784 

.683 

48 

7162 

1.396 

.698 

.174 

10 

1810 

5.526 

2.763 

.691 

49 

7016 

1.425 

.n2 

.178 

12 

1791 

5.583 

2.792 

.698 

50 

6876 

1.454 

.727 

.182 

14 

1772 

5.643 

2.821 

.705 

51 

6741 

1.483 

.741 

.185 

16 

1754 

5.707 

2.850 

.713 

62 

6611 

1.513 

.767 

.189 

18 

1736 

5.760 

2.880 

.720 

63 

6486 

1.542 

'  .771 

.193 

20 

1719 

5.817 

2.908 

.727 

54 

6366 

1.571 

.786 

.197 

22 

1702 

5.875 

2.937 

.734 

55 

6251 

1.600 

.800 

.200 

24 

1685 

5.935 

2.967 

.742 

56 

6139 

1.629 

.815 

.204 

26 

1669 

5.992 

2.996 

.749 

67 

6031 

1.658 

.829 

.207 

28 

1653 

6.050 

3.025 

.766 

58 

5927 

1.687 

.844 

.211 

30 

1637 

6.108 

3.054 

.764 

59 

5827 

1.716 

.858 

.214 

32 

1622 

6.166 

3.083 

.771 

1 

5730 

1.745 

.872 

.218 

34 

1607 

6.223 

3.112 

.778 

2 

5545 

1.803 

.902 

.225 

36 

1592 

6.281 

3.140 

.786 

4 

5372 

1.862 

.931 

.233 

38 

1577 

6.341 

3.170 

.793 

6 

5209 

1.920 

.960 

.240 

40 

1563 

6.398 

3.199 

.800 

8 

5056 

1.978 

.989 

.247 

42 

1549 

6.456 

3.228 

.807 

10 

4911 

2.036 

1.018 

.255 

44 

1535 

6.515 

3.257 

.814 

12 

4775 

2.094 

1.047 

.262 

46 

1521 

6.575 

3.287 

.822 

14 

4646 

2.152 

1.076 

.269 

48 

1508 

6.631 

3.316 

.829 

16 

4523 

2.211 

1.105 

.276 

50 

1495 

6.689 

3.345 

.836 

18 

4407 

2.269 

1.134 

.284 

52 

1482 

6.748 

3.374 

.843 

20 

4297 

2.327 

1.163 

.291 

54 

1469 

6.807 

3.403 

.851 

22 

4192 

2.385 

1.192 

.298 

56 

1457 

6.863 

3.432 

.858 

24 

4093 

2.443 

1.221 

.305 

58 

1445 

6.920 

3.460 

.865 

26 

3997 

2.502 

1.251 

.313 

4 

1433 

6.980 

3.490 

.873 

28 

3907 

2.560 

1.280 

.320 

5 

1403 

7.125 

3.562 

.891 

30 

3820 

2.618 

1.309 

.327 

10 

1375 

7.271 

3.6.S5 

.909 

32 

3737 

2.676 

1.338 

.334 

15 

1348 

7.416 

3.708 

.927 

34 

3657 

2.734 

1.367 

.342 

20 

1323 

7.561 

3.781 

.945 

RAILROADS. 


785 


Table  of  Radii,  Middle  Ordinates,  Ac,  of  CarT€»8.   Chord  100  feet. 

(Continued.) 

The  Tangential  Angle  is  always  one-half  of  the  Angle  of  Deflection. 


Aug.  of 

D9fl. 

Bad. 
in  ft. 

Defl. 
Dist. 
in  ft. 

Tang, 
in  ft. 

Mid.» 
OnL 

Ang.  of 
Defl. 

Bad. 
in  ft. 

Defl. 
Dist. 
in  ft. 

Tang. 
Dist. 
in  ft. 

Mid.* 
Ord. 

4  25 

1298 

7.707 

3.854 

.963 

O     ' 

10  15 

559.7 

17.87 

8.942 

2.238 

30 

1274 

7.852 

3.927 

.982 

30 

546.4 

18.30 

9.160 

2.292 

35 

1250 

7.997 

8.999 

1.000 

45 

533.8 

18.73 

9.378 

2.34T 

40 

1228 

8.143 

4.072 

1.018 

11 

521.7 

19.17 

9,596 

2.402 

45 

1207 

8.288 

4.145 

1.036 

15 

510.1 

19.60 

9.814 

2.456 

50 

1186 

8.433 

4.218 

1.054 

30 

499.1 

20.04 

10.03 

2.511 

55 

1166 

8.579 

4.290 

1.072 

45 

488.5 

20.47 

10.25 

2,566 

5 

1146 

8.724 

4.363 

1.091 

12 

478.3 

20.91 

10.47 

2.620 

5 

1128 

8.869 

4.436 

1.109 

15 

468.6 

21.34 

10.69 

2.675 

10 

1109 

9.014 

4.508 

1.127 

30 

459.3 

21.77 

10.90 

2.730 

15 

1092 

9.160 

4.581 

1.145 

45 

450.3 

22.21 

11.12 

2.785 

20 

1075 

9.305 

4.654 

1.164 

13 

441.7 

22.64 

11.34 

2.889 

25 

1058 

.9.450 

4.727 

1.182 

15 

433.4 

23.07 

11.56 

2,894 

30 

1042 

9.596 

4.799 

1.200 

30 

425.4 

23.51 

11.77 

2.949 

35 

1027 

9.741 

4.872 

1.218 

45 

417.7 

23.94 

11.99 

3.003 

40 

1012 

9.886 

4.945 

1236 

14 

410.3 

24.37 

12.21 

3.058 

45 

996.9 

10.03 

5.017 

1.255 

15 

403.1 

24.81 

12,43 

3,113 

50 

982.6 

10.18 

5.090 

1.273 

30 

396.2 

25.24 

12,65 

3.168 

55 

968.8 

10.32 

5.163 

1.291 

45 

389.5 

25.67 

12,86 

3.223 

6 

955.4 

10.47 

5.235 

1.309 

15 

383.1 

26.11 

13.08 

3,277 

5 

942.3 

10.61 

5.308 

1.327 

15 

376.8 

26.54 

13.30 

3,332 

10 

929.6 

10.76 

5.381 

1.346 

30 

370.8 

26.97 

13,52 

3.387 

15 

917.2 

10.90 

5.453 

1.364 

45 

364.9 

27.40 

18,73 

3.442 

20 

905.1 

11.05 

5.526 

1.382 

16 

359.3 

27.84 

13.95 

3,496 

i 

893.4 

11.19 

5.599 

1.400 

30 

348.5 

28.70 

14.39 

3,606 

882.0 

11.34 

5.672 

1.418 

17 

338.3 

29.56 

14.82 

3,716 

35 

870.8 

11.48 

5.744 

1.437 

30 

328.7 

30.43 

15.26 

3.825 

40 

859.9 

11.63 

5.817 

1.455 

18 

319.6 

31.29 

15.69 

3,935 

45 

849.3 

11.77 

5.890 

1.473 

30 

811.1 

32.15 

16,13 

4,045 

50 

839.0 

11.92 

5.962 

1.491 

19 

302.9 

33.01 

16,56 

4,155 

55 

828.9 

12.07 

6.035 

1.510 

30 

295.3 

33.87 

17,00 

4,265 

7 

819.0 

12.21 

6.108 

1.528 

20 

287.9 

34,73 

17.43 

4.375 

5 

809.4 

12.36 

6.180 

1.546 

21 

274.4 

86.44 

18.30 

4,594 

10 

800.0 

12.50 

6.25S 

1.564 

22 

262.0 

38.17 

19.17 

4,815 

15 

790.8 

12.65 

6.326 

1.582 

23 

250.8 

39.87 

20.04 

5,035 

20 

781.8 

12.79 

8.398 

1.600 

24 

240.5 

41.58 

20.91 

5,255 

25 

773.1 

12.94 

6.471 

1.619 

25 

281.0 

43.29 

21.77 

5,476 

30 

764.5 

13.06 

6.544 

1.637 

26 

222.8 

44.98 

22.64 

5,696 

35 

756.1 

13.23 

6.616 

1.655 

27 

214.2 

46.69 

23.51 

5,917 

40 

747.9 

18.37 

6.689 

1.673 

38 

206.7 

48.38 

24,37 

6,139 

45 

7.39.9 

13.52 

6.762 

1.691 

29 

199.7 

50.08 

25.24 

6,361 

50 

732.0 

13.66 

6.835 

1.710 

30 

193.2 

51,76 

26.11 

6.582 

55 

724.3 

13.81 

6.907 

1.728 

31 

187.1 

53.45 

26.97 

6.805 

8 

716.8 

13.95 

6.980 

1.746 

32 

181.4 

55.18 

27.83 

7.027 

15 

695.1 

14.39 

7.198 

1.801 

83 

176.0 

56.82 

28.70 

7.252 

30 

674.7 

14.82 

7.416 

1.855 

34 

171.0 

58.48 

29.56 

7.473 

45 

655.5 

15.26 

7.634 

1.910 

35 

166.3 

60.13 

30.42 

7.695 

9 

637.3 

15.69 

7.852 

1.965 

86 

161.8 

61.80 

31.29 

7.919 

15 

620.1 

16.13 

8.070 

2.019 

37 

157.6 

63.45 

32.15 

8.142 

30 

603.8 

16.56 

8.288 

2.074 

38 

153.6 

65.10 

38.01 

8.366 

45 

588.4 

17.00 

8.506 

2.128 

39 

149.8 

66.76 

83.87 

8.591 

10 

573.7 

17.43 

8.724 

2.183 

40 

146.2 

68.40 

84.78 

8.816 

To  find  tang^ential  and  deflection  ang^les  for  any  given  rad  and 
chord..  Divide  half  the  chord  by  the  rad.  The  quot  will  be  nat  sine  of  the  tangl 
ang.     Find  this  tangl  ang  in  the  table  of  nat  sines ;  and  muU  it  by  2  for  the  def  ang. 

To  find  tlie  def  dist  for  ehords  100  ft  longr-  Div  10000  by  the 
rad  in  feet. 

To  find  the  def  dist  for  eqnal  cliords  of  any  griven  leng:tli. 
Div  chord  by  rad.    Mult  quot  by  chord.    Or  div  sq  of  chord  by  rad. 

To  find  ttie  tangl  dist  for  equal  cliords  of  any  g-iven  leng^tti. 
First  find  the  tangl  ang  as  above.  Divide  it  by  2.  Find  in  the  table  of  nat  sines 
the  nat  sine  of  the  quot.    Mult  this  nat  sine  by  the  given  chord.    Muit  prod  by  2. 

To  find  tlie  rad  to  any  g^iven  def  ang-,  for  equal  chords  of  any 
length.  Divide  the  def  ang  by  2.  Find  nat  sine  of  the  quotient.  Divide  half 
the  chord  by  this  nat  sine. 

*  The  middle  ordinate  for  a  rad  of  600  ft  or  more,  (chord  100  ft,)  may  in 

practice  be  taken  at  one-fourth  of  the  tang  dist.     Even  in  400  ft  rad  it  will  be  too  short  only  6  in  the 
third  decimal. 

60 


786 


CIRCULAR  CURVES. 


Radii,   <&c,   of   Curves;    in   metres, 
d  ekametr  es. 


Chord,   20   metres 


The  stakes,  at  the  ends  of  the  2-dekametre  chords,  should  be  numbered  2,  4,  6,  Acj 
meaning.  2,  4,  6,  &c,  dekametres.  The  tangential  angle  in  the  table  will  then  give 
the  amount  of  deflection  per  unit  (dekametre)  of  measurement. 


"5) 
1 

0°    5' 

Is 

ii 

'•5  1 

11 

S 

1 

1 

4 

1^ 

143.36 

2.790 

¥ 

1.396 

0  ii 

SS 

0°  10' 

6875.50 

.058 

.029 

.007 

8° 

0' 

4°    0' 

.349 

20 

10 

3437.75 

.116 

.058 

.015 

10 

5 

140.44 

2.848 

1.425 

.356 

30 

15 

2291.84 

.175 

.087 

.022 

20 

10 

137.63 

2.906 

1.454 

.364 

40 

20 

1718.88 

.233 

.116 

.029 

30 

15 

134.94 

2.964 

1.483 

.371 

50 

25 

1375.11 

.291 

.145 

.036 

40 

20 

132.35 

3.022 

1.512 

.378 

1°     0' 

30 

1145.93 

.349 

.175 

.044 

50 

25 

129.85 

3.080 

1.541 

.386 

10 

35 

982.23 

.407 

.204 

.051 

9° 

0' 

30 

127.45 

3.138 

1.570 

.393 

20 

40 

859.46 

.465 

.233 

.058 

10 

35 

125.14 

3.196 

1.599 

.400 

30 

45 

763.97  1     .524 

.262 

.065 

20 

40 

122.91 

3.254 

1.629 

.407 

40 

50 

687.57 

.582 

.291 

.073 

30 

45 

120.76 

3.312 

1.658 

.415 

50 

55 

625.07 

.640 

.320 

.080 

40 

50 

118.68 

3.370 

1.687 

.422 

2°    0' 

1°     0! 

572.99 

.698 

.349 

.087 

50 

55 

116.68 

3.428 

1.716 

.429 

10 

5 

528.92 

.756 

.378 

.095 

10° 

0' 

5°    0' 

114.74 

3.486 

1.745 

.437 

20 

10 

491.14 

.814 

.407 

.102 

20 

10 

111.05 

3.602 

1.803 

.451 

30 

15 

458.40 

.873 

.436 

.109 

40 

20 

107.58 

3.718 

1.861 

.466 

40 

20 

429.76 

.931 

.465 

.116 

11° 

0^ 

80 

104.33 

3.834 

1.919 

.480 

50 

25 

404.48 

.989 

.494 

.124 

20 

40 

101.28 

3.950 

1.977 

.495 

3°    0' 

30 

382.02 

1.047 

.524 

.131 

40 

50 

98.39 

4.065 

2.035 

.509 

10 

35 

361.91 

1.105 

.553 

.138 

12° 

0' 

6°    0' 

95.67 

4.181 

2.093 

.524 

20 

40 

343.82 

1.163 

.582 

.145 

20 

10 

93.09 

4.297 

2.152 

.539 

30 

45 

327.46 

1.222 

.611 

.153 

40 

20 

90.65 

4.41S 

2.210 

.553 

40 

50 

312.58 

1.280 

.640 

.160 

13° 

0' 

30 

88.34 

4.528 

2.268 

.568 

50 

55 

298.99 

1.338 

.669 

.167 

20 

40 

86.14 

4.644 

'4.326 

.582 

4°    0' 

2°    0' 

286.54 

1.396 

.698 

.175 

40 

50 

84.05 

4.759 

2.384 

.597 

10 

5 

275.08 

1.454 

.727 

.182 

14° 

0' 

7°    0' 

82.06 

4.875 

2.442 

.612 

20 

10 

264.51 

1.512 

.756 

.189 

20 

10 

80.16 

4.990 

2.500 

.626 

30 

15 

254.71 

1.570 

.785 

.196 

40 

20 

78.34 

5.106 

2.558 

.641 

40 

20 

245.62 

1.629 

.814 

.204 

15° 

0' 

30 

76.61 

5.221 

2.616 

.655 

50 

25 

237.16 

1.687 

.844 

.211 

20 

40 

74.96 

5.336 

2.674 

.670 

5°    0' 

30 

229.26 

1.745 

.873 

.218 

40 

50 

73.37 

5.452 

2.732 

.685 

10 

35 

221.87 

1.803 

.902 

.225 

16° 

0' 

8°    0' 

71.85 

5.567 

2.790 

.699 

20 

40 

214.94 

1.861 

.931 

.233 

20 

10 

70.40 

5.682 

2.848 

.714 

30 

45 

208.43 

1.919 

.960 

.240 

40 

20 

69.00 

5.797 

2.906 

.729 

40 

50 

202.30 

1.977 

.989 

.247 

17° 

0' 

30 

67.65 

5.912 

2.964 

.743 

50 

55 

196.53 

2.035 

1.018 

.255 

20 

40 

66.36 

6  027 

3.022 

.758 

6°    0' 

3°    0' 

191.07 

2.093 

1.047 

.262 

40 

50 

65.12 

6.142 

3.080 

.772 

10 

5 

185.91 

2.152 

1.076 

.269 

18° 

0' 

9°    0' 

63.92 

6.257 

3.138 

.787 

20 

10 

181.03 

2.210 

1.105 

.276 

20 

10 

62.77 

6.372 

3.196 

.802 

30 

15 

176.39 

2.268 

1.134 

.284 

40 

20 

61.66 

6.487 

3.2.^4 

.816 

40 

20 

171.98 

2.326 

1.163 

.291 

19° 

0' 

30 

60.59 

6.602 

3.312 

.831 

50 

25 

167.79 

2.384 

1.192 

.298 

20 

40 

59.55 

6.71? 

3.370 

.846 

70    0' 

30 

163.80 

2.442 

1.222 

.306 

40 

50 

58.55 

6,831 

3.428 

.860 

10 

35 

160.00 

2.500 

1.251 

.313 

20° 

0' 

10°    0' 

57.59 

b.946 

3.486 

.875 

20 

40 

156.37 

2.558 

1.280 

.320 

21° 

0' 

30 

54.87 

7.289 

3.660 

.919 

30 

45 

152.90 

,  2.616 

1.309 

.327 

22° 

0' 

11°    0' 

52.41 

7.632 

3.834 

.963 

40 

50 

149.58 

2.674 

1.338 

.335 

23° 

0' 

30 

50.16 

7.975 

4.008 

1.007 

60 

55 

146.40 

2.732 

1.367 

.342 

24° 

0' 

12°    0' 

48.10 

8.316 

4.181 

1.051 

25° 

0' 

30 

46.20 

8.658 

4.355 

1.09* 

Radins  = 


Half  the  chord Half  the  chord  ^  cosecant  of  tangential 

Sine  of  tangential  angle  ^  angle. 


Deflection  dist : 


Square  of  chord 


==  Twice  the  chord  X 


sine  of  tangential 


Radius  "  "  ^""  """  """^ "  ^  angle. 

Tangential  dist  =  Twice  the  chord  X  sine  of  half  the  tangential  angle. 

Middle  ord  =  Radius  X  (1  —  cosine  of  tangential  angle)  =  Half  the  chord  X 
tangent  of  half  the  tangential  angle. 

For  curves  of  60  metres,  or  greater,  radius,  the  ordinate  at  5  metres  from 
the  end  of  the  20-metre  chord,  or  midway  between  the  end  of  the  chord  and  the  mid- 
dle ordinate,  may  be  taken  at  three-fourths  of  the  middle  ordinate. 


•    TABLE  OF  LONG  CHORDS. 


787 


Table  of  L.ongr  Chords. 

Lengths  of  Chord  in  ft,  required  to  subtend  from  1  to  4  stations  of  100  ft  each. 


Ang. 
of 

ISta. 

2  Sta. 

3Sta. 

t 
4Sta. 

^c^^- 

ISta. 

2  Sta. 

3  Sta. 

4  Sta. 

Defl. 

Defl. 

1° 

100 

200.0 

300.0 

400.0 

% 

100 

199.7 

298.9 

397.5 

\i 

100 

200.0 

300.0 

399.9 

6° 

100 

199.7 

298.8 

397.3 

i 

100 

200.0 

300.0 

399.9 

K 

100 

199.7 

298.7 

397.0 

100 

200.0 

300.0 

399.8 

'i 

100 

199.7^ 

298.6 

396.7 

2° 

100 

200.0 

299.9 

399.7 

100 

199.6 

298.5 

396.5 

1 

100 

200.0 

299.9 

399.6 

70  * 

100 

199.6 

298.4 

396.2 

100 

200.0 

299.8 

399.5 

/4 

100 

199.6 

298.3 

396.0 

100 

200.0 

299.8 

399.4 

100 

199.6 

298.2 

396.7 

3° 

100 

200.0 

299.7 

399.3 

100 

199.6 

298.1 

395.4 

i 

100 

200.0 

299.7 

399.2 

80 

100 

199.6 

298.0 

395.1 

100 

200.0 

299.6 

399.1 

,^ 

100 

199.5 

297.9 

394.8 

100 

200.0 

299.6 

399.0 

/* 

100 

199.5 

297.8 

394.5 

4° 

100 

199.9 

299.6 

398.9 

100 

199.4 

297.7 

394.3 

74 

100 

199.9 

299.5 

398.7 

9° 

100 

199.4 

297.5 

394.1 

100 

199.9 

299.4 

398.5 

i 

100 

199.4 

297.4 

393.7 

100 

199.9 

299.3 

398.3 

100 

199.3 

297.3 

393.2 

100 

199.9 

299.2 

398.0 

/4 

100 

199.2 

297.2 

392.8 

1 

100 

199  8 

299:1 

397.8 

10° 

100 

199.2 

297.0 

392.4 

100 

199.8 

299.0 

397.6 

Elevation  of  outer  rail  in  curves  theoretically  is  equal  in  ins  to  (square 
of  vel  in  ft  per  sec  X  gauge  in  ins)  -=-  (-Rad  of  curve  in  ft  X  32.2).  Experience 
has  shown  that  half  an  inch  for  each  degree  of  def  angle  (100  ft  chords)  does  very- 
wed  for  4  ft  8.5  ins  gauge  up  to  40  miles  per  hour.  At  60  miles  use  1  inch  per  deg. 
In  dangerous  places  this  may  be  increased  for  safety  against  high  winds  Ap- 
proaching the  curve  raise  the  outer  rail  at  the  rate  of  1  inch  in  about  60  or  80  ft. 
When  the  ends  of  a  curve  are  tapered  off  by  transition  curves,  the  rise  is  made 
upon  the  latter. 

Relation  of  radius  to  leng^th  of  wbeel-base. 
Mr.  A.  M.  Wellington  *  found  by  experimeuts  with  models  that  a  rigid  truck 
passing  around  a  curve,  whether  alone  or 
coupled   with    another    truck,    assumes 
the  position  shown  in  this  Fig.,t  i.e.,  the 
flange  of  the  outer  front  wheel  presses 
against  the  outer  rail,  and  the  rear  axle 
coincides  with  a  radius  to  the  curve. 
Then,  for  the    angle  A,  between    that 
radius  and  the  radius  K  which  passes 
through  the  center  of  the  front  axle ; 
„  .       wheel-base  B 

sine  of  A  = Tr"""^T; —  ; 

radius  R 

and  the  space  d  between  the  flange  of  the  outer  hind  wheel  and  the  outer  rail  is, 
d  =  radius  R  X  versed  sine  of  angle  A,  very  nearly. 
For  a  given  wheel-base  B,  we  have,  approximately, 

(I  =  [d  for  a  1°  curve)  X  degree  of  curvature ; 
and  the  inner  hind  wheel  will  touch  the  inner  rail  when  d  becomes  equal  to  the 
total  room  for  play  left  between  the  wheel-flanges  and  the  rails,  L  e.,  when 

total  play 

degree  of  curvature  =  -5-^ -^ ■. 

d  for  a  1"  curve 
This  commonly  occurs  on  European  railways,  where  the  cars  have  rigid  wheel- 
bases  much  longer  than  our  pivoted  trucks,  and  where  d  for  a  given  radius  is 
therefore  much  greater  than  with  us.    Hence  the  inner  rails  on  curves  are  more 
generally  worn  there  than  here. 

For  a  wheel-base  5  ft.  long,  "  (/  for  a  1°  curve  "  is  0.0022  ft.  It  varies  (nearly)  as 
the  square  of  the  length  of  the  wheel-base,    d  is  independent  of  the  gauge. 

*  The  Economic  Theory  of  Railway  Location,  New  York,  John  Wiley  &  Sons,  1887. 

t  In  our  figure,  necessarily  mnch  exaggerated,  we  omit,  for  simplicity,  the  treadg 
of  the  wheels,  all  of  which  are  supposed  to  rest  on  the  rails,  and  show  only  so  much 
of  their  flanges  as  extends  below  the  top  of  the  rail. 


788 


TABLE   OF   ORDINATES. 


Table  of  Ordinate^  5  ft  apart.    Chord  100  ft. 

For  Railroad  Onrves. 


Ordinates  for  angles  intermediate  of  those  in  the  table  can  at  once  be  found  bj 
eimple  proportion. 

Distances  of  the  Ordinates  from  the  end  of  the  100  feet  Chord. 


Ang.  of 
Defl. 

Mid. 
50  ft. 

46  ft. 

40  ft. 

85  ft. 

30  ft. 

25  ft. 

20  ft. 

15  ft. 

10  ft. 

5  ft. 

4 

.014 

• 
.014 

.014 

.013 

.012 

.010 

.008 

.008 

.006 

.003 

8 

.029 

.029 

.028 

.026 

.024 

.022 

.018 

.015 

.010 

.005 

12 

.043 

.043 

.041 

.038 

.037 

.033 

.028 

.022 

.015 

.008 

16 

.058 

.058 

.056 

.052 

.049 

.044 

.037 

,080 

.020 

.011 

20 

.073 

.072 

.070 

.066 

.061 

.055 

.047 

.087 

.026 

.014 

24 

.087 

.086 

.083 

.077 

.074 

.066 

.056 

.045 

.031 

.017 

28 

.102 

.101 

.098 

.092 

.086 

.077 

.065 

.052 

.036 

.019 

32 

.116 

.115 

.112 

.106 

.098 

.088 

.075 

.058 

,042 

.022 

36 

.131 

.130 

.126 

.119 

.110 

.099 

.084 

.066 

.047 

.024 

40 

.145 

.144 

.140 

.133 

.123 

.110 

.093 

.074 

.062 

.027 

44 

.160 

.158 

.153 

.145 

.135 

.121 

.103 

.081 

.067 

.030 

48 

.174 

.172 

.167 

.158 

.147 

.182 

.112 

.088 

.062 

.033 

52 

.189 

.187 

.181 

.171 

.159 

.143 

.122 

.096 

.068 

035 

56 

.204 

.202 

.195 

.185 

.171 

.154 

.131 

.103 

.073 

.038 

1 

.218 

.216 

.209 

.198 

.183 

.164 

.140 

.111 

.078 

.041 

4 

.233 

.231 

.223 

.211 

.196 

.176 

.150 

.118 

.083 

.043 

8 

.247 

.245 

.237 

.224 

.208 

.186 

.159 

.125 

.088 

.046 

12 

.262 

.260 

.252 

.237 

.220 

.196 

.168 

.133 

.094 

.049 

16 

.276 

.274 

.265 

.251 

.232 

.207 

.177 

.140 

.099 

.052 

20 

.291 

.288 

.279 

.264 

.244 

.218 

.187 

.148 

.104 

.055 

24 

.306 

.303 

.293 

.277 

.256 

.229 

.197 

.155 

.109 

.057 

28 

.320 

.317 

.307 

.291 

.269 

.240 

.206 

.163 

.114 

.060 

32 

.,334 

.331 

.321 

..304 

.281 

.251 

.215 

.171 

.120 

.063 

36 

.349 

.345 

.335 

.317 

.293 

.262 

.224 

.178 

.125 

.066 

40 

.364 

.360 

.349 

.330 

.305 

.273 

.233 

.186 

.130 

.069 

44 

.378 

.374 

.363 

.343 

.318 

.284 

.242 

.192 

.135 

.072 

48 

.393 

.389 

.377 

.356 

.330 

.295 

,251 

.200 

.141 

.075 

52 

.407 

.403 

.391 

.370 

.342 

.305 

.261 

.208 

.147 

.077 

56 

.422 

.418 

.405 

.383 

.354 

.316 

.270 

.216 

.152 

.080 

2 

.436 

.432 

.419 

.397 

.366 

.327 

.280 

.222 

.157 

.083 

4 

.451 

.446 

.433 

.409 

.379 

.338 

.289 

.230 

.162 

,086 

8 

.465 

.461 

.447 

.425 

.391 

.349 

,298 

,237 

.167 

,088 

12 

.480 

.475 

.461 

.437 

.403 

.360 

.308 

.245 

.173 

.090 

16 

.495 

.490 

.475 

.450 

.415 

.371 

.317 

.252 

.178 

,093 

20 

.509 

.504 

.489 

.463 

.428 

.382 

.326 

.260 

.183 

.096 

24 

.522 

.518 

.503 

.476 

.440 

.393 

.334 

.267 

.188 

.099 

28 

.538 

.533 

.517 

.489 

.452 

.404 

.346 

.275 

.194 

.102 

32 

.552 

.547 

.531 

.503 

.465 

,415 

..355 

.282 

.199 

,104 

36 

.567 

.562 

.545 

.516 

.477 

.426 

.364 

.289 

.204 

.107 

40 

.582 

.576 

.559 

.529 

.489 

.436 

.373 

.297 

.209 

.110 

44 

.596 

.590 

.573 

.542 

.501 

.447 

.382 

.304 

.214 

.113 

48 

.611 

.605 

.587 

.555 

.513 

.458 

.391 

.312 

.219 

.116 

52 

.625 

.619 

.601 

.569 

.526 

.469 

.401 

.319 

.225 

.118 

56 

.640 

.634 

.615 

.582 

.538 

.480 

.410 

.326 

.230 

.121 

8 

.654 

.648 

.629 

.595 

.550 

.491 

.419 

.334 

.235 

.124 

4 

.669 

.662 

.643 

.608 

.562 

.502 

.428 

.341 

.240 

.127 

8 

.683 

.677 

.657 

.621 

.574 

.512 

.438 

.349 

.246 

.130 

12 

.698 

.691 

.671 

.635 

.587 

.523 

.448 

.357 

.251 

.132 

16 

.713 

.705 

.685 

.649 

.599 

.534 

.457 

.364 

.257 

.135 

20 

.727 

.720 

.699 

.662 

.611 

.545 

.466 

.371 

.262 

.138 

24 

.742 

.734 

.713 

.675 

.623 

.556 

.475 

.378 

.267 

.141 

28 

.756 

.749 

.727 

.688 

.635 

.567 

.485 

.386 

.272 

.144 

32 

.771 

-763 

.741 

.702 

.648 

.578 

.494 

.394 

.278 

.146 

36 

.786 

.777 

.755 

.715 

.660 

.589 

.508 

.401 

.283 

.149 

40 

.800 

.792 

.769 

.728 

.673 

.600 

.512 

.408 

,288 

.152 

44 

.814 

.806 

.783 

.741 

.685 

.611 

•521 

.415 

,293 

.155 

48 

.829 

.821 

.797 

.754 

.697 

.621 

.531 

.423 

.298 

.158 

52 

.843 

.835 

.811 

.768 

.709 

.632 

.541 

.431 

.304 

,160 

56 

.858 

.850 

.825 

.781 

.721 

.643 

.550 

.438 

.309 

.163 

4 

.873 

.864 

.839 

.794 

.734 

.655 

.559 

.445 

.314 

.166 

10 

.909 

.900 

.874 

.827 

.764 

.682 

.682 

.464 

.327 

.173 

20 

.945 

.936 

.909 

.860 

.795 

.709 

.606 

.482 

.340 

.179 

30 

.981 

.972 

.944 

.893 

.825 

.736 

.629 

.501 

.354 

.186 

40 

1.017 

1.008 

.979 

.926 

.855 

.764 

.652 

.519 

.367 

.193 

SO 

1.054 

1.044 

1.014 

.959 

.886 

.791 

.676 

.538 

.380 

.199 

5 

1.091 

1.080 

1.048 

.993 

.917 

.818 

.699 

.557 

.393 

.207 

10 

1.127 

1.116 

1.083 

1.026 

.947 

.845 

.722 

.576 

.406 

.214 

20 

1.164 

1.152 

1.118 

1.058 

.978 

.^2 

.746 

.594 

.419 

.230 

t    30 

1.200 

1.188 

1.153 

1.092 

1.009 

.m 

.769 

.613 

.432 

.226 

TABLE  OF  OKDINATES. 


789 


Table  of  Ordinates  5  ft  apart. —(Continued.) 


Distances  of  the  Ordinates  from  the  end  of  the  100  feet  Chord. 

Aug.  of 
Dea. 

Mid. 
50  ft. 

45  ft. 

40  ft. 

35  ft. 

30  ft. 

25  ft. 

20  ft. 

15  ft. 

10  ft. 

5  ft. 

o    • 
6  40 

1.236 

1.224 

1.188 

1.124 

1.039 

.927 

.792 

.631 

.445 

.236 

50 

1.273 

1.260 

1.223 

1.157 

1.070 

.954 

.816 

.649 

.458 

.241 

6 

1.309 

1.296 

1.258 

1.191 

1.100 

.982 

.839 

.668 

.472 

.248 

11 

1.345 

1.332 

1.293 

1.224 

1.130 

1.009 

.862 

.686 

.485 

.255 

20 

1.382 

1.368 

1.328 

1.256 

1.161 

1.036 

.886 

.705 

.498 

.262 

30 

1.419 

1.404 

1.362 

1.290 

1.192 

1.064 

.909 

.724 

.511 

.269 

40 

1.455 

1-440 

1.397 

1.323 

1.222 

1.091 

.932 

.742 

.524 

.276 

50 

1.491 

1.476 

1.432 

1.355 

1.253 

1.118 

.956 

.761 

.537 

.288 

7 

1.528 

1.512 

1.467 

1.389 

1.284 

1.146 

.979 

.779 

.551 

.290 

10 

1.564 

1.548 

1.502 

1.422 

1.314 

1.173 

1.002 

.798 

.564 

.297 

20 

1.600 

1.584 

1.537 

1.454 

1.345 

1.200 

1.026 

.816 

.576 

.304 

30 

1.637 

1.620 

1.572 

1.488 

1.375 

1.228 

1.048 

.835 

.590 

.311 

40 

1.673 

1.656 

1.607 

1.521 

1.405 

1.255 

1.071 

.854 

.603 

.318 

50 

1.710 

1.692 

1.641 

1.553 

1.436 

1.282 

1.095 

.872 

.616 

.324 

8 

1.746 

1.728 

1.677 

1.587 

1.467 

1.310 

1.118 

.891 

.629 

.332 

30 

1.855 

1.836 

1.782 

1.687 

1.559 

1.392 

1.188 

.946 

.669 

.353 

G 

1.965 

1.944 

1.886 

1.787 

1.651 

1.474 

l.iJ58 

1.002 

.708 

.373 

30 

2.074 

2.052 

1.991 

1.887 

1.742 

1.556 

1.328 

1.057 

.748 

.394 

10 

2.183 

•2.161 

2.096 

1.987 

1.834 

1.637 

1.398 

1.114 

.787 

.415 

SO 

2.292 

2.269 

2.201 

2.087 

1.926 

1.719 

1.468 

1.170 

.827 

.436 

11 

2.401 

2.377 

2.306 

2.186 

2,018 

1.802 

1.538 

1.226 

.866 

.457 

30 

2.511 

3.486 

2.411 

2.286 

2.110 

1.884 

1.609 

1.282 

.906 

.478 

12 

2.620 

2.594 

2.516 

2.386 

2.203 

1.967 

1.6a 

1.339 

.946 

.499 

30 

2.730 

2.703 

2.621 

2.485 

2.295 

2.049 

1.750 

1.395 

.985 

.520 

IS 

2.839 

2.811 

2.726 

2.585 

2.387 

2.132 

1.820 

1.451 

1.025 

.541 

30 

2.949 

2.920 

2.832 

2.685 

2.479 

2.214 

1.891 

1.507 

1.065 

.562 

14 

3.058 

3.028 

2.937 

2.785 

2.571 

2.297 

1.961 

1.564 

1.105 

.583 

30 

3.168 

3.136 

3.042 

2.884 

2.664 

2.379 

2.031 

1.620 

1.144 

.604 

15 

3.277 

3.246 

3.147 

2.984 

2.756 

2.462 

2.102 

1.676 

1.184 

.625 

30 

3..387 

3.354 

3.252 

3.084 

2.848 

2.544 

2.172 

1.732 

1.2-24 

.646 

16 

3.496 

3.462 

3.358 

3.184 

2.941 

2.627 

2.243 

1,789 

1.264 

.667 

17 

3.716 

3.680 

3.569 

3.384 

3.125 

2.792 

2.384 

1.902 

1.344 

.709 

18 

3.935 

3.897 

3.779 

3.584 

3.310 

2.958 

2.525 

2.014 

1.424 

.751 

19 

4.155 

4.115 

3.990 

3.784 

3.495 

3.123 

2.666 

2.127 

1.504 

.793 

20 

4.375 

4.332 

4.201 

3.984 

3.680 

3.288 

2.808 

2.240 

1.583 

.836 

22 

4.815 

4.768 

4.624 

4.386 

4.050 

3.620 

3.093 

2.467 

1744 

.922 

24 

5.255 

5.204 

5.048 

4.789 

4.423 

3.952 

3.379 

2.695 

1.905 

1.008 

26 

5.697 

5.642 

5.473 

5.192 

4.798 

4.286 

3.665 

2.924 

2.068 

1.094 

28 

6.139 

6  079 

5.898 

5.595 

5.171 

4.622 

3.952 

3.154 

2.232 

1.181 

30 

6  582 

6.517 

6.323 

5.999 

5.544 

4.958 

4.239 

3.385 

2.396 

1.268 

32 

7-027 

6.957 

6.751 

6.406 

5.922 

5.297 

4.530 

3.619 

2.565 

1.35«» 

34 

7.472 

7.398 

7.179 

6.813 

6.300 

5.637 

4.822 

3.854 

2.733 

1.445 

36 

7.918 

7.841 

7.609 

7.222 

6.679 

5.978 

5.115 

4.090 

2.901 

1.535 

38 

8.367 

8.286 

8.041 

7.633 

7.060 

6.320 

5.410 

4.327 

3.069 

1.626 

40 

8.816 

8.731 

8.474 

8.044 

7.442 

6.663 

5.705 

4.566 

3.238 

1.718 

Oaug;>e  on  curves.  Let  R  =  radius  of  wheel  from  center 

to  tread,  F  —  depth  of  flange  of  wheel,  and  L  =  the 
length  of  that  portion  of  th^e  wheel-flange  which  ex- 
tends below  the  top  of  the  rail,  =  2  >/(R+Fp  — R^^ 
all  in  inches.  Then  if,  on  the  curve,  we  widen  the 
gauge  by  a  quantity,  in  inches,  = 
^^      _.    .      ^    L,/ee/ 4- length  of  rigid  wheel-base,  ft. 

>       •  ^  gauge,  ft.  -1-  2  X  rad.  of  the  curve,  ft. 
the  wheels  will  have  approximately  the  same  play 
on  the  curve  as  on  the  tangent.    For  a  rigid  wheel- 
bace  14  ft.  long  and  drivers  4  ft.  diam.,  with  1]4  inch 
flanges,  Q  is  about  0.02  inch  (=  one-fiftieth  of  an  in.)  for  each  degree  of  curvature. 

Many  roads  use  the  sarnie  gauge  on  curves  as  on  tangents.  Others  widen  the 
gauge  on  curves  by  from  one  thirty-second  to  one-eighth  inch  for  each  degree  of 
curvature,  seldom,  however,  exceeding  1  inch  as  a  maximum.  In  Philadelphia 
the  Pennsylvania  Railroad  1 
(same  on  curves  as  on  t 


ilroad  has  freight-car  sidings  of  60  feet  radius ;  track  gauge 
I  tangents),  4  ft.  9  ins. ;  standard  wheel-gauge,  4  ft.  834  ii^s. 


*  Except  for  very  sharp  curves  and  for  very  short  wheel-bases  with  large  wheels 
and  deep  flanges,  it  is  amply  approximate  to  say 

^  .     .     ,  nr  .     .     ,       .X      wheel-base  in  feet 

Q  in  inches  =  L  in  inches  X  „ 


2  X  rad.  of  curve  in  ft.* 


790 


LEVEL  CUTTINGS. 


To  prepare  a  Table,  T,  of  L.evel  Cuttiiig:s,  for  every  Jg  of  a 
foot  of  heigrbtf  or  deptli. 


c^T 


-y^ 


4~llw 


Let  the  fig  represent  the  cutting ;  or,  if  inverted, 
the  filling;  in  which  the  horizontal  lines  are  sup- 
posed to  be  yL  foot  apart.  First  calculate  the 
area  in  square  feet,  of  the  layer  ab  co,  adjoining 
the  roadway  a  b.  Then  find  how  many  cubic 
yards  that  area  gives  in  a  distance  of  100  feet. 
i>  ci  cL  These  cubic  yards  we  will  call  Y ;  they  form  the 

first  amount  to  be  put  into  the  Table  T. 
Next  calculate  the  area  in  square  feet  of  the  triangle  an  o.    Multiply  this  area  by  4.     Find  how 
many  cubic  yards  this  increased  area  gives  in  a  distance  of  100  feet.     Or  they  will  be  found  ready 
calculated  below.  We  will  call  them  y.     This  is  all  the  preparation  that  is  needed  before 

commencing  the  table. 

Exam.— Let  the  roadbed  a  ft  be  18  feet,  and  the  side-slopes  1}4  to  1.  Then  for  the  area  of  a  6  c  o : 
since  the  side-slopes  are  1}4  to  1 ;  and  a  t  iB  .1  foot;  c  o  must  be  18.3  feet ;  and  the  mean  length  of 
ab  CO  must  be  18.15  feet.    Consequently,  the  area  is  18.15  X  .1—  1.815  square  feet;  which,  in  a 

distance  of  100 feet, gives  181.5  cubic  feet;  whichisequal  to -—^  =  6,7222  cubic  yards;  or  Y. 

Next,  as  to  the  triangle  ano:  its  height  a  n  being  .1  foot,  and  its  base  n  o  .15  feet ;  its  area 
=  ' — — ^—  = "  ——.0075  square  ft.  This  multiplied  by  4,  gives  .03  square  feet ;  which,  in  a  distance  of 
100  feet,  gives  .03  X  100  = 

Having  thus  found  Y  and  w,  proceed  to  make  out  the  table  in  the  manner  following,  which  is  to 
plain  as  to  require  no  explanation.  The  work  should  be  tested  about  every  5  feet,  by  calculating  the 
area  of  the  full  depth  arrived  at ;  multiply  it  by  100,  and  divide  the  product  by  27  for  the  cubic  yards 
The  cubic  yards  thus  found  should  agree  with  the  table. 


Y.... 

...  6.7222 
...  .1111 

y.... 

6.8333 
...  .1111 

jf.... 

6.9444 
...  .1111 

y.... 

7.0555 
...  .1111 

v.... 

7.1666 
...  .1111 

,  Y.  6.722     .1 


7.2777 


Height. 
Feet. 


.1  . 
.2  . 
.3  , 
.4  . 
.5  . 
.6  . 


Cub.  Yds. 


&c. 


6.72  Y. 
13.6 
20.5 
27.6 
34.7 
42.0 


The  following  table  contains  y,  ready  calculated  for  different  side-slopes.  It  plainlj 
remains  the  same  for  all  widths  of  roadbed. 


Side-slope. 

y 

Side-slope. 

y 

1^  to  1  

0185 

0370 

1^  to  1 

1296 

1^  to  1  

2   to  1 

1482 

3Z  to  1   

0556 

0741 

21/;  to  1 c 

1667 

1   to  1  

fA  to  1 

3  tol 

4  tol.. 

1852 

11^  to  1  

0926 

1111 

2222 

1^  to  1  

2963 

RAILROADS. 


Table  1.     lievel  Catting^s.* 

Roadway  14  feet  wide,  side-slopes  IJ^  to  1. 
For  single-track  embankment. 


Height 
in  Ft. 

.0 

.1 

.2 

.3 

.4 

.5 

.6 

.7 

.8 

.9 

Cu.Yds. 

Ou.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

0 

5.24 

10.6 

16.1 

21.6 

27.3 

33.1 

390 

45.0 

51.2 

1 

57.4 

63.8 

70.2 

76.8 

83.5 

90.3 

97.2 

104.2 

111.3 

118.6 

2 

125  9 

133.4 

141.0 

148.6 

156.4 

164.4 

172.4 

180.5 

1887 

197.1 

3 

205.6 

214.1 

222.8 

2.31.6 

240.5 

249.5 

258.7 

267.9 

277.3 

286.7 

4 

296.3 

306.0 

315.8 

325.7 

335.7 

345.8 

356.1 

366.4 

376.9 

387,5 

5 

398.1 

408.9 

419.9 

430.9 

442.0 

453.2 

464.6 

476.1 

487.6 

499.3 

6 

511.1 

523.0 

535.0 

547.2 

559.4 

571.8 

584.2 

596.8 

609.5 

622.3 

7 

635.2 

648.2 

661.3 

674.6 

687.9 

701.4 

714.9 

728.6 

742.4 

756.3 

8 

770.3 

784.5 

798.7 

813.1 

827.5 

842.1 

856.8 

871.6 

886.5 

901.5 

9 

916.7 

931.9 

947.3 

962.7 

978.3 

994.0 

1010 

1026 

1042 

1058 

10 

1074 

1090 

1107 

1123 

1140 

1157 

1174 

1191 

1208 

1225 

11 

1243 

1260 

1278 

1295 

1313 

1331 

1349 

1367 

1385 

1404 

12 

1422 

1441 

1459 

1478 

1497 

1516 

1535 

1554 

1574 

1593 

13 

1613 

1633 

1652 

1672 

1692 

1712 

1733 

1753 

1773 

1794 

14 

1815 

1835 

1856 

1877 

1898 

1920 

1941 

1962 

1984 

2006 

15 

2028 

2050 

2072 

2094 

2116 

2138 

2161 

2183 

2206 

2229 

16 

2252 

2275 

2298 

2321 

2344 

2368 

2391 

2415 

2439 

2463 

17 

2487 

2511 

2535 

2559 

2584 

2608 

2633 

2658 

2683 

2708 

18 

2733 

2759 

2784 

2809 

2835 

2861 

2886 

2912 

2938 

2964 

19 

2991 

3017 

3044 

3070 

3097 

3124 

3151 

3178 

3205 

3232 

20 

3259 

3287 

3314 

3342 

3370 

3398 

3426 

3454 

3482 

3510 

21 

3539 

3567 

3596 

3625 

3654 

3683 

3712 

3741 

3771 

3800 

22 

3830 

3859 

3889 

3919 

3949 

3979 

4009 

4040 

4070 

4101 

23 

4132 

4162 

4193 

4224 

4255 

4287 

4318 

4349 

4381 

4413 

24 

4444 

4476 

4608 

4541 

4573 

4605 

4638 

4670 

4703 

473fl 

25 

4769 

4802 

4835 

4868 

4901 

4935 

4968 

5002 

5036 

5070 

26 

5104 

5138 

5172 

5206 

5241 

5275 

5310 

5345 

5380 

5415 

27 

5450 

5485 

5521 

5556 

5592 

5627 

5663 

5699 

5735 

5771 

28 

5807 

5844 

5880 

5917 

5953 

5990 

6027 

6064 

6101 

6139 

29 

6176 

6213 

6251 

6289 

6326 

6364 

6402 

6440 

6479 

6517 

30 

6556 

6594 

6633 

6672 

6711 

6750 

6789 

6828 

6867 

6907 

31 

6946 

6986 

7026 

7066 

7106 

7146 

7186 

7226 

7267 

7307 

35? 

7348 

7389 

7430 

7471 

7512 

7553 

7595 

7836 

7678 

7719 

33 

7761 

7803 

7845 

7887 

7929 

7972 

8014 

8057 

8099 

8142 

34 

8185 

8228 

8271 

8315 

8358 

8401 

8445 

8489 

8532 

8576 

35 

8620 

8664 

8709 

8753 

8798 

8842 

8887 

8932 

8976 

9022 

36 

9067 

9112 

9157 

9203 

9248 

9294 

9340 

9386 

9432 

9478 

37 

9524 

9570 

9617 

9663 

9710 

9757 

9804 

98.51 

9898 

9945 

38 

9993 

10040 

10088 

10135 

10183 

10231 

10279 

10327 

10375 

10424 

39 

10472 

10521 

10569 

10618 

10667 

10716 

10765 

10815 

10864 

10913 

40 

10963 

11013 

11062 

11112 

11162 

11212 

11263 

11313 

11364 

11414 

41 

11465 

11516 

11567 

U618 

11669 

11720 

11771 

11823 

11874 

11926 

42 

11978 

12029 

12081 

12134 

12186 

12238 

12291 

12343 

12396 

12449 

43 

12502 

12555 

12608 

12661 

12715 

12768 

12822 

12875 

12929 

12983 

44 

13037 

13091 

1.3145 

13200 

13254 

13309 

13363 

13418 

13473 

13528 

45 

13583 

13639 

13694 

13749 

13805 

13861 

13916 

13972 

14028 

14084 

46 

14141 

14197 

14-254 

14310 

14367 

14424 

14480 

14537 

14595 

14652 

47 

14709 

14767 

14824 

14882 

14940 

14998 

15056 

15114 

15172 

15230 

48 

15289 

15347 

15406 

15465 

15524 

15583 

15642 

15701 

15761 

1582S 

49 

15880 

15939 

15999 

16059 

16119 

16179 

16239 

16300 

16360 

16421 

60 

16481 

16542 

16603 

16664 

16725 

16787 

16848 

16909 

16971 

17033 

61 

17094 

17156 

17218 

17280 

17343 

17405 

17467 

17530 

17593 

17656 

52 

17719 

17782 

17845 

17908 

17971 

18035 

18098 

18162 

18226 

18290 

63 

18354 

18418 

18482 

18546 

18611 

18675 

18740 

18805 

18870 

18935 

54 

19000 

19065 

19131 

19196 

19262 

19327 

19393 

19459 

19525 

19.591 

65 

19657 

19724. 

19790 

19857 

19923 

19990 

20057 

20124 

20191 

20259 

56 

120326 

20393 

20461 

20529 

20596 

20664 

20732 

20800 

20869 

20937 

57 

121005 

21074 

21143 

21212 

21280 

21349 

21419 

21488 

21557 

21627 

58 

21696 

21766 

21836 

21906 

21970 

22046 

22116 

22186 

22257 

22327 

89 

22398 

22469 

22540 

22611 

22682 

22753 

22825 

22896 

22968 

23039 

60 

23111 

23183 

23255 

23327 

23399 

23472  123544 

23617 

23689 

23762 

•i  From  the  Authois  "  Measurement  and  Cost  of  Earthwork." 


792 


RAILROADS. 


Table  2.     I^evel  Cuttingrs. 

Roadway  24  feet  wide,  side-slopes  1}/^  to  1. 
For  double-track  embankment. 


Height 

.0 

.1 

.2 

.3 

.4 

.5 

.6 

.7 

.8 

.9 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.YdB. 

Cu.Yds. 

Cu.Yds. 

Cu.YdB. 

Cu.Yds. 

Cu.Yds. 

Cu.Ydi. 

0 

8.94 

18.0 

27.2 

36.4 

45.8 

55.3 

64.9 

74.7 

84.5 

1 

94.4 

104.5 

114.7 

124.9 

135.3 

145.8 

156.4 

167.2 

178.0 

188.9 

2 

200.0 

211.2 

222.4 

233.8 

245.3 

256.9 

268.6 

280.5 

292.4 

304.4 

3 

316.6 

328.9 

341.2 

353.7 

366.3 

379.0 

391.9 

404.8 

417.8 

431.0 

4 

444.4 

457.8 

471.3 

484.9 

498.6 

512.4 

526.4 

540.4 

554.6 

568.8 

6 

583.3 

597.8 

612.4 

627.1 

642.0 

656.9 

671.9 

687.1 

702.3 

717.7 

6 

733.3 

748.9 

764.7 

780.5 

796.4 

812.5 

828.7 

844.9 

861.3 

877.8 

7 

894.4 

911.2 

928.0 

944.9 

962.0 

979.2 

996.4 

1014 

1031 

1049 

8 

1067 

1085 

1102 

1121 

1139 

1157 

1175 

1194 

1212 

1231 

9 

1250 

1269 

1288 

1307 

1326 

1346 

1366 

1385 

1405 

1425 

10 

1444 

1465 

1485 

1505 

1525 

1546 

1566 

1587 

1608 

1629 

11 

1650 

1671 

1692 

1714 

1735 

1757 

1779 

1800 

1822 

1846 

12 

1867 

1889 

1911 

19.34 

1956 

1979 

2002 

2025 

2048 

2071 

13 

2094 

2118 

2141 

2166 

2189 

2213 

2236 

2261 

2285 

2309 

14 

2333 

2358 

2382 

2407 

2432 

2457 

2482 

2607 

2532 

2558 

15 

2583 

2609 

2635 

2661 

2686 

2713 

2739 

2765 

2791 

2818 

16 

2844 

2871 

2898 

2925 

2952 

2979 

3006 

3034 

3061 

3086 

17 

3117 

3145 

3172 

3201 

3229 

3267 

3285 

3314 

3342 

3371 

18 

3400 

3429 

3458 

3487 

3516 

3546 

3575 

3605 

3635 

3665 

19 

3694 

3725 

3755 

3785 

3815 

3846 

3876 

3907 

3938 

39«0 

20 

4000 

4031 

4062 

4094 

4125 

4157 

4189 

4221 

4252 

4285 

21 

4317 

4349 

4381 

4414 

4446 

4479 

4512 

4546 

4578 

4611 

22 

4644 

4678 

4711 

4745 

4779 

4813 

4846 

4881 

4916 

4949 

23 

4983 

5018 

5052 

5087 

5122 

5167 

5192 

5227 

5262 

5298 

24 

6333 

5369 

5405 

6441 

5476 

5513 

6549 

5585 

5621 

6658 

25 

5694 

5731 

5768 

5806 

6842 

6879 

5916 

5964 

6991 

6029 

26 

6067 

6106 

6142 

6181 

6219 

6257 

6295 

6334 

6372 

6411 

27 

6450 

6489 

6528 

6567 

6606 

6646 

6685 

6725 

6765 

6805 

28 

6844 

6885 

6925 

6965 

7005 

7046 

7086 

7127 

7168 

7209 

29 

7250 

7291 

7332 

7374 

7415 

7457 

7499 

7641 

7582 

7625 

30 

7667 

7709 

7751 

7794 

7836 

7879 

7922 

7965 

8008 

8061 

31 

8094 

8138 

8181 

8225 

8269 

8313 

8356 

8401 

8445 

8489 

32 

8533 

8678 

8622 

8667 

8712 

8767 

8802 

8847 

8892 

8938 

33 

8983 

9029 

9075 

9121 

9166 

9212 

9259 

9305 

9351 

9398 

34 

9444 

9491 

9538 

9585 

9632 

9679 

9726 

9774 

9821 

9869 

35 

9917 

9965 

10012 

10061 

10109 

10157 

10205 

10254 

10.302 

10351 

36 

10400 

10449 

10498 

10547 

10596 

10646 

10695 

10745 

10795 

10846 

37 

10894 

10945 

10995 

11045 

11095 

11146 

11196 

11247 

11298 

11349 

38 

11400 

11451 

11502 

11554 

11605 

11657 

11709 

11761 

11812 

11865 

39 

11917 

11969 

12021 

12074 

12126 

12179 

12232 

12285 

12338 

12391 

40 

12444 

12498 

12551 

12605 

12659 

12713 

12766 

12821 

12875 

12929 

41 

12983 

13038 

13092 

13147 

13202 

13257 

13312 

13367 

13422 

1347P 

42 

13533 

13589 

13645 

13701 

13756 

13813 

13869 

13925 

13981 

1403M 

43 

14094 

14151 

14208 

14265 

14322 

14379 

14436 

14494 

14551 

14609 

44 

14667 

14725 

14782 

14840 

14899 

14957 

15015 

15074 

15132 

15191 

45 

15250 

15309 

15368 

15427 

15486 

15546 

15605 

15665 

15725 

15785 

46 

15844 

15905 

15965 

16025 

16085 

16146 

16206 

16267 

16328 

16389 

47 

16450 

16511 

16572 

16634 

16695 

16757 

16819 

16881 

16942 

17005 

48 

17067 

17129 

17191 

17254 

17316 

17379 

17442 

17505 

17568 

17631 

49 

17694 

17758 

17821 

17885 

17949 

18013 

18076 

18141 

18205 

18269 

60 

18333 

18398 

18462 

18527 

18592 

18657 

18722 

18787 

18852 

18918 

61 

18983 

19049 

19115 

19181 

19246 

19313 

19379 

19445 

19511 

19578 

52 

19644 

19711 

19778 

19845 

19912 

19979 

20046 

20114 

20181 

20249 

53 

20317 

20385 

20452 

20521 

20589 

20657 

20725 

20794 

20862 

20931 

54 

21000 

21069 

21138 

21207 

21276 

21346 

21415 

21485 

21555 

21625 

55 

21694 

21765 

21835 

21905 

21975 

22046 

22116 

22187 

22258 

22329 

56 

22400 

22471 

22542 

22614 

22685 

22757 

22829 

22901 

22972 

23045 

57 

23117 

23189 

23261 

23334 

23406 

23479 

23552 

23625 

23698 

23771 

58 

23844 

23918 

23991 

24065 

24139 

24213 

24286 

24361 

24436 

24509 

59 

24583 

24658 

24732 

24807 

24882 

24957 

25032 

25107 

25182 

25258 

60 

25333 

25409 

25485 

25561 

25636 

25713 

25789 

25865 

25941 

26018 

For  oontinuauou  w  lUU  twet,  see  Tablb  7. 


BAILBOADS. 


793 


Table  3.      liCvel  Cuttingrs* 

Roadway  18  feet  wide,  side-slopes  1  to  1. 
For  single-track  excavation. 


Depth 
in  Ft. 

.0 

.1 

.2 

.3 

.4 

.5 

.6 

.7 

.8 

.9 

Cu.Yd». 

Ca.Yd8. 

Cu.Yds. 

Ou.Yds. 

Cu.Yds. 

Cu.Yds. 

Ou.Yd8. 

Cu.Yd8. 

Cu.Yds. 

Cu.Ydi. 

0 

6.70 

13.5 

203 

27.3 

34.3 

41.3 

48.5 

55.7 

63.0 

1 

70.4 

77.8 

85.3 

929 

100.6 

108.3 

116.1 

124.0 

132.0 

140.0 

2 

148.1 

156.3 

164.6 

172.9 

181.3 

189.8 

198.4 

207.0 

215.7 

224.5 

3 

233.3 

242.3 

251.3 

260.3 

269.5 

278.7 

288.0 

297.4 

306.8 

316.3 

4 

325.9 

335.6 

345.3 

355.1 

365.0 

375.0 

385.0 

395.1 

405.3 

415.6 

5 

425.9 

436.3 

446.8 

457.4 

468.0 

478.7 

489.5 

600.3 

611.3 

522.3 

6 

533.3 

544.5 

655.7 

567.0 

678.4 

689.8 

601.3 

612.9 

624.6 

636.3 

7 

648.1 

660.0 

672.0 

684.0 

696.1 

708.3 

720.6 

732.9 

745.3 

757.8 

8 

770.4 

783.0 

795.7 

808.5 

821.3 

834.3 

847.3 

860.3 

873.5 

886.7 

9 

900.0 

913.4 

926.8 

940.3 

953.9 

967.6 

981.3 

995.1 

1009 

1023 

10 

1037 

1051 

1065 

1080 

1094 

1108 

1123 

1137 

1152 

1167 

11 

1181 

1196 

1211 

1226 

1241 

1256 

1272 

1287 

1.302 

1318 

12 

1333 

1349 

1365 

1380 

1396 

1412 

1428 

1444 

1460 

1476 

13 

1493 

1509 

1525 

1542 

1558 

1575 

1592 

1608 

1625 

1642 

14 

1659 

1676 

1693 

1711 

1728 

1745 

1763 

1780 

1798 

1816 

15 

1833 

1851 

1869 

1887 

1905 

1923 

1941 

1960 

1978 

1996 

16 

2015 

2033 

2052 

2071 

2089 

2108 

2127 

2146 

2165 

2184 

17 

2204 

2223 

2242 

2262 

2281 

2301 

2321 

2340 

2360 

2380 

18 

2400 

2420 

2440 

2460 

2481 

2501 

2521 

2542 

2562 

2583 

19 

2604 

2624 

2645 

2666 

2687 

2708 

2729 

2751 

2772 

2793 

20 

2815 

2836 

2858 

2880 

2901 

2923 

2945 

2967 

29S9 

3011 

21 

3033 

3056 

3078 

3100 

3123 

3146 

3168 

3191 

3213 

3236 

22 

3259 

3282 

3305 

3328 

3352 

3376 

3398 

3422 

3445 

3469 

23 

3493 

3516 

3540 

3564 

3588 

3612 

3636 

3660 

3685 

3709 

24 

3733 

3758 

3782 

3807 

3832 

3856 

3881 

3906 

3931 

3966 

26 

3981 

4007 

4032 

4057 

4083 

4108 

4134 

4160 

4185 

4211 

26 

4237 

4263 

4289 

4315 

4341 

4368 

4394 

4420 

4447 

4473 

27 

4500 

4527 

4563 

4580 

4607 

4634 

4661 

4688 

4716 

4743 

28 

4770 

4798 

4825 

4863 

4881 

4908 

4936 

4964 

4992 

5020 

29 

5048 

5076 

6105 

6133 

5161 

5190 

5218 

5247 

6276 

6304 

30 

5333 

5362 

6391 

5420 

5449 

5479 

5508 

5537 

6567 

5596 

31 

5626 

5656 

6685 

5716 

5746 

5776 

6805 

5836 

6865 

5S96 

32 

5926 

5956 

5987 

6017 

6048 

6079 

6109 

6140 

6171 

6202 

33 

6233 

6264 

6296 

6327 

6358 

6390 

6421 

6453 

6486 

6516 

34 

6548 

6580 

6612 

6644 

6676 

6708 

6741 

6773 

6805 

6838 

35 

6870 

6903 

6936 

6968 

7001 

7034 

7067 

7100 

7133 

7167 

36 

7200 

7233 

7267 

7300 

7334 

7368 

7401 

74:56 

7469 

7503 

37 

7537 

7571 

7605 

7640 

7674 

7708 

7743 

7777 

7812 

7847 

38 

7881 

7916 

7951 

7986 

8021 

8056 

8092 

8127 

8162 

8198 

39 

8233 

8269 

8305 

8340 

8376 

8412 

8448 

8484 

8620 

8556 

40 

8593 

8629 

8665 

8702 

8738 

8775 

8812 

8848 

8885 

8922 

41 

8959 

8996 

9033" 

9071 

9108 

9145 

9183 

9220 

9258 

9296 

42 

9333 

9371 

9409 

9447 

9485 

9523 

9561 

9600 

9638 

9676 

43 

9715 

9753 

9792 

9831 

9869 

9908 

9947 

9986 

10025 

10064 

44 

10104 

10143 

10182 

10222 

10261 

10301 

10341 

10380 

10420 

10460 

45 

10500 

10540 

10580 

10620 

10661 

10701 

10741 

10782 

10822 

10863 

46 

10904 

10944 

10985 

11026 

11067 

11108 

11149 

11191 

11232 

11273 

47 

11315 

11356 

11398 

11440 

11481 

11623 

11565 

11607 

11649 

11691 

48 

11733 

11776 

11818 

11860 

11903 

11945 

11988 

12031 

12073 

12116 

49 

12159 

12202 

12245 

12288 

12332 

12376 

12418 

12462 

12505 

12549 

50 

12593 

12636 

12680 

12724 

12768 

12812 

12856 

12900 

12945 

12989 

51 

13033 

13078 

13122 

13167 

13212 

13256 

13301 

13346 

13391 

13436 

52 

13481  • 

13527 

13572 

13617 

13663 

13708 

13754 

13^00 

13845 

13891 

53 

13937 

13983 

14029 

14075 

14121 

14168 

14214 

14-260 

14307 

14363 

54 

14400 

14447 

14493 

14540 

14587 

14634 

14681 

14728 

14776 

14823 

55 

14870 

14918 

14965 

15013 

15061 

15108 

15156 

15204 

15252 

15300 

56 

15348 

16396 

15445 

15493 

15541 

15590 

15638 

15687 

15736 

15784 

67 

15833 

15882 

15931 

15980 

16029 

16079 

16128 

17177 

16227 

16276 

58 

16326 

16376 

16425 

16475 

16525 

16575 

16625 

16676 

16725 

16776 

69 

16826 

16876 

16927 

16977 

17028 

17079 

17129 

17180 

17231 

17282 

60 

17333 

17384 

17436  . 

17487 

17538 

17590 

17641 

17693 

17746 

17796 

For  contiDnation  to  IW  Ie«t  Aeep,  see  Table  7. 


794 


KAILROADS. 


Table  4.     liCvel  Cuttingis. 

Roadway  18  feet,  side-slopes  l^^^  to  1. 
For  single-track  exoavatioa. 


Depth 

in  Ft. 

.0 

.1 

.2 

.3 

.4 

.5 

.6 

.7 

.8 

.9 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Ydi. 

Cu.Yds. 

0 

6.72 

13.6 

20.5 

27.6 

34.7 

42.0 

49.4 

56.9 

64.5 

1 

72.2 

80.1 

88.0 

96.1 

104.2 

112.5 

120.9 

129.4 

138.0 

146.7 

2 

155.5 

164.5 

173.5 

182.7 

191.9 

201.3 

210.8 

220.4 

230.1 

240.0 

3 

249.9 

260.0 

270.1 

280.4 

290.8 

301.3 

311.9 

322.6 

333.4 

344.5 

4 

355.5 

366.7 

378.0 

389.4 

400.9 

412.5 

424.2 

436  0 

448.0 

460.0 

5 

472.2 

484.5 

496.9 

509.4 

522.0 

534.7 

647.6 

560.5 

673.6 

586.7 

6 

600.0 

613.4 

626.9 

640.5 

654.2 

668.1 

682.0 

696.1 

710.2 

724.5 

7 

738.9 

753.4 

768.0 

782.7 

797.6 

812.5 

827.6 

842.7 

858.0 

873.4 

8^ 

888.9 

904.5 

920.2 

936.1 

952.0 

968.1 

984.2 

1001 

1017 

1033 

9 

1050 

1067 

1084 

1101 

1118 

1135 

1152 

1169 

1187 

1205 

10 

1222 

1240 

1258 

1276 

1294 

1313 

1331 

1349 

1368 

1387 

11 

1406 

1425 

1444 

1463 

1482 

1501 

1521 

1541 

1560 

1580 

12 

1600 

1620 

1640 

1661 

1681 

1701 

1722 

1743 

1764 

1785 

13 

1806 

1827 

1848 

1869 

1891 

1913 

1934 

1956 

1978 

2000 

14 

2022 

2045 

2067 

2089 

2112 

2135 

2158 

2181 

2204 

2227 

15 

2250 

2273 

2-297 

2321 

2344 

2368 

2392 

2416 

2440 

2465 

16 

2489 

2513 

2538 

2563 

2588 

2613 

2638 

2663 

2688 

2713 

17 

2739 

2765 

2790 

2816 

2842 

2868 

2894 

2921 

2947 

2973 

18 

3000 

3027 

3054 

3081 

3108 

3135 

3162 

3189 

3217 

3245 

19 

3272 

3300 

3328 

3356 

3384 

3413 

3441 

3469 

3498 

3527 

20 

3556 

3585 

3614 

3643 

3672 

3701 

3731 

3761 

3790 

3820 

21 

3850 

3880 

3910 

3941 

3971 

4001 

4032 

4063 

4094 

4125 

22 

4156 

4187 

4218 

4249 

4281 

4313 

4344 

4376 

4408 

4440 

23 

4472 

4505 

4537 

4569 

4602 

4635 

4668 

4701 

4734 

4767 

24 

4800 

4833 

4867 

4901 

4934 

4968 

5002 

5036 

5070 

5105 

25 

5139 

5173 

5208 

5243 

5278 

5313 

53^18 

5383 

5418 

5453 

26 

5489 

5525 

5560 

5596 

5632 

5668 

5704 

5741 

5777 

5813 

27 

5850 

5887 

5924 

5961 

5998 

6035 

6072 

6109 

6147 

6185 

28 

6222 

6260 

6298 

6336 

6374 

6413 

6451 

6489 

6528 

6667 

29 

6606 

6645 

6684 

6723 

6762 

6801 

6841 

6881 

6920 

6960 

30 

7000 

7040 

7080 

7121 

7161 

7201 

7242 

7283 

7324 

7365 

31 

7406 

7447 

7488 

7529 

7571 

7613 

.7654 

7696 

7738 

7780 

32 

7822 

7865 

7907 

7949 

7992 

8035 

8078 

8121 

8164 

8207 

33 

8250 

8293 

8337 

8381 

8424 

8468 

8512 

8556 

8600 

8645 

34 

8689 

8733 

8778 

8823 

8868 

8913 

8958 

9003 

9048 

9093 

35 

9139 

9185 

9230 

9276 

9322 

9368 

9414 

9461 

9507 

9553 

36 

9600 

9647 

9694 

9741 

9788 

9835 

9882 

9929 

9977 

10025 

37 

10072 

10120 

10168 

10216 

10264 

10313 

10361 

10409 

10458 

10507 

38 

10556 

10605 

10654 

10703 

10752 

10801 

10851 

10901 

10950 

11000 

39 

11050 

11100 

11150 

11200 

11251 

11301 

11352 

11403 

11454 

11505 

40 

11556 

11607 

11658 

11709 

11761 

11813 

11864 

11916 

11968 

12020 

41 

12072 

12125 

12177 

12229 

12282 

12335 

12388 

12441 

12494 

12547 

42 

12600 

12653 

12707 

12761 

12814 

12868 

12922 

12976 

13030 

13085 

43 

13139 

13193 

13248 

13303 

13358 

13413 

13468 

13523 

13578 

13633 

44 

13689 

13745 

13800 

13856 

13912 

13968 

14024 

14081 

14137 

14193 

45 

14250 

14307 

14364 

14421 

14478 

14535 

14592 

14C49 

14707 

14765 

46 

14822 

14880 

14938 

14996 

15054 

15113 

15171 

15229 

15288 

15347 

47 

15406 

15465 

15524 

15583 

15642 

15701 

15761 

15821 

15880 

15940 

48 

16000 

16060 

16120 

16181 

16241 

16301 

16362 

16423 

16484 

16545 

49 

16606 

16667 

16728 

16789 

16851 

16913 

16974 

17036 

17098 

17160 

60 

17222 

17285 

17347 

17409 

17472 

17536 

17598 

17661 

17724 

17787 

51 

17850 

17913 

17977 

18041 

18104 

18168 

18232 

18296 

18360 

18425 

62 

18489 

18553 

18618 

18683 

18748 

18813 

18878 

18943 

19008 

19073 

63 

19139 

19205 

19270 

19336 

19402 

19468 

19534 

1^601 

19667 

19733 

64 

19800 

19867 

19934 

20000 

20068 

20135 

20202 

20269 

20337 

20405 

65 

20472 

20540 

20608 

20676 

20744 

20813 

20881 

20949 

21018 

21087 

66 

21156 

21225 

21294 

21363 

21432 

21501 

21571 

21641 

21710 

21780 

67 

21850 

21920 

21990 

2-2061 

22131 

22201 

22272 

22343 

22414 

22485 

68 

22556 

22627 

22698 

22769 

22841 

22913 

22984 

23056 

23128 

23200 

69 

23272 

23345 

23417 

23489 

23562 

23635 

23708 

23781 

23854 

23927 

60 

24000 

24073 

24147 

24221 

24294 

24368  . 

24442 

24516 

24590 

24665 

For  continuation  to  100  M«(  d««i^*  see  Table  7. 


RAILROADS. 


795 


Table  5.     licvel  Cutting. 

Roadway  28  feet  wide,  side-slopes  1  to  1. 
For  double-track  excavation. 


I»epth 

i 

laFt. 

.0 

.1 

.2 

.3 

.4 

.5 

.6 

.7 

.8 

.9 

"^ 

Cu.Xda. 

Cu.Yds. 

Cu.Yds. 

Ou.Yda. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds, 

0 

10.4 

20.9 

31.4 

42.1 

52.8 

63.6 

74.4 

85.3 

96.3 

1 

107.4 

118.6 

129.8 

141.1 

152.4 

163.9 

175.4 

187.0 

198.7 

210.4 

2 

222.2 

234.1 

246.1 

258.1 

270.2 

282.4 

294.7 

307.0 

319.4 

331.9 

3 

344.4 

357.1 

369.8 

382.6 

395.4 

408.3 

421.3 

434.4 

447.6 

460.8 

4 

474.1 

487.4 

500.9 

514.4 

528.0 

541.7 

555.4 

569.2 

583.1 

597.1 

6 

611.1 

625.2 

639;4 

653.7 

668.0 

682.4 

696.9 

711.4 

726.1 

740.8 

6 

755.6 

770.4 

785.4 

800.4 

815.5 

830.6 

845.8 

861.1 

876.5 

891.9 

7 

907.5 

923.0 

938.7 

954.5 

970.3 

986.2 

1002 

1018 

1034 

1050 

8 

1067 

1083 

1099 

1116 

1132 

1149 

1166 

1182 

1199 

1216 

9 

1233 

1250 

1267 

1285 

1302 

1319 

1337 

1354 

1372 

1390 

10 

1407 

1425 

1443 

1461 

1479 

1497 

1515 

1534 

1552 

1570 

11 

1589 

1607 

1626 

1645 

1664 

1682 

1701 

1720 

1739 

1759 

12 

1778 

1797 

1816 

1836 

1855 

1875 

1895 

1914 

1934 

1954 

13 

1974 

1994 

2014 

2034 

2055 

2075 

2095 

2116 

2136 

2157 

14 

2178 

2199 

2219 

2240 

2261 

2282 

2304 

2325 

2346 

2367 

15 

2389 

2410 

2432 

•2454 

2475 

2497 

2519 

2541 

2563 

2585 

16 

2607 

2630 

2652 

2674 

2697 

2719 

2742 

2765 

2788 

2810 

17 

2833 

2856 

2879 

2903 

2926 

2949 

2972 

2996 

3019 

3043 

18 

3067 

3090 

3114 

3138 

3162 

3186 

3210 

3234 

3259 

3283 

19 

3307 

3332 

3356 

3381 

3406 

3431 

3455 

3480 

3505 

3530 

20 

3556 

3581 

3606 

3631 

3657 

3682 

3708 

3734 

3759 

3785 

21 

3811 

3837 

3863 

3889 

3915 

3942 

3968 

3994 

4021 

4047 

22 

4074 

4101 

4128 

4154 

4181 

4208 

4235 

4263 

4290 

4317 

23 

4344 

4372 

4399 

4427 

4455 

4482 

4510 

4538 

4566 

4594 

24 

4622 

4650 

4679 

4707 

4735 

4764 

4792 

4821 

4850 

4879 

25 

4907 

4936 

4965 

4994 

5024 

5053 

5082 

5111 

5141 

5170 

26 

5200 

5230 

5259 

5289 

5319 

5349 

5379 

5409 

5439 

5470 

27 

5560 

5530 

5561 

5591 

5622 

5653 

5684 

5714 

5745 

5776 

28 

5807 

5839 

5870 

5901 

5932 

5964 

5995 

6027 

6059 

6090 

29 

6122 

6154 

6186 

6218 

6250 

6282 

6315 

6347 

6379 

6412 

30 

6444 

6477 

6510 

6543 

6575 

6608 

6641 

6674 

6708 

6741 

31 

6774 

6807 

6841 

6874 

6908 

6942 

6975 

7009 

7043 

7077 

32 

7111 

7145 

7179 

7214 

7248 

7282 

7317 

7351 

7386 

7421 

33 

7456 

7490 

7525 

7560 

7595 

7631 

7666 

7701 

7736 

7772 

34 

7807 

7843 

7879 

7914 

7950 

7986 

8022 

8058 

8094 

8130 

35 

8167 

8203 

8239 

8276 

8312 

8349 

8386 

8423 

8459 

8496 

36 

8533 

8570 

8608 

8645 

8682 

8719 

8757 

8794 

8832 

8870 

37 

8907 

8945 

8983 

9021 

9059 

9097 

9135 

9174 

9212 

9250 

38 

9289 

9327 

9366 

9405 

9444 

9482 

9521 

9560 

9599 

9639 

39 

9678 

9717 

9756 

9796 

9835 

9875 

9915 

9954 

9994 

10034 

40 

10074 

10114 

10154 

10194 

10235 

10275 

li3315 

10356 

10396 

10437 

41 

10478 

10519 

10559 

10600 

10641 

10682 

10724 

10765 

10806 

10847 

42 

10889 

10930 

10972 

11014 

11055 

11097 

11139 

11181 

11223 

11265  . 

43 

11307 

11350 

11392 

11434 

11477 

11519 

11562 

11605 

11648 

11690 

44 

11733 

11776 

11819 

11863 

11906 

11949 

11992 

12036 

12079 

12123 

45 

12167 

12210 

12254 

12298 

12342 

12386 

12430 

12474 

12519 

12563 

46 

12607 

12652 

12696 

12741 

12786 

12831 

12875 

12920 

12965 

13010 

47 

13056 

13101 

13146 

13191 

13237 

13282 

13328 

13374 

13419 

13465 

48 

13511 

13587 

13603 

13649 

13695 

13742 

13788 

13834 

13881 

13927 

49 

13974 

14021 

14068 

14114 

14161 

14208 

14255 

14303 

14350 

14397 

50 

14444 

14492 

14539 

14587 

14635 

14682 

14730 

14778 

14826 

14874 

61 

14922 

14970 

15019 

15067 

15115 

15164 

15212 

15261 

15310 

15359 

62 

15407 

15456 

15505 

15054 

15604 

15653 

15702 

15751 

15801 

15850 

63 

15900 

15950 

15999 

16049 

16099 

16149 

16199 

16249 

16299 

16350 

64 

16400 

16450 

16501 

16551 

16602 

16653 

16704 

16754 

16805 

16856 

55 

16907 

16959 

17010 

17061 

17112 

17164 

17215 

17267 

17319 

17370 

66 

17422 

17474 

17526 

17578 

17630 

17682 

17735 

17787 

17839 

17892 

67 

17944 

17997 

18050 

18103 

18155 

18208 

18261 

18314 

18368 

18421 

58 

18474 

18527 

18581 

18634 

18688 

18742 

18795 

18849 

18903 

18957 

59 

19 -n 

19065 

19119 

19174 

19228 

19282 

19337 

19391 

19446 

19501 

60 

19556 

19610 

19665 

19720 

19775 

19831 

19886 

19941 

19996. 

20052 

For  continuation  to  100  feet,  see  Table  7. 


RAILROADS. 


Table  6.      liOTel  Cuttingrs. 

Roadway  28  ft  wide,  side-slopes  1}^  to  1. 
For  donble-track  excavation. 


Depth 

• 

" 

in  Ft. 

.0 

.1 

.2 

.3 

.4 

.5 

.6 

.7 

.8 

.9 

Cu.Yds. 

Cu.Yds. 

Cu.Yd8. 

Cu.Yds. 

Cu.Yds. 

Cu.Yd8. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

0 

10.4 

21.0 

31.6 

42.4 

63.2 

64.2 

75.3 

86.5 

97.9 

1 

109.3 

120.8 

132.5 

144.3 

156.1 

168.1 

180.2 

192.4 

204.8 

217.2 

2 

229.6 

242.3 

255.0 

267.9 

280.9 

294.0 

307.2 

320.5 

334.0 

347.5 

3 

361.2 

374.9 

388.8 

402.8 

416.9 

431.1 

445.4 

459.9 

474.4 

489.1 

4 

503.7 

518.6 

533.6 

548.6 

563.9 

579.3 

594.7 

610.2 

625.8 

641.6 

6 

657.5 

673.4 

689.5 

705.7 

722.1 

738.5 

755.0 

771.7 

788.4 

806.3 

6 

822.2 

839.3 

856.5 

873.8 

891.2 

908.8 

926.4 

944.2 

962.0 

980.0 

7 

998.1 

1016 

1035 

1053 

1072 

1090 

1109 

1128 

1147 

1166 

8 

1186 

1204 

1224 

1243 

1263 

1283 

1303 

1322 

1343 

1363 

9 

1383 

1403 

1424 

1445 

1465 

1486 

1507 

1528 

1549 

1671 

10 

1592 

1614 

1635 

1657 

1679 

1701 

1723 

1745 

1767 

1790 

11 

1812 

1835 

1858 

1881 

1904 

1927 

1950 

1973 

1997 

2020 

12 

2044 

2068 

2092 

2116 

2140 

2164 

2189 

2213 

2238 

2262 

13 

2287 

2312 

2337 

2362 

2387 

2413 

2438 

2464 

2489 

2616 

14 

2541 

2567 

2593 

2619 

2645 

2672 

2698 

2725 

2752 

2779 

15 

2806 

2833 

2860 

2887 

2915 

2942 

2970 

2997 

3025 

3063 

16 

3081 

3109 

3138 

3166 

3195 

3223 

3252 

3281 

3310 

3339 

17 

3368 

3397 

3427 

3456 

3486 

3516 

3546 

3676 

3606 

3636 

18 

3667 

3697 

3728 

3758 

3789 

3820 

3851 

3882 

3913 

3944 

19 

3976 

4007 

4039 

4070 

4102 

4134 

4166 

4198 

4231 

4263 

20 

4296 

4328 

4361 

4394 

4427 

4460 

4493 

4627 

4560 

4694 

21 

4627 

4661 

4695 

4729 

4763 

4797 

4832 

4866 

4900 

4935 

22 

4970 

5005 

5040 

5075 

5111 

5146 

5181 

5217 

5253 

6288 

23 

5324 

5360 

5396 

6432 

5469 

5505 

5542 

5578 

5615 

5652 

24 

6689 

5726 

5763 

5800 

5838 

6875 

5913 

5951 

5989 

6027 

25 

6065 

6103 

6141 

6179 

6218 

6257 

6296 

6334 

6373 

6412 

26 

6451 

6491 

6530 

6570 

6609 

6649 

6689 

6729 

6769 

6809 

27 

6850 

6890 

6931 

6971 

7012 

7053 

7094 

7135 

7176 

7217 

28 

7259 

7300 

7342 

7384 

7426 

7468 

7510 

7552 

7594 

7637 

29 

7680 

7722 

7765 

7808 

7851 

7894 

7937 

7981 

8024 

8067 

30 

8111 

8156 

8199 

8243 

8287 

8331 

8375 

8420 

8464 

8509 

31 

8554 

8598 

8643 

8688 

8734 

8779 

8824 

8870 

8915 

8961 

32 

9007 

9053 

9099 

9145 

9191 

9238 

9284 

9331 

9378 

9426 

33 

9472 

9519 

9566 

9613 

9661 

9708 

9756 

9804 

9851 

9900 

34 

9948 

9997 

10045 

10093 

10142 

10190 

10239 

10288 

10337 

10386 

36 

10435 

10484 

10534 

10583 

10633 

10683 

10732 

10782 

10832 

10882 

36 

10933 

10983 

11034 

11084 

11135 

11186 

11237 

11288 

11339 

11391 

37 

11443 

11494 

11546 

11598 

11649 

11701 

11753 

11806 

11858 

11910 

38 

11963 

12016 

12068 

12121 

12174 

12227 

12281 

12334 

12387 

12441 

39 

12494 

12548 

12602 

12656 

12710 

12764 

12819 

12873 

12928 

12982 

40 

13037 

13092 

1314X 

13202 

13257 

13312 

13368 

13423 

13479 

13535 

41 

13591 

13647 

13703 

13759 

13815 

13872 

13928 

13985 

14042 

14099 

42 

14156 

14213 

14270 

14327 

14385 

14442 

14500 

14558 

14616 

14673 

43 

14731 

14790 

14848 

14906 

14965 

15024 

15082 

15141 

15200 

16259 

44 

15318 

15378 

15437 

15497 

15556 

15616 

15676 

15736 

15796 

15856 

45 

15917 

15977 

16038 

16098 

16159 

16220 

16281 

16342 

16403 

16466 

46 

16526 

16587 

16649 

16711 

16773 

16835 

16897 

16969 

17021 

17084 

47 

17146 

17209 

17272 

17335 

17398 

17461 

17524 

17587 

17651 

17714 

48 

17778 

17842 

17905 

17969 

18033 

18098 

18162 

18226 

18291 

18356 

49 

18420 

18485 

18550  ' 

18615 

18680 

18746 

18811 

18877 

18942 

19008 

50 

19074 

19140 

19206 

19272 

19339 

19405 

19472 

19538 

19605 

19672 

51 

19739 

19806 

19873 

19940 

20008 

20075 

20143 

20211 

20279 

20347 

62 

20415 

20483 

20551 

20620 

20688 

20757 

20826 

20894 

20963 

21032 

53 

21102 

21171 

21241 

21310 

213S0 

21450 

21519 

21589 

21669 

21730 

64 

21800 

21870 

21941 

22012 

22082 

22153 

22224 

22295 

22366 

22438 

56 

22509 

22581 

22652 

22724 

22796 

22868 

22940 

23012 

23085 

23157 

56 

23230 

23302 

23375 

23448 

23521 

23594 

23667 

23741 

23814 

23888 

57 

23961 

24035 

24109 

24183 

24257 

24331 

24405 

24480 

24654 

24629 

68 

24704 

24779 

24854 

24929 

25004 

25079 

25155 

25230 

25306 

25381 

59 

25457 

25533 

25609 

25686 

25762 

25838 

25915 

25992 

26068 

26146 

60 

26222 

26299 

26376 

26454 

26531 

26609 

26686 

26764 

26842 

26920 

For  continuation  to  100  feet,  see  Table  7. 


EAILEOADS. 


797 


Table  7.      lievel  Cattingrs. 

Oontlnnation  of  the  six  foregoing  T&bles  of  Cubic  Contents,  to  100  feet  of  height  or  depth. 


Height 

Tftble 

Table 

Table 

Table 

Table 

Table 

or  Depth 

in  Feet. 

1 

2 

3 

4 

5 

6 

Cu.  Yds. 

Cu.  Yds. 

Cu.  Yds. 

Cu.  Yds". 

Cu.  Yds. 

Cu.  Yds, 

61 

23835 

26094 

17848 

24739 

20107 

26998 

.5 

24201 

26479 

18108 

25113 

20386 

27390 

62 

24570 

26867 

18370 

25489 

20667 

27785 

.5 

24942 

27257 

18634 

25868 

20949 

28183 

63 

25317 

27650 

18900 

26250 

21233 

28583 

.5 

25694 

28046 

19168 

26635 

21519 

28986 

64 

26074 

28444 

19437 

27022 

21807 

29393 

.5 

26457 

28846 

19708 

27413 

22097 

29801 

65 

26843 

29250 

19981 

27806 

22389 

30213 

.5 

27231 

29657 

20256 

28201 

22682 

30627 

66 

27622 

30067 

20533 

28600 

22978 

31044 

.5 

28016 

30479 

20812 

29001 

23275 

31464 

67 

28413 

30894 

21093 

29406 

23574 

31887 

.5 

28812 

31313 

21375 

29813 

23875 

32312 

68 

29216 

31733 

21659 

30222 

24178 

32741 

.5 

29620 

32157 

21945 

30635 

24482 

33172 

69 

30028 

32583 

22233 

31050 

24789 

33605 

.5 

30438 

33013 

22523 

31468 

25097 

34042 

70 

30852 

33444 

22814 

31889 

25407 

34481 

.5 

31268 

33879 

23108 

33313 

25719 

34924 

71 

31687 

34317 

23404 

32739 

26033 

35369 

.5 

32108 

34757 

23701 

33168 

26349 

35816 

72 

32533 

35200 

24000 

33600 

26667 

36267 

.5 

32960 

35646 

24301 

34035 

26986 

36720 

73 

33390 

36094 

24604 

34472 

27307 

37176 

.5 

33823 

36546 

24907 

34913 

27631 

37635 

74 

34259 

37000 

25214 

35356 

27956 

38096 

.5 

34697 

37457 

25522 

35801 

28282 

38561 

75 

35139   • 

37917 

25832 

36250 

28611 

39028 

.5 

35582 

38379 

26144 

36701 

28942 

39498 

76 

36029 

38844 

26458 

37156 

29174 

39970 

.5 

36479 

39313 

26774 

37613 

29608 

40446 

77 

36931 

39783 

27092 

38072 

29944 

40924 

.5 

37386 

40257 

27411 

38535 

30282 

41405 

78 

37844 

40733 

27733 

39000 

30622 

41889 

.5 

38305 

41213 

28056 

39468 

30964 

42375 

79 

38768 

41694 

28381 

39939 

31307 

42865 

.5 

39235 

42179 

28708 

40413 

31653 

43357 

80 

39704 

42667 

29037 

40889  ' 

32000 

43852 

81 

40650 

43650 

29700 

41850 

32700 

44850 

82 

41607 

44644 

30370 

42822 

33407 

45859 

83 

42576 

45650 

31048 

43806 

34122 

46880 

84 

43555 

46667 

31733 

44800 

34844 

47911 

85 

44546 

47694 

32426 

45806 

35574 

48954 

86 

45548 

48733 

33126 

46822 

36311 

60008 

87 

46561 

49783 

33833 

47850 

37056 

51072 

88 

47585 

50844 

34548 

48889 

-^7807 

52148 

89 

48620 

51917 

35270 

49939 

38567 

53235 

90 

49667 

53000 

36000 

51000 

39333 

54333 

91 

50724 

54094 

36737 

52072 

40107 

55443 

92 

51793 

55200 

37481 

53156 

40889 

56563 

93 

52872 

56:U7 

38233 

64250 

41678 

57694 

94 

53963 

57444 

38993 

55356 

42474 

58837 

•   95 

55065 

58583 

39759 

56472 

43278 

59990 

96 

56178 

69733 

40533 

57600 

44089 

61155 

97 

57302 

60894 

41315 

58739 

44907 

62331 

98 

58437 

62067 

42104 

59889 

45733 

63518 

99 

59583 

63250 

42900 

61050 

46567 

64716 

100 

60741 

64444 

43704 

62222 

47407 

65926 

798 


RAILROADS. 


Table  8, 

Of  Cubic  Yards  in  a  100-foot  station  of  level  cutting  or  filling,  to  be  added  to,  or  sub* 
tracted  from,  the  quantities  in  the  preceding  seven  tables,  in  case  the  excaya* 
tioxie  or  enxbankments  should  be  increased  or  diminished  2  feet  in  width. 

Cubic  Yards  in  a  length  of  100  feet ;  breadtli  2  feet;  and  of  different  depths. 


Height  or 
Dopth 
in  Feet. 

Cubic 
Yards. 

Height  or 
Depth 
in  Feet. 

.Cubic 
Yards. 

Height  or 
Depth 
in  Feet. 

Cubic 

Yards. 

Height  or 
Depth 
in  Feet. 

Cubic 
Yards. 

Height  or 

Depth 
in  Feet. 

Cubic 
Yarda. 

.5 

3.70 

.5 

152 

.5 

300 

.5 

448 

.5 

596 

1 

7.41 

21 

156 

41 

304 

61 

452 

81 

600 

.5 

n.i 

.5 

159 

.5 

307 

.6 

456 

.5 

604 

2 

14.8 

22 

163 

42 

311 

62 

459 

82 

607 

.5 

18.5 

.5 

167 

.5 

315 

.5 

463 

.6 

611 

3 

22  2 

23 

170 

43 

319 

63 

467 

83 

61S 

.5 

25'.9 

.5 

174 

.5 

322 

.5 

470 

.5 

619 

4 

29.6 

24 

178 

44 

326 

64 

474 

84 

622 

.5 

33.3 

.5 

181 

.5 

330 

.5 

478 

.5 

626 

5 

37.0 

25 

185 

45 

333 

65 

481 

86 

630 

.5 

40.7 

.5 

189 

.5 

337 

.5 

485 

.5 

633 

6 

44.4 

26 

193 

46 

341 

66 

489 

86 

637 

.5 

48.1 

.5 

196 

.5 

344 

.5 

493 

.5 

641 

7 

61.9 

27 

200 

47 

348 

67 

496 

87 

644 

.5 

65.6 

.5 

204 

.5 

352 

.5 

500 

.5 

648 

8 

59.3 

28 

207 

48 

356 

68 

504 

88 

652 

.5 

63.0 

.5 

211 

.5 

359 

.5 

607 

.5 

656 

9 

66.7 

29 

215 

49 

363 

69 

511 

89 

659 

.6 

70.4 

.5 

219 

.5 

367 

.5 

515 

.5 

663 

10 

74  1 

30 

222 

50 

370 

70 

519 

90 

667 

.5 

77.8 

.5 

226 

.5 

374 

.5 

522 

.5 

670 

11 

81.5 

31 

230 

51 

378 

71 

526 

91 

674 

.5 

85  2 

.5 

233 

.5 

381 

.5 

530 

.5 

678 

12 

88.9 

32 

237 

52 

•  385 

72 

533 

92 

681 

.5 

92.6 

.5 

241 

.5 

389 

".5 

537 

.5 

686 

13 

96.3 

33 

244 

53 

393 

73 

541 

93 

689 

.5 

100 

.5 

248 

.5 

396 

.5 

644 

.5 

693 

14 

104 

34 

252 

54 

400 

74 

■648 

94 

696 

.5 

107 

.5 

256 

.5 

404 

.5 

652 

.5 

700 

15 

111 

35 

259 

55 

407 

75 

556 

95 

704 

.5 

115 

.5 

263 

.5 

411 

.5 

659 

.5 

707 

16 

119 

36 

267 

56 

415 

76 

563 

96 

711 

.5 

122 

.5 

270 

.5 

419 

.5 

567 

.6 

715 

17 

126 

87 

274 

67 

422 

77 

570 

97 

719 

.5 

130 

.5 

278 

.5 

426 

.5 

674 

.5 

722 

18 

133 

38 

281 

58 

430 

78 

678 

98 

726 

.5 

137 

.5 

285 

.5 

433 

.5 

681 

.5 

730. 

19 

141 

39 

289 

59 

437 

79 

685 

99 

733 

.5 

144 

.5 

293 

.5 

441 

.5 

689 

.5 

737 

20 

148 

40 

296 

60 

444 

80 

593 

100 

741 

Remark.  Thie  foregroing-  tables  of  level  cnttingrs  may  also  be 
used  for  i^idtbs  of  roadway  g^reater  than  those  at  the  heads 
of  the  tables.  Thus,  suppose  we  wish  to  use  Table  1,  for  a  roadbed  m  n,  16  ft 
•wide,  instead  of  c  &,  which  is  only  14  ft,  and  for  which  the  table  was  calculated.  It 
is  only  necessary  first  to  find  the  vert  dist  s  a,  between  these  two  roadbeds  ;  and  to 
add  it  mcntaUy  to  each  height  t  s,  of  the  given  embkt,  when  taking  out  from  the 


BAILBOADS. 


799 


table  the  numbers  of  cub  yds  corresponding  to  the  heights.  By  this  means  we  obtain 
the  contents  of  the  embk.t  chop,  for  any  required  dist.  Next,  from  these  content* 
subtract  that  corresponding  to  the  height  s  a,  for  the  same  dist.  The  remainder  will 
plainly  be  the  embkt  mn  op. 

In  practice  it  will  be  sufficiently  correct  to  take  *a  to  the  nearest  tenth  of  a 
.  foot,  which  will  sav'e  trouble  in  adding  it  mentally  to  the  heights  in  the  tables. 

If  tlie  roadbed  is  narrower  tlian  the  table,  as,  for  instance,  if  mn  be 
the  width  in  the  table,  but  we  wish  to  find  the  contents  for  the  width  cb,  then 
fii'st  find  sa,  and  calculate  the  cubic  yards  in  100  feet  length  of  cbmn.  Then, 
in  taking  out  the  cubic  yards  from  the  table,  first  subtract  5  a  mentally  from 
each  height;  and  to  the  cubic  yards  taken  out  for  each  100  feet,  opposite  this 
redicced  height,  add  the  cubic  yards  in  100  feet  of  cbmn. 

To  avoid  trouble  with  contractors  about  the  measurement  of  rock 
cuts,  stipulate  in  the  contract,  either  that  it  shall  conform  with  the  theoretical 
cross  section ;  or  that  an  extra  allowance  of  say  about  2  feet  of  width  of  cut 
will  be  made,  to  cover  the  unavoidable  irregularities  of  the  sides. 

Slirinkag^e  of  Embankment.  Although  earth,  when  first  dug,  and 
loosely  tiirowri  out,  swells  about  ^  part,  so  that  a  cubic  yard  in  place  averages 
about  IJ  or  1.2  cubic  yards  when  dug:  or  1  cubic  yard  diig  is  equal  to  f,  or  to 
.8333  of  a  cubic  yard  'in  place  ;  yet  when  made  into  embankment  it  gradually 
subsides,  settles,  or  shrinks,  into  a  less  bulk  than  it  occupied  before  being  dug. 

The  following  are  approximate  averages  of  the  shrinkage;  or,  in  other  words, 
the  earth  measured  in  place  in  a  cut,  will,  when  made  into  embankment,  occupy 
a  bulk  less  than  before  by  about  the  following  proportions: 

Gravel  or  sand , about    8  per  ct;  or  1  in  123/^  less. 

Clay "      10  per  ct;  or  1  in  10     less. 

Loam "      12  per  ct;  or  1  in    83/^  less. 

Loose  vegetable  surface  soil "      lo  per  ct;  or  1  in    Q%  less, 

Puddled  clay "   •  25  per  ct;  or  1  in    4     less. 

The  writer  thinks,  from  some  trials  of  his  own,  that  1  cubic  yard  of  any  hard 
rock  in  place,  will  make  from  \%  to  1%  cubic  yards  of  embankment;  say  on  an 
average  1.7  cubic  yards.  Or  that  1  cubic  yard  of  rock  embankment  requires 
.5882  of  a  cubic  yard  in  place.  He  found  that  a  solid  cubic  yard  when  broken 
into  fragments,  made  about  as  follows 


Cubic 
yards. 

In  loose  heap 1.9 

Carelessly  piled 1.75 

Carefully  piled , 1.6 

Rubble,  very  carelessly  scabbled 1.5 

Rubble,  somewhat  carefully  scabbled.....    1.25 


Of  which  there  were 


Solid 
52.6  per  cent. 
57 


67 
80 


Voids 
47.4  per  cent- 
43         " 
87 

33         " 
20         " 


800 


COST   OF   EARTHWORK. 


COST  OF  EARTHWORK. 


Art.  1.  It  is  advisable  to  pay  for  this  kind  of  work  by  the  cubic  yard  of  excavation  only ;  Ji> 
■tead  of  allowing  separate  prices  for  excavation  and  embankment.  By  this  means  we  get  rid  of  th« 
difficulty  of  measurements,  as  well  as  the  controversies  and  lawsuits  which  often  attend  the  deter- 
mination of  the  allowance  to  be  made  for  the  settlement  or  subsidence  of  the  embankments. 

It  is,  moreover,  our  opinion  that  justice  to  the  contractor  should  lead  to  the  Eng:llsll  praC" 

tice  of  pay i  11$^  the  laborers  by  the  cubic  yard,  instead  of  by  the  day. 

Experience  fully  proves  that  when  laborers  are  scarce  and  wages  high,  men  can  scarcely  be  depended 
upon  to  do  three-fourths  of  the  work  which  they  readily  accomplish  when  wages  are  low,  and  when 
fresh  hands  are  waiting  to  be  hired  in  case  any  are  discharged.  The  contractor  is  thus  placed  at  the 
loercy  of  his  men.  The  writer  has  known  the  most  satisfactory  results  to  attend  a  system  of  task- 
work, accompanied  by  liberal  premiums  for  all  overwork.  By  this  means  the  interests  of  the  laborer* 
are  identified  with  that  of  the  contractor  ;  and  every  man  takes  care  that  the  others  shall  do  their 
fair  share  of  the  task. 

Ellwood  Morris,  C  E,  of  Philadelphia,  was,  we  believe,  the  first  person  who  properly  Investigated 
the  elements  of  cost  of  earthwork,  and  reduced  them  to  such  a  form  as  to  enable  us  to  calculate  the 
total  with  a  considerable  degree  of  accuracy.  He  published  his  results  in  the  Journal  of  the  Franklin 
Institute  in  1841.  His  paper  forms  the  basis  on  which,  with  some  variations,  we  shall  consider  the 
matter:  and  on  which  we  shall  extend  it  to  wheelbarrows,  as  well  as  to  carts.  Throughout  this  paper 
we  speak  of  a  cubic  yard  considered  only  as  solid  in  its  place,  or  before  it  is  loosened  for  removal.  It 
is  scarcely  necessary  to  add  that  the  various  items  can  of  course  only  be  regarded  as  tolerably  close 
approximations,  or  averages.  As  before  stated,  the  men  do  less  work  when  wages  are  high  ;  and  more 
when  they  are  low.  A  great  deal  besides  depends  on  the  skill,  observation,  and  energy  of  the  con- 
tractor and  his  superintendents.  It  is  no  unusual  thing  to  see  two  contrMtors  working  at  the  same 
prices,  in  precisely  similar  material,  where  one  is  making  money,  and  the  other  losing  it,  from  a  want 
of  tact  in  the  proper  distribution  of  his  forces,  keeping  his  roads  in  order,  having  his  oarts  and  bar- 
rows well  filled,  &c,  Ac,  Uncommonly  long  spells  of  wet  weather  may  seriously  affect  the  cost  of  exe- 
cuting earthwork,  by  making  it  more  difficult  to  loosen,  load,  or  empty ;  besides  keeping  the  roads  In 
bad  order  for  hauling. 

The  aggregate  cost  of  excavating  and  removing  earth  is  made  up  by  the  following  items,  namely: 

1st,    Loosening  the  earth  ready  for  the  shovellers. 

2d.     Loading  it  hy  shovels  into  the  carts  or  harrows. 

3d.     Hauling,  or  wheeling  it  away,  including  emptyinq  and  returning. 

4:th.   Spreading  it  out  into  successive  layers  on  the  emhankment. 

6th.  Keeping  the  hauling-road  for  carts,  or  the  plank  gangways  for  harrows,  in  good  order. 

6th.  Wear,  sharpening,  depreciation,  and  interest  on  cost  of  tools. 

7th.   Superintendence,  and  water-carriers. 

8th.  Profit  to  the  contractor. 
We  will  consider  these  items  a  little  in  detail,  basing  our  calculations  on  the  assumption  that  com. 
jnon  labor  costs  $1  per  day,  of  10  working  hours.     The  results  in  our  tables  must  therefore  be  in- 
creased or  diminished  in  about  the  same  proportion  as  common  labor  costs  more  or  less  than  this. 

Art.  2.    liOoseninji:  the  earth  ready  for  the  shovellers.   This  !■ 

jenerally  done  either  by  ploughs  or  by  picks  ;  more  cheaply  by  the  first.  A  plough  with  two  horses, 
and  two  men  to  manage  them,  at  $1  per  day  for  labor,'  75  cents  per  day  for  each  horse,  and  37  cents 
ter  day  for  plough,  including  harness,  wear,  repairs,  &c,  or  a  total  of  $3.87,  will  loosen,  of  strong 
leavy  soils,  from  200  to  300  cubic  yards  a  day,  at  from  1.93  to  1.29  cents  per  yard;  or  of  ordinary 
loam,  from  40u  to  600  cubic  yards  a  day,  at  from  .97  to  .64  of  a  cent  per  yard.  Therefore,  as  an  ordi- 
nary average,  we  may  assume  the  actual  cost  to  the  contractor  for  loosening  by  the  plough,  as  fol- 
lows: strong  heavy  soils,  1.6  cents  ;  common  loam,  .8  cent;  light  sandy  soils,  .4  cent.  Very  stiff  pure 
clay,  or  obstinate  cemented  gravel,  may  be  set  down  at  2.5  cents ;  they  require  three  or  four  horses. 
By  the  pick,  a  fair  day's  work  is  about  14  yards  of  stiff  pure  clay,  or  of  cemented  gravel ;  25  yards 
of  strong  heavy  soils;  40  yards  of  common  loam;  60  yards  of  light  sandy  soils  —  all  measured  In 
place;  which,  at  $1  per  day  for  labor,  gives,  for  stiff  clay,  7  cents;  heavy  soils,  4  cents;  loam,  2.5 
cents ;  light  sandy  soil,  1.666  cents.  Pure  sand  requires  but  very  little  labor  for  loosening;  .5  of  a 
cent  will  cover  it. 

Art.  3.    tShoTelllngr  the  loosened  earth  into  carts.   The  amount 

shovelled  per  day  depends  partly  upon  the  weight  of  the  material,  but  more  upon  so  proportioning 
the  number  of  pickers  and  of  carts  to  that  of  shovellers,  as  not  to  keep  the  latter  waiting  for  either 
material  or  carts.  In  fairly  regulated  gangs,  the  shovellers  into  carts  are  not  actually  engaged  In 
shovelling  for  more  than  six-tenths  of  their  time,  thus  being  unoccupied  but  four-tenths  of  it;  while, 
under  bad  management,  they  lose  considerably  more  than  one-half  of  it.  A  shoveller  can  readily 
load  into  a  cart  one-third  of  a  cubic  yard  measured  in  place  (and  which  is  an  average  working  cart- 
load), of  sandy  soil,  in  five  minutes  ;  of  loam,  in  six  minutes  :  and  of  any  of  the  heavy  soils,  in  seven 
minutes.  This  would  give,  for  a  day  of  10  working  hours,  120  loads,  or  40  cubic  yards  of  light  sandy 
soil ;  100  loads,  or  33^  cubic  yards  of  loam  ;  or  86  loads,  or  28.7  yards  of  the  heavy  soils.  But  from 
these  amounts  we  must  deduct  four-tenths  for  time  necessarily  lost;  thus  reducing  the  actual  work- 
ing quantities  to  24  yards  of  light  sandy  soil,  20  yards  of  loam,  17.2  yards  of  the  heavy  soils.  "When 
the  shovellers  do  less  than  this,  there  is  some  mismanagement. 

Assuming  these  as  fair  quantities,  then,  at  $1  per  day  for  labor,  the  actual  cost  to  the  contractor 
for  shovelling  per  cubic  yard  measured  in  place,  will  be,  for  sandy  soils,  4.167  cents  ;  loam,  5  cents; 
heavy  soils,  clays,  <kc,  5.81  cents. 

In  practice,  the  carts  are  not  usually  loaded  to  any  less  extent  with  the  heavier  soils  than  with  the 
lighter  ones.  Nor,  indeed,  is  there  any  necessity  for  so  doing,  inasmuch  as  the  difference  of  weight 
of  a  cart  and  one-third  of  a  cubic  yard  of  the  various  soils  is  too  slight  to  need  any  attention  ;  espe- 
cially when  the  cart-road  is  kept  in  good  order,  as  it  will  be  by  any  contractor  who  understands  hlf 


COST   OF    EARTHWORK.  801 

own  interest.  Neither  is  it  necessary  to  modify  the  load  on  account  of  any  alight  incli/naHona  whioh 
may  occur  in  the  grading  of  roads.     An  earth-cart  weighs  by  itself  about  14  a  ton. 

Art.  4.    Hauling  away  the  earth;  dumping,  or  empty ing^; 

and  returning"  to  reload.  The  average  speed  of  horses  in  hauling  is  about  23^  miles 
per  hour,  or  200  feet  per  minute ;  which  is  equal  to  100  feet  of  trip  each  way  ;  or  to  100  feet  of  lead, 
a«  the  distance  to  which  the  earth  is  hauled  is  technically  called.*  Besides  this,  there  is  a  loss  of 
about  four  minutes  in  every  trip,  whether  long  or  short,  in  waiting  to  load,  dumping,  turning,  Ac. 
Hence,  every  trip  will  occupy  as  many  minutes  as  there  are  lengths  of  100  feet  each  in  the  lead;  and 
four  minutes  besides.  Therefore,  to  find  the  number  of  trips  per  day  over  any  given  average  lead,  w« 
divide  the  number  of  minutes  in  a  working  day  by  the  sum  of  4  added  to  the  number  of  100  feet 
ISBgths  contained  in  the  distance  to  which  the  earth  has  to  be  removed ;  that  is. 

The  number  (600)  of  minutes  in  a  working  day  _  ^ ^g  number  of  trips,  or  loads 
4  -|-  the  number  of  IQO/eel  lengths  in  the  lead   ~  removed  per  day,  per  cart. 
And  since  3^  of  a  cubic  yard  measured  before  being  loosened,  makes  an  average  cart-load,  the  aum- 
bar  of  loads,  divided  by  3,  will  give  the  number  of  cubic  yards  removed  per  day  by  each  cart;  and 
th»  cubio  yards  divided  into  the  total  expense  of  a  «art  per  day,  will  ^ive  the  cost  per  cubio  yard  for 
hauling. 

Remark.    When  removing  loose  rock,  which  requires  more  time  for  loading,  say, 
No.  of  minutes  (600)  in  a  working  day  _  ^^  ^j,  i^^^  removed, 
6  4-  ^o.  of  100-/«et  Itngtha  of  lead  per  day,  per  cart. 

In  kMds  of  ordinary  length  one  driver  can  attend  t«  4  carts ;  which,  at  $1  per  day,  is  %  oanti  ptr 
eart.  When  labor  is  $1  per  day,  the  expense  of  a  horse  is  usually  about  75  cents;  and  that  of  the 
•art,  including  harness,  tar,  repairs,  &c,  25  cents,  making  the  total  daily  cost  per  cart  $1.25.  The 
expense  of  the  horse  is  the  same  on  Sundays  and  on  rainy  days,  as  when  at  work  ;  and  this  consid- 
eration is  included  in  the  75  cants.  Some  contractors  employ  a  greater  number  of  drivers,  who  also 
help  to  load  the  carts,  so  that  the  expense  is  about  the  same  in  either  case. 

ExAMPLK.  How  many  cubic  yards  of  loam,  meaaured  in  the  cut,  can  be  hauled  by  a  horse  and  cart 
in  a  day  of  10  working  hours,  (600  minutes j  the  lead,  or  length  of  haul  of  earth  being  1000  feet,  (or 
10  lengths  of  100  feet,)  and  what  will  be  the  expense  to  the  contractor  for  hauling,  per  cubidyard, 
assuming  the  total  cost  of  cart,  horse,  and  driver,  at  $1.25? 

„  QOQ  minutes  600         .„  ,      ,  .     ,  ^i loads        ,,„ 

Here,    .  ,  , .  , — — ^.^,     ■  =  ^rr  —  *^  loads..       And —  =  14.3  cuinc yards. 

^-\-lO  lengths  of  100  feet,         14  3 

^    ^       125  cento  .  „        . 

And   — — . =  8.74  cents  per  cubic  yard. 

14.3  cub  yds 

In  this  manner  the  2d  and  3d  columns  of  the  following  tables  have  been  calculated. 

Art.  5.    Spreading-,  or  levelTing  oflf  the  earth  into  reg^ular 

thin  layers  on  the  embankment,  a  bankman  win  spread  from  50  to  lOOcubie 
yards  of  either  common  loam,  or  any  of  the  heavier  soils,  clays,  &c,  depending  on  their  dryness. 
This,  at  $1  per  day,  is  1  to  2  cents  per  cubic  yard;  and  we  may  assume  IJ^  cents  as  a  fair  average 
for  suoh  soils ;  while  1  cent  will  suffice  for  light  sandy  soils. 

This  expense  for  spreading  is  saved  when  the  earth  is  either  dumped  over  the  end  of  the  embank- 
ment, or  is  wasted;  still,  about  i^  cent  per  yari  should  be  allowed  in  either  case  for  keeping  the 
dumping-places  clear  and  in  order. 

Art.  6.  Keeping?  the  cart-road  in  g-ood  order  for  hanlingr* 

No  ruts  or  puddles  should  be  allowed  to  remain  unfilled ;  rain  should  at  once  be  led  off  by  shallow 
litches  ;  and  the  road  be  carefully  kept  in  good  order;  otherwise  the  labor  of  the  horses,  and  the  wear 
of  carts,  will  be  very  greatly  increased.  It  is  usual  to  allow  so  much  per  cubic  yard  for  road  repairs; 
hut  we  suggest  so  much  per  cubic  yard,  per  100  feet  of  lead ;  say  -r^  of  a  cent. 

Art.  7.  l¥ear,  sharpening:,  and  depreciation  of  pic|c9  and 

shovels.    Experience  shows  that  about  J4  of  a  cent  per  cubic  yard  will  flover  this  item., 

Superintendence  and  ivater-carriers.    These  expenses  win  vary  with 

local  circumstances ;  but  we  agree  with  Mr.  Morris,  that  IJ^  cents  per  cubic  yard  will,  under  ordinary 
eircumstances,  cover  both  of  them.    An  allowance  of  about  J4  cent  may  in  justice  be  added  for  extra 
trouble  in  digging  the  side-ditches  ;  levelling  oflF  the  bottom  of  the  cut  to  grade ;  and  general  trimming 
tip.     In  very  light  cuttings  this  may  be  increased  to  }4  cent  per  cubic  yard. 
At  }4  OBnt,  all  the  items  in  this  article  amount  to  2  cents  per  cubic  yard  of  cut. 

Art.  8.  Profit  to  the  contractor.  TWs  may  generally  be  set  down  at  from  6to 
15  per  cent,  according  to  the  magnitude  of  the  work,  the  risks  incurred,  and  various  incidental  cir- 
cumstances. Out  of  this  item  the  contractor  generally  has  to  pay  clerks,  storekeepers,  and  other 
agents,  aa  well  as  the  expenses  of  shanties,  &c  ;  although  these  are  in  most  cases  repaid  by  the  profit* 
of  the  stores;  and  by  the  rates  of  boarding  and  lodging  paid  to  the  contractors  by  the  laborers. 

Art.  9.  A  knowIed$re  of  the  foreg-oing-  items  enables  us  to 
calculate  with  tolerable  accuracy  the  cost  of  removing*  earth. 

For  example,  let  it  be  required  to  ascertain  the  cost  per  cubic  yard  of  excavating  commoa  loam,  meas- 
ured in  place;  and  of  removing  it  into  embankment,  with  an  average  haul  or  lead  of  1000  feet;  the 
wages  of  laborers  being  $1  per  day  of  10  working  hours  ;  a  horse  75  cts  a  day ;  and  a  cart  25  cts.  One 
driver  to  four  carts. 


-X-  When  an  entire  cut  is  made  into  an  embankment,  the  iMaii  hanl  is  the  dist  between  eeaters 
of  gravity  of  the  cut  and  embkt. 

51 


802 


COST   OF   EARTHWORK. 


Cenis, 

Mere  we  have  coat  of  loosening,  say  by  pick,  Art  2,  per  cubic  yard,  say,  2.50 

Loading  into  carts,  Art.  3,  "  "  5.00 

Hauling  1000  feet,  as  calculated  previously  in  example.  Art.  i,  "  8.74 

Spreading  into  layers,  Art.  5,  "  1.50 

Keeping  cart-road  in  repair,  Art.  6,  10  lengths  of  100  ft,  1.00 

Various  items  in  .  Art.  7,  2.00 


Total  cost  to  contractor. 

Add  contracior'a  profit,  say  10  per  cent, 


20.74 
2.074 


Total  cost  per  cubic  yard  to  the  company,  22.814 
It  is  easy  to  construct  a  table  like  the  following,  of  costs  per  cubic  yard,  for  different  lengths  of  lead. 
Columns  2  and  3  are  tirst  obtained  by  the  Rule  in  Article  4 ;  then  to  each  amount  in  column  3  is  added 
the  variable  quantity  of  -r^  of  ^  cent  for  every  100  feet  length  of  lead,  for  keeping  the  road  in  order ; 
and  the  constant  quantity  (for  any  given  kind  of  soil)  composed  of  the  prices  per  cubic  yard,  for 
loosening,  loading,  spreading,  or  wasting,  &c,  either  taken  from  the  preceding  articles;'  or  modified 
to  suit  particular  circumstances.    In  this  manner  the  tables  have  been  prepared. 

By  Carts.    I^abor  $1  per  day,  of  10  working-  hours. 


II 


47.0 

44.4 

42.1 

40.0 

36.4 

33.3 

28.6 

25.0 

22.2 

20.0 

18.2 

16.7 

15.4 

14.3 

13.3 

12.5 

11.8 

11.1 

10.5 

JO.O 
9.52 
9.09 
8.70 
8.33 
7.54 
6.90 
6.58 
5.88 
5.48 
5.13 
4.82 
4.54 
4.30 
4.08 
8.88 
3.70 
3.52 
2.86 
2.40 
2.07 
1.82 


Cts. 
2.66 
2.81 
2.97 
3.12 
3.43 
3.75 
4.37 
5.00 
5.63 
6.25 
6.87 
7.48 
8.12 
8.74 
9.40 
10.0 
10.6 
11.2 
11.9 
12.5 
13.1 
13.7 
14.4 
15.0 
16.6 
18.1 
19.0 
21.2 
22.8 
24.3 
25.9 
27.5 
29.1 
30.6 
32.2 
33.8 
35.5 
43.8 
52.1 
60.4 
68.7 


Gommon  Loam. 


TOTAL  COST  PER  CUBIC 
YARD,  EXCLUSIVE  OF 
PROFIT  TO  CONTRACTOR. 


Cts. 

13.69 

13.86 

14.05 

14.22 

14.58 

14.95 

15.67 

16.40 

17.13 

17.85 

18.57 

19.28 

19.92 

20.74 

21.50 

22.20 

22.90 

23.60 

24.40 

25.10 

25.80 

26.50 

27.30 

28.00 

29.85 

31.60 

32.64 

35.20 

37.05 

38.80 

40.65 

42.50 

44.35 

46.10 

47.95 

49.80 

51.78 

61.40 

71.02 

80.64 


^     ^ 


Cts. 

12.44 
12.61 
12.80 
12.97 
13.33 
13.70 
14.42 
15.15 
15.88 
16.60 
17.32 
18.03 
18.67 
19.49 
20.25 
20.95 
21.65 
22.35 
23.15 
23.85 
24.55 
25.25 
26.05 
26.75 
28.60 
30.35 
31.39 
33.95 
35.80 
37.55 
39.40 
41.25 
43.10 
44.85 
46.70 
48.55 
50.53 
60.15 
69.77 
79.39 
89.01 


Cts. 

11.99 

12.16 

12.35 

12.52 

12.88 

13.25 

13.97 

14.70 

15.4^ 

16.15 

16.87 

17.58 

18.22 

19.04 

19.80 

20.50 

21.20 

21.90 

23.70 

23.40 

24.10 

24.80 

25.60 

26.30 

28.15 

29.90 

30.94 

33.50 

35.35 

37.10 

38.95 

40.80 

42.65 

44.40 

46.25 

48.10 

50.08 

59.70 

69.32 

78.94 

88.56 


Cts. 

10.74 

10.91 

11.10 

11.27 

11.63 

12.00 

12.72 

13.45 

14.18 

14.90 

15.62 

16.33 

16.97 

17.79 

18.55 

19.25 

19.95 

20.65 

21.45 

22.15 

22.85 

23.55 

24.35 

25.05 

26.90 

28.65 

29.69 

32.25 

34.10 

35.85 

37.70 

39.55 

41.40 

43.15 

45.00 

46.85 

48.83 

58.45 

68.07 

77.69 

87.31 


Strong  HeaTy  Soils. 


TOTAL  COST  PER  CUBIC 
YARD,  EXCLUSIVE  OF 
PROFIT  TO  CONTRACTOR. 


J  a  £ 


Cts. 
16.00 
16.17 
16.36 
16.53 
16.89 
17.26 
17.98 
18.71 
19.44 
20.16 
20.88 
21.59 
22.23 
23.05 
23.81 
24.51 
25.21 
25.91 
26.71 
27.41 
28.11 
28.81 
29.61 
30.31 
32.16 
33.91 
34.95 
37.51 
39.36 
41.11 
42.96 
44.81 
46.66 
48.41 
50.26 
52.11 
54.09 
63.71 
73.53 
82.95 
92.5T 


Cts. 

14.75 
14.92 
15.11 
15.28 
15.64 
16.01 
16.73 
17.46 
18.19 
18.91 
19.63 
20.34 
20.98 
21.80 
22.56 
23.26 
23.96 
24.66 
25.46 
26.16 
26.86 
27.56 
28.36 
29.06 
30.91 
32.66 
33.70 
36.26 
38.11 
39.86 
41.71 
43.56 
45.41 
47.16 
49.01 
50.86 
52.84 
62.46 
72.08 
81.70 
91.32 


Cts. 

13.50 

13.67 

13.86 

14.03 

14.39 

14.76 

15.48 

16.21 

16.94 

17.66 

18.38 

19.09 

19.73 

20.55 

21.31 

22.01 

22.71 

23.41 

24.21 

24.91 

25.61 

26.31 

27.11 

27.81 

29.66 

31.41 

32.45 

35.01 

36.86 

38.61 

40.46 

42.31 

44.16 

45.91 

47.76 

49.61 

51.59 

61.21 

70.83 

80.45 

90.07 


=  2  2 

"ctT" 

12.25 
12.42 
12.61 
12.78 
13.14 
13.51 
14.2» 
14.96 
15.69 
16.41 
17.13 
17.84 
18.48 
19.30 
20.06 
20.76 
21.46 
22.16 
22.96 
23.66 
24.36 
25.06 
25.86 
26.56 
28.41 
30.16 
31.20 
33.76 
35.61 
37.36 
39.21 
41.06 
42.91 
44.66 
46.51 
48.36 
50.34 
59.96 
69.58 
79.20 
88.82 


COST    OF   EARTHWORK. 


803 


By  Carts.    liBbar  $1  per  da^,  of  10  workinir  boars. 


1 

Ij 

1 

Pure  stiff  Clay,  or  cemented 
Gravel. 

Light  Sandy  Soils. 

ll 

—  J3 

!l 

^i 

TOTAL     COST 

PER 

CUBIC 

TOTAL 

COST 

PER 

CUBIC 

t,^ 

if 

YARD,     EXCLUSIVE    X)F 

YARD,     EXCLUSIVE     OF 

.sS* 

PROFIT  TO  CONTRACTOR. 

PROFIT  TO  CONTRACTOR. 

r. 

It 

1} 

^1 

■o 

rJ 

^ 

^s 

s 

Ill 

1-1 

0)       rs 

hi 

111 

7eet. 

Cu.Yds. 

Cts. 

Cts. 

Cts. 

Cts. 

Cts. 

Cts. 

Cts. 

Cts. 

Cts. 

25 

47.0 

2.66 

19.00 

17.75 

14.50 

13.25 

11.52 

10.77 

•  10.25 

9.50 

50 

44.4 

2.81 

19.17 

17.92 

14.67 

13.42 

11.69 

10.94 

10.42 

9.67 

75 

42.1 

2.97 

19.36 

18.11 

14.86 

13.61 

11.88 

11.13 

10.61 

9.86 

100 

40.0 

3.12 

19.53 

18.28 

15.03 

13.78 

12.05 

11.30 

10.78 

10.03 

150 

36.4 

3.43 

19.89 

18.64 

15.39 

14.14 

12.41 

11.66 

11.14 

10.39 

200 

33.3 

3.75 

20.26 

19.01 

15.76 

14.51 

12.78 

12.03 

11.51 

10.76 

300 

28.6 

4.37 

20.98 

19.73 

15.48 

15.23 

13.50 

12.75 

12.23 

11.48 

400 

25.0 

5.00 

21.71 

20.46 

17.21 

15.96 

14.23 

13.48 

12.46 

12.21 

500 

22.2 

5.63 

22.44 

21.19 

17.94 

16.69 

14.96 

14.21 

13.69 

12.94 

600 

20.0 

6.25 

23.16 

21.91 

18.66 

17.41 

15.68 

14.93 

14.41 

13.66 

700 

18.2 

.6.87 

23.88 

22.63 

19.38 

18.13 

16.40 

15.65 

15.13 

14.38 

800 

16.7 

7.48 

24.59 

23.34 

20.09 

18.84 

17.11 

16.36 
if.OO 

15.84 

15.0» 

900 

15.4 

8.12 

25.23 

23.98 

20.73 

19.48 

17.75 

16.48 

15.73 

1000 

14.3 

8.74 

26.05 

24.80 

21.55 

20.30 

18.57 

17.82 

17.30 

16.55 

1100 
1200 

13  3 

9.40 

26  81 

25.56 

22.31 

21  06 

19.33 

18.58 
19.28 

18.06 

17.31 
18.01 

1215 

10.0 

27.51 

26.26 

23.01 

21.76 

20.03 

18^76 

1300 

11.8 

10.6 

28.21 

26.96 

23.71 

22.46 

20.73 

19.98 

19.46 

18.71 

1400 

11.1 

11.2 

28.91 

27.66 

24.41 

23.16 

21.43 

20.68 

20.16 

19.41 

1500 

10.5 

11.9 

29.71 

28.46 

25.21 

23.96 

22.23 

21.48 

20.96 

20.21 

1600 

10.0 

12.5 

30.41 

29.16 

25.91 

24.66 

22.93 

22.18 

21.66 

20.91 

1700 

9.52 

13.1 

31.11 

29.86 

26.61 

25.36 

23.63 

22.88 

22.36 

21.61 

1800 

9.09 

13.7 

31.81 

30.56 

27.31 

26.06 

24.33 

23.58 

23.06  . 

22.31 

1900 

8.70 

14.4 

32.61 

31.36 

28.11 

26.86 

25.13 

24.38 

23.86 

23.11 

2000 

8.33 

15.0 

33.31 

32.06 

28.31 

27.56 

25.83 

25.08 

24.56 

23.81 

2250 

7.54 

16.6 

.35.16 

33.91 

30.66 

29.41 

27.68 

26.93 

26.41 

25.66 

2500 

6.90 

18.1 

36.91 

35.66 

32.41 

31.16 

29.43 

28.68 

28.16 

27.41 

H  mile 

6.58 

19.0 

37.95 

36.70 

33.45 

32.20 

30.47 

29.72 

29.20 

28.4S 

3000 

5.88 

21.2 

40.51 

39.26 

36.01 

34.76 

33.03 

32.28 

31.76 

31.01 

3250 

5.48 

22.8 

42.36 

41.11 

37.86 

36.61 

34.88 

34.13 

33.61 

82.86 

3500 

5.13 

24.3 

44.11 

42.86 

39.61 

38.36 

36.63 

35.88 

35.36 

34.61 

3750 

4.82 

25.9 

45.96 

44.71 

41.46 

40.21 

38.48 

37.73 

37.21 

36.4« 

4000 

4.54 

27.5 

47.81 

46.56 

43.31 

42.06 

40.33 

39.58 

39.06 

38.31 

4250 

4.30 

29.1 

49.66 

48.41 

45.16 

43,91 

42.18 

41.45 

40.93 

40.18 

4500 

4.08 

30.6 

51.41 

50.16 

46.91 

45.66 

43.93 

43.18 

42.66 

41.91 

4750 

3.88 

32.2 

53.26 

52.01 

48.76 

47.51 

45.78 

45.03 

44  51 

43.76 

5000  . 

3.70 

33.8 

55.11 

53.86 

50.61 

49.36 

47.63 

46.88 

46.36 

45.61 

1  mile 

3.52 

35.5 

57.09 

55.84 

52.59 

51.34 

49.61 

48.86 

48.34 

47.59 

l^m. 

2.86 

43.8 

66.91 

65.46 

62.21 

60.96 

59.23 

58.48 

57.96 

57.21 

l^m. 

2.40 

52.1 

76.33 

75.08 

71.83 

70.58 

68.85 

68.10 

67.58 

66.83 

IHm. 

2.07 

60.4 

85.95 

84.70 

81.45 

80.20 

78.47 

77.72 

77.20 
85.82 

76.45 

2     m. 

1.82 

68.7 

95.57 

94.32 

91.07 

89.82 

88.09 

87.34 

86.07 

Art.  10.  By  wbeelbarrOWS.  The  cost  by  barrows  may  be  estimated  in  tbe  same 
manner  as  by  carts.  See  Articles  1,  &c.  Men  in  wheeling  move  at  about  the  same  average  rate  as 
horses  do  in  hauling,  that  is,  23^  miles  an  hour,  or  200  feet  per  minute,  or  1  minute  per  every  100-feek 
length  of  lead.  The  time  occupied  in  loading,  emptying,  &c  (when,  as  is  usual,  the  wheeler  loads  hi* 
own  barrow,)  is  about  1.25  minutes,  without  regard  to  length  of  lead ;  besides  which,  the  time  lost  in 
occasional  short  rests,  in  adjusting  the  wheeling-plank,  and  in  other  incidental  causes,  amounts  to 
about  Jy-  part  of  his  whole  time ;  so  that  we  must  in  practice  consider  him  as  actually  working  but 
9  hours  out  of  his  10  working  ones.    Therefore 

The  number  of  minutee  in  a  working  day  X  .9  _  the  numher  of  trips  or  of  load* 
1.25  +  the  numbii^^OO-feet  lengths  of  lead     ~    removed  per  day  per  barrow. 

See  Remark,  next  page.  . 

The  number  of  loads  divided  by  14  will  give  the  number  of  cub  yards,  since  a  cub  yard,  meaaurea 
in  place,  averages  about  14  loads.  And  the  cost  of  a  wheeler  and  barrow  per  day,  (say  $1  per  man, 
and  5  cents  per  barrow,)  divided  by  the  number  of  cub  yards,  will  jive  the  cost  per  yard  for  loading 
wheeling,  and  emptying. 


804 


COST   OF   EAETHWOEK. 


JEX.  How  many  cubic  yards  of  common  loam,  measured  in  place,  will  one  man  load,  wheel, 
and  empty,  per  day  of  10  working  hours,  (or  600  minutes ;)  the  lead,  or  distance  to  which  the  earth  i* 
removed  being  1000  feet,  (or  10  lengths  of  100  feet ;)  and  what  will  be  the  expense  per  yard,  luppoiiBt 
tbe  laborer  and  barrow  to  cost  $1.05  per  day  ? 

_         600  minutes  X   .9  540 

Sere,  — -  =  — —    =  48  trips,  or  loads  per  day. 

1.25  4-  10  lengths         11.25  f  a 

i    J  *^        n  1.,       T     J  ■,  .     ,       105  cents 

And  -—  =  3.43  cub  yds  per  day.     And r  =  30.6  cents 

14  "  3.43  cub  yds 

per  cub  yard  for  loading,  wheeling  away,  emptying,  and  returning.     This  would  be  increased  almost 


Rem.  For  roc*,  which  requires  more  time  for  loading,  say 

No  of  minutes  in  a  working  day  X  -9    _  No  of  loads  rmnoved 
1.6 -i- No  of  100- feet  langths  of  lead     ~  pe^  day,  per  barrow. 

A.Tt»  11*  The  following  tables  are  calculated  as  in  the  case  of  carts,  by  first  finding  columns  2 
and  3  by  means  of  the  Rule  in  Art  10, and  then  adding  to  each  sum  in  column  3,  thu  variable  quantity 
of  .1  of  a  cent  per  cubic  yard  per  100  feet  of  lead  for  keeping  the  wheeling-planks  in  order ;  and  the 
prices  of  loosening,  spreading,  superintendence,  water-carrying,  &c,  per  cubic  yard,  as  given  in  thf> 
preceding  Articles  2  to  7. 

By  H^lieelbarrows.      I^abor  $1  per  day,  of  10  workings  liours* 


3« 

P 

1'^ 

Oommon  Loam. 

Strong,  Heavy  Soils. 

a  « 

*  e* 

§,2 

•S  o. 

*i 

l-s 

"2  "2 

«-« 

TOTAL     COST 

PER 

CUBIC 

TOTAL     COST     PER 

CUBIC 

^% 

^1 

^§ 

YARD,     EXCLUSIVE     OF 

YARD,     EXCLUSIVE     OF 

o  ^ 

J3 

t.^ 

PROFIT  TO  CONTRACTOR. 

PROFIT  TO  CONTRACTOR. 

-f 

in 

Hi 

II 
1 

T3       -d 

111 

u 

hi 

Ill 

%4 

Feet. 

Cu.Yds. 

Cts. 

Cts. 

Cts. 

cts. 

Cts. 

Cts. 

Cts. 

Cts. 

Cts. 

25 

25.7 

4.09 

10.12 

8.87 

8.42 

7.17 

11.62 

10.37 

9.12 

7.87 

50 

22.1 

4.75 

10.80 

9.55 

9.10 

7.86 

12.30 

11.05 

9.80 

8.56 

75 

19.3 

5.44 

11.52 

10.27 

9.82 

8.57 

13.02 

11.77 

10.52 

9.27 

100 

17.1 

6.14 

12.24 

10.99 

10.54 

9.29 

13.74 

12.49 

11.24 

9.99 

150 

14.0 

7.50 

13.65 

12.40 

11.95 

10.70 

15.15 

13.90 

12.65 

11.40 

200 

11.9 

8.82 

15.02 

13.77 

13.32 

12.07 

16.52 

15.27 

14.02 

12.77 

250 

10.3 

10.2 

16.45 

15.20 

14.75 

13.50 

17.95 

16.70 

15.45 

14.20 

•300 

9.07 

11.6 

17.90 

16.65 

16.20 

14.95 

19.40 

18.15 

16.90 

15.65 

350 

8.14 

12.9 

19.25 

18.00 

17.55 

16.30 

20.75 

19.50 

18.25 

17.00 

400 

7.36 

14.3 

20.70 

19.45 

19.00 

17.75 

22.20 

20.95 

19.70 

18.45 

450 

6.71 

15.6 

22.05 

20.80 

20..35 

19.10 

23.55 

22.30 

21.05 

19.80 

500 

6.17 

17.0 

23.50 

22.25 

21.80 

20.55 

25.00 

23.75 

22.50 

21.25 

SOO 

5.32 

19.7 

26.30 

25.05 

24.60 

23.35 

27.80 

26.55 

25.30 

24.05 

700 

4.67 

22.5 

29.20 

27.95 

27.50 

26.25 

30.70 

29.45 

28.20 

26.95 

800 

4.17 

25.2 

32.00 

30.75 

30.30 

29.05 

33.50 

32  25 

31.00 

29.75 

900 

3.76 

27.9 

34.80 

33.55 

33.10 

31.85 

36.30 

35.05 

33.80 

32.55 

1000 

3.43 

30.6 

37.60 

36.35 

35.90 

34.65 

39.10 

37.85 

36.60 

35,35 

1200 

2.91 

36.1 

43.30 

42.05 

41.60 

40.35 

44.80 

43.55 

42.30 

41.05 

1400 

2.53 

41.5 

48.90 

47.65 

47.20 

45.95 

50.40 

49.15 

47.90 

46,65 

1600 

2.24 

46.9 

54.50 

53.45 

52.80 

51.55 

56.00 

54.75 

53.50 

52.25 

1800 

2.00 

52.5 

60.30 

59.05 

58.60 

57.35 

61.80 

60.55 

59.30 

58.06 

3000 

1.81 

58.0 

66.00 

64.75 

64.30 

63.05 

67.50 

66.25 

65.00 

63.75 

2200 

1.66 

63.3 

71.50 

70.25 

69.80 

68.55 

73.00 

71.75 

70.50 

69.25 

2400 

1.53 

68.6 

77.00 

75.75 

75.30 

74.05 

78.50 

77.25 

76.00 

74.75 

H  mile. 

1.39 

75.5 

84.14 

82.89 

8J.44 

81.19 

85.64 

84.39 

gS.14 

81.W 

COST   OF   EARTHWORK. 


805 


By  Wbeelbarrows.      liabor  $1  per  day,  of  10  working:  liour*, 


5S. 

1^ 
It 

Pure  Stiff  Clay,  or  Ce- 
mented Gravel. 

TAght  Sandy  Soils. 

II 

••3-f 

Il 

il 

TOTAL     COST     PER    CUBIC 

TOTAL     COST     PER 

CUBIC 

^i 

YARD,     EXCLUSIVE     OF 

YARD,     EXCLUSIVE     OF 

1! 

J3 

PROFIT  TO  CONTRACTOR. 

PROFIT  TO  CONTRACTOR. 

hi 

5 

1 

Hi 

lit 

111 

il 

1-1 

fi! 

hi 

="9  3 

Peet. 

Ca.Yds. 

Cts. 

Cts. 

cts. 

Cts. 

cts. 

Cts. 

cts. 

Cts. 

Cts. 

25 

25.7 

4.09 

14.62 

13.37 

10.12 

8.87 

8.79 

8.04 

7.52 

6.7T 

50 

22.1 

4.75 

15.30 

14.05 

10.80 

9.55 

9.47 

8.72 

8.20 

7.45 

75 

19.3 

5.44 

16.02 

14.77 

11.52 

10.27 

10.19 

9.44 

8.92 

8.17 

100 

17.1 

6.14 

16.74 

15.49 

12.24 

10.99 

10.91 

10.16 

9.64 

8.89 

150 

14.0 

7.50 

18.15 

16.90 

13.66 

12.40 

12.32 

11.57 

11.05 

10.30 

aoo 

11.9 

8.82 

19.52 

18.27 

15.02 

13.77 

13.69 

12.94 

12.42 

11.67 

250 

10.3 

10.2 

ao.95 

19.70 

16.45 

15.20 

15.12 

14.37 

13.86 

13.10 

•      300 

9.07 

11.6 

22.40 

21.15 

17.90 

16.65 

16.57 

15.82 

15.30 

14.55 

350 

8.14 

12.9 

23.75 

22.50 

19.25 

18.00 

17.92 

17.17 

16.65 

15.90 

400 

7.36 

14.3 

25.20 

23.95 

20.70 

19.45 

19.37 

18.62 

18.10 

17.35 

450 

6.71 

15.6 

26.55 

25.30 

22.05 

20.80 

20.72 

19.97 

19.45 

18.70 

500 

6.17 

17.0 

28.00 

26.75 

23.50 

22.25 

22.17 

21.42 

20.90 

20.1S 

600 

5.32 

19.7 

30.80 

29.55 

36.30 

25.05 

24.97 

24.22 

23.70 

22.96 

700 

4.67 

22.5 

33.70 

32.45 

29.20 

27.95 

27.87 

27.12 

26.60 

25.85 

800 

4.17 

25.2 

36.50 

35.25 

32.00 

30.75 

30.67 

29.92 

29.40 

28.65 

900 

3.76 

27.9 

39.30 

38.05 

34.80 

33.55 

33.47 

32.72 

32.20 

31.45 

1000 

3.43 

30.6 

42.10 

40.86 

37.60 

36.35 

36.27 

35.52 

35.00 

34.25 

1200 

2.91 

36.1 

47.80 

46.55 

43.30 

42.05 

41.97 

41.22 

40.70 

39.90 

1400 

2.53 

41.5 

53.40 

52.15 

48.90 

47.65 

47.57 

46.82 

46.30 

45.55 

1600 

2.24 

46.9 

59.00 

57.75 

54.50 

53.25 

53.17 

52.42 

51.90 

51.15 

1800 

2.00 

52.5 

64.80 

63.55 

60.30 

59.05 

58.97 

58.22 

57.70 

56.95 

2000 

1.81 

58.0 

70.50 

69.25 

66.00 

64.75 

64.67 

63.92 

63.40 

62.65 

2200 

1.66 

63.3 

76.00 

74.75        71.50 

70.25 

70.17 

69.42 

68.90 

68.15 

2400 

1.53 

68.6 

81.50 

80.25        77.00 

75.75 

75.67 

74.92 

74.40 

73.65 

3imil«. 

1.S9 

75.5 

88.64 

87.39        84.14 

82.89 

82.81 

82.06 

81.54 

80.W 

Art.  12.  By  ivlieeled  iscrapers  and  drag:  scrapers.  The  body 
of  the  wheeled  scraper  is  a  box  of  smooth  sheet-steel  about  'd\^  ft  square  by  16  ins 
daep,  containing  about  ^  cubic  yard  of  earth  when  "even  full."  The  box  is  open 
in  front  (in  some  machines  it  is  closed  by  an  "  end  gate  "  when  full),  and  can  be  raised 
and  lowered,  and  revolved  on  a  horizontal  axis.  To  fill  the  box,  it  is  lowered  into, 
and  held  down  in,  the  earth,  while  the  team  draws  the  machine  forward.  When  full, 
it  is  raised  to  about  a  foot  above  ground ;  and,  on  reaching  the  dump,  is  unloaded  by 
being  overturned  on  its  axis.  All  the  movements  of  the  box  are  made  by  means  of 
levers,  and  without  stopping  the  team,  which  thus  travels  constantly.  The  wheels 
have  broad  tires,  to  prevent  them  from  cutting  into  the  ground. 

In  the  drag  scraper  the  box,  owing  to  the  greater  resistance  to  traction,  is  made 
much  smaller.  It  contains  about  .15  to  .25  cubic  yard  in  place,  and  is  always  open  in 
front.  The  operation  of  the  drag  scraper  is  similar  to  that  of  the  wheeled  scraper, 
except  that  the  box,  when  filled,  rests  upon  the  ground  and  is  dragged  over  it  by  the 
team. 

Each  scraper  ("wheeled"  or  "drag")  requires  the  constant  use  of  a  team  of  two 
horaee  with  a  driver.  Besides,  a  number  of  men,  depending  on  the  shortness  of  the 
lead  and  the  number  of  scrapers,  are  required  in  the  pit  and  at  the  dump  to  load  the 
scrapers  (by  holding  the  box  down  into  the  earth)  and  unload  them  (by  tipping  the 
box).  Except  in  sand,  or  in  very  soft  soil,  it  is  economical  to  use  a  plow  before 
scraping. 

The  severest  work  for  the  team  is  the  filling  of  the  box  ;  and  this  occurs  oftenest 
where  the  lead  is  shortest.  Hence  smaller  scrapers  are  used  on  short  than  on  long 
hauls.    We  baae  our  calculations  on  the  following  loads : 

For  drag  scrapers  (used  only  on  short  hauls) 2    cubic  yard 

For  wheeled  scrapers 

lead  lem  than  100  feet 33  " 

"    100  to  300  feet 4  " 

"    400  to  500  feet 6 

**    over  600  feet 6  " 


806 


COST   OF   EARTHWORK. 


The  daily  expense  per  scraper,  for  driver's  wages  and  the  use  of  a  2-horse  team,  is 
about  $8.50.  For  leads  of  400  feet  and  over,  we  add  60  cts  per  day  for  use  of  "snatch 
team  "  to  help  load  the  larger  scrapers  then  used.  One  snatch  team  generally  serves 
a  number  of  scrapers. 

Owing  to  the  fact  that  the  teams  are  constantly  in  motion  without  rest,  they  travel 
•omewhat  more  slowly  than  with  carts.  We  take  150  ft  per  minute  (or  75  ft  of  lead 
per  minutej  as  an  average. 

In  loading  and  unloading,  the  teams  not  only  go  out  of  their  way  in  order  to  turn 
around,  but  travel  more  slowly  than  when  simply  hauling.  To  cover  this  we  make 
an  addition  of  25  ft  to  each  length  of  lead,  whether  long  or  short,  for  wheeled  scrap' 
«rs ;  and  15  feet  for  drag  scrapers. 

We  add  1  cent  per  cubic  yard  for  the  cost  of  loading  and  dumping  the  scrapers;  and 
estimate  the  approximate  cost  of  the  other  items  as  follows : 

Repairs  of  cart-road  ^  ct  per  cub  yd  in  place  for  each  100  ft  of  lead 

Light  Soils  Heavy  Soils 

Loosening  cts  per  cub  yd  in  place    cts  per  cub  yd  in  place 

by  pick *    6. 

by  shovel *   2. 

Spreading 1 1.5 

Superintendence,  wear  and  tear  etc 1 1. 

We  repeat  that  our  figures  are  to  be  regarded  merely  as  tolerable  approximations, 
and  subject  to  great  variations  according  to  skill  of  contractor  and  superintendent, 
strength  of  teams,  character  of  material  moved,  state  of  weather  etc  etc. 

No.  of  trips  per  day  _  No.  ( 600)  of  mins  in  a  working  day 
per  wheeled  scraper  =  No.  of  75  ft  lengths  in  (lead  -f  25  ft) 

No.  of  trips  per  day  _  No.  (600)  of  mins  in  a  working  day 
per  drag  scraper     -  No.  of  75  ft  lengths^hi  (lead  +  15  ft) 

No.  of  cub  yds  in  place  moved  _  No.  of  trips  per  ^  No.  of  cub  yds  in  place, 
per  day  by  each  scraper  day  per  scraper  ^      per  scraper  per  trip 

^ffr&Tnl\'auVng,'^-  =    Daily  expens^ofone  scraper  i  ct  for  loading 

dumping  and  returning        No.  of  cub  yds  in  place,  moved  and  dumping 

per  day  by  each  scraper 

Total  cost  per        Cost  per  cub  yd        .1  ct  per  cub  yd  ^    ^           ^  ^  .      j 

cubic  yard    in        m     place,     for        m  place  for  each  of   loosening     snreadine 

place  exclusive  -  loading,     haul-  +  100    ft    of   lead,  -f  f    wSS    and  suner? 

of  a)ntractor's        ing     dumping,        for     repairs    of  StendfnceV       ^ 
profit                       and  returning          road 

By  "Wheeled  Scrapers.    Labor  ^1  per  day  of  10  working  hours. 


3" 

f^ 

Total  cost, 

per  cubic 

yard  in  place,  exclusive  of  contractor'a  profit. 

*5 

ill 
ill 

Cost    per   cub    yd 
place,    for    load 
hauling,     dump] 
and  returning. 

Light  Soils 

Heavy  Soils 

1'' 

Spread 

"Wasted 

Picke 
Spread 

dand 
Wasted 

Plow 
Spread 

»dand 
Wa8t«i 

Feet 

cub  vds 

Cts 

Cts 

Cts 

Cts 

Cts 

Cts 

cts 

50 

200 

2.8 

4.9 

3.9 

10 

8.5 

7.3 

5.8 

100 

140 

3.4 

5.5 

4.5 

11 

9.5 

8 

6.5 

150 

105 

4.3 

6.5 

5.5 

12 

11 

9 

7.5 

200 

80 

5.4 

7.6 

6.6  . 

13 

12 

10 

8.5 

300 

56 

7.3 

9.6 

8.6 

15 

14 

12 

11 

400 

50 

8.5 

11 

10 

16 

15 

13 

12 

600 

43 

10 

13 

12 

18 

17 

15 

14 

800 

33 

13 

16 

15 

21 

20 

18 

17 

1000 

27 

16 

19 

18 

25 

24 

22 

21 

*  Light  soils  can  generally  be  advantageously  loosened  by  the  scrapers  themselves  in  the  act  af 
loading. 


COST   OF    EARTHWORK. 


807 


By  Brag:  Scrapers. 

Labor  Si 

per  day  of  10  working  hou 

rs. 

■«    B 

. 

a  -  s« 

« 

1-^ 

"-  Sfa 

Total  cost 

per  cubic  yard  in  place,  exclusive  of  contractor's  profit. 

u5       . 

c.^ 

■^IL 

°.s 

•o 

1 

i; 

Light  Soils 

Heavy  Soils 

^-•s 

ill 

fill 

Spread 

Wasted 

Picked  and 
Spread       Wasted 

Plowed  and 
Spread         Wasted 

Feet 

cub  yds 

cts 

Cts 

Cts 

Cts 

Cts 

Cts 

cts 

less) 

than  y 

220 

2.6 

4.6 

3.6 

10 

8.5 

7 

5.6 

40  J 

50 

140 

3.5 

5.5 

4.5 

11 

9.5 

8 

6.5 

75 

100 

4.5 

6.6 

5.6 

12 

11 

9 

8 

100 

80 

5.4 

7.5 

6.5 

13 

12 

10 

9 

150 

54 

7.5 
9.3 

9.6 

8.6 

15 

14 

12 

11 

200 

42 

12. 

11 

17            16 

14 

13 

Art.  13.  By  cars  and  locomotive,  on  level  track.  We  have  based 
our  calculations  upon  the  following  assumptions:  Trains  of  10  cars,  each  car  con- 
taining 13^  cubic  yards  of  earth  measured  in  place.  Average  speed  of  trains, 
including  starting  and  stopping,  but  not  standing,  10  miles  per  hour,  =  5  miles 
of  lead  per  hour.  Labor  |l  per  day  of  10  working  hours.  Loosening,  loading 
(by  shovelers),  spreading,  wear  Ac  of  tools,  superintendence,  &c,  the  same  as 
with  carts.  Arts  2,  3,  5,  and  7.  Loss  of  time  in  each  trip  for  loading,  unloading, 
&c,  9  minutes,  =  0.15  hour.     Therefore 

Number  of  trips  per  )   _     The  number  (10)  of  hours  in  a  working  day 
day,  per  train       j        .15  -f-  the  number  of  5-mile  lengths  in  the  lead 

Number  of  cubici  Number  of  Number  (10)  Number  (1.5)  of  cubic 
yards  in  place,  per  >•  =  trips  per  day  X  of  cars  in  a  X  yards  in  place  in  each 
day  per  train  J  ^     per  train  train  car 

fo/^haulin^'^'dum 'i'n"  ^^andl  =  0"^  ^ay^s  train  expenses  +  1  day>8  cost  of  track 
returning     '  *  j        Number  of  cubic  yards  in  place  per  day  per  traia 

One  day's  train  expenses: 

Cost  of  10  cars  @  $100 - ~ $1000 

**       locomotive « «.. 3000 

$4000 

One  day's  interest  at  6  per  cent,  on  cost  of  train $0.67 

Wages  of  engine  driver  (who  fires  his  own  engine) 2.00 

**        foreman  at  dump ^ 2.00 

**        3  men  at  dump  at  $1 3.00 

Fuel ., 2.00 

Water 1.00 

Repairs  of  locomotive  and  cars 2.33 

Total  daily  expense  of  one  train - ...$13.00 

The  daily  expense  of  track,  for  interest  and  repairs,  may  be  taken 
at  $3  for  each  mile  of  lead. 
Therefore, 


808 


COST   OF   EARTHWORK. 


Cent  t)«r  cubic  yard  in  place, 
for  hauling,  dumping,  and 
returniiig 


Total  cost  per  cubic 
yard  in  place,  ex- 
clnsiTS  of  contrac- 
tor's profit 


'}' 


$13  +  ($3  for  each  mile  of  lead)  

'  Number  of  Number  (10)  Number  (1.5)  o< 
trips  per  day  X  of  cars  in  a  X  cubic  yards  in 
per  train  train  each  car 

Cost  per  cub  yd  in  Cost  per  cubic  yard,  in  place,  for 

place    for  hauling,  ,   loosening,    loading,    spreading    of 

'  dumping,    and    re-  •  wasting,  and  superintendence,  Ac 

turning  (Arts  2,  3,  5,  and  7.) 


By  Cars  and  L.ocomotive 

,    Labor  $1  per 

day  of 

10  working  houre. 

5ls 

11! 

111 

Total  cost  per  cubic  yard,  in  place 

,  exclusive  of  contractor's  profit. 

Light  Sandy  Soils. 

StroBff  Heavy  Soils. 

^»3 

•o 

« 

•o 

« 

per  cub 
ace,  for 
mping, 
rning. 

•S2 

s 

11 

II 

It 

■|1 

J.S 

1-^ 

r 

i> 

S£*. 

§«* 

.2^ 

Miles. 

Cu  yds. 

Cts. 

Cts. 

cts. 

cts. 

Cts. 

Cts. 

Cts. 

cts. 

Cts. 

0.125 

855 

1.56 

10.4 

9.6 

9.1 

8.4 

14.9 

13.6 

11.4 

11.1 

0.250 

750 

1.83 

10.7 

9.9 

9.4 

8.6 

15.1 

13.9 

12.6 

11.4 

0.375 

660 

2.14 

11.0 

10.2 

9.7 

9.0 

15.4 

14.2 

12.9 

11.7 

0.500 

600 

2.42 

11.3 

10.5 

10.0 

9.2 

15.7 

14.5 

13.2 

12.0 

0.750 

495 

3.08 

11.9 

11.2 

10.6 

9.9 

16.4 

15.1 

13.9 

12.6 

1.000 

420 

3.81 

12.6 

11.9 

11.4 

10.6 

17.1 

15.9 

14.6 

13.4 

2.000 

270 

7.04 

15.9 

15.1 

14.6 

13.9 

20.3 

19.1 

17.8 

16.6 

3.000 

195 

11.30 

20.1 

19.4 

18.9 

18.1 

24.6 

23'.4 

22.1 

20.9 

4.000 

150 

16.70 

25.5 

24.8 

24.3 

23.5 

30.0 

28.8 

27.5 

26.3 

Where  large  amounts  of  work  are  to  be  done,  the  steam  excavator,  land 
dredg'e  or  steam  shovel  generally  economizes  time  and  money.  Where  the 
depth  of  cutting  is  less  than  10  ft,  so  much  time  is  lost  in  moving  from  place  to  place 
that  the  excavators  do  not  work  to  advantage.  In  stiff  soils,  cuttings  may  be  made 
about  from  17  to  20  ft  deep  without  changing  the  level  of  the  machine.  For  greater 
depths  in  such  soils  the  work  is  done  in  two  levels,  since  the  bucket  or  dipper  cannot 
reach  so  high.  But  in  sand  and  looie  gravel,  much  deeper  cuts  may  be  made  from  a 
single  level. 

The  excavator  resembles  a  dredging  machine  in  its  appearance  and  operation.  A 
large  plate-steel  bucket,  like  a  dredging  bucket,  with  a  flat  hinged  bottom,  and  pro- 
vided with  steel  cutting  teeth,  is  forced  into  and  dragged  through  the  earth  by 
eteam  power.  It  dumps  its  load,  by  means  of  the  hinged  bottom,  either  into  cars 
for  transportation,  or  upon  the  waste  bank,  as  desired. 

Each  machine  is  mounted  on  a  car  of  standard  gauge,  which  can  be  coupled  in  an 
ordinary  freight  train.  The  car  is  made  of  wood  or  iron,  as  desired,  and  is  provided 
with  a  locomotive  attachment,  by  which  it  can  be  moved  frcA  point  to  point  aa  the 
work  proceeds.  The  machines  can  be  used  as  nrrecking^  or  derrick  cars. 
Each  machine  has  a  water  tank,  holding  from  300  to  550  gallons,  for  the  supply  of 
its  boiler. 

Before  beginning  to  excavate,  the  end  of  the  car  nearest  the  work  is  lifted  from 
the  track  by  hydraulic  or  screw  jacks,  upon  which  it  rests  M'hile  working. 

In  stiff  soils  the  excavator  leaves  the  sides  of  the  cut  nearly  vertical ;  and  the  de- 
sired slope  is  afterwards  given  by  pick  and  shovel.  When  the  soil  is  hard  or  much 
frozen,  it  may  be  loosened  by  blasting  in  advance  of  the  excavator. 


COST  OP  EARTHWORK.  809 

The  excavator  has  to  be  moved  forward  (as  the  work  advances)  abt  8  ft  at  a  time. 
As  regularly  made,  it  can  dig  at  a  distance  of  17  ft,  horizomtally,  from  the  center  of 
the  car  in  any  direction,  and  can  dump  12  ft  above  the  track.  In  sand  or  gravel  it 
takes  out,  while  actually  digging,  3  dipperfuls  (=  4^^  to  6  cub  yards  in  the  dipper, 
aa  3.75  to  5  cubic  yards  in  place)  per  minute;  in  stiff  clay,  2  dipperfuls  per  minute 
(=  3  to  4  cub  yards  in  the  dipper,  =  2.5  to  3.33  cubic  yards  in  place).  An  average 
day's  work  (10  hours)  for  a  **No  1'*  machine,  including  time  lost  in  moving  the  ma- 
chine, &c,  is  about  500  cubic  yards  in  "  hard-pan,"  and  from  1200  to  1500  in  sand  and 
gravel.  This  allows  for.the  usual  and  generally  unavoidable  delays  in  having  cars 
ready  for  the  excavator. 

The  excavators  carry  about  80  to  90  lbs  of  steam.  They  burn  from  100  to  150  lbs 
of  good  hard  or  soft  coal  per  hour;  and  require  one  engineer,  one  fireman,  one 
cranesman,  and  5  to  10  pitmen,  including  a  boss.  The  pitmen  are  laborers,  who 
attend .  to  the  jacks,  lay  track  for  the  excavator  and  for  the  dump  cars,  assist  in 
moving  the  latter,  bring  or  pump  water,  &c,  &c. 

After  reaching  the  site  of  the  work,  about  30  minutes  are  required  for  getting  the 
excavator  into  working  condition ;  and  an  equal  length  of  time,  after  completioa 
of  the  work,  in  getting  it  ready  for  transportation. 

.  The  following  figures  are  taken  from  the  records  of  work  done  by  a  No  1  machine, 
fSrom  May  to  Nov,  1883.  The  material  was  hard  clay  with  pockets  of  sand.  The 
expenses  per  day  of  12  working  hours,  at  $1.50  per  such  day  for  labor,  were 

Water  (a  very  high  allowance) $  6.00 

Coal,  l]4  tons  bituminous 10.00 

Wages  of  engineer 4.00 

**      "  fireman 1.50 

*•      •*  cranesman  or  dipper-tender  ~ 2.50 

«•      «*  pit  boss 3.00 

•*      «  8  pitmen  at  $1.50 12.00 

Oil,  waste,  repairs,  Ac  (estimated) 5.00 

Interest  on  cost  ($7500)  of  machine.^. ^ 1.25 

$44.26 

Reduced  to  our  standard  of  $1  for  labor  per  day  of  10  working  hours,  this  would 
be  say  $30.00  per  day.  Reduced  to  the  same  standard,  and  allowing  for  the  greater 
proportional  loss  of  time  in  stopping  at  evening  and  starting  in  the  morning:  the 
average  daily  quantity  excavated,  measured  in  place,  was,  in  shallow  cutting,  530 
cubic  yards;  in  deep  cutting,  1200  cubic  yards;  average  of  whole  operation,  800 
cubic  yards.  This  would  make  the  cost,  per  cubic  yard  measured  in  place,  for 
loosening  and  loading  into  cars,  5.67  cts,  2.5  cts,  and  3.75  cts  respectively ;  while  the 
cost  by  ploughing  and  shoveling,  in  strong  heavy  soils,  by  Arts  2  and  8,  is  7.4  cts ;  and 
by  picking  and  shoveling,  say  10  cts. 


810 


COST    OF    EARTHWORK. 


Art.    14.    Removinj^    rock    excavation    by    wheelbarrows. 

A  cubic  yard  of  hard  rock,  in  place,  or  before  being  blasted,  will  weigh  about 
1.8  tons,  if  sandstone  or  conglomerate,  (150  flbs  per  cubic  foot ;)  or  2  tons  if  good 
compact  granite,  gneiss,  limestone,  or  marble,  (168  lbs  per  cubic  foot.)  So  that, 
near  enough  for  practice  in  the  case  before  us,  we  may  assume  the  weight  of  any 
of  them  to  be  about  1.9  tons,  or  4256  lbs  per  cubic  yard,  in  place ;  or  158  fts  per 
cubic  foot. 

Now,  a  solid  cubic  yard,  fVben  broken  up  by  blasting  for  removal  by  wheel- 
barrows or  carts,  will  occupy  a  space  of  about  1.8,  or  1^  cubic  yards;* whereas  average  earth,  when 
loosened,  swells  to  but  about  1.2,  or  \\  of  its  original  bulk  in  place ;  although,  after  being  made  into 
embauKment,  it  eventually  shrinks  into  less  than  its  original  bulk.  In  estimating  for  earth,  it  is 
assumed  that  ^^  cubic  yard,  in  place,  is  a  fair  load  for  a  wheelbarrow.    Such  a  cubic  yard  will  weigh 

on  an  average  2430  fi)s,  or  1.09  tons;   therefore,  -rr-  =  174'  tts,  is  the  weight  of  a  barrow-load,  of 

2.31  cubic  feet  of  loose  earth.    Assuming  that  a  barrow  of  loose  rock  should  weigh  about  the  same 

4256 
as  one  of  earth,  we  may  take  it  at  ^^  of  a  cubic  yard  ;  which  gives  -rj-  =  177  ft>s  per  load  of  loose 

rock,  occupying  2  cubic  feet  of  space. 

In  the  following  table,  columns  2  and  3  are  prepared  on  the  same  principle  as  for  earth,  as  directed 
in  Article  10,  Column  4  is  made  up  by  adding  to  each  amount  in  column  3,  .2  of  A  cent  for  each  100 
feet  length  of  lead,  for  keeping  the  wheeling-planks  in  order  ;  and  45  cents  per  cubic  yard,  in  place, 


as  the  actual  cost  for  loosening,  including  tools,  driUing,  powder,  &c  ;  as  well  as  moderate  drainage, 
and  every  ordinary  contingency  not  embraced  in  column  3.     Cor" 
here  included. 


Contractor's  profits,  of  course,  are  not 


Ample  experience  shows  that  when  labor  is  at  %\  per  day,  the  foregoing  45  cents  per  cubic  yard,  in 
place,  is  a  sufficiently  liberal  allowance  for  loosening  hard  rock  under  all  ordinary  circumstances. 
In  practice  it  will  generally  range  between  30  and  60  cents ;  depending  on  the  position  of  the  strata, 
hardness,  toughness,  water,  and  other  considerations.  Soft  shales,  and  other  allied  rocks,  may  fre- 
quently be  loosened  by  pick  and  plough,  as  low  as  15  to  20  cents;  while,  on  the  other  hand,  shallow 
cuttings  of  very  tough  rock,  with  an  unfavorable  position  of  strata,  especially  in  the  bottoms  of  ex- 
cavations, may  cost  $1,  or  even  considerably  more.  These,  however,  sire  exceptional  cases,  of  com- 
paratively rare  occurrence.  The  quarrying  of  average  hard  rock  requires  about  %\o  %^  of  powder 
per  cubic  yard,  in  place ;  but  the  nature  of  the  rock,  the  position  of  the  strata,  &c,  may  increase  it 
to  J^  ft,  or  more.  Soft  rock  frequently  requires  more  powder  than  hard.  A  good  churn-driller  will 
drill  8  to  10  feet  in  depth,  of  holes  about  23^  feet  deep,  and  2  inches  diameter,  per  day,  in  average 
hard  rock,  at  from  12  to  18  cents  per  foot.    Drillers  receive  higher  wages  than  common  laborers. 


Hard  Rock,  by  IVlieelbarrows. 

Labor  §1  per  day,  of  10  working  hours. 


Length  of 
Lead,  or  dis- 

Numberof 
cubic  yards, 

Cost  per 

cubic  yard, 

in  place. 

Total  cost 
per  cubic 
yard,  in 

Length  of 
Lead.ordis- 

Number  of 
cubic  yards, 

Cost  per 

cubic  yard, 

in  place, 

Total  cost 
per  cubie 
yard,  in 

tance  to 
which  the 

rock  is 
wheeled. 

in  place, 
wheeled  per 
day  by  each 

barrow. 

for  loading, 
wheeling, 

and 
emptying. 

place,  ex- 
clusive of 
profit  to 
contractor. 

tance  to 

which  the 

rock  Is 

wheeled. 

in  place, 
wheeled  per 
day  by  each 

barrow. 

for  loading, 
wheeling, 

and 
emptying. 

place,  ex- 
clusive of 
profit  to 
contractor 

Feet. 

Cubic  Yds. 

Cents. 

Cents. 

Feet. 

Cubic  Yds 

Cents. 

Cents. 

25 

12.2 

.    8.64 

53.7 

600 

2.96 

35.5 

81.7 

50 

10.7 

9.81 

54.9 

700 

2.62 

40.1 

86.5 

75 

9.58 

11.0 

56.2 

800 

2.34     ' 

44.8 

91.4 

100 

8.66 

12.1 

57.3 

900 

2.12 

49.5 

96.3 

150 

7.26 

14.5 

59.8 

1000 

1.94 

54.1 

101.1 

200 

6.25 

16.8 

62.2 

1200 

1.65 

63.6 

115.0 

250 

5.49 

19.1 

61.6 

1400 

1.44 

72.9 

120.7 

300 

4.89 

21.5 

67.1 

1600 

1.28 

82.2 

130.4 

350 

4.41 

23.8 

69.5 

1800 

1.15 

91.5 

140  1 

400 

4.02 

26.1 

71.9 

2000 

1.04 

100.8 

149.8 

450 

3.69 

28.5 

74.4 

2^00 

.953 

110.2 

159,6 

500 

3.41 

30.8 

76.8 

2400 

.879 

119.5 

169.3 

Art.  15.  Removing*  rock  excavation  by  carts.  A  cart-load  of 
rock  may  be  taken  at -^  of  a  cubic  yard,  in  place.  This  will  weigh,  on  an  average, 
851  ft)s  ;  or  but  41  lbs  more  than  a  cart-load  of  average  soil.  Since  the  cart  itself  will 
weigh  about  3^  a  ton,  the  total  loads  are  very  nearly  equal  in  both  case.s.  Columns 
2  and  3  of  the  following  table  are  prepared  on  thfe  same  principle  as  for  earth,  as 
directed  in  Art.  4,  Column  4  is  made  up  by  adding  to  each  amount  in  column  3, 
the  following  items:  For  blasting,  (and  for  everything  except  those  in  column  3;. 
loading,  and  repairs  of  cart-road,)  45  cents  per  cubic  yard,  in  place ;  for  loading, 
8  cents,  per  cubic  yard,  in  place ;  and  for  repairs  of  road,  .2,  or  ^  of  a  cent  for  each 
100-feet  length  of  lead.     Contractor's  profit  not  included. 


COST  OF   EAETHWORK. 


811 


Hard  Bock,  by  Carts. 

Labor  $1  per  day,  of  10  working  hours. 


Length  of 

Number  of 

Cost  per 

Total  cost 
per  cubic 
yard,  in 
place,  ex- 

Length  of 

Number  of 

Cost  per 

Total  ooot 
per  cubic 
yard,  in 
place,  ex- 
clusive of 
profit  to 
contractor 

Liead,  or  dis- 

cubic yards, 

cubic  yard, 

Lead,  or  dis- 

cubic yards, 

cubic  yard. 

tance  to 

in  place, 

in  place. 

tance  to 

in  place, 

in  place,  for 

which  the 

hauled  per 

for  hauling, 

which  the 

hauled  per 

hauling. 

rock  is 

day,  by  each 

and 

profit  to 
contractor. 

rock  is 

day,  by  each 

and 

hanled. 

cart. 

emptying. 

hauled. 

cart. 

emptying. 

Feet. 

Cubic  Yda. 

Cents. 

Cents. 

Feet. 

Cubic  Yds. 

Cents. 

Cents 

25 

19.2 

6.51 

69.6 

1800 

6.00 

25.0 

81.6 

50 

18.5 

«.77 

59.9 

1900 

4.80 

26.0 

82.8 

75 

17.8 

7.03 

60.2 

2000 

4.62 

27.1 

84.1 

100 

17.1 

7.29 

60.5 

2250 

4.21 

29.7 

87.2 

150 

16.0 

7.81 

61.1 

2500 

3.87 

32.3 

90.3 

200 

15.0 

8.33 

61.7 

i^mile 

3.70 

33.7 

92.0 

800 

13.3 

9.37 

63.0 

3000 

3.33 

37.5 

96.5 

400 

12.0 

10.4 

64.3 

3250 

3.12 

40.1 

99.6 

600 

10.9 

11.5 

65.5 

3500 

2.92 

42.8 

102.8 

€00 

10.0 

12.5 

66.7 

3750 

2.76 

45.3 

105.8 

700 

9.23 

13.6 

68.0 

4000 

2.61 

47.9 

108.9 

800 

8.57 

14.6 

69.2 

4250 

2.47 

50.6 

112.1 

90» 

8.00 

15.6 

70.4 

4500 

2.35 

53.2 

115.2 

1000 

7.50 

16.7 

71.7 

4750 

2.24 

55.8 

118.S 

1100 

7.0C 

17.7 

72.9 

5000 

2.14 

58.4 

121.4 

1200 

6.67 

18.7 

74.1 

Imile 

2.04 

61.2 

124.8 

1300 

6.32 

19.8 

75.4 

134  " 

1.67 

75.0 

141.2 

1400 

6.00 

20.8 

76.6 

1^" 

1.41 

88.8 

157.6 

1500 

5.71 

21.9 

77.9 

IH" 

1.22 

102.5 

174.0 

1600 

5.45 

22.9 

79.1 

2      " 

1.08 

116.3 

190.4 

1700 

5.22 

24,0 

80.4 

2}i" 

.962 

13<i.O 

206.8 

"  liOOse  rock  "  will  cost  about  30  cts  per  yd  less;  and  even  solid  rock  will 

mverage  about  10  cts  less  than  the  tables. 

Art.  16.  Removing  rock  excavation  by  cars  and  locomo- 
tive, on  level  track.  Our  calculations  are  based  upon  the  following  assump- 
tions :  Trains  of  10  cars,  each  car  containing  1  cubic  yard  of  rock  measured 
in  place.  Average  speed  of  trains,  including  starting  and  stopping,  but  not 
standing,  10  miles  per  hour  =  5  miles  of  lead  per  hour.  Labor  $1  per  day  of 
10  working  hours.  Loosening,  45  cts  per  cubic  yard  in  place.  Loading,  8 
cts  per  cubic  yard  in  place.  Cost  of  track,  for  interest  and  repairs,  $3  per 
day  per  mile  of  lead.  The  calculations  are  the  same,  in  principle,  as  those 
in  Art.  13. 

Hard  Rock,  by  €ars  and  liOcomotive. 

Labor  $1  per  day  of  10  working  hours. 

Length  of  lead,  or  distance  to  which  the  rock 

is  nauled miles         1  3         5  7        10 

Number  of  cubic  yards,  in  place,  hauled  per 

day  by  each  train .- 2900      1300      800       600      400 

Cost,  per  cubic  yard  in  place,  for  hauling, 

dumping,  and  returning cents        .6        1.7      3.5      5.7      10.8 

Total  cost,  per  cubic  yard  in  place,  exclusive 

of  contractor's  profit cents     53.6      54.7     56.5     58.7      63.8 


812  TUNNELS, 


TUNNELS. 

Tniiiiels  for  railroads  should,  if  possible,  be  straigbt,  espe- 
ciallj  when  there  is  but  a  single  track  ;  inasmuch  as  collisions  or  other  accidents 
in  a  tunnel  would  be  peculiarly  disastrous.  A  tunnel  will  rarely  be  expedient 
before  the  depth  of  cutting  exceeds  60  feet.  Firm  rock  of  moderate  hardness, 
and  of  a  durable  nature,  is  tlie  most  favorable  material  for  a  tunnel; 
especially  if  free  from  springs,  and  lying  in  horizontal  strata.  In  soft  rock,  or 
in  shales  (even  if  hard  and  firm  at  first),  or  in  earth,  a  lining  of  hard  brick  or 
masonry  in  cement,  is  necessary.  A  tunnel  should  have  a  g'rade  or  incli- 
nation in  one  direction,  for  ease  of  future  drainage  and  ventilation.  No 
special  arrangement  is  essential  for  ventilation  either  during  construction^ 
pr  after,  if  the  length  does  not  exceed  about  1000  feet;  but  beyond  that,  gen^ 
'erally  during  construction  either  shafts  are  resorted  to,  or  means  provided  for 
forcing  air  into  the  tunnel  through  pipes  from  its  ends.  But  after  the  work  i» 
finished,  except  under  peculiar  circumstances,  nothing  of  the  kind  is  necessary. 
Shafts  often  draw  air  downwards;  and  frequently,  even  when  aided  by  a  steep, 
uniform  grade,  do  not  secure  ventilation.  The  Mont  Cenis  tunnel  under  the 
Alps,  completed  in  1871,  is  7^  miles  long,  and  has  no  shafts,  although  it  grades 
up  from  each  end,  which  is  the  most  unfavorable  of  all  conditions  for  ventila- 
tion without  shafts.  It  was  made  so  for  facilitating  drainage.  Its  ventilation 
is  maintained  by  air  forced  in  from  the  ends.  The  Hoosac  tunnel,  Mass,  4i<^ 
miles  long,  has  shafts  ;  one  of  them  1030  feet  deep ;  but  they  were  for  expediting 
the  work.  Sbafts  gr^nerally  cost  from  V^  to  3  times  as  much  per  cubic 
y»rd  as  the  main  tunnel,  owing  to  the  greater  difficulty  of  excavating  and  re- 
moving the  material,  and  getting  rid  of  the  water,  all  of  which  must  be  done 
by  hoisting.  When  through  earth,  they  must  be  lined  as  well  as  the  tunnel; 
and  the  lining  must  usually  be  an  under-pinning  process.  Or  the  lining  may 
,  first  be  built  over  the  intended  shaft,  and  then  sunk  by  undermin^ig  it  grad- 
ually. Their  sectional  area  commonly  varies  from  about  40  to 
100  square  feet.  They  have  the  great  advantage  of  expediting  the  work  by  in- 
creasing the  number  of  points  at  which  it  can  be  carried  on  ;  but  if  placed  too 
close  together,  their  cost  more  than  compensates  for  this.  The  air  in  some 
tunnels,  while  being  constructed,  is  much  more  foul  than  in  others;  so  that 
after  the  work  has  been  commenced,  shafts  with  forced  air  may  be  expedient 
where  they  were  not  anticipated.  In  excavating  the  tunnel  itself,  a  heading 
or  passage-way,  5  or  8  feet  high,  and  3  to  12  feet  wide,  is  driven  and  maintain^ 
a  short  distance  (10  to  100  feet,  or  more,  according  to  the  firmness  of  the  ma- 
terial) in  advance  of  the  main  work.  In  rock,  the  heading  is  just  below  the 
top  of  the  tunnel,  so  that  the  men  can  conveniently  drill  holes 'in  its  floor  for 
blasting;  but  in  earth,  the  heading  is  driven  along  the  bottom  of  the  tunnel- 
that  being  the  most  convenient  for  enlarging  the  aperture  to  the  full  tunnel 
size,  by  undermining  the  earth,  and  letting  it  fall.  In  earth,  the  top  and  sides 
of  the  heading,  as  well  as  of  the  tunnel,  must  be  carefully  prevented  from 
caving  in  before  the  lining  is  built ;  and  this  is  done  by  means  of  rows  of  verti- 
cal rough  timber  props,  and  horizontal  caps  or  overhead  pieces,  between  which 
land  the  earth  rough  boards  are  placed  to  form  temporary  supporting  sides  and 
ceiling  to  the  excavation.  The  props  and  caps  are  placed  first ;  and  the  boards 
are  then  driven  in  between  them  and  the  earthen  sides  of  the  excavation. 
These  are  gradually  removed  as  the  lining  is  carried  forward.  Tbe  lining:, 
when  of  brick,  is  usually  from  2  to  3  bricks  thick  (17  to  26  inches)  at  bottom, 
and  from  11^  to  2}4  bricks  thick  at  top;  and  when  of  rough  rubble  in  cement, 
about  half  again  as  thick.  It  is  important  that  the  bricks  or  stone  should  be 
of  excellent  hard  quality,  and  laid  in  good  cement.  The  bricks  should  bo 
moulded  to  the  shape  of  the  arch.  As  the  lining  is  finished  in  short  lengths, 
and  before  the  centers  are  removed,  any  cavities  or  voids  between  it  and 
the  earth  should  be  carefully  and  compactly  filled  up.  Even  in  rock,  if  much 
fissured,  or  if  not  of  durable  character,  as  common  shale,  lining  is  necessary, 
Tbe  cross-section  of  a  single-track  railroad  tunnel,  in  the  clear  of  every- 
thing, and  for  cars  of  11  feet  extreme  width,  should  not  be  less  than  about  15 
feet  wide,  by  18  feet  high ;  nor  a  double-track  one,  less  than  27  feet  wide,  by  24 
feet  high ;  unless  in  the  last  case  the  material  is  firm  rock,  in  which  a  high  arch 
is  not  necessary  for  lining.  The  roof  may  then  be  much  flatter,  so  that  a  height 
of  20  feet  may  answer.  With  cars  of  10  feet  extreme  width,  the  width  of  the 
tunnel  may  be  reduced  to  25  feet ;  or  with  9  feet  cars,  to  23  feet.  Many  have 
been  made  22  feet.  The  Mont  Cenis  is  26  feet  wide,  by  25  high.  Tbe  rate  of 
daily  progress  from  each  face  of  a  tunnel  varies  from  18  inches  to  9  feet  of 
length  per  24  hours,  with  three  relays  of  workmen.   On  the  Mont  Cenis  the  e»« 


TRESTLES. 


813 


tremes  were  about  4  to  9  feet  daily  for  a  whole  year,  from  each  face.  Drills 
worked  by  compressed  air  were  employed  in  the'headiugs,  which  were  12  feet 
wide  by  8  feet  high.  Ordinarily,  from  1^  to  3  feet  may  be  taken  as  averages. 
The  difference  of  rate  of  progress  between  a  single  and  a  double  track  tunnel 
is  not  so  great  as  might  be  supposed  ;  inasmuch  as  a  larger  force  can  be  era- 

{)loyed  on  the  wider  one.  If  the  tunnel  is  in  earth,  the  construction  of  the 
ining  about  makes  up  for  the  slower  excavation  of  one  in  rock.  In  rock,  with 
labor  at  31  per  day,  tne  cost  will  usually  vary  with  the  character  of  the  rock, 
from  S2  to  $5  per  cubic  yard  for  the  main  tunnel ;  and  from  $3  to  $10  for  the 
heading;  while  shafts  will  average  about  50  per  cent,  more  than  heading.  The 
coal  of  a  single-track  tunnel,  when  common  labor  is  $1  per  day,  will  generally 
range  between  $30  and  $75  per  foot  of  length.  Tunnel  work,  however,  is  liable 
to  serious  contingencies  which  cannot  be  foreseen.  Since  the  sides  and  roof  are 
rough  as  blasted,  the  width  and  height  should  each  be  estimated  to  the  con- 
tractor as  about  \S  inches  or  2  feet  greater  than  the  established  clear  ones.  At 
»ny  rate,  the  mode  of  measurement  should  be  clearly  stated  in  the  specifications 
for  the  work.  When  a  tunnel  is  made  with  a  uniform  grade,  the  work  gen- 
erally progresses  in  a  more  satisfactory  manner  from  the  lower  end,  because 
the  descent  favors  the  drainage  of  the  spring  water  that  is  usually  met  with ; 
whereas,  at  the  upper  end,  it  must  be  removed  by  pumps  or  by  bailing.  The 
upper  end  has,  however,  the  advantage  of  sooner  getting  rid  of  the  smoke  in 
blasting.  Before  commencing  a  tunnel,  or  even  deciding  upon  one,  trial 
shafts  should  be  sunk  to  ascertain  the  nature  of  the  material.  In  long  ones, 
the  greatest  care  and  accuracy  are  necessary  for  preserving  the  line  of  direc- 
tion, so  that  the  work  from  both  ends  shall  meet  properly  at  the  center. 

In  the  heading  of  the  Vosburg  (Pa)  tunnel  of  the  Lehigh  Valley  R  R,  built 
1884,  cross-section  7\^  feet  X  26  feet,  the  averag^e  progress  per  working 
day  of  24  hours  with  two  shifts  of  12  hours  each,  was  as  follows;  by  hand 
drilling  2.8  feet  and  2.4  feet  respectively  from  each  end;  by  machine  drills 
(two  rival  drills  in  competition)  5.6  feet  and  7.8  feet.  The  material  was  hard 
gray  sandstone.    For  the  whole  tunnel  the  rate  was  about  2  feet  per  day. 

For  further  information  respecting  tunnels,  the  reader  is  referred  to  Mr. 
H.  S.  Drinker's  very  full  treatise  on  the  subject,  published  by  the  Messrs  Wiley, 


TBESTLES. 

3  ,— 


4Pie8  1,  2,  8,  5,  6,  7,  are  elevations  of  trestles  ?  taken  across  the  track  or 
roadway.    We  may  consider  Fig  1  as  adapted  to  a  height  of  about  10  to  20  ft;  Figs  2 


814  TRESTLES. 

and  3,  to  heights  from  20  to  30  ft ;  Fig  5,  from  30  to  40  ft;  Pig  6,  from  40  to  00  ft;  u 
rough  approximations  merely.  A  single  framework,  such  as  that  shown  in  each  of 
these  six  figures,  is  called  a  "bent/*  These  bents  of  course  admit  of  many  modifi* 
cations.  They  are  usually  supported  by  bases  of  masonry,  as  in  the  figures.  These 
preserve  the  lower  timbers  from  contact  with  the  earth,  which  would  hasten  their 
decay.  It  is  adrisable  to  make  these  bases  high  enough  to  prevent  injury  from  cattle, 
orpassing  vehicles,  Ac.  Up  to  heights  of  about  40  or  60  ft,  a  single  row  of  posta  or  up- 
rights, a,  a,  a,  Figs  1  to  9,  as  shown  at  ee  under  Figs  1  and  6,  will  answer,  fiat  as  the 
height  becomes  greater,  more  posts  should  be  introduced,  as  shown  At  xx  under  Fig 
6;  or  two  entire  rows  of  them  ;  or  three  rows,  as  under  Fig  7 ;  and  as  also  in  Fig  8, 
which  is  an  end  view  of  Fig  7.  Figs  7  and  8  bear  much  resemblance  to  the  trestle* 
190  ft  high,  with  masonry  bases  30  ft  high  (S.  Seymour,  C.  E.),  which  carried  the 
Erie  Rway  (now  the  N  Y,  Lake  Erie  &  West'n  R  R)  over  the  Oenesee  River  aft 
Portagre,  N  Y«  There  each  bent  had  21  posts  14  ins  square,  at  its  base  ;  and  \b 
posts  of  12  X  12,  at  its  top.  The  other  timbers  were  6  X  12 ;  many  of  them  were  in 
pairs,  embracing  the  posts.  This  single-track  viaduct  was  begun  July  1,  1851,  and 
completed  Aug.  14, 1852.  It  contained  1,602,000  ft  (B  M)  of  timber,  and  108,862  lbs 
of  iron.  In  the  foundations  were  9200  cub  yds  of  masonry.  The  entire  cost  wae 
about  $140,000,  It  was  burned  down  in  1875,  and  was  replaced,  in  less  than  3  moa, 
with  a  single-track  viaduct  of  -wroug-lit-iron  trestles,  containing,  in 
all,  1,340,000  lbs  of  iron,  and  130,600  ft  (B  M)  of  timber;  and  costing,  complete, 
above  the  masonry,  about  895,000.  Frequently  the  posts  of  trestles  are  in  pairs; 
and  the  other  timbers  pass  between  ;  all  bolted  together. 

In  Fig  4,  the  posts  a,  a,  a,  are  end  views  of  three  trestles  or  bents,  such  as  Fig  3; 
Ktidtt  are  diag  braces  extending  from  trestle  to  trestle;  the  two  outer  ones  inclining 
in  one  direction;  and  the  central  one  crossing  them.  These  may  be  placed  either 
intermediate  of  the  posts,  as  in  Fig  3;  with  the  heads  of  the  two  outer  ones  confined 
to  the  cap  c  c  of  one  trestle;  and  their  feet  to  the  sill  y  y  of  the  next  one ;  or  they 
may  all  be  spiked  or  bolted  to  the  posts  themselves,  as  in  Fig  4.  The  last  is  the  best, 
as  it  serves  also  directly  to  stiffen  the  posts:  as  do  also  the  braces  oo,nn.  Fig  2, 
Such  bracing  is  too  frequently  omitted.  During  the  passage  of  trains,  the  backward 
pressure  of  the  steam,  exerted  through  the  driving  wheels  against  the  track,  pro- 
duces a  serious  strain  lengthwise  of  the  road,  and  tending  to  upset  the  trestles;  and 
the  sudden  application  of  brakes  to  a  moving  train,  produces  a  similar  strain  in  the 
opposite  direction.  These  strains  become  moredangeron-<as  the  ht  increases.  Hence 
the  need  for  such  braces.  Usually  the  outer  posts  may  lean  1.5  to  2.5  ins  to  a  ft. 

The  posts  should  not  be  less  than  about  12  ins  square,  except  in  quite  low  trestles; 
and  even  then  not  less  than  about  10  X  10-  The  diag  bracing  may  generally  be  about 
as  wide  as  the  posts ;  and  half  as  thick.  The  dist  apart  of  the  bents,  when  the  road* 
way  is  supported  by  simple  longitudinal  beams,  should  not  exceed  10  or  12  ft,  for 
railroads.  But  if  these  beams  receive  support  from  braces  beneath,  like  s«,  Fig  8 ;  or 
from  iron  truss  rods,  the  dist  may  be  extended  to  16  or  20  or 

more  ft.  But  when  the  trestles  become  very  high,  and  contain  a  great  deal  of  tim* 
ber,  it  becomes  cheaper  to  place  them  farther  apart,  say  30  to  60  ft;  and  to  carry 
the  railway  upon  regular  framed  trusses,  as  at  u  u.  Figs  7  and  8 ;  as  in  a  bridge  with 
stone  piers.    In  the  Genesee  viaduct,  the  trestles  were  50  ft  apart,  center  to  centerc 

When  such  a  trestle  as  Fig  8  becomes  very  narrow  in  proportion  to  its  height,  we 
may  add  to  its  stability  by  introducing  beams  w,  extending  from  trestle  to  trestle; 
and  still  further  by  inserting  diag  braces  v  v,  as  in  the  old  Genesee  viaduct. 

As  far  as  practicable,  arrange  the  pieces  so  that  any  one  may  be  removed  if  it 
becomes  decayed ;  and  another  put  in  its  place. 

On  curves,  additional  strength  should  be  given  on  the  convex  side;  as  sug- 
gested by  the  dotted  lines  in  Fig  5.  On  very  high  trestles  especially  (as  well  as 
on  bridges),  wheel-guards,  g  g,  Fig  10,  either  inside  or  outside  of  the  rails, 
should  never  be  omitted. 

Jn  marshy  ground^  piles  may  be  driven  to  support  the  trestles;  or  may  be  left  so 
far  above  ground,  as  themselves  to  constitute  the  posts.  Such  trestles  may  often  be 
used  advantageously,  even  when  to  be  afterward  filled  in  by  embkt.  They  then  sus- 
tain the  rails  at  their  proper  level  until  the  embkt  has  reached  it    final  settlement. 

They  are  generally  used  to  avoid  the  expense  of  embkt;  especially  when  earth  can 
only  be  obtained  from  a  great  dist.  Even  when  earth  and  timber  are  equally  con- 
venient, they  will  rarely  much  exceed  about  half  the  cost  of  embkt;  even  when  but 
about  30  ft  high;  but  owing  to  their  liability  to  decay,  they  should  be  resorted  t9 
only  in  case  of  necessity;  or  as  a  temporary  expedient. 


RAILROAD   CONSTRUCTION. 


815 


BALIiAST. 

Table  of  cubic  yar^s  of  ballast  per  mile  <vf  road. 

Side-slope  of  the  ballast  1  to  1.  Width  in  clear  between.~2  tracks  6  ft.  The  ties 
and  rails  may  be  laid  first,  for  carrying  the  ballast  along  the  line;  then  raised  a 
few  ft  of  length  at  a  time,  and  the  ballast  placed  under  them.  Deduct  for  ties» 
as  below. 


Depth 
in 
Ins. 

Top  width, 
SncGLK  Tkack 

Top  width, 
Double  Traok. 

10  Ft. 

11  Ft. 

12  Ft. 

21  Ft. 

22  Ft. 

23  Ft. 

12 
18 
24 
SO 

Cub.  Y. 
2152 
3374 
4694 
6111 

Cob.  Y. 
2347 
3667 
5085 
6600 

Cub.  Y. 

2543 
3960 

5474 
7087 

Cub.  Y. 
4303 
6600 
8996 

11490 

Cub.  Y. 
4499 
6894 
9388 
11980 

Cub.  Y. 
4695 
7188 
9780 

12470 

A  man  can  break  3  to  4  cubic  yards  per  day,  of  hard  quarried  stone,  to  a  siza 
suitable  for  ballast;  say  averaging  cubes  of  3  inches  on  an  edge.  Where  other 
ballast  cannot  be  had,  hard-burnt  clay  is  a  good  substitute.  The  slag  from  iron 
furnaces  is  excellent.  The  ties  decay  more  rapidly  when  gravel  or  sand  is  use(} 
instead  of  broken  stone,  because  these  do  not  drain  off  the  rain,  but  keep  the  ties 
damp  longer. 


TIES. 

In  the  United  States  tbe  life  of  a  tie  is  about  as  followi: 


Average, 

Artragt 

Years. 

Tears. 

Years. 

Years. 

Chestnut, 

5  to  12 

7 

White  Oak, 

5  to  12 

7 

Cedar, 

6  to  15 

9 

Spruce  Pine, 

4  to    7 

5 

Hemlock, 

3  to    8 

5 

It  will  often,  especially  in  the  case  of  the  softer  and  more  perishable  woods,  b» 
true  economy  to  preserve  ties  by  the  injection  of  creosote.  Creosote 

preserves  the  spikes. 

The  writer  believes  that  most  of  the  fault  usually  ascribed  to  cross-ties,  as  well  as 
to  rail-joints,  is  in  reality  diie  to  imperfect  drainage  of  the  roadbed.  Hence,  he  does 
not  agree  with  those  who  advocate  very  long  ties;  but  considers  that  with  good 
ballast,  on  a  well-drained  roadbed,  83^  ft  is  as  good  as  more ;  and  that  S}/^  ft,  by  9 
ins,  by  7  ins ;  and  2^  ft  apart  from  center  to  center,  is  suflBcient  for  the  heaviest 
traflQc.  On  many  important  roads  they  are  but  8  ft;  and  on  some  only  7^  ft  long ; 
track  4  ft  8]/^.  On  narrow-gauge  roads  the  ties  are  generally  from  6  to  7  ft  long. 
The  actual  cost  of  cutting  down  the  trees,  lopping  off  the  branches,  and  hewing 
the  ties  ready  for  hauling  away  to  be  laid,  is  about  6  to  9  cts  per  tie,  at  $1.75  p«r 
day  per  hewer. 

The  narrow  bases  of  rails  resting  immediately  on  the  cross-ties,  without  chain, 
frequently  produce  in'time  such  an  amount  of  crushing  in  the  ties  as  to  injure  them 
materially  even  before  decay  begins.  Burnetised  ties  rust  the  spikes  away  rapidly. 
Creosoted  ones  preserve  them. 

Cross-ties  of  8}^  feet,  by  9  inches,  by  7  inches,  contain  3.719  cubic  feet  each; 
and  if  placed  2}^  feet  apart  from  center  to  center,  there  will  be  2112  of  them  per 
mile,  amounting  to  291  cubic  yards.  Therefore,  if  they  are  completely  embedded 
in  the  ballast,  they  will  diminish  its  quantity  by  that  amount.  At  2  feet  apart  there 
will  be  2640  of  them,  occupying  364  cubic  yards;  and  at  3  feet  apart,  1760  of  them; 
243  cubic  yards. 


816 


RAILROAD   TIES. 


Cubic  feet  contained  in  cross-aties  of  diflferent  sizes. 


Dimensions. 

Contents. 

Dimensions. 

Contents. 

Ft.    Ins.    Ins. 

Cub.  Ft. 

Ft.    Ins.    Ins. 

Cub.  Ft. 

8     by  8  by  6 

2.667 

8%  by  10  by  7 
8M        10       8 

4.132 

8           9        6 

3.000 

4.722 

8           9        7 

3.500 

8)1       12       8 

5.667 

8         10       6 

3.333 

9             8       6- 

3.000 

8         10        7 

3.889 

9             9        6 

3.375 

8         10       8 

4.444 

9             9        7 

3.938 

8         12       8 

5.333 

9           10       6 

3.750 

8K       8        6 

2.833 

9           10       7 

4.375 

8>|       9        6 

3.188 

9           10       8 

5.000 

8>|       9        7 
SJ4      10       6 

3.719 

9           12        8 

6.000 

3.542 

TIE    PI.ATES, 

Where  the  rails  bear  directly  upon  the  ties,  the  great  unit  pressure  of  the 
narrow  rail  base,  the  churning  action  of  the  rail  under  passing  wheels,  and  the 
hastening  of  decay  by  the  bruising  of  the  wood  fibres,  cause  rapid  wear  of  the 
tie  immediately  under  the  rail. 

Among  prominent  forms  of  tie  plates  are  the  Servis,  the  Goldie,  the  Church- 
ward, and  the  Wolhaupter. 


Servis. 


Goldie. 


All  consist  essentially  of  a  flat  iron  or  steel  plate,  laid  on  the  tie  immediately 
under  the  rail.  Spikes,  holding  the  plate  and  the  rail  in  place,  are  driven  into 
the  tie  through  holes  in  the  plate.  Nearly  all  successful  forms  have  two  or  more 
ribs  on  the  lower  side.  These  ribs  stiffen  the  plate,  but  their  principal  use  is, 
by  cutting  into  the  upper  face  of  the  tie,  to  prevent  motion  of  the  plate  and  con- 
•equent  abrasion  of  the  tie.  In  some  forms  the  ribs  run  across  the  fibres  of  the 
tie ;  but  ribs  running  with  the  fibres,  as  in  the  Servis  plate,  are  usually  pre- 
ferred, as  being  more  easily  imbedded,  more  difficult  to  displace,  and  less  in- 
jurious to  the  fibres.  Most  forms  have  also  a  shoulder  on  the  upper  side,  to  assist 
the  spikes  in  preventing  spreading  of  the  rails,  and,  in  some  cases,  to  act  as  a 
rail  brace;  but  this  shoulder  is  seldom  considered  essential. 

Tie  plates  greatly  lengthen  the  life  of  the  tie.  On  curves  and  bridges  the  sav- 
ing in  a  number  of  cases  has  been  estimated  at  50  per  cent,  in  cost,  and  60  to  75 
per  cent,  in  labor.  The  tie  plate  has  often  displaced  small  gangs  of  men  whose 
sole  duty  it  was  to  replace  ties. 

Tie  plates  cost  from  5  to  15  cents  each  ;  and  placing  them  costs  from  3^  to  1% 
cents  each. 

Most  roads  use  tie  plates  only  under  rail  joints;  on  curves,  heavy  grades, 
bridges,  and  trestles  ;  in  tunnels  where  there  is  much  dampness ;  at  switches,  at 
stations,  and  at  street  and  road  crossings ;  in  yards ;  and  where  sand  is  much 
used  by  the  locomotives. 


BAILROADS. 


817 


Table  of  Middle  Ordinates,  to  be  used  for  the  bending  of  rails  of  different 
lengths,  so  as  to  form  portions  of  curves  of  different  radii.  Ordinates  for  lengthi 
or  radii  intermediate  of  those  in  the  table,  may  be  found  by  simple  proportion. 


Def. 

LENGTHS  OP  RAILS. 

Ang. 

Radius. 

_ 

30 

38 

36 

34 

33 

30 

18 

16 

14 

13 

10 

8 

6 

Deg. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

F^ 

.5 

11460. 

.010 

.008 

.006 

.005 

.004 

.004 

.003 

.002 

.002 

.001 

.001 

.000 

.000 

1. 

5730. 

.020 

.016 

.013 

.011 

.009 

.008 

.006 

.005 

.004 

.003 

.002 

,001 

.001 

1.5 

3820. 

.029 

.026 

.021 

.018 

.016 

.013 

.010 

.008 

.006 

.004 

.003 

.002 

.001 

2. 

2865. 

.038 

.034 

.029 

.025 

.021 

.017 

.014 

.OU 

.008 

.006 

.004 

.003 

.001 

2.5 

2292. 

.049 

.043 

.037 

.031 

.027 

.022 

.018 

.014 

.010 

.007 

.005 

.003 

.002 

.3. 

1910. 

.058 

.051 

.044 

.037 

.031 

.026 

.022 

.017 

.012 

.009 

.006 

.004 

.002 

3.5 

1637. 

.070 

.061 

.052 

.043 

.037 

.031 

.025 

.020 

.015 

.011 

.008 

.005 

.008 

4. 

1433. 

.079 

.069 

.060 

.050 

.042 

.035 

.029 

.023 

.018 

.013 

.009 

.006 

.008 

4.5 

1274. 

.0»8 

.077 

.067 

.056 

.J47 

.039 

.032 

.026 

.020 

.015 

.010 

.007 

.004 

5. 

1146. 

.099 

.086 

.074 

.063 

.053 

.044 

.035 

.029 

.022 

.016 

.011 

.007 

.004 

5.5 

1042. 

.108 

.094 

.082 

.070 

.059 

.048 

.039 

.032 

.024 

.018 

.012 

.008 

.004 

6. 

955.4 

.117 

.102 

.088 

.076 

.064 

.052 

.042 

.034 

.026 

.019 

.013 

.008 

.005 

6.5 

882. 

.128 

.112 

.097 

.082 

.069 

.057 

.046 

.037 

.028 

.021 

.014 

.009 

.005 

7. 

819. 

.137 

.120 

.104 

.088 

.074 

.061 

.049 

.039 

.030 

.022 

.015 

.010 

.005 

7.5 

764.5 

.146 

.127 

.111 

.094 

.079 

.065 

.053 

.042 

.032 

.024 

.016 

.010 

.006 

8. 

716.8 

.158 

.137 

.119 

.100 

.085 

.070 

.056 

.045 

.034 

.025 

.017 

.011 

.006 

8.5 

674.6 

.166 

.145 

.126 

.106 

.090 

.074 

.060 

.048 

.036 

.027 

.018 

.012 

.007 

9. 

637.3 

.175 

.153 

.133 

.112 

.095 

.078 

.063 

.050 

.038 

.029 

.019 

.012 

.007 

9.5 

603.8 

.187 

.163 

.141 

.119 

.101 

.083 

.067 

.054 

.042 

.031 

.021 

.013 

.006 

10 

573.7 

.196 

.171 

.148 

.125 

.106 

.087 

.071 

.057 

.045 

.032 

.022 

.014 

.008 

11 

521.7 

.216 

.188 

.163 

.139 

.117 

.096 

.078 

.063 

.049 

.036 

.024 

.016 

.009 

12 

478.3 

.236 

.206 

.179 

.151 

.128 

.105 

.085 

.069 

.053 

.039 

.026 

.017 

.010 

13 

441.7 

.254 

.222 

.192 

.163 

.138 

.113 

.092 

.075 

.057 

.042 

.028 

.019 

.010 

14 

410.3 

.275 

.239 

.207 

.175 

.148 

.122 

.099 

.080 

.061 

.045 

.030 

.020 

.011 

15 

383.1 

.295 

.257 

.223 

.188 

.159 

.131 

.106 

.085 

.065 

.049 

.033 

.021 

.012 

16 

359.3 

.313 

.273 

.236 

.200 

.170 

.139 

.113 

.091 

.070 

.052 

.035 

.023 

.013 

17 

338.3 

.333 

.290 

.252 

.213 

.180 

.148 

.120 

.096 

.074 

.055 

.0:37 

.024 

.014 

18 

319.6 

.351 

.306 

.265 

.225 

.190 

.156 

.127 

.102 

.078 

.058 

.039 

.025 

.014 

19 

302.9 

.371 

.324 

.280 

.238 

.201 

.165 

.134 

.108 

.082 

.061 

.041 

.027 

.015 

20 

287.9 

.392 

.341 

.296 

.250 

.212 

.174 

.141 

.114 

.087 

.066 

.044 

.028 

.016 

21 

274.4 

.410 

.357 

.309 

.262 

.222 

.182 

.148 

.120 

.091 

.069 

.046 

.030 

.017 

22 

262. 

.430 

.375 

.325 

.275 

.233 

.191 

.155 

.126 

.096 

.072 

.048 

.031 

.018 

23 

250.8 

.450 

.390 

.3.38 

.287 

.243 

.199 

.162 

.131 

.100 

.075 

.050 

.033 

.019 

24 

240.5 

.469 

.408 

.354 

.299 

.253 

.208 

.169 

.137 

.104 

.078 

.052 

.034 

.019 

25 

231. 

.486 

.424 

.367 

.311 

.263 

.216 

.176 

.142 

.108 

.081 

.054 

.035 

.020 

26 

222.3 

.506 

.441 

.382 

.323 

.274 

.225 

.183 

.148 

.112 

.084 

.056 

.037 

.021 

27 

214.2 

.524 

.457 

.396 

.335 

.284 

.233 

.190 

.153 

.116 

.087 

.058 

.038 

.022 

i8 

206.7 

.545 

.475 

.411 

.348 

.294 

.242 

.197 

.158 

.120 

.090 

.060 

.039 

.022 

19 

199.7 

.564 

.491 

.424 

.361 

.303 

.250 

.203 

.163 

.124 

.093 

.062 

.041 

.023 

RAIL.S. 


Tons  (2240  lbs.)  of  rail  _  1 1   y  weight  of  rail      /gxact) 
per  mile  of  single  track        ~T        in  ft)s.  per  yard.  ^  ' 

Brery  sq  inch  of  sectional  area  of  rail,  corresponds  to  10  lbs  per  yard  of  a  single 
rail ;  or  to  15.7143  tons  per  mile  of  single-track  road.    Consequently, 


Wt  in  S)s  per  yd  of  rail, 

lo 


Wt  in  tons  per  mile 

of  single-track 

15.7143 


Area  of  rail 
in  sq  ins. 


Thus,  a  rail  of  100  tons  per  mile  of  single  track,  will  have  a  section  of  6.364  sq 
ins  ;  and  will  weigh  63.64  fts  per  yd  of  single  rail.  Add  for  turnouts,  sidings,  road- 
crossings,  and  a  trifle  for  waste  in  cutting.  When  the  ties  are  in  place,  and  the  rails 
distributed  in  piles  at  short  intervals,  a  gang  of  6  men  can  lay  }^  a  mile  of  rails  per 
day,  of  single  track;  or  after  the  ballast  is  in  place,  a  gang  of  15  men  will  lay  about 
one  mile  of  complete  single-track  superstructure  per  week. 

Steel  rails  last  from  9  to  25  years ;  average  15  years. 


52 


818 


RAILROAD   SPIKES. 


RAII.ROAD  SPIKES. 

The  book-beaded  spikes  /,  commonly  used  for  confining  rails  to 
the  cross-ties,  vary  within  the  limits  of  the  following  table ;  the  lightest 
ones  for  light  rails  on  short  local  branches ;  and  the  heaviest  ones  for 
heavy  rails  on  first-class  roads.  The  spikes  are  sold  in  kegs  usually  of 
150  ft)s.  For  the  weight  of  spikes  of  larger  dimensions,  we  may  near 
enough  take  that  of  a  square  bar  of  the  same  length.  What  is  saved  at 
the  point  suffices  for  the  addition  at  the  head. 


Size  in  ins. 

No.  per  keg 

No.  per 

100  fi)s. 

Size  in  ins. 

No.  per  keg 

No.  per 
100  fi)s. 

'Length,  i 

Side. 

of  150  lbs. 

Length.  1  Side. 

of  150  lbs. 

4J^X 

7 

526 

350 

51/^  X      ^ 

5^X      k 
6      X      K 

350 

233 

4MX 

400 

266 

289 

193 

5      X 

% 

705 

470 

218 

146 

5      X 

il 

488 

325 

310 

207 

5      X 

390 

260 

6     X      ^ 
6      X      % 

262 

175 

5      X 

i 

295 

197 

196 

130 

5      X 

257 

171 

A  mile  of  slng'le-track  road,  with  2640  cross-ties,  2  feet  apart  from 
center  to  center;  and  with  rails  of  the  ordinary  length  of  80  feet,  or  fifteen  ties 
to  a  rail ;  will  have  352  rail-joints  per  mile ;  and,  with  4  spikes  to  each  tie,  will 
require  10560  spikes,  or  nearly  37  kegs  (5500  Bbs.)  of  5^^  X  A,  a  size  in  very  com- 
mon use,  which  weighs  a  trifle  more  than  ^  ib,  per  spike. 

But  an  allowance  must  be  made  for  rail-guards  at  road-crossings,  which  we 
may  assume  to  be  30  feet  wide,  or  the  length  of  a  rail.  A  guard  will  usually  con- 
sist of  4  extra  rails  for  protecting  the  track-rails,  and  spiked  to  the  15  ties  by 
which  said  track-rails  are  sustained.  Consequently  such  a  crossing  requires 
15  X  8  =  120  spikes.  For  turnouts,  sidings,  loss,  etc.,  we  may  roughly  average 
700*  spikes  more  per  mile;  thus  making  in  all  (if  we  assume' one  road-crossing 
per  mile)  10560  -f  120  -f  700  =  11380  spikes  per  mile;  or  say  6000  lbs.  or  40  kegs  of 
150ft)s. 

Adhesion  of  Spikes.  Professor  W.  R.  Johnson  found  that  a  plain  spike 
.375,  or  ^  inch  square,  driven  3%  ins.  into  seasoned  Jersey  yellow  pine  or  un- 
seasoned chestnut,  required  about  2000  Bbs.  force  to  extract  it;  from  seasoned 
white  oak,  about  4000 ;  and  from  well-seasoned  locust,  about  6000  lbs.  Bevan 
found  that  a  6-penny  nail,  driven  one  inch,  required  the  following  forces  to  ex- 
tract it :  Seasoned  beech,  667  lbs ;  oak,  507  ;  elm,  327 ;  pine,  187. 

Very  careful  experiments  in  Hanover,  Germany,  by  Engineer  Funk 
give  from  2465  to  3940  R)s.  (mean  of  many  experiments,  about  3000  lbs.), 
as  the  force  necessary  to  extract  a  plain  }/^  inch  square  iron  spike,  6 
inches  long,  wedge-pointed  for  1  inch  (twice  the  thickness  of  the^spike), 
and  driven  4^^  inches  into  white  or  yellow  pine.  When  driven  5  inches, 
the  force  required  was  about  y^^  part  greater.  Similar  spikes,  f^  inch 
square,  7  inches  long,  driven  6  inches  deep,  required  from  3700  to  6745 
lbs.  to  extract  them  from  pine  ;  the  mean  of  the  results  being  4873  Bbs. 
In  all  cases  about  twice  as  much  force  was  required  to  extract  tMm  from  oak.  The 
spikes  were  all  driven  across  the  grain  of  the  wood.  Experience  shows  that  when 
driven  with  the  grain,  spikes  or  nails  do  not  hold  with  much  more  than  half  as 
much  force. 

Jagged  spikes,  or  twisted  ones  (like  an  auger),  or  those  which  were  either 
swelled  or  diminished  near  the  middle  of  their  length,  all  proved  inferior  to 
plain,  square  ones.  When  the  length  of  the  wedge  point  was  increased  to  4 
times  the  thickness  of  the  spike,  the  resistance  to  drawing  out  was  a  trifle  less. 
But  see  "Jag-spike"  in  Glossary. 

When  the  length  of  the  spike  is  fixed,  there  is  probably  no  better  shape  than 
the  plain  square  cross-section,  with  a  wedge-point  twice  as  long  as  the  width  of 
the  spike,  as  per  this  fig. 

*  This  allows  that  turnouts  and  sidings  amount  to  about  1  mile  of  extra  track  ob 
15  miles  of  road. 


RAIL- JOINTS.  819 

RAII..JOINTS. 

Art.  1.  A  track,  being  weakest  at  the  joints  between  the  rails,  where  they 
Sjre  deprived  of  their  vertical  strength,  has  of  course  a  greater  tendency  to  bend  at 
ihose  points ;  and  this  bending  produces  an  irregularity  in  the  movement  of  the 
train,  which  is  detrimental  to  both  rolling-stock  and  track.  Moreover,  that  end  of  a 
rail  upon  which  a  loaded  wheel  is  moving,  bends  more  than  the  adjacent  unloaded 
end  of  the  next  rail ;  so  that  when  the  wheel  arrives  at  said  second  rail,  it  imparts  to 
its  end  a  severe  blow,  which  injures  it.  Thus,  the  ends  of  the  rails  are  exposed  to 
far  more  injury  than  its  other  portions.  Numerous  devices  have  been  resorted  to  for 
strengthening  the  joints  of  the  rails,  with  a  view  of  preventing  this  bending  entirely ; 
or,  at  least,  of  causing  the  two  adjacent  rail-ends  to  bend  equally,  and  together;  so 
as  to  avoid  the  blows  alluded  to.  None  of  these  joint-fastenings,  known  as  chairs, 
fish-plates,  wooden  blocks,  «S;c,  have  proved  entirely  satisfactory. 

Much  of  the  deficiency  ascribed  to  the  fastenings,  is,  however,  really  due  to  want 
of  stability  in  tlie  cross-ties  at  the  joints,  and  more  attention  must  be  directed  to 
this  latter  consideration,  before  an  etficient  fastening  can  be  obtained.  Observation 
shows  thac  when  the  joint-tius  are  very  firmly  bedded,  almost  any  of  the  ordinary 
fastenings  will  (if  the  joint  is  placed  between  two  ties,  instead  of  resting  upon  a  tie),* 
answer  very  well;  whereas,  when  the  cross-ties  are  so  insecurely  bedded  as  to  play 
up  and  down  for  half  an  inch  or  more  under  the  driving-wheels  of  the  engines,  the 
strongest  and  most  effective  fastenings  soon  become  comparatively  inoperative.  All 
the  parts  of  the  best  of  them  will  in  that  case  become  gradually  loosened,  warped, 
bent,  or  broken. 

Experience  has  established  the  superiority  of  suspended  joints  over  supported 
ones.  Long  fastenings,  perhaps,  possess  but  little  superiority  over  short  ones,  where 
the  track  is  not  kept  in  good  repair;  for  the  great  bearing  of  the  former,  although 
imparting  increased  firmness  on  a  good  track,  becomes  converted  into  a  powerful 
leverage,  by  which  it  accelerates  its  own  destruction,  in  a  bad  one.  An  element  in 
the  injury  of  joints,  is  the  omission  of  proper  fastenings  at  the  center  of  the  rails. 
Each  rail  should  be  so  firmly  attached  to  the  cross-ties  at  and  near  its  center,  as  tw 
compel  the  contraction  and  expansion  to  take  place  equally  from  that  point,  toward 
each  end.  It  would  probably  be  somewhat  difficult  to  accomplish  this  perfectly. 
The  attempts  hitherto  made  have  failed. 

Under  the  extremes  of  temperature  in  the  United  States,  bar  iron  expands 
•r  contracts  about  1  part  in  916;  or  1  inch  in  1Q}/^  feet;  consequently,  a  rail 
30  ft  long  will  vary  /g  inch  ;  and  one  20  ft  long  fully  ^  inch. 

Beside  ttois,  tlie  rails  are  very  liable  to  move  or  creep 
bodily  in  the  direction  of  the  heaviest  trade,  especially  when  the 
grade  descends  in  the  same  direction;  and  by  this  process  also  the  joint-fastenings 
are  exposed  to  additional  strain  and  derangement.* 

All  rails  appear  to  become  elongated  very  slightly  at  their  ends  by  use;  and  thi» 
renders  a  full  allowance  for  contraction  and  expansion  the  more  necessary. 

Art.  2.  £ven  joints  and  broken  joints.  If,  in  the  two  lines  of 
rails  forming  a  track,  the  joints  are  placed  opposite  to  each  other,  they  are  called 
"even  joints;"  while  "staggered"  or  "broken"  joints  are  those  where  each  joini 
in  one  of  the  lines  of  rails  is  opposite  to  the  middle  of  a  rail  in  the  other  line.  Iff 
the  latter  case,  the  jar  of  passing  from  rail  to  rail  is  less  severe,  but  of  course  more 
frequent,  than  where  both  wheels  make  that  passage  at  the  same  time. 

Art.  3.  Beveled,  or  mitred  joints.  To  lessen  this  jar,  Mr.  Sayre  sug. 
gests  cutting  the  rails  so  that  the  vertical  plane  forming  the  rail  end  shall  make  an 
angle  t>f  45°  to  60°  with  the  longitudinal  vert  plane  of  the  web  of  the  rail,  instead  of 
the  usual  right  angle.  This  would  permit  the  use  of  longer  rails  than  are  now  laid, 
as  the  great  space  (^  inch  or  more)  between  the  ends  of  such  long  rails  in  cold 
weather,  would  not  be  so  serious  an  objection  when  the  ends  were  thus  cut  obliquely. 
This  method  of  cutting  the  rails  has  been  tried,  with  good  results,  but  has  not  yet 
come  into  general  use.  It  is  claimed  that  a  comparatively  inexpensive  change  in  the 
arrangement  of  the  saws  at  the  rolling  mill,  would  permit  the  rails  to  be  cut  with 
ends  at  any  angle,  as  readily  as  with  square  ends,  and  without  further  increate  in 
tiie  cost  of  sawing. 


one. 


*In  the  first  case  the  joint  is  called  a  suspended  one;  in  the  last  a  supported 


820 


RAIL-JOINTS. 


Art.  4.  Fisli-plates,  Fig.  1,  and  Aii^le-plates,  Figs  2, 3, 4,  and  5,  have 
nearly  supplanted  all  other  forms  of  joint  on  the  prin- 
cipal railroads  of  the  U.  S.  They  are  rolled  in  long 
bars,  and  cut  off  in  any  desired  length,  generally  about 
2  ft;  and  are  bolted  together,  and  to  the  rails,  by  4 
or  6  bolts,  2  or  3  in  each  rail  end. 

Art.  5.    The  fisb-plate  joint  was  one  of  the 

earliest  suggested.  It  was  introduced  upon  the  New- 
castle and  Frenchtown  R  K,  in  Delaware,  by  Robt. 
H.  Barr,  in  1843.  The  weight  of  a  complete  fish-plate 
joint,  including  bolts  and  nuts,  is  about  20  ft)s. 

Fig  1,  one-fifth  of  real  size,  represents  a  fish-plate 
joint  with  a  steel  rail  weighing  50  lbs  per  yard.     The  JFi^.l, 

thickness  of  plate  at/  is  ^»g  inch.     The  plates  weigh  *'* 

434  fibs  per  lineal  foot  of  a  single  plate. 

Art.  6.  The  principal  advantage  of  the  ang-le-plates  is  that  the  spikes, 
which  are  driven  through  slots  in  their  flanges  to  confine  them  to  the  cross-ties, 
serve  also  to  counteract  the  tendency  of  the  rails  to  "creep."*  (See  Art  1.) 
Where  ^A-plates  are  used,  the  flanges  of  the  rails  have  to  be  slotted  for  this 
purpose. 

A  further  advantage  of  the  angle  plate  is  that  it  transfers  at  least  a  part  of  the 
load  directly  to  the  ties.  It  thus  supports  the  rail  better  than  the  fish-plate  can 
do,  and  may  continue  to  give  some  support  even  if  the  bolts  should  become 
somewhat  loose,  provided  the  spikes  hold  firm.  Moreover,  the  spreading  base  of 
the  angle  joint  adds  greatly  to  the  lateral  strength  of  the  rail  at  the  joint. 


INCHES 


Fig.  2. A. 


Art.  7.    The  following  are  usual  dimensions  of  angle  plates : 


For  rails. 

Height 

Thickness  on 

center  line  of 

bolt. 

Weight  of 
one  bar. 

60  fibs  &  70  ft)s 
Softs 

SYs  ins. 
3^^    - 

%  inch 

261^  lbs 
30}^  " 

*0n  the  St.  Louis  bridge  (steel  arches)  and  its  eastern  approach  (plate  girders 
on  iron  columns)  the  rails  creep,  in  the  direction  of  the  traffic,  about  a  foot  per 
day,  both  up  and  down  a  grade  of  80  feet  per  mile,  and  with  such  force  that, 
although  various  fastenings  were  used,  in  order  to  prevent  the  creeping,  none 
proved  effectual,  and  the  track  was  adjusted  daily  to  accommodate  the  creeping. 
For  a  very  interesting  account  of  this  case,  see  a  paper  by  Prof.  J.  B.  Johnson 
in  the  Journal  of  the  Association  of  Engineering  Societies,  Vol.  IV,  No.  1,  Nov. 
1884,  and  abstract,  with  editorial  discussion,  in  Railroad  Gazette,  Jan.  2d,  1885. 
Prof.  Johnson  attributes  the  creeping  to  a  wave  motion  of  the  rail  caused  by  the 
passage  of  trains. 


BAIL- JOINTS. 


821 


Art.  8.  The  wheel-tread  and  76  lb  steel  rail,  shown  (one-fifth  of  real  size)  in 
Fig  3,  are  those  designed  by  Robt.  H.  Sayre, 
C.  E.,  and  used  by  the  liehig-h  Valley  R  R* 
under  very  heavy  tratfic.  Tlie  angle-plate 
Joint  was  designed  by  Mr.  Jolin  Fritz, 
Supt  Bethlehem  Iron  Co,  Bethlehem,  Pa,  and 
Mr.  Sayre. 


These  forms  of  wheel-tread,  rail,  and  splice^ 
are  the  result  of  careful  study,  and  each  detail 
has  been  modified  from  time  to  time  as  experi- 
ence dictated.  The  stems  of  the  two  plates  are 
placed  wide  apart,  thus  giving  the  joint  greater 
lateral  strength  ;  at  the  same  time  adding  to  its 
vertical  strength  by  the  support  given  to  the  lower 
side  of  the  rail-head  by  the  upper  enlargement  c  ; 
while  the  lower  one  a  secures  a  full  bearing  on 
the  flange  of  the  rail.  The  joint,  for  76  ft)  rail, 
complete,  2  ft  long,  with  4  bolts  %  inch  diam, 
weighs  40  to  48  ft)s,  depending  upon  the  thick- 
ness of  the  angle  plate.  The  drilled  bolt-holes 
in  the  stem  of  the  rail,  are  1  inch  diam,  to  allow 
the  rails  to  contract  and  expand. 


2        3       4 

INCHES 

Fig.3. 


Art.  9.    Figs  4  and  5  (one-fifth  of  actual  size)   show  an  ang^le-plate 

joint  made  by  Cambria  Steel  Co,  and 
furnished  with  their  patent  nut-lock, 

which  consists  of  a  small  piece,  or  "  key," 
JO,  of  Bessemer  steel,  semi-circular  in  cross- 
section  at  one  end,  and  tapered  to  a  hori- 
zontal edge  at  the  other.  After  the  nut 
has  been  screwed  to  its  place,  the  key  is 
driven  close  up  to  it,  and  then  the  pointed 
end  of  the  key  is  bent  up  (as  shown  in 
Fig  5)  by  a  special  tool  with  a  lever 
attached.  The  key  is  prevented  from 
falling  out  sideways  by  the  edge  of  the 
longitudinal  groove,  Fig  4,  in  the  angle- 
plate,  into  which  it  fits. 


Fig.4. 


3        4 
INCHES 


^ig.5. 


Art.  10.  Both  fish-  and  angle-plates  are  apt  to  crack  vertically  about  the 
middle  of  their  length,  or  opposite  to  the  joint  in  the  rail.  To  obviate  this,  the 
"Samson-bar"  (made  either  of  fish  or  of  angle  form)  was  rolled  about 
half  inch  thicker  at  the  middle  than  at  its  ends.  The  thickened  portion  was 
about  8  ins  long,  extending  say  4  ins  each  way  from  the  joint;  but  the  upper 
edge  of  the  bar,  upon  which  the  head  of  the  rail  rested,  and  in  ^^-bars  the 
lower  edge  also,  were  made  of  this  increased  thickness  throughout  their  length. 

Art.  11.  Fish- and  angle-plates,  of  all  the  patterns  shown,  and  others,  are 
rolled  to  suit  different  sizes  and  shapes  of  rails.  The  bolt 
beads  are  usually  round,  and  the  shoulders  of  the  bolts,  immediately 
under  the  heads,  are  therefore  made  of  oval  cross-section,  fitting  into  corre- 
sponding oval  holes  in  the  fish-  or  angle-plate.  The  bolt  is  thus  prevented  from 
turning  when  the  nut  is  screwed  on,  and  afterwards.  Many  devices  have  been 
tried,  with  a  view  to  preventing  the  nuts  from  wearing*  loose 

The  plates  are  frequently  rolled  with  a  longitudinal  groove,  as  wide  as 
the  head  or  nut  of  the  bolt,  and  about  %  inch  deep,  running  their  entire  length. 
This  groove  receives  either  the  head  of  the  bolt,  which  in  such  cases  is  made 
square  or  oblong  and  inserted  first,  and  the  nut  afterwards  screwed  on  ;  or  else 
the  nut  is  first  placed  in  the  groove,  and  the  bolt  afterwards  screwed  into  it. 
This  is  intended  to  prevent  the  unscrewing  of  the  nut,  but  cannot  be  relied  upon 
to  do  so. 


It  is  well  to  have  the  slots  in  the  flanges  of  rails  or  of  angle-bars  so  spaced  that 


822 


RAIL-JOINTS. 


the  two  spikes  of  a  joint,  driven  into  the  same  cross-tie^ 
shall  not  be  directly  opposite  to  each  other,  but  "  staggered,"  so  as  ta 
diminish  the  danger  of  splitting  the  tie. 
Joints  are  frequently  laid  with  one  fish-  and  one  ang^le-plate. 


Art.  12.  It  will  be  noticed  that  both  fish-  and  angle-plates  act  by  placing  a 
support  under  the  head  of  the  rail.  The  Fisher  bridge-joint.  Figs  6  to  9, 
made  by  Mr.  Clark  Fisher,  Trenton,  N  J,  applies  the  support  under  the  base  of  the 
rail. 

The  principal  feature  of  this  joint  is  a  flanged  beam,  Fig  6,  about  6  ins  wide  and 
22  ins  long,  which  extends  across,  and  is  spiked  to,  the  two  joint-ties,  as  in  Fig  7. 
The  holes  for  the  spikes  are  placed  so  that  the  two  spikes  in  the  same  tie  are  not 
opposite  to  each  other ;  and  the  flanges  F  F  also  are  staggered,  so  as  not  to  interfere 
with  the  driving  of  the  spikes.  The  joint-ties  T  T  are  placed  7  inches  apart  in  the 
clear.    The  beam  has  an  upward  camber  of  about  one-eighth  of  an  inch.    The  two 


Fig.  7. 


Fig.  8. 


rail-ends,  forming  the  joint,  rest  upon  the  beam,  and  meet  at  the  middle  of  iti 
length.  They  are  held  down  to  it  by  a  single  U-shaped  bolt  B,  of  1  inch  diam, 
with  a  nut  on  each  leg.  These  nuts  bear  directly  upon  the  horizontal  upper  sides 
of  the  "  fore-locks  "  L  L,  one  of  which  is  shown  separately  in  Fig  9.  The  fore-locki 
are  rolled  to  fit  accurately  to  the  rail-flanges.  The  legs  of  the  U-bolt  pass  first 
through  the  circular  holes  h  h,  in  the  beam,  Fig  6;  next  through  rounded  notchei 
cut  in  the  corners  of  the  rail-flanges ;  then  through  the  holes  in  the  fore-locks ;  and 
lastly  through  the  nuts.  Between  the  U-bolt  and  the  bottom  of  the  beam  is  placed 
a  small  piece  s,  of  spring  steel,  slightly  cambered  downward,  and  having  two  semi-cir- 
cular notches  for  the  legs  of  the  U-bolt,  which  hold  it  in  place.  This  is  in  tended  to  keep 
the  joint  elastic,  to  take  up  any  loose  space  produced  by  the  wear  of  the  surfaces  in 
contact,  to  render  less  abrupt  the  strains  on  the  bolt,  and,  by  keeping  the  threads 
of  the  nut  pressed  against  those  of  the  bolt,  to  prevent  the  nuts  from  becoming 
loose.  The  joints  are  shipped  from  the  factory  complete,  and  with  all  the  parts 
bolted  together ;  the  nuts  being  screwed  down  to  within  about  two  threads  of  their 
final  places,  so  that  the  ends  of  the  rail-flanges  can  be  easily  slid  into  place  under 
the  fore-locks. 

As  an  additional  precaution  against  creeping  of  the  rails,  the  rail-flanges  may  be 
slotted  near  their  ends,  as  in  cases  where  fiSi-plates  are  used,  and  spikes  driven 
through  these  slots.  For  such  cases  the  beams  are  punched,  at  the  mill,  with  four 
additional  square  holes  a  little  further  from  the  edges  of  the  beam  than  the  others. 
Unlike  the  angle-  and  fish-plate  joints,  the  Fisher  may  be  used  with  any  section  of 
T-rail ;  and  the  head  of  the  rail  may  be  made  stronger  by  being  rolled  pear-shaped, 
which  is  inadmissible  with  fish-  and  angle-joints,  because  these  require  a  nearly 


RAIL-JOINTS. 


823 


horizontal  bearing  on  the  under  side  of  the  head.  The  *'  Fisher"  requires  no  drilV 
ing  or  punching  of  the  stem  of  the  rail.  It  costs  about  25  per  cent  more  than  a 
fish-  or  angle-joint  for  the  same  rail.  Its  iveig^bt,  complete,  for  65-S)  rail,  is  about 
32  ft»s. 

Mr.  Fisher  makes  also  an  extra  strong  Joint  with  three  U-bolti., 
for  heavy  curves  and  for  places  liable  to  wash-outs.  It  is  intended  to  support  the 
joint,  even  if  the  ballast  is  removed  from  under  the  joint-ties.  Either  of  the  Fisher 
joints  can  be  made  of  any  desired  weight.  The  "Fisher"  is  largely  used  on  some 
of  the  principal  eastern  roads,  and  with  very  satisfactory  results. 


Vig.  lO 


Art*  13.  Tlie  Bonzano  rail-joint.  Figs.  10, 11,  and  12,  invented  and 
patented  by  Mr.  Adolphus  Bonzano,  C.  E.,  of  Philadelphia,  is  essentially  an 
angle-plate  joint  (Art.  6),  but,  in  the  Bonzano  joint,  the  horizontal  flange  of  the 
angle  bar  is  rolled  about  3  inches  wider  than  usual,  and,  after  rolling,  and  cut- 
ting to  the  proper  lengths,  its  middle  portion  is  pressed  downward  by  dies,form- 
ing  a  girder,  G,  which  projects  downward  between  the  two  joint  ties. 

The  broad  horizontal  dange,  being  level  with  that  of  the  rail,  affords  greatly 
increased  bearing  upon  the  ties,  and  adds  to  the  lateral  stiffness  of  the  joii)t ;  the 
girder,  G,  between  the  joint  ties  increases  the  vertical  strength  of  the  joint ;  and 
the  triangular  gussets,  6,  Figs.  10  and  11,  securely  hold  the  horizontal  flange  and 
the  downward-projecting  girder  in  their  relative  positions. 

The  two  splice-bars,  in  the  Bonzano  joint,  have  a  combined  cross-section 
about  1.2  times  that  of  the  rail,  and,  as  shown  by  tests  made  by  Prof.  Henry  T. 
Bovey,  at  McGill  University,  Montreal,  equal  strength  with  the  rail,  while  the 
ordinary  angle  bar  joint  has  but  one-third  that  strength. 

The  joints  are  made  either  30  inches  long  for  6  bolts,  or  from  24  to  26  inches 
long  for  4  bolts.  For  80  ft),  rails,  the  splice-bars  for  the  30  inch  and  24  inch  joints 
weigh  together  about  59  ft)s.  and  73  ft)s.  respectively. 

The  price  is  about  0.3  cent  per  pound  higher  than  that  of  the  ordinary  angle 
bar  joint. 


824 


TURNOUTS. 


TUENOUTS. 


Art.  1.    To  enable  an  engine  and  train  to  pass  from  one  track,  A  B,  Fig  1,  t# 
another,  A  D,  a  turnout  is  introduced.     Tbis  consists  essentially  of  a 


eTA. 


sif  itch,  qmp  s,&  frog,/,  and  two  fixed  g>iiar<l- rails,  g  and  g\  If  a  switch 
is  made  to  serve  for  two  turnouts,  A  D  and  A  D',  Fig  2,  one  on  each  side  of  the  main 
track,  A  B,  it  is  called  a  tbree-throw  sirltell. 


V 


Fiff.  2. 


Fig.  3. 


Art.  2.  When  a  train  approaches  a  switch  in  the  direction  of  either  arrow,  Figj 
1 ;  or  so  that  it  passes  the  frog  before  reaching  the  switch,  it  is  said  to  "  trail  *' 
the  switch.  When  it  approaches  in  the  opposite  direction,  passing  the  switch  before 
reaching  the  frog,  it  is  said  to  "  face"  the  switch.  Fig  3  represents  a  portion  of 
a  double-track  road  in  which  the  trains  keep  to  the  right,  as  shown  by  the  arrows. 
In  this  fig,  V  and  W  are  ^'' trailing''^  switches;  and  X  and  Y  are  ^\facing''^  switches. 
In  order  to  leave  the  main  track  by  a  trailing  switch,  a  train  must  move  in  a  direc- 
tion contrary  to  the  proper  one  on  said  track. 


Art.  3.  Misplaced  switches.  A  moving  train, /acinar  any  switch,  must 
plainly  go  as  the  switch  is  set,  whether  right  or  wrong.  If  wrong,  serious  accident 
may  result.  For  instance,  the  train  may  run  upon,  and  over  the  end  of,  a  short 
trestle  siding,  or  may  collide  with  a  train  standing  or  moving  upon  the  turnout. 
Safety  suritclies.  such  as  the  Lorenz,  Arts  13,  &c,  and  Wharton,  Arts  18,  Ac, 
are  so  arranged  that  trains  trailing  them  can  pass  them  safely,  even  if  the  switch 
is  misplaced.  But  in  the  case  of  the  plain  stiib-suritcji.  Art  4,  when  mis- 
placed, a  trailing  train  will  leave  the  rails  at  h  and  r,  or  e  and  w,  Fig  4,  and  run 
upon  the  ties. 

Stub-switches  are  frequently  provided  with  ''safety-casting's"  of  iron, 
bolted  to  their  sides,  and  reaching  from  their  toes  m  and  s.  Fig  4.  several  feet  toward 
p  and  q.  These,  in  case  of  misplacement  of  the  switch,  receive  the  flanges  of  th« 
wheels  of  a  trailing  train,  and  guide  the  wheels  safely  on  to  the  switch-rails  gm  and 
ps.    Tbe  '* Tyler"  suritcb  is  arranged  in  this  way. 


TURNOUTS. 


825 


Art.  4.  The  common  blimt-ended  or  stnb-switcli  consists 
essentially  of  two  rails,  q  m  and  p  s.  Figs  1  and  4.  The  ends,  q  and  p,  of  these 
rails,  where  they  are  fixed  in  line  with  the  main  track,  form  the  "'heel"  of  the 
Rwitch.    Their  other  ends,  m  and  s,  form  the  '*  toe,"  and  are  free  to  move  from  d 


and  n  (where  they  are  in  line  with  the  main-track  rails,  h  h  and  r  z)  to  m  and  f 
(where  tiiey  are  in  line  with  the  turnout-rails,  ex  and  uy). 

On  main  lines  of  road,  the  switch-rails  are  usually  from  18  to  26  feet  long  from 
heel  to  toe.  Formerly  their  heels,  qr  and  p,  were  fixed  by  being  confined  in  the  same 
chairs  which  held  the  adjoining  ends  of  the  main-line  rails  ;  or  by  being  connected 
with  said  rails  by  short  fish-plates.  In  either  case  they  remained  prac- 

tically straight,  even  when  set  for  the  turnout.    Now,  however,  they  are  generally 
made  long  enough  to  extend  4  to  6  feet  back  from  their  heels,  toward  A ;  and  thifl 


IGiflt-S 


additional  length  is  spiked  nnyieldingly  to  the  ties;  so  that,  when  set  for  the  turnont, 
the  switch-rails  bend  so  as  to  form  (at  least  approximately)  a  part  of  the  turnout 
curve.    Their  toes,  m  and  *,  in  any  case,  rest  and  slide  upon  iron  **liead-platea,'* 

P.  P.,  Fig.  4,  shown  in  detail  at  Fig.  5,  which  represents  a  riveted  steel  plate, 
made  by  the  Weir  Frog  Co.  These  head  plates  also  receive  the  ends,  b  e  and 
r  u,  Fig.  4,  of  the  main-track  and  turnout  rails.  In  three-throio  switches,  the 
head-plates  must  of  course  be  longer,  to  give  room  for  the  three  rail-ends  side  by 
side. 

Frequently  a  plain  strip  of  iron,  about  3  inches  wide  by  half-inch  thick,  is  fastened 
to  the  upper  surface  of  each  tie,  under  the  base  of  the  rail,  for  the  latter  to  slide  on. 

The  switch-rails  are  connected  together  by  from  3  to  5  transverse  wrought-iron 
clamp-rods^  R  R  R%  Fig  4,  full  1^  ins  diam.  These  are  fastened  to  the  rails  in 
various  ways ;  generally  as  shown  in  Fig  6.    The  clamp-bars  should  be  placed,  if 

possible,  at  least  as  low  as  the  tops 
of  the  cross-ties,  so  as  to  avoid 
danger  of  their  coming  into  con- 
tact with  any  portions  of  cars  or 
engines  that  may  become  par- 
tially detached  and  drag  on  the 
track. 

One  of  the  clamp-bars,  R',  is 
near  the  toes,  m  and  s,  Fig  4.  It  projects  beyond  the  track,  and  is  jointed,  as  shown 
at  Figs  7  and  8,  and  connected  with  the  lever,  L,  by  which  the  switch  is  moved. 
The  tie,  T,  Fig  4,  to  which  the  head-plates  are  fastened,  is  made  longer  than  th© 
others,  in  order  to  give  room  for  the  switch-stand,  M,  Figs  4, 7,  and  8,  which  is  bolted 
to  its  upper  surface.  This  tie  should  also  be  of  larger  cross-section  than  the  others, 
perfectly  sound,  and  well  bedded;  because  upon  it  come  severe  strains  due  to  the 
passage  of  cars  and  engines  across  the  space  between  the  rail-ends,  m  and  e,  s  and  u, 
etc,  Fig  4. 


Fig.  6. 


826 


TURNOUTSc 


Art.  5.    The  switch .leTers,  and  the  switch-stands  to  which  they  arc 

attached,  are  made  in  a  great  variety  of  forms.  See  Figs  7,  H,  9,  14, 15,  and  17.  That 
shown  in  Figs  7  and  8  is  the  **  Tuinbling--lever  stand  "  or  •'  Oround-leTcr 
stand,"  and,  in  its  numerous  modifications,  is  very  largely  used.  It  is  so  arranged, 
that,  whichever  way  the  switch  is  set,  the  crank,  C,  is  on  the  dead  center,  so  iha,t 
the  lateral  strains  of  passing  cars  or  engines  can  exert  no  tendency  to  turn  it. 


Tumbling  switches  are  convenient  because  they  occupy  but  little  space.  By  t 
of  a  target  or  lantern,  connected  with  the  switch,  they  may  be  made  to  indicate  to 
the  engine  driver  the  position  of  the  switch. 

When  the  switch  is  set  either  way,  the  lever  is  padlocked  to  a  staple  driven  into 
the  tie  and  passing  up  through  the  slot  in  the  handle  of  the  lever.  The  lever  is  fre- 
quently made  with  a  weight  of  say  20  lbs  on  its  free  end,  to  aid  in  bringing  it  down 
to  its  proper  position. 

Art.  6.  Fig  9  represents  a  common  form  of  the  nprig-ht  lever  and  stand. 

The  switch-rod,  R'  Figs  4,  7  and  8,  is  generally  attached  at  the  lower  end.  A,  of  the  lever. 
The  cast-iron  frame,  F,  is  fastened  to  the  long  tie,  T,  Fig  4,  by  large  screws  or 
Bpikes,  which  pass  through  its  broad  feet  or  flanges,  B  B.  The  top  of  the  frame  is 
provided  with  two  notches,  N  N,  and  staples,  to  which  the  lever  is  secured  by  a 
padlof.k.  When  this  stand  is  to  be  used  for  a  three-throw  switch,  the  frame  has 
three  notches  and  three  staples.  The  upright  stand  may  be  used  wherever  it  will 
not  be  in  the  way  of  passing  trains.  The  target,  T,  at  the  top  of  the  lever,  by 
showing  the  position  of  the  latter,  indicates  to  the  driver  of  an  approaching  engine 
\rhioh  way  the  switch  is  set. 


Fig.  ©• 


iTLc  Ke  s 
MONKEY  SWITCH 

Fig.  10. 

Art.  7.  In  the  **  MonlLey-switch,''  Fiff  10,  the  crank,  o  t,  is  moved  hori« 
Bontally  through  an  arc  of  a  circle  by  means  of  the  lever,  h  h,  about  3  ft  long,  which 
fits  upon  the  square  head,  s,  of  the  vertical  spindle  or  pin,  «  o.  The  switch-rod,  E'  Figs 
4,  7  and  8,  is  attached  to  the  pin,  iv. 

Many  modifications  of  the  monkey-switch  are  in  use.  The  spindle,  5  o,  is  fre- 
qnently  made  long  enough  to  bring  the  lever  to  about  the  level  of  the  hand;  and 
the  lever  is  permanently  attached  to  the  stand,  and  hinged  near  the  spindle  so  aa 
to  hang  down,  out  of  the  way,  when  not  in  use.  To  the  top  of  the  spindle  is  fre- 
quently attached  a  vertical  rod  of  any  desired  length,  and  carrying  at  its  top  a  target 
Which  turns  as  the  spindle  does,  and  thus  indicates  the  position  of  the  switch. 


TURNOUTS. 


827 


Art.  8.  All  parts  of  the  switch-stand,  and  the  tie  upon  which  it  rests,  should 
be  perfectly  rigid,  because  it  is  very  important  that  they  should  hold  the  ends  of 
the  switch-rails  exactly  in  line  with  those  of  the  main  line  and  turnout.  They 
therefore,  in  view  of  the  great  strains  to  which  they  are  subjected,  must  be  strongly 
constructed,  and  frequently  looked  after. 


Fig.  11. 


Art.  9.  In  Figs  1  and  4,  dqm  ia  called  the  switcll-angi'le.  The  dist,  dm. 
Figs  1,  4,  and  11,  required  for  the  motion  of  the  toes,  is  called  the  throw  of  the 
switch.  It  must  be  equal  at  least  to  the  width,  d  w,  Fig  11,  of  the  top  of  the  rail, 
in  addition  to  a  width,  w  m,  sufScient  to  allow  the  flanges  of  the  wheels  to  pass 
along  readily  between  b  and  e,  Fig  1,  and  between  r  and  «.  The  tops  of  the  rails 
are  generally  between  2  and  2}/^  ins  wide ;  and  about  1  %  to  2}^  ins  suffice  for  the 
flanges.    The  throw,  d  m,  however,  is  commonly  about  5  ins. 

The  g'aug'e.  Fig  6,  of  a  railroad  track,  is  the  distance  between  the  inner 

sides  G  G'  of  the  heads  of  its  two  rails.  Hence  these  inner  sides  are  called  the  g^ailg'e 
sides  of  the  rails. 

Art.  10.  The  stub-switch  is  cheaper  in  first  cost  than  the  improved  safety 
switches.  Arts  13, 18,  etc,  but  is  less  economical  in  the  long  run. 

As  it  is  very  essential  that  the  toes  of  the  switch-rails  should  never  come  into 
contact  with  the  adjoining  rail-ends,  a  space  of  about  an  inch  must  be  allowed 
at  the  toes  for  expansion,  and  for  "creeping"  This  renders  the  blows 

of  passing  trains  very  severe,  and  injurious  to  rolling  stock,  and  to  the  rail-ends. 
The  latter  are  worn  away  rapidly  and  must  be  frequently  renewed.  From  the  same 
cause  the  tie  under  the  head-plate  is  apt  to  become  loose  in  its  bed. 


828 


TUKNOUTS 


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TURNOUTS. 


829 


Art.  12.    Tlie  '*  Automatic  "  switch-stand  is  shown  in  horizontal 
section  by  Fig  14,  and  in  vertical  section  by  Fig  15.    It  is  designed  especially 


Fig  15 

for  split  switches  like  those  shown 
in  Figs  12  and  13.  It  is  operated 
by  a  tumbling-lever,  L,  Fig  14. 
To  the  horizontal  axis,  A,  of 
the  lever,  is  fastened  the  beveled 
pinion,  P.  This  engages  in  the 
teeth  of  the  quadrant,  Q,  and 
moves  it  horizontally  through  a 
I  quarter  of  a  circle,  when  the  lever  L  is  thrown  from  its 
)  position  L  to  that  of  17,  shown  by  the  dotted  lines.  The  rod, 
K',  from  the  switch,  is  attached  at  X,  or  at  X',  to  this  quad- 
rant; and  the  movement  of  the  latter  thus  sets  the  switch, 
and  at  the  same  time  turns  the  target,  T,  Fig  15,  or  lantern,  fixed  to  the  vertical 
spindle,  S,  thus  indicating  the  position  of  the  switch.  The  gearing  is  inclosed 
in  a  cast-iron  case,  as  shown  in  Fig  15.  If  a  train  on  the  turnout,  moving  in  the 
direction  of  the  arrow,  Fig.  1  (or  "trailing"),  approaches  a  split-switch  pro- 
vided with  the  automatic  stand,  and  set  ^through  oversight  or  otherwise)  for 
the  mai7i  track,  as  in  Fig  12,  the  flange  of  the  first  wheel,  pressing  between 
rails  X'  and  S'  V,  will  push  the  switch-rails  into  the  proper  position.  Fig  13,  for 
the  turnout ;  at  the  same  time  necessarily  throwing  the  weighted  lever  over  into 
the  reverse  position,  and  turning  the  target  so  as  to  indicate  that  the  switch  is 
set  for  the  turnout.  A  similar  movement  of  the  switch,  in  the  opposite  direction, 
takes  place  if  a  trailing  train  on  the  main  track  approaches  the  switch  when  set 
for  the  turnout,  as  in  Fig  13.  Hence  the  term  "automatic,"  as  applied  to  this 
stand.    The  switch  is  thus-  made  a  safety  switch. 


Art.  13.  In  tlie  liOrenas  safe ty-S'vr itch,  the  connecting-bar,  W,  near- 
est to  the  toes,  is  provided  with  a  spring',  S,  Fig  16,  placed  sometimes  between 
the  rails,  as  there  shown  ;  sometimes  outside  of  the  track.  This  spring  permits 
the  moving  of  the  switch-rails  by  the  wheels  of  a  trailing  train,  as  does  the 
automatic  switch-stand.  Figs  14  and  15;  but  after  the  passage  of  each  wheel, 
the  spring  returns  the  switch  rails  to  their  original  position.    The  blow  of  the 


830  TURNOUTS. 

iwitch-rafis  f^afnit  the  utoA-fflg.  thus  ooeMioned,  is  Injnriotis  to  both,  and  Habit  t« 
break  the  former  On  tbe  other  hand,  the  oompressioo  of  the  spring,  during  th» 
paasaff"*  of  the  rrafn  throueh  the  switch,  sometimes  impairs  its  elasticit^y.-  so  that  it 
then  fills  trv  rntitrn  the  nwltch-rail  to  its  proper  position  in  contact  with  the  stock- 
mil,  and  allows  it  to  remain  half  an  inch  or  mo^e  away  from  it,  and  in  dansjer  of 
being  struck  by  the  wheel-flanares  of  approaching  trains  •'  facing  "  the  switch.  A 
Bimilar  accident  may  happen  during  the  ordinary  working  of  the  switch,  if  an 
obstacle,  as  a  small  stone,  becomes  lodged  between  the  switch-rail  and  the  stock* 
rail ;  for  the  spring  may  permit  the  switchman  to  force  the  switch-lever  homo  to  itt 
place  without  bringing  the  two  rails  pr<^rlj  into  contact.    See  Art.  14. 

Art.  14.  I>e  Vout's  safety  switch-stand.  Fig  17,  made  by  Penna 
Steel  Co,  is  designed  to  remedy  this.  In  this  stand,  the  spring  is  placed  in,  and  se- 
cured to,  a  semi-cylindrical  iron  spring-case  or  box,  B;  to  the  opposite  sides  of  which 
ire  fi.xed  two  hor  axles.    One  of  these  is  shown  at  A.    This  axle  passes  through  the 


Bwitch-lever,  L,  near  its  fulcrum,  F.  It  also  passes  through  tbe  inverted  T-shaped 
slot,  H,  in  the  rigid  bar,  S,  which,  together  with  the  bar,  W,  attached  to  the  spring- 
case,  is  jointed,  at  J,  to  the  switch-rod,  R'.  When  the  switch  is  properly  set,  either 
for  the  main  line  or  for  the  turnout,  the  axle.  A,  is  in  the  hor  part  of  the  slot,  H, 
and  immediately  under  the  vert  part,  so  that  there  is  no  obstruction  to  the  move- 
ment of  the  switch,  and  a  trailing  train  will  open  a  misplaced  switch  as  explained 
in  Arts  12  and  13.  But  when  the  lever  is  raised,  for  the  purpose  of  setting  the  switch 
in  the  other  position,  the  axle.  A,  rises  into  the  vert  part  of  tiie  slot,  as  in  the  fig, 
lifting  the  spring-case  with  it.  If  now  any  obstruction  prevents  the  switch-rail 
from  being  pressed  home,  the  rigid  bar,  S,  by  means  of  the  axle,  A,  prevents  the 
lever,  L,  from  moving  farther. 

Art.  15.  Theory  would  require  that  ttie  lengths  of  the  switch-rails, 

in  split-switches,  should  vary  with  the  radius  of  the  turnout  curve,  and  formerly 
they  were  so  made.  Where  this  radius  is  such  that  a  No  10  frog  (see  Art  26)  is  re- 
quired, the  switch-rails  should,  theoretically,  be  28  ft  long.  But  in  practice  a  uni- 
form length  of  15  ft  (just  half  the  usual  length  of  the  steel  rail  from  which  the 
switch-rails  are  cut)  for  all  turnouts,  gives  the  best  results,  combining  economy  of 
manufacture  with  greater  strength,  and  greater  ease  of  handling,  than  are  possible 
with  much  longer  rails. 

Art.  16.  It  will  be  noticed  that  in  point-switches  (as  also  in  the  Wharton  switch, 
Arts  18,  &c)  there  can  be  no  such  jar  as  that  occasioned  in  the  stub-switch  by  the 
long  space  between  the  toes  of  the  switch-rails  and  the  ends  of  the  adjoining  rails. 

Art.  17.  It  is  important  that  the  thin  portions  of  each  switch-rail  should  bo 
carefully  shaped  so  as  to  receive  throughout  a  firm  lateral  support  from  the  stock- 
rail  when  in  contact  with  it.  Otherwise  the  switch-rails  are  in  danger  of  bending 
under  the  lateral  pressure  of  passing  trains.  This  might  throw  the  point  out  from 
the  stock-rail,  endangering  the  train. 

Art.  IT  a.  Fig.  17  a  shows  a  three-tHro-vr  point  «\Titch  made  by  Ths 
Weir  Frog  Co.,  Cincinnati.  Ohio.  It  has  the  usual  stock  rails.  C  and  Z,  and  four 
switch  rails,  A,  B,  X  and  Y.  The  switch  rails  all  slide  upon  the  same  set  of  iron 
"  friction  plates,"  which  are  spiked  to  the  ties  under  the  rails,  but  are  not  shown  in 
the  figure.  Rails  A  and  B,  are  held  rigidly  together  by  four  connecting  bars 
m,a,a^a,  while  X  and  Y  are  similarly  connected  ky  the  other  four  connecting  bars 


TURNOUTSi 


831 


7     8      9     10 


i     5 

FEET. 

r,  X,  a,  X.  Each  pair  of  switch  rails,  thus  formed,  moves  independently  of  the  other; 
the  switch-rod  R  a  operating  the  pair  A-B,  and  R  y  operating  the  pair  X-Y.  Out 
figure  shows  the  switch  as  arranged  for  two  separate  switch-stands,  one  on  the  near 
side  operating  the  rod  R  y  acd  the  rails  X-Y,  and  one  on  the  farther  side,  operating 
the  rod  R  a  and  the  rails  A-B ;  but  it  may  be  arranged,  instead,  so  that  both  pairs 
of  rails  may  be  operated  by  means  of  a  single  stand,  placed  on  one  side  of  the  switch. 

The  figure  shows  the  switch  set  for  the  track  X  Z. 

To  set  it  for  the  track  A  Y,  the  rod  R  y  is  pulled,  and  draws  rails  X-Y  over, 
bringing  Y  into  contact  with  Z,  so  that  wheel  K  may  run  upon  rail  Y,  and  leaving  a 


F'ig.  ir  6 

SECTION  ATM-  O 


space  between  rails  X  and  A  for  the  passage  of  the  flang,e  of  wheel  L,  which  whMl 
than  runs  upon  rail  A. 

To  set  the  switch  for  the  track  C  B,  the  rod  R  a  is  then  pushed,  and  throws  rails 
A-B  over,  bringing  B  again  into  contact  with  Y,  so  that  wheel  K  may  run  upon  rail 
B,  and  leaving  a  space  between  rails  C  and  A  for  the  flange  of  wheel  L,  which  wheel 
then  runs  upon  rail  C. 

As  in  all  point  switches,  it  is  impossible  to  spike  the  inner  flanges  of  the  stock 
rails  C  and  Z  to  the  ties  along  that  portion  of  their  length  (some  12  feet  from  the 
switch  point)  where  they  come  in  contact  with  the  switch-rails.  As  a  subititute 
they  are  provided  with  special  supporting  blocks  S.  S.  on  the  outer  side.  In  the 
Weir  switch,  each  of  these  is  made  of  one  piece  of  flat  bar  iron,  bent  over  and 
twisted,  and  serving  also  as  a  sliding  plate,  as  shown  more  clearly  in  the  section  n  o, 

Fig.  17  h,  which  shows  also  the 


SECTION  ATl>.g. 


manner  of  fastening  the  end  o\ 
the  flat  connecting  bar  a  to  the 
base  of  rail  A  by  means  of  a 
malleable  casting  M,  through 
which  the  end  of  the  bar  a 
passes,  and  to  which  it  is  lolteti. 
The  end  shown  in  Fig.  17  6  is 
that  which  passes  under  rail  X 
and  is  therefore  hidden  by  it  in 
Fig.  17  a.  The  section  atp  g, 
»3  (fig.  17  c)  shows  the  attachment 
of  one  of  the  rods  x  to  rail  X. 
The  arrangement  here  is  similar  to  that  at  n  o,  except  that  in  Fig.  17  c  the  malleable 
casting  M  is  bolted  to  the  web  of  the  rail  as  shown,  while  in  Fig.  17  6  ii  is  of  dififereat 
8hapf>,  and  is  riveted  to  the  Jlange  of  the  rail. 


O  OS  "^ 


-^   s 


their 

e  wor 
-swit 

show 
re  the 
r  brok 
loes  in 

•s 

=^«-^ 

a 
1 

2«  ^C! 

aj  ^3  o  o 

.a  c  c  „ 

s 

rigrg*  of  t 
t.  AAa 
are  not 
he  place 

< 

< 

f*M 

^m 

-s 

^=H^ 

A 

omitt 
>r  the  t 
hout. 

are  at 

^ 

1-2  ill  a 

Q?  OJ  c3  W,  0^ 

.  o  o  ^-^^  • 


TUENOUTS.  833 

111  a  iiftiii  III  IB,  ii;ti  til!  ^ 

fill  t^i  i|-:iiililii  !i-ylliNl!  lyil 

Hi!  Ill  li  ll^fLP  M^i1||l|fi  ift  I 


(VI  -C      -  >i  *J    eT  fe    2    ^  -    i 


«  o  ! 


=  g  p  fc  life's  B-si^  s  5?=s^_-^  r  ..r  =^ 


tiiiriii^fisiiir  is^Esiiriiiirisx^ 


834 


TURNOUTS. 


Art.  22.  Fro^s.  The  frog  is  a  contrivance  for  allowing  the  flange  of  th« 
wheel  on  the  rail  e  tc,  Fig  1,  to  cross  the  rail  r  z ;  and  that  of  the  wheel  on  r  ^,  to  cross 
ex.  Tlie  first  contrivance  for  this  purpose  was  a  bar,  approxi- 
mately of  the  shape  of  the  rail,  pivoted  at  the  point  where  the  center  lines  of  the 
rails  e.x  and  r  z  cross  each  other,  and  free  to  move  horizontally  about  this  pivot,  so 
that  it  could  form  a  portion  oiex  when  the  train  was  passing  to  or  from  the  turnout^ 
or  a  portion  of  rz  when  the  train  was  using  the  main  track.    Sometimes  the  pivot 


Fig.  20. 


pMsed  through  one  end  of  the  bar,  as  In  Fig  20,  and  sometimes  through  Its  center, 
as  in  Fig  21.  Such  bars  were  generally  moved  by  a  rod 
(attached  at  n)  and  lever,  similar  to  those  used  for  switches ; 
and  they  then,  of  course,  required  an  attendant;  but  many 
attempts  have  been  made  to  vise  such  frogs  by  connectin^^ 
them  with  the  switch  by  means  of  rods,  &c,  so  that  the  bar 
should  move  automatically  when  the  switch  was  turned. 
Owing  to  the  considerable  distance  (80  ft,  more  or  less!  be- 
tween the  frog  and  switch,  it  has  been  found  difficult  to 
«ecure  simultaneous  movements  of  the  switch  and  frog,  and  the  contrivances  referred 
to  have  not  come  into  extensive  use.  Such  bars,  while  they  avoid  the  jar  produced 
by  wheels  passing  across  the  throat  of  the  frog  (Art  35),  labor  under  the  same  dis- 
advantage as  the  stub-switch,  Art  10,  in  requiring  a  liberal  allowance  of  space  be- 
tween their  ends  and  those  of  the  adjoining  rails,  to  avoid  any  possibility  of  theif 
coming  into  contact. 


<  «. 

Fig.  21. 

z 

Flg.22. 

Art.  23.    These  bars  were  soon  superseded  by  rigid  cast-iron  A*Ogrs,  Figf 
12  and  23.    These  were  hardened  by  chilling,  so  as  better  to  resist  the  action  of 


Fig.  23, 


passing  wheels;  but  even  with  this  precaution  they  wore  out  so  much  more  rapidly 
than  the  rails,  that  the  wings,  wm  and  tc,  and  the  ton g^ue,  P,  were  capped 
with  steel  from  J^  inch  to  1  inch  thick,  bolted  or  riveted  to  their  upper  surfaces;, 


TURNOUTS.  835 

Wie  triangle,  P,  called  the  tongue  of  the  frog,  Is  the  meeting-point  of  the  two  raila, 
/«and/a?,Figl ;  while  the  wings,  wm  and  ic,are  continuations  of  the  rails  e  andr. 
The  wings  give  support  to  the  treads  of  the  wheels  in  passina:  over  the  spaces  between 
the  point  and  w  and  z,  which  spaces  are  left  for  the  passage  of  the  flanges. 

The  channel  is  called  the  moiitb  of  the  frog  at  a,  Figs  22  and  23;  and  iti 
throat  at  the  narrowest  part,  iv  i.  That  part  of  the  tongue  back  of  w,  Fig  23,  or 
between  u  and  g,  is  called  its  lieel. 

The  channel  is  made  about  2  ins  deep  to  prevent  the  flanges  from  touching  its  bottom. 

The  projections,  1 1,  Fig  22,  are  for  bolting  ihe  frog  to  the  wooden  cross  ties. 

Although  one  side  of  the  frog  forms  a  part  of  the  turnout  curve,  its  shortness  war» 
rants  us  in  making  both  sides,  jo,  st,  Fig  23,  straight. 

Art.  24.  Giiide-rails,  or  guard-rails,  g  fir','Fig  1.  Suppose  wheels  to  be 
rolling  from  A  toward  B,  Fig  1,  on  the  main  track;  the  switch-rails  being  in  the 
dotted  positions.  On  arriving  opposite  the  frog,  some  irregularity  of  motion  might 
cause  the  flanges  of  the  wheels  running  along  the  rail,  r  2,  to  press  laterally  against 
eaid  rail.  Consequently,  after  passing  the  throat,  w  i.  Fig  22,  they  would  press  against 
the  wing,  ic,  and  passing  between  c  and  P,  they  would  leave  the  track;  or  strike 
the  sharp  end  of  P,  breaking  it,  and  endangering  the  train.  To  prevent  this,  the 
guard-rail,  g.  Fig  1,  is  placed  so  near  the  rail,  b  h  (say  1%  to  2  ins  from  it),  that  the 
flanges  at  6 //,  while  passing  between  it  and  ^r,  prevent  those  at  the  opposite  rail 
from  pressing  against  the  wing,  ic.  Fig  22,  and  from  striking  the  point;  and  guide 
them  safely  along  their  proper  channel,  im.  Similarly,  if  wheels  be  rolling  from 
A  toward  D,  Fig  1  (the  switch-rails  being  in  the  positions,  ^m,j?s),  the  centrifugal 
force  due  to  the  curve  would  cause  the  flanges  to  press  against  the  rail,  ea;,  and 
against  the  wmg,wm.  Fig  22,  thus  rendering  the  train  liable  to  the  same  kind  of 
accident  as  in  the  preceding  case.  This  is  prevented,  in  the  same  manner  as  before, 
by  the  guard-rail,  gf',  Fig  1,  which  keeps  the  flanges  in  their  proper  channel,  m?c. 
Fig  22. 

The  narrow  flangre-way  between  the  guard-rail,  g.  Pig  1,  and  the  rail,  b  h, 
should  extend  at  least  a  foot  each  way  from  a  point  directly  opposite  the 
point,/,  Fig  23,  of  the  frog.  In  a  distance  of  at  least  about  2  ft  more  at  each  of  its 
ends  the  guard-rail  should  flare  out  to  about  3  ins  from  the  rail,  b  h,  so  as  to  guide 
the  flanges  into  the  narrow  channel.    The  same  with  g'. 

Ouard-rails  have  to  resist  a  strong:  side  pressure,  and  should 
be  very  firmly  secured  to  the  wooden  cross-ties.  This  is  usually  done  by  bolting 
against  them  two  or  more<  stout  blocks  of  steel,  or  of  wrought  iron,  which,  in  turiu 
are  bolted  to  the  ties.     > 

Art.  25.  The  cast-iron  fro^,  as  first  made,  had  no  provision  for 
fastening*  it  to  the  rails;  but  was  simply  bolted  to  the  cross-ties.  It  waa 
afterwards  provided  with  a  recess  at  each  end,  of  the  exact  shape  and  size  of  the  end 
of  the  rail.  The  rail  ends  were  inserted  into  these  recesses,  and  the  frog  was  thus 
kept  in  line  with  the  rail.  In  frogs  made  of  rails,  the  same  purpose  is  served  by  fish- 
er angle-plates,  by  which  the  ends  of  the  frog  are  secured  to  those  of  the  rails. 

Art.  26.  The  leng'th,  a  g^  Fig  23,  of  a  cast-iron  frog,  usually  varies  from 
4  to  8  ft ;  and  depends  upon  the  angle,  oft,  at  which  the  rails,  e  x  and  r  z,  Figs  1  and 
23,  cross  each  other.  This  is  called  the  frog'-ang'lc*  This  angle  may  be  expressed 
either  in  degs  and  mine,  or  in  the  number  of  times  the  width  of  the  tongue  on  any 
line,  as  ot.  Fig  23,  is  contained  in  the  distance,  gf,  from  the  point./,  to  the  center, 

ff,  of  that  line.  This  number  is  called  the  fro^  number.  Thus,  if  the  angle, 
oft,  Fig  23,  is  such  that  the  length,  gf,  is  3,  4,  or  10,  &c,  times  the  width,  o  t,  the 
frog  is  called  a  No  3,  4,  or  10,  &c,  frog.  Fig  23  is  a  No  3 ;  Fig  22,  No  5.  Frogs  are 
usually  made  of  Nos  4  to  12 ;  sometimes  with  half  numbers,  as  1}^,  8J^,  &c. 

Art.  27.  Draw  two  parallel  lines,  6  b',  dd',  for  the  top  of  rail,  ex.  Fig  23,  and 
A  W,  kk',  for  that  of  rail,  rz;  crossing  each  other  at  the  required  angle.  Then  the 
Intersection,/  of  lines  dd'  {tnd  h  h'  is  the  theoretical  point  of  the  fro^.  As 
this  point  would  be  too  narrow  and  weak  for  service,  it  is  in  practice  rounded  off 
where  the  tongue  is  about  3^  inch  wide,  as  shown.  If  the  frog  is  to  be  simply  abutted 
to  the  rail-ends,  zx.  Fig  23,  as  in  some  cast-iron  frogs,  the  length,/^,  need  be  only 
great  enough  to  give  a  width,  to,  sufficient  to  accommodate  the  rail-ends,  z  and  a;, 
and  the  heads  of  the  two  spikes  at  v  which  confine  them  to  the  ties.  If  desired,  a 
portion  of  the  flange  of  each  rail-end  may  be  cut  away  so  that  the  rail-heads  come 
together;  thus  diminishing  the  width  necessary  for  to,  and,  of  course,  the  distance, 

fg.  In  the  case  of  cast-iron  frogs  provided  with  recesses  for  holding  the  rail-ends, 
as  in  Art  26,  the  width,  o  t,  and  length,  fg,  must  of  course  be  greater. 

In  frog's  made  of  rails,  the  length  must  be  such  that  the  rail-ends, 
z  and  a;,  Fig  23,  are  far  enough  apart  to  give  room  for  fitting  to  their  inner  sides  the 
splice-plates  by  which  they  are  connected  with  the  frog.  Where  an^rZc-plates  an 
used,  this  distance  must  be  greater  than  in  the  case  of  Jiah-pleiXea. 


836 


TURNOUTS. 


•^ 


rO 


I^ISs 


./  S  u  ti  d  to  ••:: " 
'^  1  S-S  si'-  ©  I  ^  -  S^^  s'"'"  ^"-d  2  iS^  «  ^ 


te  o  S:  ^  ^  c  -^  '^  :3  !n  ^    •  5  "T  fl 


«f=.^  2?"' 


dT.^  set's  ^©  >.^  ^tNfa  ^  M 

Wo  o  ^IW  o^M  g*-  bcW  ^^t>,>. 

,  d  tJD  .  ^  g  .  S^  5  .  ^>-o 
^Jo5CoJa3°-w.9  bc*::!  ■«  g  cs  ^ 
b  y«3t^  fa^'^  fa^  s.-  faC3  ®-^ 

ca  e3t3        ©53        C^S  eS        !- J2  e3 


TURNOUTS. 


837 


into  recesses  let  into  the  sides  of  the  throat-pieces  and  blocks.  The  clamps 
are  prevented  from  sliding'  by  clips  riveted  on  the  flanges  of  the  rails. 
Great  care  is  taken  to  have  all  the  adjoining  surfaces  in  full  contact 

with  each  other,  throughout,  so  as  to  diminish  the  liability  to  wear. 

Art.  32.  In  stiff  frogs  the  parts  are  held  together  by  bolts  passing  through 
the  rails  and  the  throat-pieces.  In  some  there  are  no  throat-pieces,  and  the 
four  rails  forming  the  frog  are  riveted  to  a  wrought-iron  plate.  All  of  these 
frogs  can  be  made  of  any  desired  length  and  to  anv  desired  angle. 

Art.  33.  The  standard  length  of  stiff  Yrogs  from  be  to  dz,  for  Nos 
4  to  8,  inclusive,  is  8  ft ;  Nos  9  and  10,  9  ft ;  Nos  11  and  12,  10  ft ;  No  15,  12  ft. 

Art.  34:;  In  the  AVeir  fro^  (Weir  Froa:  Co.,  Cincinnati,  Ohio),  Figs.  26  A, 
26  B  and  26  C,  the  notching  of  the  main  or  long  point,  z,  to  receive  the  short  point 
d,  is  avoided  by  forging  the  latter,  under  hydraulic  pressure,  to  fit  the  former  which 


FEET. 

I  I 


ITig.  S6A. 

is  nsed  as  the  upper  die  in  the  forging  process.  The  rail  forming  the  long  point  z  is 
thus  preserved  intact  throughout  the  space  where  •the  two  points  are  in  contact,  as 
shown  in  the  section  at  n  o,  Fig.  26  B.    The  short  point,  d,  is  held  tightly  against, 


Fig.^s'e:B 


TTig.-SeC 


SECTION-AT  tt  O. 


w  w/ 

INCHES. 
1  -  C    1     2    3  15   6 


SECTION  AT  p  q. 


and  under  the  head  of,  the  long  point  z,  by  the  tightening  of  the  nuts  on  the  bolts 
(which  are  provided  with  nut-locks),  and  thus  gives  it  additional  support. 

The  long  filUng-blocks,  < «,  are  forged  in  dies  from  wrought  iron  or  steel.     They 
are  then  planed  to  fit  the  point  rails,  drilled,  and  bolted  as  shown. 


Art.  35.    The  object  in  reducing  the  width  of  the  channels 

for  some  distance  each  way  from  the  point,  /,  Fig  23,  so  as  barely  to  admit  the 
flanges  freely,  is  to  allow  the  treads  of  the  wheels  to  have  as  much  bearing  as  pos- 
sible upon  the  wings  and  tongue  while  moving  over  the  broadest  part  of  the  chan- 
nel near  f.  In  frogs  shorter  than  about  No  4,  it  is  difficult  to  secure  sufficient 
bearing  for  the  treads,  even  with  the  utmost  allowable  contraction  of  the  channel, 
when  the  width  of  the  tires  is,  as  usual,  about  6  ins.  In  the  earliest  frogs  this  diffi- 
culty was  partiallv  overcome  by  gradually  raising  the  bottom  of  the 
channel  between  the  point  and  wings,  so  that  the  wheels,  in  traversing  that  part, 
ran  upon  their  flanges  instead  of  upon  their  treads.  The  jar  occasioned  by  the 
tread,  in  striking  against  wing  and  point,  was  thus  avoided.  This  arrangement  is 
still  used  in  a-ossinps,  where  the  tracks  cross  at  a  very  obtuse  angle,  and  where, 
consequently,  the  wings  can  give  little  or  no  support  to  the  treads.  The  flanges, 
however,  soon  cut  gutters  in  the  bottoms  of  the  channels,  and  thus  increase  thei« 
depths,  so  that  the  treads  strike  the  wings  and  point,  as  in  the  ordinary  frog. 


838 


TURNOUTS. 


©     Bom  S 

•^'^  bhf-i  iS^' 


Hogs 


5  s^-:i 


ife^.So^ 


■^  —  -S  O        O  .2  "^ 
C     -""^-^        bCrtO 

-£^^3    "lis 

fc-  2  ,i»<  i  -^  C'rT^  ^*  :ii 

Sd'^  ^  ^  £  "cD '-'  iS  o 
eS'-;^c:0OO>*-' 

^t^  F  >  o    I    ^  ^  s 
^'Si   O   c^^.^-g   ^O 

^  «o  ^  g  tc-g  _«^^ 


G 

03 

o 

g.i::^ 

1 

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1^ 

eight 

eable 
der  a 
ring-r 

=^ 

V?  rt 

:^. 

^  &<c  «  a 

0)       rt 
03      a> 


2  a 


c3  a.S^  o) 

•^   5'^'i    03 

tn       2^  03 


o  'c  ^^  ^  *:::  'c 


.5  2  3  I 
o  S,o  I 


'S  oj.S'f  *S 

^tC  03  C   ^ 
gH'H^  03 


~ti  tc*^  a  03  'C  *^  2 
■^  s  3  «  5^  §  o  «2 

,W        C3        O  ^  So  bO 


TURNOUTS. 


839 


If  the  bearing  between  the  head  of  the  spring-rail  s  and  that  of  the  point- 
rail/,  is-too  short,  a  ridge  worn  on  the  outer  edge  of  the  tread  of  a  wheel  G  pass- 
ing from  B  toward  A,  may  drop  between  the  spring-rail  and  the  point-rail  when 
it  reaches  a  place,  near/,  where  the  point  is  narrower  than  the  gutter  worn  in 
the  wheel-tread.  The  ridge  is  then  liable  to  wedge  the  spring-rail  5  away  from 
the  point/.  To  avoid  this,  the  wing  w  of  the  spring-rail,  and  its  bearing  against 
the  point-rail,  have  been  lengthened  in  the  present  pattern,  thus  giving  a  wider 
bearing  and  keeping  the  ridge  of  a  worn  wheel-tread  on  top  of  the  rails  until  the 
frog  has  been  passed.  Tlie  standartl  leng'th  for  either  style  of  springs- 
rail  frogs,  and  for  any  angle,  is  15  feet. 

Since  spring-rail  frogs  do  not  present  a  full  bearing  to  wheels  entering  or  leav- 
ing the  luimoui,  but  only  to  those  passing  to  and  fro  on  the  mai7i  track,  they  are 
most  useful  where  the  greater  part  of  the  traffic  moves  on  the  main  line,  and 
where  but  few  trains  use  the  turnout. 

Art.  37.  In  ordering:  frog^s,  give  the  frog  angle  or  number,  and  the 
exact  cross-section  of  the  rail  used  on  the  road. 

Art.  38.  For  springs-rail  fro^s,  specify  also  whether  the  turn- 
out is  to  the  right  or  left  hand.     In  the  case  of  stiff"  frogs,  this  is  not  necessary. 

Art.  39.    Tlie  layingf-ont  of  Tnrnonts. 

In  each  of  the  Figs  29,  30,  and  31,  wpz  represents  the  main  track.  The  frogs 
distance,  pf,  is  a  straight  line  drawn  from  the  theoretical  point  of  frog,  f,  to  the 
hee],p,  of  that  switch-rail  which,  when  opened,  forms  the  inner  rail  of  the  turnout. 
Formerly,  when  th«  turnout  curve  was  taken  as  starting  at  the  toe  of  the  switch,  the 
frog  dist  was  a  straight  line  from  the  theoretical  point  of  frog  to  the  toe,  m,  Fig  4, 
of  the  outer  switch-rail,  q  m,  when  opened. 

Scrupulous  accuracy  is  not  necessary  in  these  matters.  Thus,  a  deviation,  either 
way,  of  say  3  per  cent  in  the  length  of  the  turnout  radius  from  that  given  by  the 
tftble  or  the  formulas,  will  be  almost  inappreciable.  So  too,  if  a  frog  number  should 
be  used,  intermediate  of  those  in  the  first  column  of  the  table,  the  other  dimensions 
may  be  found,  approximately  enough,  by  using  quantities  similarly  intermediate. 
A  rail  almost  always  has  to  be  cut  in  two  in  order  to  fill  up  the  frog  dist;  and  the 
exact  length  of  the  piece  can  be  found  by  actual  measurement  at  the  time  of  cutting  it. 

Rem.  When  the  turnout  leaves  a  straight  track,  as  in  Fig  29,  the  frogs  angle 
is  equal  to  the  central  angle, /c  0.  When  the  main  track  is  curved,  and  the  turnout 
curves  in  the  opposite  direction  (Fig  30),  it  is  equal  to  the  sum  (vfn)  of  the  ■ 
central  angles, jrco, /no;  and  when  the  two  curve  in  th^  same  direction  (Fig 
31),  it  is  equal  to  the  diff  (nfc)  of  the  central  angle8,/c o,/ no. 

-^ 


840 


TURNOUTS. 


Art.  40.    To  Imy  out  a  turnout,  px,Fis  29,  from  a  stralgrht 
track)  jp  ;S^*     From  the  column  of  radii  in  the  table  below,  select  one^co,  suit- 
able for  the  turnout ;  together  with  the  correspond* 

— — (/ j      *~-„..^^ ing  frog  number,  frog  dist,^/,  and  switch  length. 

Q I      __^~v^ »    Place  the  frog  so  that  the  main-line  side  of  its 

^\      ^^^^"-^^  ^X^ tongue  shall  be  at  fz,  precisely  in  line  with  the 

■^  Jr  I    ■■ — ^^^^^    ^>v   ^A.        ^  inner  edge  of  the  rail,  w  z ,   and  its  theoretical 
I  ^^\/    \     \  point,  /,  at  the  tabular  frog  dist,  pf^  from  the 

I/N.     \    \^     startiug-point,  p.    Stretch  a  string  from  5  (oppo- 
y^    \    \  \       site  p)  to/;  and  from  it  lay  off  the  three  ordinates 
,  y  \  from  the  table ;  thus  finding  three  points  (in  addi- 

I         ^/       ^  tion  to  q  and/)  in  the  outer  curve.    Do  not,  hoW" 

{      /       !B!ig..  S9  ever,  drive  stakes  at  these  points;  but  as  each  oi' 

•  /  them  is  found,  measure  off  from  it,  inward,  half 

^\/  the  gauge  of  the  track;  and  there  drive  stakes. 

Do  the  same  from  7  and/.  The  five  stakes  will  all 
then  be  in  the  dotted  center  line  of  the  turnout,  Fig  29;  and  will  serve  as  guides  t© 
the  work,  without  being  liable  to  be  displaced.  The  dimensions  in  the  table  below 
are  found  by  the  following  formulas,  the  main  track  being  straight: 

half  f^T  angle     -  Oa°ge  ^  Frog  dist. 

Frog"  No =■  V  Radius  co  -5-  Twice  the  gauge. 

Or,  Frog  Hfo »■  Half  the  cotangent  of  half  the  frog  angle. 

Radius  C  O =  Twice  the  gauge  X  Square  of  frog  number. 

Or,  Radius  e  o =  (Frog  dist  pf-i-  Sine  of  frog  angle)  —  half  the  gauge. 

Or,  Radius  C  O =  (Gauge  -*-  Versed  sine  of  frog  angle)  —  half  the  gauge. 

Frog  dist  pf. =  Frog  number  X  Twice  the  gauge. 

Or,  Frog  dist  p  f. =  Gauge  pq  -i-  Tangent  of  half  the  frog  angle. 

Or,  Frog  dist  pf. =  (Rad  co  -{■  half  the  gauge)  X  Sine  of  frog  angle. 

Middle  ord =  W  gauge,  approx  enough. 

Eacb  side  ord...  =  %  mid  ord.  =  ^  (or  .188  of  the)  gauge,  approx  enough. 

Switch  I^ength  =  .     /  Throw  in  ft  X  lOOOO" 

a3)prox  enough  \/   Tangential  dist  for  chords  of  100  ft,  for  rad  coot 

^  turnout  curve. 


TABLE  OF  TURNOUTS  FROM  A  STRAIGHT  TRACK.    Fig  29. 

Gauge  4  ft  8}^  ins.    Throw  of  switch  5  ins. 
For  any  other  gauge,  the  frog  angle  for  any  given  frog  number  remains 
the  same  as  in  the  table.    The  other  items  may  be  taken,  approx  enough,  to  vary  di- 
rectly as  the  gauge. 


Fros 
Number 

Frog 
Ancle 

Turnout 
Radius 

CO 

DeflAngof 

Turnout 

Curve 

Fro» 
Dist 
pf 

Middle 
Ordinate 

Side 
Ords 

Stub 
Switolt 
Lensth 

o     r 

Feet. 

o     r 

Feet. 

Feet. 

Feet. 

Feet. 

12 

4  46 

1356 

4  14 

113.0 

1.177 

.883 

84 

UK 

4  58 

1245 

4  36 

108.3 

1.177 

.883 

32 

11 

5  12 

1139 

5    2 

103.6 

1.177 

.883 

31 

103^ 

5  28 

1038 

5  31 

98.9 

1.177 

.883 

29 

10 

5  44 

942 

6    5 

94.2 

1.177 

.883 

28 

9^ 

6    2 

850 

6  45 

89.5 

1.177 

.883 

27 

9 

6  22 

763 

7  31 

84.7 

1.177 

.883 

25 

^H 

644 

680 

8  26 

80.0 

1.177 

.883 

24 

8 

7  10 

603 

9  31 

75.3 

1.177 

.883 

22 

7K 

7  38 

630 

10  50 

70.6 

1.177 

.883 

21 

7 

8  10 

461 

12  27 

65.9 

1.177 

.883 

20 

6M 

8  48 

398 

14  26 

61.2 

1.177 

.883 

18 

6 

9  32 

339 

16  58 

56.5 

1.177 

.883 

17 

5M 

10  24 

285 

20  13 

51.8 

1.177 

.883 

15 

5 

11  26 

235 

24  32 

47.1 

1.177 

.883 

14 

4H 

12  40 

191 

30  24 

42.4 

1.177 

.883 

13 

4^ 

14  14 

151 

38  46 

37.7 

1.177 

.883 

11 

*  Shortest  length  of  stub  switch  that  will  at  the  same  time  form  part  of  the 
turnout  curve,  and  give  5  ins  throw.  Point  switches  require  only  half  this 
throw.  In  practice,  switches  are  frequently  made  much  shorter  than  the  table 
requires,  thereby  sharpening  the  beginning  of  the  curve. 


TURNOUTS. 


841 


Art.  41.    Turnout  firom  a  curved  malUi  track.    The  following  con- 
venient approximate  method  was,  we  believe,  first  published  by  Mr.  E.  A, 
Gieseler,  C.  E.,  of  New  York,  in  1878.    There  are  two  cases. 
Case  1,  Fig  30;  when  the  two  curves  deflect  in  opposite  directions. 
Case  2,  Fig  31 ;  when  the  two  curves  deflect  in  tne  same  direction. 
Having  determined  approx  upon  a  radius  for  the  turnout  curve,  take  from  the 
table,  p  784,  its  corresponding  deflection  angle,  and  that  for  the  main  curve.     In 
Case  1,  find  the  sum  of  these  two  angles.    In  Chse  2, 
/j,i\  find  their  difference.    In  the  table,  p  840,  find  the 

I  ^  deJUction  ang^e  (not  the  frog  angle)  nearest  to  the 

I  \  sum  or  difF  just  found.    The  frog  number, 

switch  lengtb  and  frog"  distance  pf,  in 
the  table,  opposite  the  deflection  angle  thus  se- 
lected, are  the  proper  ones  for  the  turnout. 
Theoretically  we  should,  in  Case  1,  add  to  the  tab- 


\ 


JFig.  3J. 
Case  2 

ular  frog  distance  pf  about  half  an  inch  per  100  feet  for  each  degree  of  deflec- 
tion angle  of  the  easier  of  the  two  curves ;  and,  in  Case  2,  deduct  it ;  but  this 
refinement  is  unnecessary. 
Deflection  ang-Ie  of' turnout  curve 

(in  Case  1)  =  Tabular  deflection  angle  — deflection  angle  of  main  curve, 

(in  Case  2)  =  Tabular  deflection  angle  +  deflection  angle  of  main  curve. 


Ex.  Rad  of  main  curve,  2865  ft.  Rad  selected  approx  for  turnout,  716.8  ft. 
Here  the  defl  angles  are,  respectively,  2°  and  8°. 

In  Case  1 ;  8°-j-  2°  =  10°.  Nearest  defl  angles  in  table,  p  840,  10^  50'  and  9°  31'. 

If  we  select  10°  50',  we  have  Frog  No,  7)^;  Switch  L'gth,  21  ft;  Frog  Di8t=  70,6 
ft  +  twice  ^  in  =  say  70.7  ft ;  Defl  Angle  for  turnout  =  10°  50'  —  2°  =  8°  50' ;  Rad 
CO  649  ft.    If  we  select  9°  31',  we  have  Frog  No,  8;  Switch  L'gth, 

22  ft:  Frog  Dist  =  75.3  ft  +  twice  ^  in  =  say  75.4  ft;  Defl  Angle  =  9°  31'  —  2°  = 
7°  31' ;  Rad  c  o,  say  763  ft. 

In  Case  2  ;  8^  —  2°  =  6°.  Tabular  defl  angles,  p  840,  6°  5'  and  5°  31'.  6°  o' 
gives  Frog  No,  10 ;  Switch  L'gth,  28  ft ;  Frog  Dist  =  94.2  ft  —  twice  ^  in  =  say  94.1 
ft;  Defl  Angle  =  6°  5'  +  2°  =  8°  5';  Rad,  say  709  ft.  5°  31'  gives  Frog  No,  10)^; 
Switch  L'gth,  29  ft;  Frog  Dist  =  98.9  ft  —  twice  K  i^  =  say  98.8  ft;  Defl  Angle  — 
6°  31'  +  2°  =  7°  31' ;  Rad,  say  763  ft. 

Tbe  frog:  dist  p  f  miiy  also  be  found  tbns  : 

»,               ^     «  ^    ««  u.              Gauge  X  Frog  No 
Tansrent  of  half /«  o  -. ^^msno. 

Frog  ]>ist  p  f==  np  X  twice  the  sine  of  half /wo. 

Place  the  frog  with  the  main-line  side  of  its  tongue  at/z,  in  line  with  the  inner 
•dge  of  the  frog-rail,  p  z,  of  the  main  line,  and  with  its  theoretical  point,/,  at  the 
dist,  p/ (found  as  above),  from  the  heel,  jj,  of  the  inner  switch-rail.  Stretch  a  string 
from/  to  the  heel,  9,  of  the  outer  switch-rail.  Measure  the  dist,  qf,  divide  it  into 
four  equal  parts,  and  lay  off  three  ordinates,  found  thus : 

Middle  ord  =  (Square  of  half  qf)  -i-  twice  the  rad  of  turnout  curve.  Each  side 
ord  =  three-fourths  of  middle  ord. 

These  three  ords.  and  the  points  q  and  /,  give  us  5  points  of  the  outer  rail  of  the 
turnout  curve ;  and  from  these  we  measure,  inward,  half  the  gauge,  and  drire  5  ce» 
ter  guide-etakes.  as  in  Art  40. 


842 


TUBNOUTS. 


Art.  43.  To  find  trog  dists,  Ac,  by  means  of  a  drawing:  t« 
KCale.  The  frog  diet  can  generally  be  found  near  enough  for  practice,  from  a 
drawing  on  a  scale  of  about  ^  or  J^  inch  to  a  foot.  And  so  in  the  many  cases  where 
turnouts  cross  tracks  in  various  directions,  in  and  about  stations,  depots,  &c. 

Figs  32  and  33  are  intended  merely  to  furnish  a  few  general  hints  in  regard  to 


wsch  drawings.  For  instance,  the  curves  of  a  main  track,  as  well  as  those  of  a  turn- 
out, generally  have  radii  too  large  to  admit  of  being  drawn  on  a  scale  of  34  itich  to 
a  foot,  by  a  pair  of  dividers  or  compasses.  But  they  may  be  managed  thus :  Draw 
any  straight  line,  ab.  Fig  32,  to  represent  by  scale  a  100-ft  chord  of  the  curve,  divid* 
it  into  twenty  6-ft  parts,  a  1,  1  2,  2  3,  &c,  and  lay  off  by  scale  the  19  corresponding 
ordinates,  1 1,  2  2,  3  3,  &c,  taken  from  the  table  on  page  730.  By  joining  the  ends 
of  these,  we  obtain  the  reqd  curve,  a  c6,  of  the  main  track ;  and  of  course  can  draw 


the  inner  line,  y  t,  distant  from  it  by  scale  the  width  of  track,  say  4  ft  8^^  ins.  Now 
M  acb  and  y  ty  Fig  33,  be  a  curved  main  track  so  drawn ;  and  let  any  point  m  be 
taken  as  the  starting-point  of  the  turnout,  m  v,  &c.  On  each  side  of  m  measure  off 
any  two  equidistant  points,  n  and  n,  in  the  same  curve;  and  through  m  draw  sg^ 
parallel  tonn.  Then  is  w^ratang  to  the  curve,  y  w»  ^,  at  w.  Having  determined 
on  the  rad  of  the  turnout  curve,  mve,  draw  that  curve  by  the  same  process  as  before ; 
first  laying  off  the  angle,  g  m  i,  equal  to  the  tangential  angle  of  the  curve,  taken 
from  the  table,  p  788.  Then,  beginning  at  m,  lay  off  5-feet  dists  along  m  i ;  and  from 
them,  as  in  Fig  32,  draw  the  ords  corresponding  to  the  turnout  curve.  Through  the 
ends  of  these  ords  draw  the  curve,  mve,  itself.  Then  the  frog  dist  will  be  th« 
straight  dist  from  c  to  »,  and  can  be  measured  by  the  scale,  within  a  few  inches ;  or 
near  enough  for  practice.  The  middle  ord  of  the  arc,  m  v,  cannot  be  found  correctly 
by  so  small  a  scale  as  ^^  inch  to  a  foot,  but  should  be  calculated  thus :  From  the 
square  of  the  rad  take  the  square  of  half  the  chord,  m  v.  Take  the  sq  rt  of  the  rem. 
Subtract  this  sq  rt  from  the  rad.  If  two  other  ords  should  be  desired,  half  way  be- 
tween m  and  v  and  the  center  one,  they  may  each  be  taken  as  ^  of  the  center  one. 
Make  the  switch-rail  long  enough  to  leave  2'^  ins  at  its  toe  between  m  n  and  m  tt. 

The  frog  angle  at  v  will  be  equal  to  the  angle,  rvd^  formed  between  the  tang,  v  r, 
to  the  curve,  acb;  and  the  tang,  v d,  to  the  curve,  m v  e.  These  tangs  are  found  in 
the  same  way  as  m^r;  namely,  for  the  tang,  vr,  lay  off  from  v  two  equidistant  points, 
h  and  h,  on  the  curve,  acb;  and  through  v  draw  v r  parallel  to  h h.  Also,  for  v  d, 
lay  off  from  v  any  equidistant  points,  u  and  w,  on  the  curve,  mve,  and  through  v 
draw  V  d  parallel  to  them.  This  angle  may  be  measured  by  a  protractor.  Or,  if  on 
the  two  tangs  we  make  v  4  and  v  4  equal  to  each  other,  and  draw  the  dotted  line  4  4; 
and  from  its  center  at  6  draw  6  v ;  then  6  v  divided  by  4  4  will  give  the  No  of  the  frog. 
With  care,  and  a  little  ingenuity,  the  young  student  will  be  able,  by  similar  proc- 
esses, to  solve  graphically  any  turnout  case  that  may  present  itself.  The  method 
by  a  drawing  has  great  advantages  over  the  tedious  and  complicated  calculations 
which  otherwise  become  necessary  in  cases  where  curved  and  straight  tracks  inter- 
sect each  other  in  various  directions.  The  drawing  serves  as  a  check  against  serious 
errors,  which  would  be  detected  at  once  by  eye.  None  of  the  graphical  measure- 
ments will  be  strictly  accurate ;  but  with  care,  none  of  the  errors  need  be  of  prac- 
tical importance. 


TUENOUT8. 


843 


Art.  43,  An  experienced  track-layer,  with  a  good  eye,  can  place  his  own  guide- 
•takes  by  trial  on  the  ground;  and  by  them  lay  his  turnouts  with  an  accuracy  as 
practically  useful  as  the  most  scrupulous  calculations  of  the  engineer  can  secure. 

The  following  example,  Fig  34,  of  a  turnout  from  a  straight  track,  Y  Z,  exhibits  a 
common  case,  in  which  all  the  work  may  be  performed  on  the  ground,  without  pre< 


Flgr.  34. 


▼ious  calculation.  Let  ivohe  the  tongue  of  a  frog,  with  which  the  assistant  has 
been  directed  to  make  a  turnout  from  Y  Z ;  and  that  he  has  received  no  instructions 
more  than  that  the  turnout  must  start  at  d,  and  terminate  in  a  track,  W,  to  be  laid 
parallel  to  Y  Z,  and  distant  from  it  r  a;  or  r  a;,  equal  to  6  ft. 

Place  the  tongue  ot  the  frog  by  guess  near  where  it  must  come,  having  its  edge, 
V  iy  precisely  in  line  with  the  inner  or  flange  edge  of  the  rail,  b  r.  Then  stretch 
a  piece  of  twine  along  the  edge,  ov,  of  the  frog,  and  extending  to  dg.  Try 
by  measure  whether  u  e  is  then  equal  to  ed;  and  if  it  is  not,  move  the  frog  along 
the  line,  b  r,  until  those  two  dists  become  equal.  Then  is  v  the  proper  place  for  the 
point  of  the  frog ;  bv  is  the  frog  dist ;  one-half  of  c  e  is  the  length  of  the  middle  ord 
of  the  turnout  curve,  dv;  and  if  two  intermediate  ords  are  needed  at  a  and  s,  e%ch 
of  them  will  be  ^  of  said  middle  one. 

The  frog  being  now  placed,  proceed  thus :  Place  two  stakes  and  tacks,  x  and  x,  at 
the  reqd  inter-track  dist,  rx  and  ra;,  of  6  ft  from  the  rails,  br.  Then  range  by 
pieces  of  twine  xx  and  v/,  to  find  the  point,  n,  of  intersection.  Then  measure  nv, 
and  make  nm  equal  to  it.  Tlien  is  m  the  end  of  the  reverse  curve,  v  m,  of  the  turn- . 
out.  The  ords  of  this  curve  may  be  found  as  before;  one-half  of  nk  being  the 
middle  one,  &c. 

Rem.  It  may  frequently  be  of  use  to  remember  that  in  any  arc,  as  tj  w,  of  a  circle ; 
vn  and  mn  being  tangs  from  the  ends  of  the  arc;  one-half  of  the  dist,  kn,  is  the 
middle  ord,  kz,  oi  the  curve ;  near  enough  for  most  practical  purposes,  whenever  the 
length  of  the  chord,  v  m,  of  the  arc  is  not  greater  than  one-half  the  rad  of  the  circh 
of  which  the  arc  is  apart.  Or,  within  the  same  limit,  vice  versa,  if  we  make  k  n  equal 
to  twice  kz,  then  will  n  be  very  approximately  the  point  at  which  two  tangs  from 
the  ends  of  the  arc  will  meet.  Also,  the  middle  ord  of  the  half  arc,  v  «  or  «  m,  may 
be  taken  m  J^  of  the  middle  ord,  k  z,  of  the  whole  arc 


844 


TUKNTABLES. 


TURNTABLES. 


845 


TURNTABLES. 


Art.  1.  A  turntable  is  a  platform,  usually  from  40  to  60  ft  long  and 
about  6  to  10  ft  wide  (see  Fig  1,)  upon  which  a  locomotive  and  its  tender  may 
be  run,  and  then  be  turned  around  hor  through  any  portion  of  a  circle ;  and  thus  be 
transferred  from  one  track  to  another  forming  any  angle  with  it.  The  table  is  sup- 
ported by  a  pivot  under  its  center ;  and  by  wheels  or  rollers  under  its  two  ends. 
Frequently  other  rollers  are  added  between  the  center  and  ends.  Beneath  the  plat- 
form is  excavated  a  circular  pit  about  4  or  5  ft  deep,  having  its  circumf  lined  with 
a  wall  of  masonry  or  brick  about  2  ft  thick,  capped  with  either  cut  stone  or  wood. 
The  diam  of  the  pit  in  clear  of  this  lining  is  about  2  ins  greater  than  the  length  of 
the  turntable.  The  lining  is  generally  built  with  a  step,  as  seen  in  Fig  1,  for  sup- 
porting the  circular  rail  on  which  the  end  rollers  travel ;  or,  instead  of  this  step,  a 
detached  support  may  be  used  for  this  circular  rail,  as  at  u,  Fig  11.  At  the  center 
of  the  pit  is  a  solid  well-founded  mass  of  masonry  or  timber,  for  the  pivot  to  rest 
on,  as  seen  in  Fig  1.  This,  as  well  as  the  step  for  the  end  rollers,  should  be  very 
firm,  and  perfectly  level ;  otherwise  the  platform  will  be  hard  to  work.  The  plat- 
form is  frequently  floored  across  for  a  width  of  6  to  10  ft  to  furnish  a  pathway 
across  the  pit,  without  stepping  down  into  it;  especially  when  under  cover  of  a 
building.  At  first  they  were  floored  over  so  as  to  cover  the  entire  circular  pit ;  but 
this  increased  not  only  their  cost  but  their  wt,  so  as  to  make  them  difficult  to  turn; 
besides  causing  much  expense  for  repairs ;  with  greater  trouble  in  making  them. 
It  is  therefore  rarely  done  at  present,  except  where  want  of  space  sometimes  ren- 
ders it  necessary  in  indoor  turntables. 

For  the  minimum  length  of  a  turntable,  add  from  1^  to  2  ft  to 
the  total  wheel  base  of  the  longest  locomotive  and  tender  for  which  it  is  to 

be  used ;  but  a  turntable  should  be  several  feet  longer  than  is  necessary  for  merely 
allowing  the  engine  and  tender  to  stand  on  it;  for  the  increased  length  enables  the 
engine-men  to  move  them  a  little  backward  or  forward,  so  as  to  balance  them  chiefly 
upon  the  central  support;  and  thus  relieve  the  end  rollers.  By  this  means  the  fric- 
tion while  turning  is  confined  as  much  as  possible  to  the  center  of  motion;  and  is  there* 


Fig.S 


Fig.  6 


846  TURNTABLES. 

fore  more  readily  overcome  than  if  it  were  allowed  to  act  at  the  circumf.    Th% 
engine-men  soon  learn,  by  feeling,  the  proper  spot  for  stopping  the  engine  so  as  thue 
to  balance  the  platform. 
Art.  2.    Figs  1  to  6  represent  the  Sellers  cast-iron  turntable  of  Wm 

Sellers  &  Co.,  Phila.  It  consists  of  two  cast-iron  girders  of  about  1%  ins  average 
thickness,  perforated  by  circular  openings  to  save  metal.  One  of  these  girders  is 
shown  in  Fig  1 ;  and  parts  of  one  in  Figs  3  and  4.  Each  girder  is  in  two  separate 
pieces,  which  are  fastened,  as  shown  in  Figs  1,  3,  and  4,  to  a  hollow  cast-iron  "  cen- 
ter-box," A  B,  Figs  2,  3,  4,  and  6,  by  means  of  2}4:  inch  screw-bolts,  at/,  Fig  3;  and 
by  hor  bars,  o  o,  of  rolled  iron  about  Z%  ins  square,  fitting  into  sunk  recesses  on 
top  of  the  boxing,  and  tightened  in  place  by  wedges,  i  i,  screw-bolted  beneath. 

Art.  3.  The  sides  of  the  center-box  are  about  1%  ins  thick.  It  is  sus- 
pended from  the  steel  cap,  C,  by  8  screw-bolts  2  ins  diam.  On  its  lower  side 
this  cap  has  a  semi-cylindrical  groove  extending  across  it,  transversely  of  the 
track,  as  shown  in  Fig  6.  This  groove  fits  over  a  corresponding  semi-cylindrical 
ridge  on  the  top  of  the  cast-iron  '*  socket,"  s  (so  called),  on  which  the  cap  tkus 
rests.  The  socket,  in  turn,  rests  upon  the  upper  one,  u,  of  two  annular  steel  plates, 
u  and  V,  which  form  a  circular  box  containing  15  steel  conical  anti- 
friction rollers,  d  d,  Figs  2,  &,  and  6.  These  are  about  3  ins  in  length,  and 
in  greatest  diam.  They  have  no  axles,  but  merely  lie  loosely  in  the  lower  part,  v, 
of  the  circular  box;  filling  its  circumf  with  the  exception  of  about  }^  inch  left  for 
play.  In  the  direction  of  their  axes  they  have  V^  inch  play.  The  lid,  u,  of  th?  cir- 
cular box,  rests  upon  the  tops  of  the  rollers,  which  separate  it  from  v  by  al-jut  }/^ 
inch.  V  rests  upon  the  top  of  the  hollow  cast-iron  post,  P,  which,  by  means 
of  its  flanges,  is  bolted  to  the  cap-stone,  M,  of  the  foundation  pier. 

In  order  to  insure  a  perfect  bearing  of  the  revolving  surfaces  upon  eacn  other, 
and  thus  diminish  the  liability  to  abrasion,  the  rollers,  d  d,  and  the  insides  of  the 
box  in  contact  with  them,  are  accurately  finished,  as  are  also  the  top  and  bottom  of 
the  roller-box,  and  the  sui'faces  of  the  socket,  s,  and  post,  P,  in  contact  with  them. 
The  rollers  are  oiled  by  means  of  the  spaces  shown  by  the  arrows  in  Fig  2. 

Art.  4.  Adjustment  of  the  beig-bt  of  the  table.  By  turning  ths 
nuts,  N  N,  of  the  8  screw-bolts  which  support  the  table,  the  latter  may  be  raised  or 
lowered  1  or  2  ins;  the  cap,  C,  socket,  s,  and  roller-box,  u  r,  remaining  stationary 
on  top  of  the  post,  P.  All  turntables  should  have  the  means  of  making  such 
adjustment.  Before  the  nuts,  N  N,  are  finally  tightened  up,  the  blocks,  nr  nr, 
of  hard  wood,  cut  to  the  proper  tliickness  for  the  desired  ht  of  the  table,  are 
inserted  between  the  cap,  C,  and  the  top  of  the  center-box,  A  B,  as  shown. 

The  ht  of  the  table  should  be  such  that  each  of  the  wheels  at  its  outer 
ends  shall  be  }/^  inch,  in  the  clear,  from  the  circular  rail  on  which  they  travel. 

Art.  5.  At  each  of  the  outer  ends  of  the  table,  the  two  girders  are  connected 
transversely  by  heavy  cast-iron  beams,  called  "cross-girts."  These  project 
beyond  the  girders,  and  carry  the  cast-iron  end-wheels,  20  ins  diam,  2  at  each 
end  of  the  platform.  The  treads  of  these  wheels  are  but  about  3  or  4  ins  below  the 
bottoms  of  the  girders,  and  the  wheels  therefore  do  not  require  any  considerable 
depth  of  pit  for  their  accommodation.  In  order  that  they  may  roll  freely,  their 
treads  are  coned,  and  their  axles  are  made  radial  to  the  circular  turntable  pit,  In« 
termediate  transverse  connection  between  the  main  girders,  is  secured  by  the 
wooden  cross-ties  notched  upon  them  to  support  the  rails,  and  frequently  10 
or  12  ft  long,  for  giving  a  wide  footway  across  the  pit.  A  lever,  8  or  10  ft  long, 
fitting  into  a  staple,  is  used  for  turning  the  table,  not  on  account  of  friction,  but  ai 
a  handle  for  the  workmen. 

Art.  6.  The  semi-cylindrical  shape  of  the  joint  between  the  cap,  C,  and  the 
socket,  s,  permits  the  slight  longitudinal  rocking  motion  of  the  turntable  which 
takes  place  when  a  locomotive  comes  upon  it  or  leaves  it ;  but  prevents  it  from  tip- 
ping sideways,  as  it  was  apt  to  do,  when,  as  formerly,  the  cap  rested  directly  upon 
the  lid  tt  of  the  roller-box,  and  the  top  of  the  post  P  was  hemispherical,  forming  a 
ball-and-socket  joint  with  a  casting  upon  which  the  roller-box  rested. 

Art.  7.  Ten  sizes  of  these  turntables  are  furnished  for  locomotive  use. 
The  diameters  are  aa  follows :  70  feet,  65  feet,  60  feet,  56  feet,  54  feet,  50  feet  (two 
patterns),  45  feet,  40  feet,  30  feet  and  12  feet.  For  prices,  weights,  dimensions  of  pit 
and  foundations,  timber  required,  etc ;  address  William  Sellers  &  Co,  1600  Hamilton 
St.,  Philadelphia.  Machinery  for  turning,  being  considered  unnecessary,  is  not  at- 
tached, unless  specially  ordered.    Its  cost  is  extra. 

The  entire  cost  of  excavating  and  lining  the  pit;  foundation  for  pivot;  circular 
rail  for  end  rollers,  &c,  complete  for  a  56-ft  turntable  will  vary  from  $1200  to  $2500 
in  addition,  depending  on  the  class  of  materials  and  workmanship ;  and  whether  the 
bottom  of  the  pit  is  paved  or  not. 

Art.  8.    The  g^irders  of  turntables  are  now  very  generally  made  of 


•roENTABLEa 


847 


rolled-Iron  plates,  each  girder  being  In  one  piece ;  and  the  two  main  girders 
•re  then  connected  with  each  other,  near  their  centers,  by  two  cross-grirders 

of  I  or  channel  beams,  or  of  plate-iron,  one  on  each  side  of  the  central  pivot.  These 
cross-girders,  and  the  sides  of  the  main  girders  between  them,  form  a  sort  of  rectan- 
gular box,  corresponding  to  the  cast-iron  center-box  of  the  Sellers  table,  Arts  2  and 
3.  The  arrangement  of  the  center  bearing  apparatus,  and  the  manner  in  which  it 
is  connected  with  the  cross-girders,  differ  in  the  tables  of  the  several  makers. 

Art.  9.  In  the  plate-iron  turntables  made  by  the  Eldg'e  Moor  Bridge  Works, 
Wilmington,  Del.,  the  cross-girders,  G  G,  Fig.  8,  are  of  plate-iron ;  and  at  their  ends 
they  have  flanges,  F  F,  of  angle-iron,  by  which  they  are  riveted  to  the  main-girders,  E. 
In  the  54-ft.  tables  the  cross-girders  are  26  ins.  apart  in  the  clear. 


Enlarged  transverse  section  of  roller- 
box  &c,  on  center  line  of  Fig  8. 


Art.  10.  The  bolts,  H,  by  which  the  table  is  suspended  from  the  cap,  C,  are 
bIx  in  number,  and  are  arranged  in  two  rows  of  three  bolts  each,  one  row  being  ou 
each  sid&  of  the  post,  P,  and  between  it  and  one  end  of  the  turntable.  To  each  cross- 
girder  is  riveted  a  short  hor  bar  of  angle-iron,  J  J.  Through  the  hor  flange  of  each 
of  these  two  bars  pass  the  3  bolts,  H,  of  one  row.  The  hor  flange,  bearing  all  the 
wt  of  its  girder  and  load,  rests  upon  the  heads  of  these  3  bolts.  To  prevent  the 
flange  from  yielding  under  its  load,  vert  struts  of  angle-iron  are  riveted  to  the  web 
of  the  crOss-girder  between  the  bolts.  Two  of  these  struts  are  shown  at  K  K,  Fig  7. 
Their  ends  abut  against  the  upper  flange  of  the  cross-girder,  and  against  J  J, 

The  6  bolts,  H  H,  pass  up  through  the  flanges  of  the  cap,  C  (three  bolts  through 
each  flange),  and  their  nuts  rest  ui)on  its  top. 

Art.  11.  The  cap,  C,  is  beld  in  place  on  the  socket,  s,  by  means  of 
flanges  which  extend  down  from  it  on  both  sides,  as  shown  in  Fig  7. 

The  rollers,  ddy  and  the  roller-box,  uv,  are  those  made  by  Wm.  Sellers  &  Co, 
Phila,  Art  3. 

The  ht  of  the  tahle  may  be  adjusted,  within  a  range  of  2  or  3  ins,  by 
means  of  the  nuts,  N  N,  as  in  the  Sellers  table.  Art  4. 

The  lower  part,  v,  of  the  roller-box.  Instead  of  resting  directly  upon  the  post,  P, 
as  in  the  Sellers  table.  Art  3,  rests  upon  an  iron  casting,  L,  which,  in  turn,  rests  upon 
the  post,  P,  and  is  held  in  place  upon  it  by  a  lug  which  projects  down  into  it. 

The  post  is  built  up  of  plate-  and  angle-irons  riveted  together,  and  may  be  a  hoI« 
low  truncated  square  pyramid,  as  shown,  or  of  other  shapes.  Those  presenting  the 
shape  of  a  cross  in  hor  section  have  the  advantage  of  being  accessible  for  painting 
and  inspection. 

Art.  12,  The  main  girders  are  braced  by  hor  diag  rods,  whose  endt 
are  siown  at  Q,  Fig  8.  These  rods  are  provided  with  sleeve-swivel  turn-buckles,  by 
which  they  may  be  tightened  or  loosened. 

The  remarks  in  Art  6  on  the  end  wheels  of  the  Sellers  table  apply  also  tm 
those  of  the  Edge  Moor,  except  that  in  the  latter  the  wheels  are  25  ins  in  diam,  and 
the  cross-girts  which  carry  them  are  of  angle-iron.  The  wheels  are  fastened  to  their 
axles,  which  turn  in  brass  bearings  enclosed  in  cast-iron  journal-boxes.  Each  box 
ia  provided  with  means  for  raising  or  lowering  it;  and  the  brasses  are  rounded  on 
top,  so  as  to  insure  a  uniform  bearing,  no  matter  what  Inclination  may  be  given  to 
the  axle  by  the  vert  adjustment  of  the  journal-boxes. 


848 


TURNTABLES. 


UTooden  turntables,  with  none  but  two  common  wheel  rollers  at  each 
end  of  the  platform,  are  sometimes  resorted  to  from  motives  of  original  cost. 
They  are,  however,  much  harder  to  turn,  generally  requiring  two  men,  aided 
by  wheelwork  ;  and  are  more  liable  to  get  out  of  order;  and  more  expensive  to 
repair.  They  are  made  of  a  great  variety  of  patterns,  both  as  regards  the 
girders,  and  the  central  pivots,  end  rollers,  &c.  Frequently  an  addition  is  made 
of  8  to  12  small  rollers  travelling  on  a  circular  rail  of  6  to  12  feet  diameter, 
around  the  pivot  as  a  center.  These  are  intended  to  sustain  the  whole  weight; 
the  end  rollers  being  so  adjusted  as  to  touch  their  rail  only  when  the  platform 
rocks  or  tilts  as  the  engine  enters  or  leaves  it.  Therefore,  there  is  less  resistance 
from  friction  than  when,  as  in  Figs  11,  there  are  only  the  end  rollers  r.  In  this 
last  case,  the  engine  and  tender  cannot  be  balanced  so  precisely  upon  the 
slender  central  pivot,  as  to  prevent  a  great  part  of  the  weight  from  being 
thrown  upon  the  end  rollers;  thus  materially  increasing  the  frictional  re- 
sistance. 


In  plan,  these  wooden  platforms  are  sometimes  in  shape  of  a  cross ;  that  is,  ia 
addition  to  the  main  platform  for  the  engine,  there  is  another  transverse  or  at 
right  angles  to  it,  also  extending  across  the  pit;  and  having  end  rollers  travel- 
ling on  the  circular  rail.  Thus,  in  Figs  11,  (which  show  one  of  the  many  modes 
of  framing  a  table  which  has  only  a  central  pivot/,  and  end  rollers  r,)  the  main 
platform  rests  en  the  girders  c,  which  are  strengthened  below  by  braces  a;  while 
the  transverse  one  rests  on  the  timbers  o,  o,  which  must  be  imagined  to  extend 
across  the  pit.  One-half,  or  one  arm,  of  this  transverse  platform  is  intended  to 
carry  the  wheelwork  Rxx,  for  turning  the  platform;  and  the  other  arm 
serves  merely  as  a  balance  to  it :  therefore,  neither  of  them  requires  to  be 
very  strong.  It  is  important  to  connect  the  four  ends  of  the  two  platforms  by 
four  beams,  as  the  whole  structure  is  thereby  materially  stiffened.  In  the 
figures  the  wheelwork  Ra;  a;  is  for  convenience  improperly  shown  as  if  it  stood 
upon  the  main  platform. 


TR.  SEC. AT  CENTER  SIDE-VIEW 


B      Figs.  11 


The  figures  need  but  little  explanation.  They  represent  ac  actual  45-foot 
platform,  which  has  been  in  use  for  some  years.  The  convex  foot  /  of  the 
central  pivot,  about  6  inches  diameter,  should  be  faced  with  steel :  and  should 
rest  on  a  steel  step  ss.  This  should  be  kept  well  oiled;  and  protected  from 
dust  by  a  leather  collar  around  p,  and  resting  on  gg.  Its  upper  part,  about  4 
inches  diameter,  is  cut  into  a  screw  with  square  threads  about  I4  inch  thick, 
for  a  distance  of  about  15  inches.  It  works  in  a  female  screw  in  the  strong 
cast-iron  nutyy;  and  serves  for  raising  the  whole  platform  when  necessary. 
When  not  in  use  for  this  purpose,  it  is  keyed  tight  to  the  platform,  (by  a  key  at 


TURNTABLES.  849 

its  head  n^)  so  as  to  revolve  with  it.    Strong  screw-bolts  i  i  connect  the  several 
timbers  at  the  center  of  the  platform. 

E  is  a  light  cast-iron  stand  supporting  two  bevel  wheels  about  1  foot  diameter, 
which  give  motion  by  means  of  an  axle  d,  \%  inches  diameter  to  two  similar 
ones  below,  shown  more  plainly  at  W  and  Y.  These  last  give  motion  by  the 
axle  X  to  the  pinion  e,  (6  inches  diameter,  and  2^^  inches  face,)  which  turns  the 
platform  by  working  into  a  circular  rack  ^,  (teetli  horizontal,  1  inch  pitch;  3i^ 
inches  face,)  which  surrounds  the  entire  pit.  This  rack  is  spiked  to  the  under 
side  of  a  continuous  wooden  curb  H,  which  is  upheld  by  pieces  F,  a  few  feet 
apart,  which  are  let  into  the  wall  J  J,  which  lines  the  pit.  The  short  beam 
M  N,  (about  6  feet,)  which  carries  the  lower  wheelwork,  is  suspended  strongly 
from  the  beams  of  the  transverse  platform  above  it.  instead  of  the  two  lower 
bevel  wheels  W  Y,  and  the  horizontal  axle  x,  a  more  simple  arrangement  is  to 
place  the  pinion  e  at  the  lower  end  of  the  vertical  axle  d]  and  let  it  work  into 
a  rack  with  vertical  teeth  at  w,  on  the  inner  face  of  the  stone  foundation  of  the 
circular  rail.  P'or  this  purpose  the  stand  R  should  be  directly  over  u.  There 
are  two  cast-iron  rollers  r,  2  feet  diameter,  3  inch  face,  under  each  end  of  the 
main  platform ;  and  one  under  each  end  of  the  secondary  one. 

Although  this  kind  of  platform  necessarily  has  much  friction,  yet  one  man 
can  generally  turn  a  45-foot  one  by  means  of  the  wheelwork,  when  loaded 
with  a  heavy  engine  and  tender.  Indeed,  he  may  do  it  with  some  difficulty 
by  hand  only,  while  all  is  new  and  in  perfect  order;  but  when  old,  and  the 
circular  railway  uneven  and  dirty,  it  requires  two  men  at  the  winches  to  do  it 
with  entire  ease. 

As  before  remarked,  the  resistance  to  turning  is  diminished  by  employing  a 
Bet  of  from  8  to  12  rollers  or  wheels  r, 
Figs  12,  at)Out  a  foot  to  15  inches  in  diam- 
eter, so  arranged  as  to  form  a  circle  8  to  12 
feet  diameter  around  the  pivot.  When  this 
is  done,  the  main  girders  of  the  platform 
are  placed  8  to  12  feet  apart;  and  long 
cross-ties  are  used  for  supporting  the 
railway  track.  Also,  the  main  girders 
are  sometimes  trussed  by  iron  rods.         FigSi  12. 


Fig  12  shows  the  arrangement  of  these  rollers  r,  which  revolve  upon  a  cir- 
cular track  s\  while  the  platform  rests  on  their  tops  by  the  track  u.  The 
rollers  r  are  beld  between  two  wrought-iron  rings  o,o,  about  3  inches  deep, 
}^  inch  thick,  which  also  are  carried  by  the  rollers.  From  each  roller  a  radial 
tie-rod  <,  1  inch  diameter,  extends  to  a  ring  ww,  which  surrounds  the  pivot  «, 
closely,  but  not  tightly,  so  as  to  revolve  independently  of  it.  These  tie-rods 
keep  the  rings  oo  at  their  proper  distance  from  the  pivot,  so  that  the  rollers 
cannot  leave  the  rails  5  and  w.  Between  each  two  rollers,  the  rings  oo  should 
be  strengthened  by  some  arrangement  like  a,  to  prevent  change  of  shape.  The 
pivot  p  may  be  as  in  Figs  11.  There  must,  of  course,  be  the  usual  two  rollers 
under  each  end  of  the  platform,  for  sustaining  the  engine  as  it  goes  on  or  oflf; 
but  during  the  act  of  turning  the  platform,  the  whole  weight  should  rest  on 
the  central  rollers.  Such  a  platform  of  50  feet  length  can,  if  carefully  made, 
be  turned,  together  with  an  engine  and  tender,  by  one  man,  by  means  of  a 
wooden  lever  12  to  15  feet  long, inserted  in  a  staple  for  that  purpose;  and  there- 
fore may  dispense  with  the  transverse  platform  for  sustaining  wheelwork. 

Such  rollers  as  have  just  been  described,  in  connection  with  friction  rollers, 
Fig  5,  fo/m  perhaps  tlie  best  arrangement  for  a  large  turning: 

bridiT^.    At  least  one  end  of  a  platform  must  be  provided  with  a  catcli  or 
stop  for  arresting  its  motion  at  the  moment  it  has  reached  the  proper  spot 
54 


850 


ENGINE-HOUSES,   ETC. 


A  common  mode  is  shown  at  Fig  13.  It  consists  of  a  wrought-iron  bar  mn,  4 
feet  long,  3  inches  wide,  and  %  thick ;  hinged  at  its  end  w,  which  is  confined  to 
the  top  of  the  platform.  Its  outer  end  n  is 
formed  into  a  ring  V  for  lifting  it.  A  strong 
casting  ee  (or  in  longitudinal  section  at  tt,) 
about  15  inches  long,  3  inches  wide,  and  1 
inch  thick,  is  also  firmly  bolted  to  the  top  of 
the  platform  ;  and  the  stop-bar  mn  rests  in  its 
recess  r,  while  the  platform  is  being  turned. 
A  similar  casting  a  a  is  well  bolted  to  the 
wooden  or  stone  coping  cc,  which  surrounds 
the  top  of  the  lining  wall  of  the  pit.  When 
the  stop-bar  reachies  this  last  casting,  as  the 
platform  revolves,  it  rises  up  one  of  Us  little 
inclined  planes  tt,  and  falls  into  the  recess  of 
a  a,  bringing  the  platform  to  a  stand.  When 
the  platform  is  to  be  started  again,  the  bar  is 
lifted  out  of  its  recess  by  the  ring  F,  until  it  passes  the  casting;  when  it  is  again 
laid  upon  the  coping  cc^  and  moves  with  the  platform;  or,  if  required,  the 
hinge  at  m  allows  it  to  be  turned  entirely  over  on  its  back.  When  there  is  a 
transverse  platform,  the  proper  place  for  the  stop  is  at  that  end  which  carries 
the  turning  gear;  as  it  is  there  handy  to  the  men  who  do  the  turning.  If  there 
is  only  a  main  platform,  the  stop  may  be  placed  midway  of  the  rails.  Some- 
times a  spring;  cateh  is  placed  at  each  end  of  the  platform;  and  each  catch 
is  loosened  from  its  hold  at  the  same  instant  by  a  long  double-acting  lever.  All 
the  details  of  a  platform  admit  of  much  variety. 

Instead  of  the  friction  rollers.  Fig  5,  friction 
halls  5  or  6  inches  diameter,  of  polished  steel,  are 
sometimes  used.  The  pivots  also  are  made  in 
many  shapes. 

Platforms  lilie  oct.  Fig  14,  revolT- 
iiig*  aronnd  one  end  o  as  a  center  of  mo- 
tion, are  sometimes  useful.  The  shaded  space  is 
the  pit.  If  an  engine  approaching  along  the  track  W,  is  intended  to  pass  on  to 
any  one  of  the  tracks  1,  2,  3,  4,  the  platform  is  first  put  into  the  required  posi- 
tion, and  the  engine  passes  at  once  without  detention.  If  the  platform  is  long, 
it  will  be  necessary  to  have  roller-wheels  not  only  under  the  moving  end  a,  but 
at  one  or  two  other  points,  as  indicated  by  the  roller  rails  cc. 


Fig.  14. 


Engine  bouses,  of  brick,  cost  from  $1000  to  #1200  per  engine  stall,  exclu- 
sive ot  the  foundations. 


Tlie  cost  of  a  complete  set  of  shops  of  brick,  for  the  thorough  re- 
pair of  about  20  locomotives,  and  of  the  corresponding  number  of  passenger 
and  other  cars;  together  with  suitable  smith  shop,  foundry,  car  shop,  boiler 
shop,  copper  and  brass  shop,  paint* shop,  store  rooms,  lumber  shed,  oflSices,  Ac; 
completely  furnished  with  steam  power,  lathes,  planing  machines,  scales,  and 
all  other  necessary  tools  and  appliances,  win  be  about  from  ^75000  to  $100000  ex- 
clusive of  ground.  A  large  yard,  of  at  least  an  acre,  should  adjoin  the  buildings. 
A  moderate  establishment,  for  the  repairs  of  a  few  engines  only,  may  be>  built 
and  furnished  for  $25000. 


WATER   STATIONS. 


851 


WATER  STATIONS. 

Water  stations  are  points  along  a  railroad,  at  which  the  engines  stop  to 
take  in  water.  Tbeir  distance  apart  varies  (like  that  of  the  fuel  sta- 
tions, which  accompany  them,)  from  about  6  miles,  on  roads  doing  a  very  large 
business:  to  15  or  20  miles  on  those  which  run  but  few  trains.  Much  depends, 
however,  upon  where  water  can  be  had.  It  has  at  times  to  be  conducted  in 
pipes  for  2  or  3  miles  or  more.  The  object  in  having  them  near  together  is  to 
prevent  delay  from  many  engines  being  obliged  to  use  the  same  station.  To 
prevent  interruption  to  travel,  they  are  frequently  placed  upon  a  side  track. 
A  supply  of  water  is  kept  on  hand  at  the  station,  usually  in  large  wooden  tubs 
or  tanks,  enclosed  in  frame  tank-houses.  The  tank-house  stands  near  the  track, 
leaving  only  about  2  to  4  feet  clearance  for  the  cars.  It  is  two  stories  high ;  the 
tank  being  in  the  upper  one  ;  and  having  its  bottom  about  10  or  12  feet  above  the 
rails.  In  the  lower  story  is  usually  the  pump  for  pumping  up  the  water  into  the 
tank;  and  a  stove  for  preventing  the  water  from  freezing  in  winter.* 

The  tanks  are  usually  circular;  and  a  few  inches  greater  in  diameter  at  the 
bottom  than  at  the  top,  so  that  the  iron  hoops  may  drive  tight.  Their 
capacity  generally  varies  from  6000  to  40000  gallons,  (rarely  80000  or  mone,) 
depending  on  the  number  of  engines  to  be  supplied.  A  tender-tank  holds 
from  1000  to  3000  gallons ;  and  an  engine  evaporates  from  20  to  150  gal- 
lons per  mile,  depending  on  the  class  of  engine;  weight  of  train  ;  steepness  of 
grade,  &c.  Perhaps  40  gallons  will  be  a  tolerably  full  average  for  passenger,  and 
80  for  freight  engines.  The  follonring  are  the  contents  of  tanks 
of  different  inner  diameters,  and  depths  of  water.  U.  S.  gallons  of  231  cubic 
inches;  or  7.4805  gallons  to  a  cubic  foot. 


Diara. 

Depth. 

Contents. 

Diam. 

Depth. 

Contents. 

Ft. 

Ft. 

Gallons. 

Cub.  Ft. 

Ft. 

Ft. 

Gallons. 

Cub.  Ft. 

12 

8 

6767 

905 

24 

12 

40607 

5429 

14 

9 

10363 

1385 

26 

13 

51628 

6902 

16 

9 

13535 

1810 

28 

14 

64481 

8621 

18 

10 

19034 

2545 

30 

15 

79310 

10603 

20 

10 

23499 

3142 

32 

16 

96253 

12868 

22 

11 

31277 

4181 

34 

17 

115451 

15435 

Cypress  or  any  of  the  pines  answer  very  well  for  tanks.  The  staves 
may  be  about  23^  inches  thick  for  the  smaller  ones;  to  4  or  5  inches  for  the 
largest.  The  bottoms  may  be  the  same.  The  staves  should  be  planed  by  ma- 
chinery to  suit  the  curve  precisely.  Nothing  is  then  needed  between  the  staves 
to  produce  tightness.  A  single  wooden  dowel  is  inserted  between  each  two  near 
the  top,  merely  to  hold  them  in  place  while  being  put  together.  The  bottom  is 
dowelled  together ;  and  simply  inserted  into  a  groove  very  accurately  cut,  about 
an  inch  deep,  around  the  inner  circumference  of  the  tub,  at  a  few  inches  above 
the  bottoms  of  the  staves. 

One  of  20  feet  diameter,  and  12  feet  deep,  may  have  9  hoops  of  good  iron ;  placed 
several  inches  nearer  together  at  the  bottom  of  the  tank  than  at  the  top.  Their 
width  3  inches ;  the  thickness  of  the  lower  two,  i^  inch  ;  thence  gradually  dimin- 
ishing until  the  top  one  is  but  half  as  thick.  The  lower  two  are  driven  close 
together.  These  dimensions  will  allow  for  the  rivet-holes  for  riveting  together 
the  overlapping  ends;  and  for  a  moderate  strain  in  driving  the  hoops  firmly 
into  p>lace.t  Three  rivets  of  3^  inch  diameter,  and  3  inches  apart,  in  line,  are 
sufficient  for  a  joint  of  a  lower  hoop.  One  of  34  feet  diameter,  17  deep,  may 
have  12  hoops;  the  lower  ones  4  inches  by  3^;  with  three  ^-inch  rivets  to  a 
lower  hoop-joint. 

The  bottom  planks  of  the  tank  must  bear  firmly  upon  their  supporting  joists, 
or  bearers. 

A  tank  must  have  an  inlet-pipe  by  which  the  water  may  enter  it;  a  ivaste- 
pipe  for  preventing  overflow;  and' a  discharge  or'feed-pipe  7  or  8 
inches  diameter,  in  or  near  the  bottom;  through  which  the  water  flows  out  to 
the  tender.  The  inner  end  of  the  discharge-pipe  is  covered  by  a  valve,  to  be 
opened  at  will  by  the  engine  man,  by  means  of  an  outside  cord  and  lever.    To 

*  A  frame  tank-house,  18  feet  square,  with  stone  foundations  for  both  house 
ir^d  tank,  will  by  itself  cost  $400  to  $600.  A  brick  or  stone  one  somewhat  more. 
fSuch  a  tank,  put  up  in  ht  place,  will  cost  about  $400. 


852 


WATER  STATIONS. 


fts  outer  end  is  generally  attached  a  flexible  canvas  and  gum-elastic  hose  about 
7  or  8  inches  diameter,  and  8  or  10  feet  long,  through  which  the  water  enters  the 
tender-tank.  Or,  instead  of  a  hose,  the  feed-pipe  may  be  prolonged  by  a,  metal- 
lic pipe,  or  nozzle,  sufficiently  long  to  reach  the  tender ;  and  so  jointed  as,  wheu 
not  in  use,  to  swing  to  one  side,  or  to  be  raised  to  a  vertical  position,  (in  the  last 
case  it  is  called  a  drop,)  so  as  not  to  be  in  the  way  of  passing  trains. 

The  same  tank  may  supply  two  engines  on  different  tracks,  at  once.  The 
tanks  are  very  durable. 

The  patent  frost-proof  tank  of  John  Burnham,  Batavia, 
Illinois,  is  simply  an  ordinary  tank,  in  which  the  water  is  prevented  from 
freezing  by  means,  1st,  of  a  circular  roof  which  protects  a  ceiling  of  joists,  be- 
tween which  is  a  layer  of  mortar ;  2d,  by  an  air-space  obtained  by  a  similar  ceil- 
ing beneath  the  timbers  on  which  the  tank  rests.  Although  the  sides  are  en< 
tireiy  unprotected,  no  house  is  necessary ;  but  merely  strong  posts  and  beams 
on  a  stone  foundation,  for  the  support  of  the  tank.*  The  supply  pipes  are  ia 
boxes  made  of  boards  and  tar-paper. 

Tanks  are  frequently  made  rectang-ular,  with  vertical  sides  of 
posts  lined  with  plank,  and  braced  across  in  both  directions  by  iron  rods.  They 
are  more  apt  to  leak  than  circular  ones.  They  have  been  made  of  iron ;  but 
wood  seems  to  be  preferred. 

The  i¥ater  for  supplying^  the  tanks,  may  be  pumped  by  hand,  steam^ 
horse,  wind,  hydraulic  ram,  or  otherwise,  from  a  running  stream;  from  a  pond 
made  by  damming  the  stream  if  very  small  or  irregular;  from  a  cistern  below 
the  tank;  or  from  a  common  well.  Many  roads  doing  a  business  of  10  or  12 
engines  daily  in  each  direction,  depend  entirely  upon  wells ;  and  pump  by  hand ; 
generally  two  men  to  a  pump.  Those  doing  a  very  large  business,  when  the 
supply  cannot  be  obtained  by  gravity,  mostly  use  steam.  The  i¥inclniill  is 
the  mostj,ecouomical  power;  and  when  well  made,  is  very  little  liable  to  get  out 
of  order.  Of  course  it  will  not  work  during  a  calm  j  but  this  objection  may  be 
obviated  in  most  cases  by  having  the  tanks  large  enough  to  hold  a  supply  for 
several  days.      Steam,  however,  is  most  reliable. 

The  follon  in^  table  will  give  some  idea  of  the  poi¥er  required  in 
a  steam  eng-ine  for  the  pumping*.  In  ordering  an  engine,  specify  not 
its  number  of  iiorse-powers,  but  the  number  of  gallons  it  must  raise  in  a  given 
number  of  hours,  to  a  given  height;  with  a  given  steam  pressure,  (say  about  60 
to  80  lbs  per  square  inch.)  The  pump  should  be  sufficiently  powerful  not  to  have 
to  work  at  night ;  and  should  be  capable  of  performing  at  least  25  per  cent,  more 
than  its  required  duty. 

A  fair  average  horse  should  pump  in  8  hours  the  quantities 
contained  in  the  first  3  columns;  to  the  height  in  the  4th  column;  or  sufficient 
to  supply  the  number  of  locomotives  in  the  5th  column,  with  about  2000  gallons 
each.    Two  men  should  do  about  one-third  as  much. 


Cub.  Ft. 

Lbs. 

Gals. 

Ht.Pt. 

No.  of 

Locos. 

Cub.  Ft. 

Lbs. 

Gals. 

Ht.Ft. 

No.  of 
Loooi. 

1600 

100000 

11968 

100 

6 

4571 

285714 

34194 

35 

17 

2000 

125000 

14960 

80 

IV. 

5333 

333333 

39893 

30 

20 

2667 

166666 

19946 

60 

10 

6400 

400000 

47872 

25 

24 

3200 

200000 

23936 

50 

12 

8000 

500000 

69840 

20 

30 

3555 

222222 

26596 

45 

13i< 

10667 

666667 

79787 

15 

40 

4000 

250000 

29920 

•  40 

15 

16000 

1000000 

119680 

10 

60 

A  reservoir,  with  a  stand-pipe,  or  urater  column,  is  preferable 
to  the  ordinary  tank,  when  the  locality  admits  of  it ;  being  less  liable  than  the 
pump  to  get  out  of  order;  and  being  cheaper  in  the  end.  The  reservoir  is  sup- 
posed to  be  filled  by  water  flowing  into  it  by  gravity ;  and  to  have  its  bottom  at 

*The  cost  of  windmill  alone,  for  railway  stations,  varies  from  about  ^50  for 
18  feet  diameter ;  to  gloOO  for  36  feet  diameter,  at  factory. 


WATER   STATIONS. 


853 


least  about  8  feet  above  the  rails;  or  at  any  greater  height  whatever  that  the 
ground  and  the  height  of  the  water  may  require.  It  may  be  excavated  in  the 
ground;  lined  with  brick  or  masonry  in  cement;  with  a  bottom  of  concrete; 
or  it  may  be  built  above  ground,  according  to  the  locality.  ^  It  may  be  roofed 
and  covered  in,  or  not ;  and  it  may  be  near  the  tracks,  or  at  a  considerable  dis- 
tance from  them,  according  to  circumstances.*  From  its  bottom,  an  iron  pipe 
from  8  to  12  inches  diameter,  is  carried  (generally  underground,)  to  within  a  few 
feet  of  the  track.  At  that  point  it  turns  vertically  upward  to  about  8  or  10  feet 
above  the  track,  forming  a  stand-pipe,  or  water-colunin ;  from  the 
upper  end  of  which  the  water  flows  (through  either  a  hose  or  a  jointed  nozele,) 
as  in  the  case  of  a  tank.  Several  such  pipes,  or  one  larger  one,  may  be  laid,  for 
the  supply  of  two  or  more  engines  at  once,  through  as  many  stand-pipes.  Where 
the  pipe  makes  its  bend,  and  becomes  vertical,  is  a  valve  for  opening  and  closing 
It;  and  which  may  be  worked  by  a  hand-wheel  placed  at  such  a  height  as  to  be 
easily  reached  by  the  engine  man. 


On  some  of  the  more  important  lines,  the  tenders  of  fast  trains  scoop 
np  water,  while  running,  from  a  long  trougb,  or  ''track  tank'' 

laid  between  the  rails.    The  tanks  are  about  i<^  mile  long.    They  must  of  course  be 
level,  and  they  therefore  require  a  level  track. 

As  originally  introduced  in  England,  by  Ramsbottom,  the  trough  was  of  cast- 
iron,  in  lengths  of  about  6  ft.  These  were  bolted  together  by  means  of  flanges  at 
their  ends.  The  ends  were  not  in  contact  with  each  other,  but  were  separated  by 
a  strip  of  vulcanized  rubber. 


Our  figure  shows  a  track  tank  of  ^g  inch  rolled  plate-iron,  the  sheets  of  which 
are  62  ins  long.  The  lengths  overlap  each  other  2  ins ;  leaving  5  ft  as  their  showing 
length.  The  sheets  are  cut  slightly  tapering,  so  that  at  one  end  of  each  length 
the  trough  is  ^g  in  deeper  than  at  the  other,  and  the  tops  are  thus  kept  flush  with 
each  other  throughout.  The  joints  are  double  riveted  with  %  inch  rivets,  about 
13^  ins  from  center  to  center,  and  staggered.  At  each  end  of  the  trough,  the 
bottom  slopes  upward,  and  in  a  length  of  6  ft,  comes  to  the  level  of  the  tops  of 
the  sides.  The  cross-ties  are  notched,  as  shown,  to  receive  the  trough,  which  is 
loosely  held  to  them  by  two  spikes,  S  and  S,  in  each  tie.  The  heads  of  the  spikes 
fit  over  the  horizontal  flanges  of  the  13^  X  1/^  inch  angle  bars,  A  and  A.  M  and 
M  are  mouldings  of  l}^  X  K  i"ch  bar-iron.  The  angles  and  the  mouldings  are 
in  lengths  of  15  ft,  and  are  riveted  to  the  sides  of  the  trough  continuously 
throughout  its  length. 

The  scoop  on  the  tender  is  lowered  into  the  trough,  and  raised  from  it, 
by  means  of  a  lever  on  the  fireman's  platform,  and  is  not  permitted  to  touch  the 
bottom  of  the  trough. 

The  trong^b  is  supplied  with  water  by  means  of  pipes  leading  from  an  ad- 
jacent tank.     The  supply  is  regulated  by  a  man  in  charge. 

To  prevent  the  water  from  freezing'  in  ivinter,  steam  is  led  to  the  trough 
from  the  boiler  of  the  pumping  engine,  through  iron  pipes  laid  under  ground  along 
Bide  of  the  track.  These  pipes  are  provided  with  branches  which  introduce  the 
steam  to  the  trough  at  every  40  ft  of  its  length.  The  steam-pipes  are  protected  bf 
■wooden  boxes,  and  are  furnished  with  valves  for  regulating  the  supply  of  steam.  ^ 


*  An  uncovered  rrservoib  50  ft  diam  by  12  ft  deep,  lined  with  brick  0 
from  $2500  to  $3500,  according  to  circumstances. 


■  masonry,  will  usually  cost 


854 


FENCES,   ETC. 


Evaporation  from  I^ocomotives.  The  evaporation  is 
usually  from  6  to  7  Bbs  of  water  to  1  ft  of  fair  coal.  Hence  if  we  take  the  average 
at  63^  Bbs,  or  say  .8  of  a  gallon  of  water  to  1  ft)  of  coal,  and  assume,  as  on  page 
800,  that  a  passenj^er  engine  evaporates  an  average  of  40  gallons  per  mile,  and 
a  freight  engine  80  gallons,  we  shall  have  very  nearly  2%  tons  of  coal  consumed 
per  100  miles  by  the  former ;  and  4^  tons  by  the  latter.  The  evaporation  from 
a  heavily  tasked  powerful  engine  may  amount  to  150  gallons  or  more  per  mile;  • 
but  such  is  an  exceptional  case. 

Theoretical  thickness  near  bottom  of  sheet-iron  water 
tanks,  single  riveted;  safety  4;  ultimate  strength  of  the  iron  40000  ft)s  per 
square  inch,  but  reduced  say  one-half  by  punching  the  rivet  holes.  Although 
safe  against  the  pressure  of  the  water,  some  are  plainly  far  too  thin  for  handling. 


Depth  in 
Feet. 


INNER  DIAMETER  IN  FEET. 
10     I      15     I     20     i     25     I     30     I     35 


40 


THICKNESS  IN  INCHES. 


1 

.0026 

.0052 

.0078 

.0104 

.0130 

.0156 

.0182 

.0208 

5 

.0130 

.0260 

.0391 

.0520 

.0651 

.0781 

.0911 

.1042 

10 

.0260 

.0521 

.0781 

.1042 

.1302 

.1562 

.1823 

.2083 

15 

.0391 

.0781 

.1172 

.1562 

.1953 

.2344 

.2734 

.3125 

20 

.0521 

.1042 

.1562 

.2084 

.2604 

.3125 

.3645 

.4.166 

25 

.0651 

.1302 

.1953 

.2604 

.3255 

.3906 

.4557 

.5208 

30 

.0781 

.1562 

.2344 

.3124 

.3906 

.4687 

.5470 

.6250 

Railroad  track  scales.    The  capacities  are  in  tons  of  2000  lbs  or  2240 
lbs,  as  may  be  desired. 


Capac- 

Length. 

Capac- 

Length. 

ity. 

ft. 

ity. 

ft. 

10 

12 

65 

40  to  65 

15 

12  to  15 

75 

40  to  85 

20 

12  to  16 

100 

50  to  112 

30 

20  to  32 

150 

60  to  123 

40 

30  to  40 

150 

100  to  150 

50 

40  to  50 

Post-an<l-rail  fences,  in  panels  8i^  ft  long;  5  rails ;  usually  cost  between 
40  to  100  cents  per  panel,  including  tlie  putting  up  ;  or  from  $512  to  $1280  per  mile 
of  road  fenced  on  both  sides,  with  1280  panels. 

Fence-posts  are  usually  of  chestnut,  cedar,  or  white  oak.  They  last  about  10  years 
on  an  average.  The  usual  size  is  2  to  3  ins  thick  X  6  to  7  ins  wide,  8  ft  long,  5  ft 
above  ground.  Their  cost  varies  greatly;  say  from  5}^  to  25  cts  each;  average,  10 
to  15. 

Worm  fences  seven  rails  high,  with  two  rails  on  end  at  each  angle,  cost  a'.'out 
J^^th  less.  Labor  $1.75  per  day.  The  scarcity  or  abundance  of  timber  chiefly  in- 
fluences the  price  ;  as  is  also  the  case  with  ties. 

Barbed  Steel  ivlre  fence  costs,  pw  mile  of  single  row  of  fence,  put  up, 
including  the  wooden  posts  and  all  labor,  from  $160  to  $250,  depending  on  the 
height  of  fence,  the  varying  market  price  of  wire,  labor,  &c, 

A  ^way-station  House,  30  X  60  feet,  surrounded  by  a  platform  12  feet  wide, 
protected  by  projecting  roof;  for  passengers  and  freight;  will  cost  from  $6000  to 
§10,000,  according  to  finish  and  completeness,  at  eastern  city  prices. 


COST  OF  RAILROADS.  855 

Approximate  averag^e  estimate  for  a  mile  of  sing^le-traek 
railnray.    Labor  $1.75  per  day. 

Grubbing  and  clearing,  {average  of  entire  road,)  3  acres  at  $50 $  150 

Grading ;  20000  cub  yds  of  earth  excavation,  at  35  cts 7000 

"  2000  cub  yds  of  rock  excavation,  at  $1.00 2000 

Masonry  of  culverts,  drains,  abutments  of  smaU  bridges,  retaining-walls,  dbc ; 

^00  cub  yds,  at$S,  average 3200 

Ballast;  3000  cub  yds  broken  stone,  at  $1.00 3000 

Cross-ties;  2640,  at  60  cts,  delivered 1584 

Mails;  (60  lbs  to  a  yard;)  96  tons,  at  $30,  delivered 2880 

Spikes 150 

Mail-joints.  < «.. 300 

Sub-delivery  of  materials  along  the  line. 300 

Laying  track 600 

Fencing  ;  (average  of  entire  road,)  supposing  only  ^  of  its  length  to  be  fenced..      450 

Small  wooden  bridges,  trestles,  sidings,  road-crossings,  cattle  guards,  (£c,  dc 1000 

Land  damages 1000 

Engineering,  superintendence,  officers  of  Co,  stationery,  instruments,  rents, 

printing,  law  expenses,  and  other  incidentals 2386 

Total $26000 

Add  for  depots,  shops.  Engine-houses.  Passenger  and  Freight  Stations,  Platforms, 
Wood  Sheds,  Water  Stations  with  the^r  tanks  and  pumps,  Telegraph,  Engines,  Cars,  Wfligh  Scalesb 
Vsota,  be,  &Q.    Also  for  large  bridges,  tunnels,  Turnouts,  ftc. 


856 


LOCOMOTIVES. 


ROLLING  STOCK. 


I.OCOMOTIVES. 
Dimensions,  Weights,  «£:c. 

Lists  of  some  of  the  principal  locomotives  made  by  the  Baldwin  Locomotive 
Works,  Philadelphia,  1902. 

In  the  designation  of  the  class,  the  first  number  (8, 10,  &c)  is  the  total  num- 
ber of  wheels  of  the  locomotive.  The  second  (30,  32,  &c)  gives  the  diameter  of  the 
cylinders,  thus  :  diam.  in  inches  ==  n/2  +  3,  where  n  =  this  second  number.  The 
letter  (C,  D,  or  E)  indicates  the  number  (4, 6,  or  8,  respectively)  of  driving-yrheeU, 

The  ivlieel-base  is  tlie  distance  from  center  to  center  of  the  front  "and  back 
wheels.  For  minimum  leng-th  of  turntable,  add  1>^  to  2  ft  to  the  total 
wheel-base  of  locomotive  and  tender,  which  is  =  wheel-base  of  locomotive  -f- 


Since  1890,  the  compound  locomotive  has  come  into  extensive  use  in 
many  branches  of  service.  The  Baldwin  (Vauclain)  type  has  a  high-pressure 
and  a  low-pressure  cylinder  on  each  side.  Those  of  other  makers  usually  have 
but  two  cylinders,  the  high-pressure  cylinder  being  on  one  side  and  the  low- 
pressure  cylinder  on  the  other.  The  dimensions  and  weights  of  compound 
engines  do  not  differ  materially  from  those  of  the  corresponding  classes  of  simple 
engines,  as  given  below. 


For  ^aug>e  of  4  ft  8  1-2  ins. 


i 

\ 

lin- 

Driving 
Wheels. 

Wheel-)}ass. 

<»  S  »-' 
3      "^ 
S  8  c 

Height 
of  top 

of 
stack 

Locomotive. 

Ten- 
der. 

Loco, 
and 

2- 

Class. 

S 

d 

> 
1 

2 

6 

|Z5 

5 

Driv's 

Total. 

ten- 
der. 

above 
rail. 

In  a. 

Ins. 

Ins. 

Ft.  Ins. 

Ft.  Ins. 

Ft.  Ins. 

Ft.  Ins. 

Ft.  Ins. 

Ft.  Ins. 

Ft.  Ins. 

8.30  C 

p 

18 

24 

4 

66-72 

7    6  [21    8 

15    0 

49    3 

65    3 

9     0 

14    6 

8.34  C 

" 

20 

24 

4 

72-78 

7    6  |21  11 

15    6 

50    0 

66    0 

10     0 

14    6 

10.34  D 

p 

F 

20 

24 

6 

56-68  11     9    22  11 

15    0 

50    6 

66    6 

10    0 

14    6 

10.38  D 

22 

26 

6 

62-72 

12    6    24    2 

15    6 

52    3 

68    3 

10    0 

15    0 

8.34  D 

F 

20 

26 

6 

50-56 

14    0  ,22    6 

15    0 

50    0 

Q&    0 

10    0 

14    6 

8.38  D 

" 

22 

30 

6 

56-62 

14    0  !22  11 

15    6 

51     0 

67    0 

10    0 

15    0 

10.34  E 

" 

20 

26 

8 

50-56 

14  10  !22    8 

15    0 

50    3 

66    3 

9    6 

14    6 

10.38  E 

** 

22 

30 

8 

50-56 

15    0 

23    2 

15    6 

51    3 

67    3 

10    0 

15    0 

4.32  C 

S 

19 

24 

4 

50 

7    6 

7    6 

15    1 

38    0 

53  10 

9    0 

14    6 

6.36  D 

21 

26 

6 

50 

11     0 

11     0 

15    7 

40    1 

55    9 

10    0 

14    6 

Weight  in  working  order,  in  pounds. 

Capacity  of  tender. 

Service. 

Type. 

Locomotive. 

o  «  ^* 

Coal, 

Class. 

'-J 

A 

Water. 

s 
s 

'2 

3  8"§ 

CO. 

tons. 

Gals.        lbs. 

8.30  C 

Pass'ger 

Amer. 

38000 

74000 

106000 

66000 

172000 

5>^ 

3300        27500 

8.34  C 

u° 

<< 

44000 

86000 

122000 

80000 

202000 

7 

4000        33333 

10.34  D 

p.  &F. 

lOWh'l  '36000 

100000  1360001  72000 

208000 

6 

3600        30000 

10.38  D 

'< 

** 

44000 

1240001 164000    80000 

244000 

7 

4000        33333 

8.34  D 

Freight 

Mogul 

42000 

120000  139000    72000 

211000 

6 

3600        30000 

8.38  D 

" 

52000 

150000! 171000 

90000 

261000 

8         4500        37600 

10.34  E 

<« 

Consol. 

34000 

125000  140000 

68000 

2080001     6       1  3400        28333 

10.38  E 

" 

•< 

45000 

170000 

188000 

90000 

278000 

8         4500        37500 

4.32  C 

Switch 

53000 
48000 

104000 
140000 

104000 
140000 

56000 
76000 

160000 
216000 

5       !  2800        23333 

6.36  D 

Q%  1  3800        31666 

1 

LOCOMOTIVES. 


857 


For 

graa§re  of  3  ft. 

it 

»  A 

10 

w 
^  M 

C 

Cylin- 
ders. 

Driving 
Wheels. 

Wheel-hase. 

—  a  ^ 
a>  cs  a> 

S   •'^ 

Height 
of  top 

Class.    , 

s 

i 

d 

a 

ft 

Locomotive. 

Ten- 
der. 

Loco, 
and 
ten- 
der. 

of 
stack 

1 

Driv- 
ers. 

Total. 

from 
rail. 

8.22  c  r 
10.24  D  I 

8.24  D  P 
10.26  E  ' 

4.20  C   S 

Ins. 
14 
15 
15 
16 
13 
15 

Ins. 
18 
20 
18 
20 
18 
18 

4 
6 
6 
8 
4 
6 

Ins. 
41-45 
45 
42 
38 
37 
41 

Ft.  Ins. 

8  2 
12    5 
12    6 
11    0 

6    0 

9  6 

Ft.  Ins. 

20  1 

21  5 
19    0 
18    2 

6    0 
9    6 

Ft.  Ins. 
13     4 
13    4 
13    4 
13    4 

Ft.  Ins. 

39  5 

40  9 
38    4 
40    6 

Ft.  Ins 

48  5 

49  9 
47    4 
49    6 

Ft.  Ins. 
8     3 

8    3 
8    3 
8    3 
8    0 
8    0 

Ft.  Ins- 
13    6 
13    6 
13    6 
13    6 
13    6 

6.24  D  ' 

13    6 

Service. 

Type. 

Weight  in  working  order,  in  pounds. 

Capacity  of  tender. 

Locomotive. 

'i 

8*55 

•3- 

H  OS 

ii 

Glass. 

5  1 

Total. 

Water. 

Gals.       ft)s. 

8.22  C 
10.24  D 

8.24  D 
10.26  E 

4.20  C 

Pass'ger 
P  and  F 
Freight 

Switch 

Amer. 
lOWh'l 

Mogul 
Consol. 

17000 
17000 
17000 
17000 
23000 
19000 

33000 
48000 
48000 
64000 
44000 
56000 

52000 
64000 
57000 
73000 
44000 
56000 

36000 
40000 
36000 
40000 

88000 
104000 

93000 
113000 

1600      13333 
1800      15000 
1600      13333 
1800      1500a 

6.24  D 



Selected  Penna.  R. 

R.  iStandards, 

1901 

.    Oau^e  4  ft  9 

ins. 

Cylin- 
ders. 

Driving 
Wheels 

Wheel-base. 

»d^ 

ai'H 

c 

11 

'ht  of  top 
chimney 
ve  top  of 
rail. 

Class. 

a 

eS 
ft 

6 
o 

d 

Locomotive. 

8-1 

as 
ft 

Driv's 

Total. 

gol 

D  16,  Passenger 
G  4,  Passenger 
F   1,  Freight 
H  6,  Freight 

Ins. 

20 
20 
22 

Ins. 
26 
28 
28 
28 

4 

6 
6 
8 

Ins. 
68 
72 
62 
56 

Ft.  Ins. 

7    9 

13  10 

14  6 
16  6K 

Ft.  Ins. 

22  9K 
25  ik 

23  4 

24  9 

Ft.  Ins. 

16  1 

17  6 

16  1 

17  6 

Ft.  Ins. 

50  31^ 

54  5K 

51  1% 

55  4}4 

Ft.  Ins. 

60  33^ 
66    % 

61  3>| 
64  7>i 

Ft. 

10 
10 
10 
10 

Ft.     Ins. 

14  111^ 

15  0 
14      8}^ 
14    11>| 

Weights  in  working  order. 

Capacity  of  tender. 

Locomotive. 

a^ 
3§ 

^- 

Class. 

Hi 

Total. 

Water. 

D  16,  Passenger 
G   4,  Passenger 
F    1,  Freight 
H  6,  Freight 

lbs. 
48300 
48500 
44650 
43800 

fts. 
93600 
140000 
128100 
166400 

Bs. 
135300 
185000 
146100 
186500 

lbs. 

82000 
134700 

76800 
111900 

lbs. 
217300 
319700 
222900 
298400 

Tons 
6.7 
8.9 
6.7 
9.8 

Gals. 
3600 
7000 
3600 
6000 

lbs. 
29900 
68300 
29900 
50000 

858  LOCOMOTIVES. 

Selected  Erie  Railroad  Standards,  1903.    Gauge  4  ft  8  1-3  ins. 


Class 


E  O  OOo  Ob. 


^O  O  O    o^ 


Class 


OOOO  ot:^  g 
OOOOo  n^x 
OOOOO    Qh.^ 


Cylinders. 

Driving 
Whepi« 

Wheel-base. 

2i  S  a 

Height 
of  top 

Locomotive. 

Ten- 

Loco. 

and 

tender. 

of 

Class. 

d 

i 

chim. 
above 

cs 

2 

d 

der. 

^si 

top  of 

Q 

m 

^ 

q 

Driv's 

Total. 

H 

H 

rail. 

Ins. 

Ins. 

Ins. 

Ft.  Ins 

Ft.  Ins. 

Ft.  Ins. 

Ft.  Ins. 

Ft.  Ins. 

Ft.  Ins. 

Ft.  Ins. 

B-  2 

19 

26 

6 

52 

10  10 

10  10 

18  0 

42  10 

57  11 

10  2 

15  4% 
13  1>| 
13  434 

D—  1 

18 

22 

4 

68 

8  6 

23  oy. 

16  1 

46   4^ 

56  0% 
55  8% 

8  10 

D—  2 

18 

22 

4 

68 

8  6 

22  11 

16  1 

45  11 

9  6 

D— 12 

18 

26 

4 

78 

8  6 

23  ly, 

23   65^ 

18  0 

51  7%  62  105| 

9  9 

15  9K 
15  934 
15  103^ 

D— 13 

19 

26 

4 

72 

8  6 

t  16  11 

49   8341 
52  9% 

47   034 

60  d% 

64  0% 

9  113^ 
9  113^ 

E—  1 

13-22 

26 

4 

76 

6  7  |24   9 

18  0 

F—  1 

18 

24 

6. 

56 

15  0 

22  1U 

22  sy 

16  1 

57  11 

9  10 

12  9% 

F—  3 

20 

26 

6 

64 

15  0 

16  11 

48  4y^ 

59  &Ys 

9  11^ 

15  9% 

G—  5 

19 

24 

6 

62 

12  10 

23   6 

16  1 

49   0 

59  4% 
62  3% 
58  0% 

9  UK 
9  11>| 

14  9 

G-  8 

21 

26 

6 

62 

12  0 

22  10 

16  11 

51  10>^ 

15  11 

H—  1 

20 

24 

8 

50 

14  0 

22  10 

16  1 

47   2 

9  10 

13  13^ 

H— 11 

21 

28 

8 

57 

15  9 

24   0 

16  9 

52   3 

63  334 

10  1 

15  4i| 

J^  1 

16-27 

28 

10 

50 

18  10 

27   3 

16  8 

53   51^ 

64  3K 

10  4>^ 

15  6 

Weight  in  working  order,  in  ponnds. 

r  ^: 

Capa 

City  of  t 

ender. 

Locomotivei     1 

1 

hM 

Class. 

a 

J 

1 

t^i- 

Coal, 
tons 

Ws 

iter. 

1 

^  ft  o  « 

of 
2000 

ft)S. 

Gals. 

fts. 

B—  2 

52500 

145100 

145100 

94050 

239150 

187.7 

6 

4500 

37552 

D-  1 

26500 

51500 

80700 

77000 

157700 

104.8 

8 

3600 

30042 

D—  2 

28300 

55400 

84400 

77000 

161400 

104.8 

8 

3600 

30042 

D— 12 

44000 

87325 

135525 

116800 

252325 

108.0 

12 

6000 

50070 

D— 13 

51280 

96060 

136930 

87900 

224830 

130.4 

9 

4500 

37552 

E—  1 

44230 

84290 

151000 

116800 

267800 

115.6 

12 

6000 

50070 

F—  1 

26500 

75300 

88400 

77000 

165400 

138.8 

8 

3600 

30042 

F-  3 

43080 

125330 

142840 

87900 

230740 

162.-5 

9 

4500 

37552 

G—  5 

34600 

99800 

122500 

77000 

199500 

139.7 

8 

3600 

30042 

G—  8 

42200 

118500 

144500 

87900 

232400 

184.9 

9 

4500 

37552 

H—  1 

25300 

88700 

103400 

77000 

180400 

192.0. 

8 

3600 

30042 

H— 11 

43500 

170000 

190000 

131600 

321600 

216.6 

12 

6000 

50070 

J-  1 

39300 

173700 

200550 

90100 

290550 

197.6 

10 

4500 

37552 

LOCOMOTIVES. 


'^2 

O  ^  r:^  .';3  2 

CmPhP3 


be 

CO       . 


^c  s  a  a 


oo  ooo  oo 

o  as  o  o  to  «D  lo 

Ci  oo  oo  O  oo  1-1  OS 
M  O  l-H  Oi-H  o  lO 
Tf  00  t- (M  05  (M 


:^ 


5  o>  •^  .^  «o     .^ 


o  »o  loo  »o  oo 

COUS"^  O'^  o  o 

■rH  T-H  CO  O  CO  C<)  -"J* 


.i?-as  CO  -^  c^  „ 


i-4T-t^C^ 


c  8  *  ^ 

-iJM  O  CIS 

'S  s  g  "  ^ 


2-'   w  "^"^"^\5< 

r'^       i  »0  -iJ  -fci  -<J   -1-5 


oo  o  oooo 

p  o  oooo  o 
•^  c^  CO  T-t -"a^  CO  »o 


►.00 


^::!^  :5^ 


:;v;<s     cc 


OCOQOpO  o  o 
t^(M  oo  0Tf<0 
r-i  C^  lO  to  O  <M  t- 
U50(M  (M  lO(M 


^a 


t>»  ■^  -^  0>        Oi"^ 


\e*  ooooooo 

i-N.^  o  o  o  o  o  o  CO 

cr>  «o  t^ -<ti  00  c^  o  «o  CO 

C^  CO-*  00  CO  ^ 


boa  ^  o 

Si    O   t-  pQ 


:  >» 

:  a> 

:  a 
IS 

•2 


U^  be 


G   O  ••  0)  2  G    I 

2  o  "-  o^ 


^i 


t  t.  a  o 
;OOEh 


03    O 


'5-a  o 


3^0 

s  a  ®  ^* 


860 


LOCOMOTIVES. 


Performance  of  liOcomotives. 

Passenger  engines  usually  carry  fuel  and  water  sufficient  for  40  or 

50  miles;  some,  60  to  70.  Freight  trains,  enough  for  20  to  25  miles.  Eoads, 
or  divisions,  with  steep  grades  require  the  fuel  and  water  stations  to  be  nearer 
together  than  where  the  grades  are  easy. 

The  following  gives  the  loads  (exclusive  of  locomotive  and  tender)  which 
the  above  described  Baldwin  eng^ines  will  haul,  at  their  usual  speeds,  on  a 
straight  track  and  on  different  g-rades  varying  from  a  level  to  3  ft.  per  100 
ft.,  or  158.4  ft.  per  mile.  The  loads  are  based  upon  the  assamiption  that  the 
so-called  ^'adbesion  "  of  the  locomotive  is  nine-fortieths  of  the  weight  on  all 
the  drivers,  and  that  the  condition  of  road  and  cars  is  such  that  the  frictional 
resistance  of  the  cars  does  not  exceed  7  Bbs.  per  ton  of  2240  Bbs.  of  their 
weight.  These  are  ordinarily  favorable  conditions.  The  adhesion  is  seldom 
less  than  one-fifth,  or  more  than  one-third,  of  the  weight  on  the  drivers. 

!Loads  in  tons  of  2240  lbs.  (exclusive  of  locomotive  and  tender). 
Oangre  4  ft.  8  1-2  ins. 


Ser- 

Type. 

On 

a  §;rade  of 

Class.* 

Oper 

Kper 

Iper 

IK  per 

2  per 

2%  per 

3  per 

vice.* 

ct.  = 

cent.  = 

cent.= 

cent.= 

cent.= 

cent.  = 

cent.  = 

Oft. 

26.4  ft. 

52.8  ft. 

79.2  ft. 

105.6  ft 

132  ft. 

154.4  ft 

per 

per 

per 

per 

per 

per 

per 

mile. 

mile. 

mile. 

mile. 

mile. 

mile. 

mile. 

8— 30— C 

P 

Amer. 

1980 

790 

470 

320 

235 

180 

140 

8— 34^-C 

•♦ 

*< 

2300 

920 

545 

375 

275 

210 

165 

10— 34— D 

PF 

10  Wh'l 

2675 

1085 

645 

445 

330 

255 

205 

10—38— D 

(( 

«' 

3325 

1350 

810 

660 

415 

325 

260 

8-34— D 

F 

Mogul 

3225 

1315 

795 

655 

415 

325 

260 

8— 38— D 

" 

** 

4050 

1650 

995 

695 

520 

410    . 

330 

10_34_E 

*' 

Consol. 

3375 

1375 

830 

680 

435 

345 

280 

10_38_E 

" 

«* 

4600 

1875 

1135 

795 

600 

470 

380 

4— S2— C 

.  S 

2825 

1150 

700 

490 

370 

295 

235 

6— 36— D 

« 

3776 

1550 

940 

660 

495 

395 

320 

Gan^e  3  It, 

Ser- 
vice.* 

Type. 

On  a  g^rade  of 

Class.* 

Oper 
ct.  = 

Oft. 

per 
mile. 

Kper 

cent.  = 

26.4  ft. 

per 

mile. 

1  per 

cent.= 

52.8  ft. 

per 

mile. 

l>^per 
cent.= 
79.2  ft. 
per 
mile. 

2  per 

cent.  = 

106.6  ft 

per 

mile. 

21^  per 
cent.= 

132  ft. 
per 

mile. 

3  per 

c(5nt.  =. 

154.4  ft 

per 

mile. 

8-22-C 
10_24— D 

8— 24— D 
10— 26— E 

4    20    C 

P 
P  F 

t 

S 

Amer. 
10  Wh'l 
Mogul 
Consol. 

610 
900 
915 

1225 
800 

1080 

265 
405 
410 
555 
365 
495 

160 
245 
255 
345 
230 
310 

105 
170 
175 
240 
165 
225 

75 
125 
135 
185 
125 
170 

65 
100 
105 
145 
100 
140 

50 
75 
80 
115 
80 

6— 24— D 

110 

of  a  locomotive,  _  ^^  ^  P^^^^'^  ^"^  ^'^^-      ^^""^^^  ^°  ^"^-  in  lbs.  per  sq.  inch, 

in  pounds        ~"  '*  '~^^'~~^~~~ 


Diameter  of  driving-wheel  in  inches. 


♦Seep.  856. 


LOCOMOTIVES. 


861 


From  the  tractive  force  must  be  deducted  20  to  30  per  cent,  for  internal  fric- 
tion, etc.  The  effective  tractive  force  cannot  exceed  the  adhesion,  or  say  one- 
fourth  the  weight  on  the  drivers. 

The  initial  steam  pressure  in  the  cylinders  is  always  less  than  the 
boiler  pressure ;  and  the  disproportion  increases  with  the  speed.  Thus,  with  the 
boiler  pressure  about  200  ibs.  per  square  inch,  the  cylinder  pressure,  at  8  to  10 
miles  per  hour,  may  be  from  180  to  190  ft)s.,  while  at  a  speed  of  30  or  40  miles,  it 
may  be  only  160  to  170  lbs.  The  average  cylinder  pressure  is  ascertained  by 
means  of  an  indicator  applied  to  the  cylinder ;  and  its  proportion  to  the  initial 
pressure  depends  upon  how  early  in  the  stroke  the  supply  of  steam  from  boiler 
to  cylinder  is  cut  off;  or,  in  other  words,  upon  the  extent  to  which  the  steam  is 
used  expansively. 

The  power  and  speed  of  locomotives,  and  their  consump- 
tion of  fuel  and  urater,  vary  greatly  with  circumstances,  such  as  grades 
and  curvature ;  condition  of  track  and  rolling  stock ;  number  of  cars  in  train  ; 
diameters,  number  and  distance  apart,  of  car  wheels ;  manner  of  coupling  the 
cars ;  skill  of  locomotive  runner  and  fireman,  &c,  &c. 

On  the  Phila  «fr  Reading  Ry  (Shamokin  Division) ,  between  Catawissa 
and  Lofty,  34  miles,  freig^ht  locomotives  are  assigned  train  loads  as  below : 
(according  to  conditions  of  the  engine  and  weather)  in  tons,*  including  weight 
of  cars  and  loading,  and  8.25  tons  *  for  caboose,  but  exclusive  of  locomotive 
and  tender.  Each  locomotive  is  assisted  by  a  helper  of  Class  1-5,  or  1-7.  The 
grades  and  curves  are  as  follows : 


Grades. 


Feet  per  mile.  Miles,  ^q^^^^^; 

Less  than            24.5  2.86         8.41 

Betw  30.62  and  35.05  30.72        90.27 

**      39.60  and  45.40  0.45          1.32 

Total  34.03      100.00 

"      32.94  and  33.26  22.72        66.76 


Curvature. 


Degree  of    No.  of  -p    . 
curve.       curves  ^*"^''- 


Less  than   3°    24 

a         a        90     36 

"       «     130    14 


8,203 
29,694 
26,218 
10,479 


Deflection. 

o         f 

376  5 

r440  52 

1343  56 

1099  51 


(14.13  miles)     133      74,594      4260    44 


i 

ft 

2 

s 

Train  load,  tons.* 

'to 

Drivers. 

Cylind's 

^^ 

Weight,  tons.* 

Slow 

Train 

W 

0 
6 

4 

^^ 

freight.! 

No.  82.t 

1 

6 
6 

Diam. 
ins. 

.2  c 

KB 

Total 
loco. 

Ten- 
der. 

Max. 

Min. 

Max. 

Min. 

H-1 

54 

18 

24 

120 

21.60 

44.60 

25.25 

1553 

1028 

L— 1 

6 

61 

4 

20 

24 

145 

47.00 

60.50 

34.50 

1638 

1128 

1533 

1023 

I  — 1 

8 

50.5 

2 

20 

24 

120 

48.25 

55.75 

27.50 

1793 

1198 

1633 

1093 

1-2 

8 

50.5 

2 

20 

24 

145 

52.67 

60.72 

28.00 

1898 

1268 

1743 

1163 

1  —  5 

8 

50.5 

2 

22 

28 

145 

63.00 

70.50 

32.75 

1  —  7 

8 

56 

2 

22 

28 

180 

72.00 

82.00 

46.00 

2193 

1463 

2038 

1358 

Special  test  runs. 

Class. 

No. 

Wt.of 

Time. 

Speed 

W'ter 

Coal 

Catawissa  to  Lofty. 

of 

train. 

per 
hour. 

used 

used 

Date. 

Loco. 

Help. 

cars. 

H.   M. 

gals. 

tons.* 

June  6,  1{ 

*'    7, 
March  8. 

J98 

I-l 
1-2 
L-2 

1-5 
1-5 
none. 

45 
45 
20 

1825 
1968 
578 

3      25 
3      21 
2        2 

10.2 
10.4 
17.0 

14500 
135715 
7000 

8 

7 

1901 

3.5 

*  Tons  of  2000  pounds. 

t  Run  from  Newberry  Junction  to  Tamaqua,  105  miles.    Slow  freights,  11  to  12 
hours  ;  train  No.  82,  10  to  11  hours. 


862  LOCOMOTIVES. 

The  greater  the  ratio  of  live  or  net  load,  to  dead  load  or  tare,  the  greater  is  the 
total  tonnage  (cars  and  load)  that  can  be  haulea  by  a  given  locomotive.  In  order 
to  take  account  of  this,  a  discrimination  is  made,  on  the  Shamokin  Division,  in 
favor  of  cars  having  a  capacity  of  not  less  than  80,000  lbs.  Trains  in  which  from 
10  to  19  of  the  cars  are  of  such  capacity,  are  given  100  tons  additional ;  with  20 
or  more  such  cars,  200  tons. 

On  the  Canadian  Pacific  Railway,  X  a  ratio  of  2  tons  of  net  load,  or 
'•  contents,"  to  1  ton  tare,  is  taken  as  standard ;  and  it  is  found,  on  the  eastern 
lines,  where  the  controlling  grades  are  generally  about  1  per  cent.,  that  a  train  of 
empty  cars  offers  about  30  per  cent,  greater  resistance  than  a  train  of  equal  weight 
with  net/tare  =  2/1,  A  chart  has  therefore  been  prepared,  by  which  the  engines 
are  loaded  upon  this  basis.  Heavier  grades  do  not  increase  the  frictional  resist- 
ances. Hence  the  total  resistance  increases  less  rapidly  than  the  grade,  so  that 
on  giades  steeper  than  about  1  per  cent,  an  addition  of  30  per  cent,  to  the  resist- 
ance of  full  trains,  would  give  too  high  a  resistance  for  empty  trains  of  equal 
weight,  and  vice  versa. 

On  the  Canadian  Pacific,  the  loading  of  freight  trains  is  graded  as  follows, 
according  to  speed,  weather  conditions,  &c.: 


Conditions. 

Ordinary. 

Temp.  +  10°  to  — 20°  F. 
or  bad  rail. 

Temp,  colder  than  —20° 
Fahr. 

Reduce  schedule  by 

Ordinary  trains.. 
Fast  trains 

0  per  cent. 
10    "      " 

7  per  cent. 
12    "      '* 

12  per  cent. 
15    *♦      " 

In  making  deductions  under  this  table,  the  probable  conditions  on  the  ruling 
grade,  not  those  at  the  starting  point,  are  considered.  During  snow  or  wind 
storms,  loadings  are  determined  by  the  conditions. 

On    the   Riclimond,  Fredericksburg    «&  Potomac    Railroad, 

freight  locomotives  draw  loads  as  follows : — exclusive  of  engine  and  tender. 
Cylinders  18  X  26,  62  in.  drivers,    90,000  lbs.  on  drivers,  630  tons. 
"  19  X  26,       "  "         102,000   "     "        "         700     " 

Maximum  (and  limiting)  grade  1  per  cent,  on  tangents. 

"Wood  fuel.  A  ton  (2240  lbs)  of  good  anthracite  or  bituminous  coal  is  about 
equal  to  1%  cords  of  good  dry,  hard,  mixed  woods  (chiefly  white  oak)  ;  or  to  2 
cords  of  sucn  soft  ones  as  hemlock,  white,  and  common  yellow  pine.  Much  of  the 
inferior  bituminous  coal  of  Illinois  is  hardly  equal  (per  ton)  to  a  cord  of  average 
wood. 

A  cord  is  4  X  4  X  8  ft,  or  128  cub  ft.  A  cord  of  good  dry,  white  oak  (next  to 
hickory,  the  best  wood  for  fuel)  weighs  3500  lbs  or  1.563  tons.  Dry  hemlock, 
white,  or  common  yellow  pine  (all  of  them  inferior  for  fuel),  about  .9  ton.  Per- 
fectly green  woods  generally  weigh  about  ^  to  3^  more  than  when  partially  dried 
for  locomotive  use;  in  other  words,  a  cord  of  wood,  in  its  partial  drying,  loses 
from  34  to  3^  ton  of  water,  and  still  contains  a  large  quantity  of  it.  Since  this 
water  causes  a  great  waste  of  heat,  green  wood  should  never  be  used  as  fuel.  Thie 
values  of  woods  as  fuel  are  in  nearly  the  same  proportion  as  their  weights  per 
cord  when  perfectly  dry. 

When  wood  is  used,  about  .2  cord ;  or  when  coal,  about  %  cord  of  wood,  must 
be  used  for  kindling,  and  getting  up  steam  ready  for  running  ;  and  this  item  is. 
the  same  for  a  long  run  as  for  a  short  one ;  so  that  long  roads  have  in  this  respect " 
an  advantage  over  short  ones,  in  economy  of  fuel.  Wood  has  the  disadvantage 
of  emitting  more  sparks ;  and  is,  moreover,  nearly  twice  as  heavy  as  coal,  for  the 
performance  of  equal  duty  ;  and  is,  therefore,  more  expensive  to  handle.  It  also 
occupies  4  or  5  times  as  much  space  as  coal. 

Up  grades  greatly  increase  the  consumption  of  fuel.  Thus,  on  a  road  95 
miles  long,  with  grades  mostly  of  less  than  6  ft  per  mile,  and  with  very  few  ex- 
ceeding 14  ft  per  mile,  with  coal  trains  of  734  tons  descending,  and  291  tons 
(empty)  ascending,  at  about  10  miles  per  hour  each  way,  the  coal  consumption 

t  Paper  by  Mr.  Thos.  Tait,  Manager,  Canadian  Pacific  Kailwav  Co.,  read  before 
the  New  York  Railroad  Club,  January  17,  1901.    Proceedings,  Vol.  XI,  No.  8, 


LOCOMOTIVES. 


863 


per  100  miles  for  each  ton  of  total  train  (including  engine  and  tender)  was  14.5 
ft)s  descending,  and  36.6  lbs  ascending. 

On  first-class  roads  a  passenisrer  engine  ivill  averag-e  about  35000 

miles  per  year,  or  say  100  miles  per  day  ;  a  freight  engine  25000  miles  per 
year,  or  say  70  miles  per  day. 


Coal 

burned  lbs 

§ 

o 

^ 

per  mile. 

.-; 

Aver,  net 

^-^-^ 

1^  .\.   \  |. 

.S 

sp'd  miles 

t^ 

t- 

i-t 

CO-M 

to 

0(M(M 

t^ 

CO 

o 

d 

per  hour. 

o 

g 

s 

^5 

g 

S^S 

§ 

s 

-2 

22 

(M  CO 

:::^:i? 

?  u 

o 

0^*110 

■^ 

>o 

H  S 

"^ 

00 

oo 

IC  Tti  o 

o 

00 

^ 

^% 

i 

lO 

^S 

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t^ 

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lO 

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o 

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r- 

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t-» 

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— 

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:^ 

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diam.ins. 

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stroke. 

CO 

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Inches. 

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a. 3 

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864  LOCOMOTIVES. 

liOComotive  expenses  per  100  miles  run,  will  average  about  as  follows: 

Passenger.  Freight. 

Fuel ^3.00  $6.00 

Water 1.00  2.00 

Oil,  waste,  &c 70  .90 

Repairs 4.00  8.00 

Engineer  and  fireman 5.00  6.00 

Putting  away,  cleaning,  and  getting  out....    1.50  2.00 

Locomotive  superintendence .  .30  .50 

$15.50  $25.40 

An  additional  allowance  of,  say  $2  per  100  miles,  should  be  made  annually  for  . 
clepreciation  of  eacb  eng'ine.    An  engine  in  active  service,  even  under 
a  judicious  system  of  repairs,  generally  becomes  worthless  (except  as  old  iron), 
in  say  16  years  on  an  average. 


CARS. 


865 


CABS. 

Usual  dimensions,  weigrbts  and  capacities.   Oang^e  4  ft  8^  ins. 


Length 

of  body. 

ft. 

Width. 

ft. 

Height 

above  rail. 

ft. 

Weight, 

empty. 

ibs. 

Nominal  ca- 
pacity, in 

passengers  or 

lbs. 

Passenger 

Parlor 

50  to  60 
60  to  70 

50  to  60 

28  to  36 

32  to  36 

29  to  36 
10  to  18 

10 

S}i  to  10 

83^  to   9 
8     to    9 
73^  to   S}i 

14 

11  to  123^* 
6  to    73^* 
4  to    4}ii 
8  to   9}i* 
6  to    7* 

50000  to   80000 
80000  to  100000 

40000to   60000 
23000  to  35500 
20000  to  30000 

25000  to   33000 
9000  to   11000 

50  to  64 
30  to  40 

Sleeper 

about  30 

Baggage,     mail, 

and  express  ... 

Box  and  cattle .. 

Gondola 

60000  to  80000 

Platform 

(( 

Coal 

(( 

15000  to  20000 

*  Add  6  inches  for  brake  shaft, 
t  Add  23^  to  3  feet  for  brake  shaft. 

On  narroir  g'au^e  (3  ft  and  33^  ft)  roads  there  is  but  little  uniformity  in 
car  building.  Freight  cars  are  usually  from  25  to  32  ft  long,  6%  to  8  ft  wide ; 
capacity,  30000  to  40000  lbs. 


Pressed  Steel  Cars,  built  by  the  Pressed  Steel  Car  Co.,  Pittsburg,  Pa. 

Length 

over  end 

sills. 

Width 

over  all. 

Height 

over  all. 

Weight 

in 
pounds. 

Capacity 

in 
pounds. 

Height 

over 
Brake  Mast. 

Hopper 

Ft.    Ins. 
31      6 

40      0 

11  'i 

43      3 

Ft.    Ins. 
10      0 
10      0 
10      2 
9    10>^ 
10      0 

Ft.    Ins. 
10      0 

3    10% 

13  4A 

14  ik 

7      6% 

36,000 
28,800 
32,300 
43,800 
35,100 

100,0W) 
80,000 
70,000 
60,000 

100,000 

Ft.      Ins. 
13      9 

Flat 

Box* 

Furniture  * .. 
Oondola 

14      8 
8      2% 

♦Wooden  bodies  with  steel  underframing :  others  all  steel. 


The  averag;>e  life  of  a  passenger  car  is  about  16  years.  Average  annual 
repairs,  including  painting,  $300  to  $700;  for  mail  and  express  cars,  $150  to 
$300 ;  freight  cars,  $75  to  $150. 

Allowing  125  ft)s  per  passenger,  a  full  car-load  of  passengers  (50  to  60  in  num- 
ber) would  weigh  but  from  6250  to  7500  lbs,  or  say  3  tons  ;  while  the  cars  them- 
selves weigh  say  30  tons,  or  nearly  10  tons  of  dead  load  to  1  paying  ton 
of  passeng-ers.  But,  as  a  general  rule,  passenger  trains  are  not  more  than 
hall-filled  ;  making  the  proportion  about  20  to  1.  The  foregoing  table  shows  that 
when  freig-lit  ears  are  loaded  to  their  nominal  capacity,  there  is  less  than 
about  1-3  ton  of  dead  load  per  ton  of  paying:  load ;  or,  with  cars 
half  loaded,  2  to  3.  ^ 

The  average  cost,  in  the  United  States,  of  moving  a  passenger  one  mile  is  about 
double  that  of  moving  a  ton  of  freight  one  mile,  while  the  receipts  per  pas- 
senger-mile are  nearly  three  times  those  per  freight-ton-mile. 

Tlie  resistance  of  cars  to  motion,  on  a  level  track,  and  with  cars  and 
track  in  fair  order,  is  usually  taken  at  about  from  6  to  8  lbs  per  ton  of  2240  lbs. 
With  everything  in  perfect  order,  it  may  fall  as  low  as  5,  or  even  4,  ft)s  per  ton. 
On  the  other  hand,  if  the  wheels  are  not  truly  round,  and  if  the  journals  are  not 
well  lubricated,  it  may  greatly  exceed  10  or  12  Tbs. 

55 


866  CARS. 

To  estimate  ron^bly  tlie  speed  of  a  train  in  which  one  is  riding ; 
if,  as  usual,  the  rails  are  30  feet  long.  By  means  of  the  sound  of  the  trucks  in 
passing  the  joints,  count  the  number  of  rail-lengths  passed  in  20  seconds.  This 
number  is  a  very  little  less  than  the  speed  of  the  train  in  miles  per  hour. 

For,  let  n  =  the  number  of  30-foot  rail-lengths  passed  over  in  20  seconds.   Then : 

o       ^  .        -T  1,  60?i  X  60  X  30 

Speed  in  miles  per  hour  =  — -^       ^7.7777-  =  1-023  n. 
^  ^  20  X  5280 

If  the  wheels  are  28  inches  in  diameter,  as  is  common  in  trolley  cars  and 
bicycles,  the  number  of  revolutions  in  5  seconds  (which  may  sometimes  be 
counted  by  means  of  irregularities  in  a  wheel)  will  give  very  closely  the  speed 
in  miles  per  hour. 

For,  let  n  =  the  number  of  revolutions  in  5  seconds.     Then  : 

Speed  in  miles  per  hour  =  ^^  ^  ^^  ^  IL^  =  0-9995  n. 
o   X  1^  X  0M\} 


RAILROAD   STATISTICS. 

EAILKOAD  STATISTICS. 


867 


Table  1.    IJr  THE  UNITED  JSTATES.* 


Plant. 

Miles  built  in  one  year 

Miles  ill  operation 

Rolling-  stock  in  operation. 

Number  of  locomotives 

"         passenger  cars 

"         baggage,  mail,  and  express  cars  . 

"         freight  and  other  cars 

Cost  of  road  and  equipment, 

per  mile,  in  dollars 

total,  in  millions  of  dollars 


Operation. 
For  one  year. 

Passengers  carried  one  mile,  per  mile  of  road 

Tons  of  freight  carried  one  mile,  per  mile  of  road  . 
Gross  earnings, 

per  mile  of  road,  from  passengers,     dollars 

"  "  "     freight "      

"  "  "    mails,  &c "      

"  "  total "     

per  passenger-mile,  from  passengers,      "      

"    ton-mile,  from  freight "      

passenger  earnings  -;-  total  earnings 

freight  **         -^    "  "        , 

mail,  &c,         "         -T-     "  "        A 

gross  earnings  -r-  total  investment , 

Expenses,     (For  details,  see  Table  3.) 

per  mile  of  road dollars 

expenses  -j-  gross  earnings 

Net  earnings. 

Net  earnings  -i-  total  investment 


1880. 

1890. 

7174 

5498 

87801 

166817 

17412 

33241 

12330 

22958 

4475 

7253 

455450 

1061970 

51561 

53783 

4530 

8789 

65392 

75062 

368514 

474728 

1641 

1732 

4740 

4686 

230 

528 

6611 

6946 

0.0251 

0.0218 

0.0129 

0.0093 

0.2483 

0.2519 

0.7169 

0.6817 

0.0348 

0.0664 

0.1136 

0.1015 

6174 

6792 

0.6078 

0.6854 

0.0504 

0.0340 

Table  2.    UNITED  STATES  BY  »IVISIOICS,  1899.* 


Plant. 

Miles  in  operation 

Cost  of  road  and  equipment,  per 
mile,  dollars 

Operation. 
For  one  year. 

Gross  earnings  per  mile,  3 

Expenses  per  mile,  $ 

Expenses  -;-  Gross  earnings 


Eastern 
States. 

Central 
States. 

Western 

States. 

Pacific 
States. 

52735 

64818 

59562 

10666 

68911 

' 45648 

48745 

61696 

10131 

6932 

0.6842 

6911 

4803 

0.6950 

4697 

3020 

0.6430 

6943 

4575 

0.6589 

*  Tables  1  and  2  are  based  chiefly  upon  Poor's  Manual. 


868 


RAILROAD  STATISTICS. 


Table  3.  Items  of  total  annual  expenses  for  maintenance  and 
operation  of  all  the  railroads  of  the  United  States.  Year  ending  June  30,  1899. 
From  Report  of  Interstate  Commerce  Commission,  1899. 


Total  $ 

«per 
mile  of 
road. 

Percent 
of  total. 

87,307,140 

457 

10.72 

10,767,381 

57 

1.32 

23,623,325 

126 

2.90 

19,335,860 

103 

2.38 

19,832,218 

105 

2.43 

3,968,408 

21 

0.49 

1,153,408 

8 

0.14 

3,837,314 

20 

0.47 

169,825,054 

897 

20.85 

50,555,264 

266 

6.21 

17,623,134 

93 

2.16 

57,320,521 

303 

7.04 

17,795,526 

94 

2.18 

143,294,445 

756 

17.59 

14,392,691 

76 

1.77 

86,586,108 

454 

10.64 

78,913,978 

417 

9.69 

74,196,282 

392 

9.11 

33,791,383 

178 

4.15 

15,525,232 

82 

1.90 

66,824,777 

352 

8.20 

22,256,884 

118 

2.74 

20,386,462 

108 

2.50 

14,290,195 

75 

1.76 

37,286,592 

198 

4.58 

464,450,584 

2450 

57.04 

36,819,917 

194 

4.52 

814,390,000 

4297 

100.00 

Repairs  of  roadway 

Renewals  of  rails 

"  *   ties .'. 

Repairs  and  renewals,  bridges  and  culverts 

**        **  *'         buildings,  docks,  wharves, 

"        "  **  fences,  crossings,  &c 

'*        "  "         telegraph 

Sundries 

Maintenance  of  way  and  structures 

Repairs  and  renewals,  locomotives 

*'        "  "         passenger  cars 

"        "  "  freight  cars 

Superintendence  and  sundries 

Maintenance  of  equipment 

Superintendence 

Fuel,'&c,  for  locomotives 

Engine  and  roundhouse  men 

Train  service,  supplies,  and  expenses 

Switchmen,  flagmen,  and  watchmen 

Telegraph 

Station  service  and  supplies , 

Balances.    Switching  charges,  car  mileage,  &c .... 

Yards,  elevators,  rentals 

Loss,  damage,  accidents 

Sundries 

Total  running  expenses 

Unclassified 

Aggregate  annual  expenses 


Each  of  these  items  is,  however,  subject  to  great  variation,  not  only  on  diflf  roads, 
but  on  the  same  road,  from  year  to  year.  A  road  with  many  bridges,  deep  cuts,  high 
embkts,  &c,  to  keep  in  repair,  will  have  heavier  maintenance  of  way  than  one 
which  has  but  few ;  and  this  item  may  be  but  small  one  year,  and  twice  as  great 
the  next.  Fuel  may  be  cheap  on  one  road  and  dear  on  another ;  and  this  mate- 
rially affects  the  item  of  motive  power.  And  so  with  the  other  items.  Some- 
times maintenance  of  way  exceeds  motive  power  and  cars  together ;  at  others, 
conducting  transportation  is  fully  half  the  total  expense. 

Tbe  total  annual  expenses  on  railroads  in  tbe  United 
States  usually  range  between  65  and  130  cents  per  train  mile ;  that  is,  per  mile 
actually  run  by  trains.  "When  a  road  does  a  very  large  business,  and  of  such  a 
character  that  the  trains  may  be  heavy,  and  the  cars  full  (as  in  coal-carrying 
roads),  the  expense  per  train  mile  becomes  large;  but  that  per  ton  or  passenger 
small ;  and  vice  versa,  although  on  coal  roads  half  the  train  miles  are  with 
empty  cars. 


RA.ILEOAD  STATISTICS. 


869 


Table  4.  Oross  annual  earning^s  per  mile,  per  passenger 
mile,  and  per  ton  mile,  of  some  of  thie  principal  U.  S.  rail- 
roads in  1899,    From  Poor's  Manual. 


Length, 
miles. 


Pennsylvania 

New  York  Central  &  Hudson  River, 

Baltimore  &  Ohio 

Chicago,  Burlington  &  Quincy 

Philadelphia  &  Reading 

Union  Pacific 

Wabash 

Atchison,  Topeka  &  Santa  Fe 

Total  and  averages,  United  States.... 


From  passengers 


Per  mile  Per  pass 
of  road.     mile. 


2780 
2395 
2024 
7419 
915 
2421 
2278 
7029 
187781 


^690 
5730 
2791 
1276 
4385 
1365 
1753 
1154 
1584 


$0.0194 
0.0182 
0.0174 
0.0211 
0.0162 
0.0279 
0.0189 
0.0228 
0.0200 


From  freight. 


Permile 
of  road, 


$18877 
11495 
9909 
3945 
19600 
5842 
4044 
4195 
4912 


Per  ton 
mile. 


$0.0047 
0.0059 
0.0039 
0.0086 
0.0078 
0.0154 
0.0055 
0.0102 
0,0073 


Table  5.  Annual  earnings  and  expenses  of  some  of  the 
principal  railroads  in  the  United  iStates  in  1899.  From 
Poor's  Manual. 


Pennsylvania 

New  York  Central  &  Hudson  River, 

Baltimore  &  Ohio 

Chicago,  Burlington  &  Quincy 

Philadelphia  &  Reading 

Union  Pacific 

Wabash 

Atchison,  Topeka  &  Santa  F6 

Total  and  averages,  United  States..... 


Gross 

Length, 

earnings 
per  mile 

miles. 

of  road. 

2780 

$25787 

2395 

-    19285 

2024 

14160 

7419 

5942 

915 

24540 

2421 

8181 

2278 

6320 

7029 

5763 

187781 

7161 

Expenses 
per  mile 
of  road. 


$17670 
12163 
10858 
3662 
13422 
4713 
4571 
3927 
4769 


Expenses 
-T-  gross 
earnings. 


0.6904 
0.6307 
0.7667 
0.6118 
0.5469 
0.5760 
0.7233 
0.6814 
0.6659 


Operating  expenses,  Erie  Railroad  Company ,  1900. 

Entire  system,  comprising  Erie  and  Ohio  Divisions. 


Coal  used  per  mile,* 

per 

per 

per  passenger  locomotive,  fl)s.. 

.    86.9 

Cost  per  mile 

loco. 

car 

*'    work                    "             •* .. 

.    93.7 

mile. 

mile. 

"    switching           "             " .. 

.    73.4 

Fuel                           cents 

7.29 

0.55 

"   pusher                 **             " .. 
"    freight                 "             "  .. 

.  147.7 

Repairs  and  renewals,  " 

7.45 

0.57 

.  152.2 
.    18.3 

Oil  and  waste " 

0.33 
0.42 

0.02 

"   passenger  car,                  "  .. 

Water  supply " 

0.03 

«   freight 

.      6.4 

Other  supplies " 

0.14 

0.01 

"    100  tons  t                          ".. 

.    20.5 

Engineers  and  firemen  " 

6.82 

0.52 

Oil  and  waste. 

Roundhouse  men " 

1.54 

0.12 

Cylinder  oil. 

Loco,  mileage  per  quart 

Lubricating  oil. 

.  122.5 

Total *' 

23.99 

1.82 

Loco,  mileage,  per  quart 

.     49.0 

Cost  per  100  tons  f  per  mile 

,  3.83  cents. 

Waste,  pounds  per  100  miles ... 

.      1.0 

Cost  of  coal,  per  tonf,  aver- 
age   of    anthracite    and 
bituminous 

$1.25 

*  1.5  cords  wood  taken  as  equivalent  to  1  ton  (2000  Bt>s)  coal. 
t  Tons  of  2000  fts. 


870 


IRON   AND   STEEL. 


REQUIREMENTS  FOR  IRON   AND  STEEIi. 

(See  also  Bridge  Specifications.) 
Dig-est  of  Specifications  adopted,  subject  to  letter  ballot,  at  4th  Annual 
Meeting  of  the  American  Section  of  the  International  Association 

for  Testing:   Materials,  June  29,  1901.    Adopted  by  letter  ballot,  August, 
191)1,  except  wrought  iron,  on  which  action  was  deferred. 

Process  of  Manufactare. 

Wrought  iron ;  puddled,  charcoal  hearth,  or  rolled  from  fagots  or  piles  made 
from  wrought  iron  scrap,  alone  or  with  muck  bar  added. 

Steel  castings.     Open-hearth,  crucible  or  Bessemer  process. 

Steel  forgings.     Open-hearth,  crucible  or  Bessemer  process. 

Steel  Rails.  Bessemer  or  open-hearth.  Ingots  shall  be  kept  vertical  in  pit- 
heating  furnaces.  No  bled  ingots  shall  be  used.  SuflScient  material  shall  be  dis- 
carded from  the  tops  of  the  ingots  to  insure  sound  rails. 

Steel  Splice  Bars.     Bessemer  or  open-hearth. 

Boiler  Plate  and  Rivet  Steel.     Open-hearth. 

Structural  steel  for  bridges  and  ships.    Open-hearth. 

Structural  steel  for  buildings.    Open-hearth  or  Bessemer. 

Test  Pieces. 

For  flat  plates,  the  specimen  shown  in  Fig.  J  shall  be  used. 
For  large  rounds,  test  specimen  as  shown  in  Fig.  K.     The  center  of  the  speci- 
men shall  be  half  way  between  the  center  and  the  outside  of  the  round. 
Whenever  possible,  iron  shall  be  tested  in  full  size,  as  rolled. 
Test  specimens  shall  be  cut  from  bar  as  rolled. 


A.hou€ 


12.70 


Parallel  Section  not  less  thanr 


=  228.G0  mm 


r. 


i  Ss 


mm  to  76.20  mm.  JRatL 


-About  18=457,20  mm^ 


FAece  to  he  of  same  thioktvess  as  the  plate. 
Fig.  J- 


Fiff.  K. 


IRON   AND   STEEL.  871 

Tests. 

Bi'lckin^  tests.  The  specimen  shall  be  slightly  and  evenly  nicked  on  one 
side,  and  bent  back  at  this  point  through  an  angle  of  180°  by  a  succession  of 
light  blows. 

Hot  bending  tests.  Specimens  shall  be  heated  to  a  bright  red,  and  bent 
by  pressure  or  by  a  succession  of  light  blows  and  without  hammering  on  head. 

Cold  bending'  tests.  Specimen  to  be  bent  by  pressure  or  by  a  succession 
of  light  blows. 

Yield  point.  The  yield  point  shall  be  determined  by  careful  observation 
of  the  drop  of  the  beam  or  halt  in  the  gage  of  the  testing  machine. 

Drop  tests.  The  drop  testing  machine  for  rails  shall  have  a  tup  of  2000 
pounds,  the  striking  face  of  which  shall  have  a  radius  of  not  more  than  5  ins ; 
and  the  test  rail,  not  more  than  6  feet  long,  shall  be  placed  head  upwards  on 
solid  supports  3  ft  apart.  The  anvil  block  shall  weigh  at  least  20,000  lbs,  and  the 
supports  shall  be  a  part  of,  or  firmly  secured  to,  the  anvil.  Height  of  drop  from 
15  ft  for  45  ft)  rail  to  19  ft  for  85  ft)  and  over.  One  test  piece  shall,  be  selected 
from  every  fifth  blow. 

Honiog-eneity  tests  for  fire  box  steel.  A  portion  of  the  broken 
tensile  test  specimen  is  either  nicked  or  grooved  ^^  inch  deep,  in  three  places 
about  2  ins  apart  and  on  opposite  sides.  It  is  then  clamped  in  a  vise  and  broken 
off,  by  light  hammer  blows,  bending  away  from  each  groove  in  succession.  The 
specimen  must  not  show  any  single  seam  or  cavity  more  than  %  i^^h  long. 

Notes  to  table,  pp.  872  and  873. 

(a)  To  be  bent  flat. 

(b)  Specimen  to  be  bent  about  a  bar  of  diameter  equal  to  its  own  diameter  or 
thickness. 

(c)  Specimen  to  be  bent  about  a  bar  twice  its  diameter. 

(d)  Elongation,  min,  per  cent,  in  sections  less  than  0.654  ft)S.  per  linear  ft, 
grade  A,  19;  B,  15;  C,  12. 

(e)  Nicking  test.    Max  per  cent  granular  surface,  grade  A,  10;  B,  10;  C,  15. 

(f)  Hot  bending  test.  Bar  to  be  bent  without  cracking  on  outside  of  bend. 
To  be  bent  flat  in  each  grade;  180°  in  grades  A  and  B,  and  sharply  to  90° 
in  C.  Grade  A,  heated  yellow  and  suddenly  quenched  in  water  between  80° 
and  90°  F,  to  bend  flat  180°.  Also,  heated  bright  red,  split  at  end,  and  each  part 
bent  back  180°.  Punched  and  drifted  to  hole  at  least  0.9  diam  of  rod  or  width  of  bar. 

(g)  Phosphorus,  pieces  for  physical  test,  0.05  for  each  grade, 
(h)  Sulphur,  pieces  for  physical  test,  0.05  for  each  grade. 

(i)  Bending.  Specimen  1  inch  X  14  i"ch  to  bend  cold  around  a  diam  of  1  inch 
without  fracture  on  outside  of  bent  portion. 

(j)  Bending.  Specimen  1  inch  X  %  inch  to  bend  cold,  without  fracture  on 
outside  of  bent  portion,  around  a  diameter  of  %  inch. 

(k)  Same,  around  diam  of  1}^  ins. 

(1)  Same,  around  diam  of  13^  ins  if  not  less  than  20  ins  diam ;  around  a  diam 
of  1  inch  if  less  than  20  ins  diam. 

(m)  Same,  about  a  diameter  of  1    inch. 

(n)        "  ♦♦  "         "  %     " 

<o)        "  "  "         "    1      " 

(p)  Deduct  1  per  cent  for  each  '^  inch  in  thickness  above  %  inch,  and  2%  per 
1  5 

cent  for  each  —  inch  below  —  inch, 
lb  lb    • 

(q)  Bending.  Rivet  rounds  to  be  tested  of  full  size,  as  rolled.  Plate  speci- 
mens shall  be  1%  ins  wide.  For  plates  not  over  %  inch  thick,  the  thickness 
shall  be  the  same  as  that  of  the  plate,  and  the  specimen  shall,  where  possible, 
have  the  natural  rolled  surface  on  two  opposite  sides.  For  plates  thicker  than 
%  inch,  the  specimen  may  be  ^  inch  thick.  Shall  be  subjected  to  both  cold  and 
quenched  bending  tests.  For  the  quenched  test,  the  material  is  to  be  heated  to 
a  light  cherry  red  (as  seen  in  the  dark)  and  quenched  in  water  of  temperature 
between  80°  and  90°  Fahr.  Samples  shall  bend  flat  without  fracture  on  the 
outside  of  bent  portion.     Bending  may  be  done  by  pres&ure  or  by  blows. 

(r)'For  pins,  the  elongation  shall  be  5  per  cent  less.  Center  of  test  specimen 
1  inch  from  surface. 

(s)  Eye-bars  shall  be  of  medium  steel.  Full-Sized  tests  shall  show  123^  per 
cent  elongation  in  15  ft  of  body.  Min  tensile  strength,  55,000  fbs,  per  sq  in. 
At  least  %  of  eye-bars  tested  shall  break  in  the  body. 

(t)  Sanie  as  (q)  but  omitting  quenching  test. 

(u)  See  Hoinogeneity  Test,  iu  text,  above. 


872 

See  notes,  p.  871. 


IRON   AND   STEEL. 


Requirements  for 


Metal. 

Allowable  percentage  of 

Carbon. 

Phos- 
phorus. 

Sul- 
phur. 

Mangan- 
ese. 

Nickel. 

1. 

Wrought  Iron. 

GradeA 

Max. 

Max. 

Max. 

Max.  Min. 

Max.  Min. 

Grade  B        

Grade  C 

2. 

Steel  Castings. 

Hard 

0.40 
0.40 
0.40 

0.08  g 
0.08  g 
0.08  g 

0.05  h 
0.05  h 
0.05  h 

Medium 

Soft 

Steel  Forcings. 

Soft  or  low  carbon 

0.10 
0.06 
0.04 

0.04 

0.10 
0.06 
0.04 

0.04 

Carbon,  not  annealed.... 

Carbon,  annealed 

3. 

Carbon,  oil  tempered  ... 

"4.06  3*66* 
4.00  3.00 

Nickel,  annealed 

Nickel,  oil  tempered 

Steel  Rails. 

50  to   59  lbs.  per  yard 

60  to    69   "      "      "    

70  to    79  "      '♦      "    

80  to  -89   "      "      "    

90  to  100  "      "      "    

Max.  Min. 
0.45  0.35 
0.48  0.38 
0.50  0.40 
0.53  0.43 
0.55  0.45 

0.10 
0.10 
0.10 
0.10 
0.10 

0.20 
0.20 
0.20 
0.20 
0.20 

1.00  0.70 
1.00  0.70 
1.05  0.75 
1.10  0.80 
1.10  0.80 

' 

4. 

5. 

Steel  Splice  Bars 

Max. 
0.15 

0.10 

0.60  0.30 

Open     Hearth     Boiler 

Plate  and  Rivet  Steel. 

Flange  or  Boiler  Steel ... 

acid  basic 
0.06  0.04 
0.04  0.03 

0.04  0.04 

0.05 
0.04 

0.04 

0.60  0.30 
0.50  0.30 

0.50  0.30 

6. 

Fire  Box  Steel 

Extra    Soft    Steel    for 
Boiler  Rivets.... 

7 

Structural    Steel    for 

Bridges  and  Ships. 

Rivet  Steel 

acid  basic 
0.08  0.06 
0.08  0.06 
0.08  0.06 

0.06 
0.06 
0.06 

Soft  Steel 

Medium  Steel 

8. 

Structural    Steel    for 
Buildings. 
Rivet  Steel 

Ma;x. 
0.10 
0.10 

Medium  Steel 

IRON   AND   STEEL. 


Iron  and  Steel. 


873 

See  notes,  p.  871. 


Tensile  Tests. 

Cold  Bending 
Tests. 

Strength.  lbs.  per  sq.  in. 

» 

Elastic  Limit 

and 
Yield  Point. 
ft)s.  per  sq.  in. 

Elonga- 
tion. 
Per- 
centage 
in  8  ins. 

Con- 
traction 
of  Area. 

Per- 
centage. 

Angle 
ofBeud. 

How 
Bent. 

II 

Si 

o 

Max. 

Min. 
50,000 
48,000 
48,000 

Yield  Point. 
Min. 
25,000 

Min. 
2.T  d 

Min. 

o 
180 
180 
180 

a 
b 
c 

e  f. 

25,000           1     20  d 

e  f 

25,000 

20  d 

c'f 

85,000 
70,000 
60,000 

38,250 
31,500 
27,000 

15 

18 
22 

20 
25 
30 

i 
i 
i 



90 
120 

Average. 
58,000 
75,000 
75,000 

85,000 
80,000 
90,000 

Average. 

29,000 

37,500 

37,500 

Elastic  Limit. 

Avge. 
28 
18 
23 

21.5 
24.5 
22.5 

Avge. 
35 
30 
32.5 

42.5 
42.5 
47.5 

180 
180 
180 

180 
180 
180 

i 

1 

m 
n 

n 

47,500 
60,000 

See  Drop  Test,  in  text,  above. 

64,000 

Min. 
54,000 

Min. 
32,000 

Min. 
25 

180 

a 

65,000 
62,000 

55,000 

55,000 
52,000 

45,000 

Max.  and  Min. 

1    3^  Tensile  J 
1     Strength.     ] 

25  p 

26  p 

28  p 

180 
180 

180 

a,  q 
a,  q 

a,  q 

u 

60,000 
62,000 
70,000 

50,000 
52,000 
60,000 

\  3^  Tensile    f 
r    Strength.    | 

r  s. 
26  p 
25  p 
22  p 

180 
180 
180 

a,  t 

a,  t 

b,  t 

60,000 
70,000 

50,000 
60,000 

")    >^  Tensile    f 
J     Strength.    \ 

r. 
26  p 
22  p 

180 
180 

a,  t 

b,  t 

874  IRON    AND   STEEL. 


Iron  is  weakened  bj-  extreme  cold. 

The  belief  (originating  with  Styflfof  Sweden)  is  gaining  ground  that  iron  and 
steel  are  not  rendered  more  brittle  by  intense  cold,  but  that  the  great  number  of 
breakages  of  rails,  wheels,  axles,  &c,  in  winter,  is  owing  to  the  more  sever©  blows 
incident  to  the  frozen  and  unyielding  nature  of  the  eartli  at  that  period  of  the  year. 
But  Sandberg's  experiments  show  conclusively  that  although  these  metals  may  per- 
haps bear  as  much  steady  force,  gradually  applied,  in  winter  as  in  summer,  yet  their 
resistance  to  impulse,  or  sudden  force,  is  not  more  than  3^  or  ^-^  as  great  in  severe 
cold ;  which  renders  them  less  flexible  and  less  stretchy.  It  is  probable  that  this 
fact  does  not  receive  as  much  attention  as  it  should,  in  proportioning  iron  bridges,  &c. 

Some  experiments  with  good  wrought  iron  showed  that  teven  at  23°  Fah,  or  only 
9°  colder  than  freezing  point,  there  was  a  loss  of  strength  of  from  2^^  to  4  pei 
cent. 

Malleable  Cast  Iron.  Experiments  by  Mr.  I).  L.  Barnes,  of  Chicago,  on 
a  large  number  of  samples  of  a  single  make  of  "  malleable  "  cast  iron,  gave  in 
most  cases  tensile  strengths  ranging  from  24000  to  32000  lbs.  per  square  inch, 
with  an  average  of  about  28000  lbs.  The  higher  figures  were  obtained  generally 
with  the  smallest  bars  (about  3  X  K  inch)  and  the  lower  with  the  largest  bars 
(about  3  X  1  inch).  Pieces  planed  on  all  four  sides  averaged  only  about  24000  lbs. 
per  square*inch.  This  may  explain  the  difference  in  favor  of  the  smaller  sections, 
m  which  the  original  "shell "  forms  a  larger  portion  of  the  whole  cross  section. 

CAST  IROX. 

Tensile  strength 14,000  to   20,000  lbs  *  per  sq  inch 

Compressive  strength  (average about  100,000)...  90,000  to  130,000  "         '*         " 
Transverse  strength,  bar  1  in  sq,  1  ft  span, 

center  load  2500  Bis.    Deflection,  minimum, 

0.15  inch. 

Elastic  limit..... about  6,000  Bbs  per  sq  inch 

Modulus  of  Elasticity "      15,000,000  "     "  " 

Specillcatious. 

Tensile  strength. 

Bureau  of  Water,  Philadelphia 16,000  to   20,000  lbs  per  sq  inch 

Water  Department,  St.  Louis,  Mo 18,000    "        *'         ** 

Transverse  strength. 
Bureau  of  Water,  Philadelphia. 
1  in  sq,  56  ins  span,  center  load  500  ft)S. 

1  in  sq,  36  ins  span,      "  **    750  ibs.     Deflection,  minimum,  0.4  to  0.6  in. 

Water  Department,  St.  Louis,  Mo. 
3  in  X  >2  in  (laid  flat)  18  ins  span,  center 
load  1000  to  1250  fi)s.    Minimum  deflection  0.3  to  %  inch. 

l¥eig:bt  of  Cast  Iron. 
Assnining*  450  lbs  per  cub  ft,  specific  gravity  7.2,  a  cub  inch  weighs 
0.2604+  lbs  ;  and  a  pound  contains  3.83995+  cub  ins. 

Table,  pag-e  875  :        D  =  thickness  or  diameter,  in  inches. 

Wt.  of  plate,  1  ft  square,  in  lbs  =  37.5     D  (Exact)  Log  W  =  1.574  0313  +    LogD 

"     "  sq  bar,  1  ft  long,  in  lbs  =    3.125  D2  (Exact)  Log  W  =  0.494  8500  +  2  Log  D 

'*    "  rd  bar,  1  ft  long,  in  fts  =    2.45437 D2  Log W  =£.389  9400*+ 2 LogD 

♦'    "ball,  in  lbs  =    0.136354D3  Log  W  =  1.134  6551 +  3  LogD 

Wei^bt  of  a  spherical  shell  =  weight  of  ball  having  outer  diam  of 
shell  minus  weight  of  ball  having  its  inner  diam. 

Weight  of  pattern.  A  casting  weighs  20  X  weight  of  pattern  of  per- 
fectly dry  white  pine.  If  not  perfectly  dry,  although  well  seasoned,  for  20, 
substitute  19  or  18.  . 

For  lead,  at  700  Bbs  per  cub  ft,  multiply  weight  of  cast  iron  by  1.555 ; 

For  copper,  at  550  fcs,  multiply  by  1.222 ; 

For  brass,  at  500  lbs;  multiply  by  1.111 ; 

For  Mrrouglit  iron,  at  485  fes,  multiply  by  1.0777: ; 

For  tin,  at  460  lbs,  multiply  by  1.022 ; 

Zinc,  at  450  lbs  =  cast  iron. 

♦High  grade  irons  may  reach  30,000  to  40,000  fi)s  per  sq  inch,  tensile. 


WEIGHT   OF   CAST   IRON. 


875 


TABL.E  OF  WEIGHT  OF  CAST  IROX. 

At  450  lbs  per  cubic  foot ;  specific  gravity,  7.2. 


D 

=  Thickness  or 

diameter,  in  inebe-s.     Foi 

•  equivalents  in 

feet,  see 

1221. 

I> 

Weights,  in  pou 

ids. 

» 

Weigrhts 

,  in  pounds. 

Plate 
1  ft  sq. 

Square 

bar 
1  ft  long 

Round 

bar 
1  ft  long 

Ball. 

Plate 
Iftsq. 

Square 

bar 
1  ft  long 

Round 

bar 
1  ft  long 

Ball. 

1/32 

1.172 

0.0031 

0.0024 

3M 

117.2 

30.52 

23.97 

4.161 

1/16 

2.344 

0.0122 

0.0096 

H 

121.9 

33.01 

25.92 

4.681 

3/32 

3.516 

0.0275 

0.0216 

0.0001 

37 

126.6 

35.60 

27.96 

5.242 

1/8 

4.688 

0.0488 

0.0383 

0.0003 

u 

131.2 

38.28 

30.07 

5.846 

5/32 

5.859 

0.0763 

0.0599 

0.0U05 

135.9 

41.06 

32.25 

6.495 

3/16 

7.031 

0.1099 

0.0863 

0.0009 

1 

140.6 

43.94 

34.51 

7.191 

7/32 

8.203 

0.1495 

0.1174 

0.0014 

145.3 

46.92 

36.85 

7.934 

1/4 

9.375 

0.1953 

0.1534 

0.0021 

4. 

150.0 

50.00 

39.27 

8.727 

9/32 

10.55 

0.2472 

0.1941 

0.0030 

% 

154.7 

53.17 

41.76 

9.571 

5/16 

11.72 

0.3052 

0.2397 

0.0042 

% 

159.4 

56.45 

44.33 

10.47 

11/32 

12.89 

0.3693 

0.2900 

0.0055 

% 

164.1 

59.81 

46.98 

11.42 

8/8 

14.06 

0.4394 

0.3451 

0.0072 

H 

168.8 

63.28 

49.70 

12.43 

13/32 

15.23 

0.5157 

0.4051 

0.0091 

173.4 

66.84 

52.50 

'   13.49 

7/16 

16.41 

0.5982 

0.4698 

0.0114 

i 

178.1 

70.51 

55.38 

14.61 

15/32 

17.58 

0.6866 

0.5393 

0.0140 

182.8 

74.27 

58.33 

15.80 

1/2 

18.75 

0.7812 

0.6136 

0.0170 

5. 

187.5 

78.12 

61.36 

17.04 

9/16 

21.09 

0.9888 

0.7766 

0.0243 

/a 

192.2 

82.08 

64.47 

18.35 

5/8 

23.44 

1.221 

0.9587 

0.0333 

196.9 

86.13 

67.65 

19.73 

11/16 

25.78 

1.477 

1.160 

0.0443 

'^8 

201.6 

90.28 

70.91 

21.17 

3/4 

28.12 

1.758  » 

1.381 

0.0575 

206.2 

94.53 

74.24 

22.69 

13/16 

30.47 

2.063 

1.620 

0.0731 

210.9 

98.88 

77.66 

24.27 

7/8 

32.81 

2.393 

1.879 

0.0913 

1 

215.6 

103.3 

81.15 

25.92 

15/16 

35.16 

2.747 

2.157 

0.1124 

220.3 

107.9 

84.71 

27.65 

1. 

37.50 

3.125 

2.454 

0.1363 

6. 

225.0 

112.5 

88.36 

29.45 

1/16 

39.84 

3.528 

2.771 

0.1636 

% 

234.4 

122.1 

95.87 

33.29 

1/8 

42.19 

3.955 

3.106 

0.1941 

% 

243.8 

132.0 

103.7 

37.45 

3/16 

44.53 

4.407 

3.461 

0.2283 

% 

253.1 

142.4 

111.8 

41.94 

1/4 

46.88 

4.883 

3.835 

0.2663 

7. 

262.5 

153.1 

120.3 

46.77 

5/16 

49.22 

5.383 

4.228 

0.3083- 

1/ 

271.9 

164.3 

129.0 

51.96 

3/8 

.51.56 

5.908 

4.640 

0.3545 

281.2 

175.8 

138.1 

57.52 

7/16 

53.91 

6.458 

5.072 

0.4050 

290.6 

187.7 

147.4 

63.47 

1/2 

56.25 

7.031 

5.522 

0.4602 

\ 

300.0 

200.0 

157.1 

69.81 

9/16 

58.59 

7.629 

5.992 

0.5202 

309.4 

212.7 

167.0 

76.57 

5/8 

60.94 

8.252 

6.481 

0.5851 

k 

318.8 

225.8 

177.3 

83.74 

11/16 

63.28 

8.899 

6.989 

0.6552 

328.1 

339.3 

187.9 

91.35 

3/4 

65.62 

9.570 

7.517 

0.7308 

9. 

337.5 

253.1 

198.8 

99.40 

13/16 

67.97 

10.27 

8.063 

0.8119 

1/ 

346.9 

267.4 

210.0 

107.9 

7/8 

70.31 

10.99 

8.629 

0.8988 

VA 

356.2 

282.0 

221.5 

116.9 

15/16 

72.66 

11.73 

9.213 

0.9917 

365.6 

297.1 

233.3 

126.4 

2. 

75.00 

12.50 

9.818 

1.091 

10. 

375.0 

312.5 

245.4 

136.3 

1/8 

79.69 

14.11 

11.08 

1.308 

34 

384.4 

•  328.3 

257.9 

146.8 

1/4 

84.38 

15.82 

12.43 

1.553 

1^ 

393.8 

344.5 

270.6 

157.9 

3/8 

89.06 

17.63 

13.84 

1.827 

% 

403.1 

361.1 

283.6 

169.4 

1/2 

93.75 

19.53 

15.34 

2.131    ■ 

11 

412.5 

378.1 

297.0 

181.5 

5/8 

98.44 

21.53 

16.91 

2.466 

^ 

421.9 

395.5 

310.6 

194.1 

3/4 

103.1 

23.63 

18.56 

2.836- 

^ 

431.2 

413.3 

324.6 

207.4 

7/8 

107.8 

25.83 

20.29 

3.240 

% 

440.6 

431.5 

338.9 

221.2 

3. 

112.5 

28.12 

22.09 

3.682 

12 

450.0 

450.0 

353.4 

235.6 

876 


WEIGHT   OF   CAST-IRON   PIPES. 


WEIOHT  OF  CAST-IROHr  PIPES  per  rnnning:  foot. 

Assuming  the  weight  of  cast-iron  at  450  ft)8  per  cub  ft,  or  .2604  ft)  per  cub  inch.  No 
allowance  is  here  made  for  the  spigot  and  faucet-joints  used  in  water-pipes.  As 
these  are  now  commonly  made,  they  add  to  the  weight  of 

each  length  or  section  of  pipe  of  any  size,  about  as  much  as  that  of  8  inches  iji 
length  of  the  plain  pipe  as  given  in  the  table. 

For  lead-pipe  mult  by  1.6 ;  copper,  mult  by  1.2 ;  brass,  add  1-7 »h ; 
welded,  iron,  mult  by  1.0667,  or  add  one  fifteenth  part. 


THICKNESS   OP 

PIPE    IN 

INCHES. 

H 

% 

}4 

% 

% 

% 

1 

IH 

IM 

^% 

1^ 

1%    1 

Wtin 

Wtin 

Wtin 

Wtia 

Wt  in 

Wt  in 

Wt  in 

Wt  in 

Wt  in 

Wt  in 

Wtin 

WtinI 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lba. 

Lbs. 

Lbs. 

Lbs.  i 

3.07 

5.07 

7.38 

9.99 

12.9 

16.2 

19.7 

23.5 

27.7 

82.1 

36.9 

47.4 

3.69 

6.00 

8.61 

11.5 

14.8 

18.3 

22.2 

26.3 

30.8 

35.5 

40.6 

51.7 

4.30 

6.92 

.    9.84 

13.1 

16.6 

20.5 

24.6 

29.1 

33.8 

38.9 

44.3 

56.0 

4.92 

7.84 

11.1 

14.6 

18.5 

22.6 

27.1 

31.8 

36.9 

42.3 

48.0 

60.3 

5.53 

8.76 

12.3 

16.2 

20.3 

24.8 

29.5 

34.6 

40.0 

45.7 

51.7 

64.61 

6.15 

9.69 

13.5 

17.7 

22.2 

26.9 

32.0 

87.4 

43.1 

49.0 

55.4 

68.9 

6.76 

10.6 

14.8 

19.2 

24.0 

29.1 

34.5 

40.1 

46.1 

62.4 

59.1 

73.2 

7.37 

11.5 

16.0 

20.8 

25.9 

31.2 

36.9 

42.9 

49.2 

65.8 

62.7 

77.5 

7.98 

12.5 

17.2 

22.3 

27.7 

33.4 

39.4 

45.7 

52.3 

59.2 

66.4 

81.8 

8.60 

13.4 

18.5 

23.8 

29.5 

35.5 

41.8 

48.4 

55.4 

62.3 

70.1 

86.1 

9.21 

14.3 

19.7 

25.4 

31.4 

37.7 

44.3 

51.2 

58.4 

65.9 

73.8 

90.4 

9.83 

15.2 

20.9 

26.9 

33.2 

39.8 

46.8 

54.0 

61.5 

69.3 

77.5 

94.7 

10.3 

16.1 

22.2 

28.5 

35.1 

42.0 

49.2 

56.7 

64.6 

72.7 

81.2 

99.0 

11.1 

17.1 

23.4 

30.0 

36.9 

44.1 

51.7 

59.5 

67.7 

76.1 

84.9 

103. 

11.7 

18.0 

24.6 

31.5 

38.8 

46.3 

54.1 

62.3 

70.7 

79.5 

88.6 

108. 

i;j.3 

18.9 

25.8 

33.1 

40.6 

48.5 

56.6 

65.0 

73.8 

83.9 

92.3 

112. 

12.9 

19.8 

27.1 

34.6 

42.5 

50.6 

59.1 

67.8 

76.9 

87.2 

96.0 

116. 

13.5 

20.8 

28.3 

36.1 

44.3 

52.8 

61.5 

70.6 

80.0 

90.6 

99.6 

121. 

14.2 

21.7 

29.5 

37.7 

46.1 

54.9 

64.0 

73.3 

83.0 

94.0 

103. 

125. 

14.8 

22.6 

30.8 

39.2 

48.0 

57.1 

66.4 

76.1 

86.1 

97.4 

107. 

129. 

15.4 

23.5 

32.0 

40.8 

49.8 

59.2 

68.9 

78.9 

89.2 

99.8 

111. 

134. 

16.6 

25.4 

34.5 

43.8 

53.5 

63.5 

73.8 

84.4 

95.3 

107. 

118. 

142. 

17.8 

27.2 

36.9 

46.9 

57.2 

67.8 

78.7 

89.4 

102. 

113. 

126. 

151. 

19.1 

29.1 

39.4 

50.0 

60.9 

72.1 

83.7 

95.5 

108. 

120. 

133. 

159. 

20.3 

30.9 

41.8 

53.1 

64.6 

76.4 

88.6 

101. 

114. 

127. 

140. 

168. 

21.5 

32.8 

44.3 

56.1 

68.3 

80.7 

93.5 

107. 

120.« 

134. 

148. 

177. 

22.8 

34.6 

46.8 

59.2 

72.0 

85.1 

98.4 

112. 

126. 

140. 

155. 

185. 

24.0 

36.4 

49.2 

62.3 

75.7 

89.3 

103. 

118. 

132. 

147. 

163. 

194. 

25.1 

38.3 

51.7 

65.3 

79.4 

93.6 

108. 

123. 

138. 

154. 

170. 

202. 

26.4 

40.1 

54.1 

68.4 

83.0 

97.9 

113.2 

129. 

145. 

161. 

177. 

211. 

27.6 

42.e 

56.6 

71.5 

86.7 

102. 

118. 

134. 

151. 

168. 

185. 

220. 

28.8 

43.8 

59.1 

74  6 

90.4 

107. 

123. 

140. 

157. 

174. 

192. 

228. 

30.0 

45.7 

61.5 

77.7 

94.1 

111. 

128. 

145. 

163. 

181. 

199. 

237. 

32.5 

49.4 

66.4 

83.8 

102. 

120. 

138. 

156. 

175. 

195. 

214. 

254. 

35.0 

53.1 

71.4 

89.4 

109. 

128. 

148. 

168. 

188. 

208. 

229. 

271. 

37.4 

56.7 

76.3 

96.1 

116. 

137. 

158. 

179. 

200. 

222. 

244. 

289. 

39.1 

60.4 

81.2 

102. 

124. 

145. 

167. 

190. 

212. 

235. 

258. 

306. 

42.3 

64.1 

8«.l 

108. 

131. 

154. 

177. 

201. 

225. 

249. 

273. 

323. 

44.8 

67.8 

91.0 

115. 

139. 

163. 

187. 

212. 

237. 

262. 

288. 

340. 

47.3 

71.5 

96.0 

121. 

146. 

171. 

197. 

223. 

249. 

276. 

303. 

357. 

49.7 

75.2 

101. 

127. 

153. 

180. 

207. 

234. 

261. 

289. 

317. 

375. 

52.2 

78.9 

106. 

133. 

161. 

188. 

217. 

245. 

274. 

303. 

332. 

392. 

54.6 

82.6 

111. 

139. 

168. 

196. 

227. 

256. 

286. 

316. 

847. 

409. 

57.1 

86.3 

116.  , 

145. 

175. 

206. 

236. 

267. 

298. 

330. 

362. 

426. 

59.6 

89.9 

121. 

152. 

183. 

214. 

246. 

278. 

311. 

343. 

375. 

444. 

62.0 

93.6 

126. 

158. 

190. 

223. 

256. 

289. 

323. 

357. 

391. 

461. 

64.5 

97.3 

131. 

164. 

198. 

231. 

266. 

300. 

3.35. 

370. 

406. 

478. 

66.9 

101. 

135. 

170. 

205. 

240. 

276. 

311. 

348. 

384. 

421. 

495. 

69.4 

105. 

140. 

176. 

212. 

249. 

286. 

323. 

360. 

397. 

436. 

512. 

71.8 

109. 

145. 

182. 

220. 

257. 

295. 

334. 

372. 

411. 

450. 

530. 

74.2 

112. 

150. 

188. 

227. 

256. 

305. 

345. 

384. 

424. 

465. 

547. 

76.7 

116. 

155. 

•195. 

234. 

275. 

315. 

356. 

397. 

438. 

4ao. 

564. 

79.1 

120. 

160. 

201. 

242. 

283. 

325. 

367. 

409. 

451. 

495. 

581. 

81.6 

123. 

165. 

207. 

249. 

292. 

335. 

378. 

421. 

465. 

509. 

598. 

84.1 

127. 

170. 

213. 

257. 

300. 

345. 

389. 

454. 

479. 

524. 

616. 

86.5 

131. 

175. 

219. 

264. 

309. 

354. 

400. 

446. 

492. 

539. 

633. 

89.0 

134. 

180. 

225. 

271. 

318.. 

364. 

411. 

458. 

506. 

554. 

650. 

104. 

156. 

210. 

262. 

315. 

370. 

423. 

478. 

532. 

588. 

644. 

753. 

119. 

178. 

239. 

298. 

369. 

422. 

482. 

544. 

605. 

669. 

733. 

856. 

For  warming  buildings  by  .steam  it  usually  suffices  to  allow  1  sq  ft 
of  cast  or  wrought  pipe  surface  for  each  120  cub  ft  of  space  to  be  warmed  ;  aud  1  cub 
ft  of  boiler  for  each  2000  cub  ft  of  such  space. 


WEIGHT   OF   WROLGHT   IRON   AND   STEEL. 


877 


Table  of  Weight  of  WROUOHT  IROX  and  STEEI.. 
At  485  ft)s  per  cubic  foot ;  specific  gravity,  7.76.    See  page  879. 


D  = 

=  Thickness  or  diameter 

in  inches.    For  equivalents  in 

feet,  see  p  221. 

Weigrhte 

,  in  pounds. 

I> 

Weigrbts 

,  in  pounds. 

I> 

Plate 
1  ft  sq. 

Square 

Round 
bar  . 

Ball. 

Plate 
Iftsq. 

Square 

Round 
bar 

Ball. 

1  ft  long 

1  ft  long 

1  ft  long 

1  ft  long 

1/32 

1.263 

0.0033 

0.0026 

3% 

126.3 

32.89 

25.83 

4.485 

1/16 

2.526 

0.0132 

0.0103 

131.4 

35.57 

27.94 

5.045 

3/32 

3.789 

0.0296 

0.0232 

0.0001 

3^ 

136.4 

38.36 

30.13 

5.650 

1/8 

5.052 

0.0526 

0.0413 

0.0003 

i 

141.5 

41.26 

32.40 

6.301 

6/32 

6.315 

0.0822 

0.0646 

0.0006 

146.5 

44.26 

34.76 

7.000 

3/16 

7.578 

0.1184 

0.0930 

0.0010 

% 

151.6 

47.36 

37.20 

7.750 

7/32 

8.841 

0.1612 

0.1266 

0.0015 

% 

156.6 

50.57 

39.72 

8.551 

1/4 

10:10 

0.2105 

0.1653 

0.0023 

4. 

161.7 

53.89 

42.32 

9.406 

9/32 

11.37 

0.2664 

0.2092 

0.0033 

y 

166.7 

57.31 

45.01 

10.32 

5/16 

12.63 

0.3289 

0.2583 

0.0045 

1 

171.8 

60.84 

47.78 

11.28 

11/32 

13.89 

0.3980 

0.3126 

0.0060 

176.8 

64.47 

50.63 

12.31 

3/8 

15.16 

0.4736 

0.3720 

0.0077 

181.9 

68.20 

53.57 

13.39 

13/32 

16.42 

0.5558 

0.4366 

0.0099 

tl 

186.9 

72.04 

56.58 

14.54 

7/16 

17.68 

0.6447 

0.5063 

0.0123 

% 

192.0 

75.99 

59.68 

15.75 

15/32 

18.95 

0.7400 

0.5812 

0.0151 

% 

197.0 

80.04 

62.87 

17.03 

1/2 

20  21 

0.8420 

0.6613 

0.0184 

5. 

202.1 

84.20 

66.13 

18.37 

9/16 

22.73 

1.066 

0.8370 

0.0261 

yk 

207.1 

88.46 

69.48 

19.78 

5/8 

25.26 

1.316 

1.033 

0.0359 

34 

212.2 

92.83 

72.91 

21.27 

11/16 

27.79 

1.592 

1.250 

0.0478 

217.2 

97.31 

76.42 

22.82 

3/4 

30.31 

1.895 

1.488 

0.0620 

222.3 

101.9 

80.02 

24.45 

13/J6 

32.84 

2.223 

1.746 

0.0788 

227.3 

106.6 

83.70 

26.16 

7/8 

35.36 

2.579 

2.025 

0.0985 

1 

232.4 

111.4 

87.46 

27.94 

15/16 

37.89 

2.960 

2.325 

0.1211 

237.5 

116.3 

91.30 

29.80 

1. 

40.42 

3.368 

2.645 

0.1470 

6. 

242.'5 

121.3 

95.23 

31.74 

1/16 

42.94 

3.802 

2.986 

0.1763 

Va 

252.6 

131.6 

103.3 

35.88 

1/8 

45.47 

4.263 

3.348 

0.2092 

n 

262.7 

142.3 

111.8 

40.36 

3/16 

47.99 

4.750 

3.730 

0.2461 

272.8 

153.5 

120.5 

45.20 

1/4 

50.52 

5.263 

4.133 

0.2870 

7. 

282.9 

165.0 

129.6 

50.41 

5/16 

53.05 

5.802 

4.557 

0.3323 

293.0 

177.0 

139.0 

56.00 

3/8 

55.57 

6.368 

5.001 

0.3820 

303.1 

189.5 

148.8 

62.00 

7/16 

58.10 

6.960 

5.466 

0.4365 

313.2 

202.3 

158.9 

68.41 

1/2 

60.63 

7.578 

5.952 

0.4960 

8. 

323.3 

215.6 

169.3 

75.24 

9/16 

63.15 

8.223 

6.458 

0.5606 

^ 

333.4 

229.2 

180.0 

82.52 

5/8 

65.68 

8.894 

6.985 

0.6306 

H 

343.5 

243.3 

191.1 

90,25 

11/16 

68.20 

9.591 

7.533 

0.7062 

353.6 

257.9 

202.5 

98.45 

3/4 

70.73 

10.31 

8.101 

0.7876 

9. 

363.7 

272.8 

214.3 

107.1 

13/16 

73.26 

11.06 

8.690 

0.8750 

^ 

373.9 

288.2 

226.3 

116.3 

7/8 

75.78 

11.84 

9.300 

0.9687 

§ 

384.0 

304.0 

238.7 

126.0 

15/16 

78.31 

12.64 

9.930 

1.069 

% 

394.1 

320.2 

251.5 

136.2 

2. 

80.83 

13.47 

10.58 

1.176 

10 

404.2 

336.8 

264.5 

147.0 

1/8 

85.89 

15.21 

11.95 

1.410 

X} 

414.3 

353.9 

277.9 

158.3 

1/4 

90.94 

17.05 

13..39 

1.674 

424.4 

371.3 

291.6 

170.1 

3/8 

95.99 

19.00 

14.92 

1.969 

% 

434.5 

389.2 

305.7 

182.6 

1/2 

101.0 

21.05 

16.53 

2.296 

11 

444.6 

407.5 

320.1 

195.6 

5/8 

106.1 

23.21 

18.23 

2.658 

n 

454.7 

426.3 

334.8 

209.2 

3/4 

111.1 

25.47 

20.00 

3.056 

464.8 

44,5.4 

349.8 

223.5 

7/8 

116.2 

27.84 

21.86 

3.492 

^ 

474.9 

465.0 

365.2 

238.4 

3. 

121.3 

30.31 

23.81 

3.968 

12.* 

485.0 

485.0 

380.9 

253.9 

878 


WEIGHT    OF    FLAT    IRON. 


Weigbt  of  1  ft  in  len^tb  of  Fli  AT  ROIili^D  IRO:Br,  at  480 1)>s  per 
cubic  foot.  For  cast  iron,  deduct  jV  part ;  for  steel,  add  -J^ ;  for  copper,  add 
j;  for  cast  brass,  add  yy ;  for  lead,  add  j^;  for  zinc,  deduct  jij. 


^5 

THICKNESS  IN  INCHES. 

^.2 

1-16 

% 

3-16 

>i 

I    5-16 

% 

7-16 

H 

% 

«      1       « 

1 

1. 

.2083 

.4166 

.6250 

.8333 

1.042 

1.250 

1.458 

1.666 

2.083 

2.500 

2.916 

3.338 

H 

.2344 

.4688 

.70.33 

.9375 

1.172 

1.406 

1.640 

1.875 

2.344 

2.812 

3.280 

3.75 

.2605 

.5210 

.7810 

1.042 

1.303 

1.563 

1.823 

2.083 

2.605 

3.125 

3.646 

4.16« 

% 

.2865 

.5730 

.8595 

1.146 

1.432 

1.719 

2.006 

2.292 

2.864 

3.438 

4.012 

4.583 

^i 

.3125 

.6250 

.9375 

1.250 

1.562 

1.875 

2.188 

2.500 

3.125 

3.750 

4.375 

5.00(i 

•    H 

.3385 

.6771 

1.015 

1.354 

1.692 

2.031 

2.370 

2.708 

3.384 

4.062 

4.740 

5.416 

H 

.3646 

.7292 

1.094 

1.458 

1.823 

2.188 

2.550 

2.916 

3.646 

4.375 

5.105 

5.833 

% 

..3906 

.  .7812 

1.172 

1.562 

1.953 

2.344 

2.735 

3.125 

3.906 

4.688 

5.470 

6.25 

I. 

.4166 

.8333 

1.25 

1.666 

2.083 

2.500 

2.916 

3.333 

4.166 

5.000 

5.833 

6.664 

H 

.4427 

.8855 

1.328 

1.771 

2.214 

2.656 

3.098 

3.542 

4.428 

5.312 

6.196 

7.083 

.4688 

.9375 

1.406 

1.875 

2.344 

2.812 

3.281 

3.750 

4.688 

5.624 

6.562 

7.50a 

% 

.4948 

.9895 

1.484 

1.979 

2.474 

2.968 

3.463 

3.958 

4.948 

5.936 

6.926 

7.914 

yi 

'  .5210 

1.042 

1.562 

2.083 

2.605 

3.125 

3.646 

4.166 

5.210 

6.250 

7.291 

8.333 

% 

.5470 

1.094 

1.641 

2.187 

2.735 

3.282 

3.829 

4.375 

5,470 

6.564 

7.658 

8.750 

H 

.5730" 

1.146 

1.719 

2.292 

2.865 

3.438 

4.011 

4.583 

6.730 

6.876 

8.022 

9.169 

Xi 

.5990 

1.198 

1.797 

2.396 

2.995 

3.594 

4.193 

4.792 

5.990 

7.188 

8.386 

9.583 

8. 

.625 

1.250 

1.875 

2.500 

3.125 

3.750 

4.375 

5.000 

6.250 

7.500 

8.750 

10.00 

% 

.6515 

1.303 

1.954 

2.605 

3.257 

3.908 

4.560 

5.210 

6.514 

7.816 

9.120 

10.43 

.6770 

1.354 

2.031 

2.708 

3.385 

4.062 

4.739 

5.416 

6.770 

8.124 

9.478 

10.83 

% 

.7031 

1.406 

2.109 

2.812 

3.516 

4.218 

4.921 

5.625 

7.032 

8.436 

9.842 

11.25 

yi 

.7291 

1.458 

2.188 

2.916 

3.646 

4.375 

5.105 

5.833 

7.291 

8.750 

10.21 

11.6« 

H 

.75.55 

1.511 

2.266 

3.021 

3.777 

4.533 

5.288 

6.042 

7.554 

9.066 

10.58 

12.08 

H 

.7812 

1.562 

2.343 

.3.125 

3.906 

4.686 

5.468 

6.25 

7.8ia 

9.372 

10.94 

12.50 

% 

.8070 

1.614 

2.421 

3.229 

4.035 

4.842 

5.65 

6.458 

8.070 

9.684 

11.30 

12.92 

4. 

.8333 

1.666 

2.500 

3.333 

4.166 

5.000 

5.833 

6.666 

8.333 

10.00 

11.66 

13.33 

H 

.8595 

1.719 

2.578 

3.438 

4.297 

5.156 

6.016 

6.875 

8.594 

10.31 

12.03 

13.75 

y* 

.8855 

1.771 

2.656 

3.542 

4.427 

5.312 

6.198 

7.083 

8.854 

10.62 

12.40 

14.16 

% 

.9115 

1.823 

2.734 

3.646 

4.557 

5.468 

6..380 

7.291 

9.114 

10.94 

12.76 

14.58 

H 

.9375 

1.875 

2.812 

3.750 

4.687 

5.624 

6.562 

7.500 

9.374 

11.25 

13.12 

15.90 

% 

.9636 

1.927 

2.891 

3.854 

4.818 

5.782 

6.745 

7.708 

9.636 

11.56 

13.49 

15.42 

H 

.9895 

1.979 

2.968 

3.958 

4.947 

5.936 

6.926 

7.917 

9.894 

11.87 

13.85 

15.83 

K 

1.016 

2.031 

3.048 

4.062 

5.080 

6.096 

7.112 

8.125 

10.16 

12.19 

14.22 

16.25 

6. 

1.042 

2.083 

3.125 

4.166 

5.210 

6.25 

7.291 

8.333 

10.42 

12.50 

14.58 

16.6« 

H 

1.068 

2.136 

3.204 

4.271 

5.340 

6.408 

7.476 

8.542 

10.68 

12.81 

14.95 

17.08 

1.094 

2.188 

3.282 

4.375 

5.470 

6.564 

7.658 

8.750 

10.94 

13.13 

15.31 

17.5a 

% 

1.120 

2.240 

3.360 

4.479 

5.600 

6.720 

7.840 

8.958 

11.20 

13.44 

15.68 

17.92 

^ 

1.146 

2.292 

3.438 

4.584 

5.730 

6.876 

8.022 

9.167 

11.46 

13.75 

16.04 

18.33 

% 

1.172 

2.344 

3.516 

4.687 

5.860 

7.032 

8.204 

9.375 

11.72 

14.06 

16.40 

18.75 

H 

1.198 

2.396 

3.594 

4.791 « 

5.990 

7.188 

8.386 

9.583 

11.98 

14.37 

16.77 

19.16 

H 

1.224 

2.448 

3.672 

4.896 

6.120 

7.344 

8.568 

9.792 

12.24 

14.69 

17.13 

19.58 

6. 

1.250 

2.500 

3.750 

5.000 

6.250 

7.500 

8.750 

10.00 

12.50 

15.00 

17.50 

20.00 

H 

1.276 

2.552 

3.828 

5.104 

6.380 

7.656 

8.932 

10.21 

12.76 

15.31 

17.86 

20.42 

H 

1.302 

2.604 

3.906 

5.208 

6.510 

7.812 

9.114 

10.42 

13.02 

15.62 

18.23 

20.83 

% 

1.328 

2.657 

3.984 

5.313 

6.640 

7.968 

9.297 

10.63 

13.28 

15.93 

18.59 

21.25 

}4 

1.354 

2.708 

4.063 

5.417 

6.770 

8.126 

9.480 

10.83 

13.54 

16.25 

18.96 

21.66 

% 

1.381 

2.761 

4.143 

5.521 

6.906 

8.236 

9.668 

11.04 

13.81 

16.57 

19.33 

22.08 

H 

1.406 

2.813 

4.218 

5.625 

7.030 

8.436 

9.843 

11.25 

14.06 

16.87 

19.69 

22.50 

% 

1.432 

2.864 

4.296 

5.729 

7.160 

8.592 

10.02 

11.46 

14.32 

17.18 

20.04 

22.92 

7. 

1.458 

2.916 

4.375 

5.833 

7.291 

8.750 

10.20 

11.66 

14.58 

17.50 

20.42 

23.33 

% 

1.484 

2.969 

4.452 

5.938 

7.420 

8.904 

10.39 

11.87 

14.84 

17.81 

20.78 

23.75 

M 

1.511 

3.021 

4.533 

6.042 

7.555 

9.066 

10.58 

12.08 

15.11 

18.13 

21.16 

24.16 

% 

1.536 

3.073 

4.608 

6.146 

7.680 

9.216 

10.75 

12.29 

15.36 

18.43 

21.50 

24.58 

H 

1.562 

3.125 

4.686 

6.250 

7.810 

9.372 

10.93 

12.50 

15.62 

18.74 

21.86 

25.00 

% 

1.588 

3.177 

4.764 

6.354 

7.940 

9.528 

11.12 

12.71 

15.88 

19.05 

22.24 

25.42 

% 

1.615 

3.229 

4.845 

6.458 

8.075 

9.690 

11.31 

12.92 

16.15 

19.38 

22.62 

25.83 

X 

1.841 

3.281 

4.923 

6.562 

8.205 

9.846 

11.48 

13.13 

16.41 

19.69 

22.96 

26.25 

8. 

1.666 

3.333 

5.000 

6.666 

8.333 

10.00 

11.66 

13.33 

16.66 

30.00 

23.33 

26.66 

» 

1.693 

3.386 

5.079 

6.771 

8.455 

10.15 

11.85 

13.54 

16.91 

20.30 

23.70 

27.08 

M 

1.719 

3.438 

5.157 

6.875 

8.595 

10.31 

12.03 

13.75 

17.19 

20.61 

24.06 

27.50 

% 

1.745 

3.489 

5.235 

6.979 

8.725 

10.47 

12.21 

13.96 

17.45 

20.94 

24.42 

27.92 

yi 

1.771 

3.542 

5.313 

7.083 

8.855 

10.63 

12.40 

14.17 

17.71 

21.26 

24.80 

28.33 

% 

1.797 

3.594 

5.391 

7.188 

8.985 

10.78 

12.58 

14.37 

17.97 

21.56 

25.16 

28.75 

H 

1.823 

3.646 

5.469 

7.292 

9.115 

10.94 

12.76 

14.58 

18.23 

21.88 

25.52 

29.17 

% 

1.849 

3.698 

5.547 

7.396 

9.245 

11.09 

12.94 

14.79 

18.49 

22.18 

25.88 

29.58 

9. 

1.875 

3.750 

5.625 

7.500 

9.375 

11.25 

13.12 

15.00 

18.75 

22.50 

26.24 

30.00 

H 

1.901 

3.802 

5.703 

7.604 

9.505 

11.41 

13.31 

15.21 

19.00 

22.81 

26.62 

30.42 

1.927 

3.854 

5.781 

7.708 

9.635 

11.56 

13.49 

15.42 

19.27 

23.12 

26.98 

30.83 

1.953 

3.906 

5.859 

7.812 

9.765 

11.72 

13.67 

15.62 

19.53 

23.44 

27.34 

31.25 

34 

1.979 

3.958 

5.937  ' 

7.916 

9.895 

11.87 

13.85 

15.84 

19.79 

23.74 

27.70 

31.67 

H 

2.005 

4.010 

6.015 

8.021 

10.02 

12.03 

14.04 

16.04 

20.04 

24.06 

28.08 

32.08 

H 

2.031 

4.062 

6.093 

8.125 

10.16 

12.18 

14.21 

16.25 

20.32 

24.36 

28.42 

32.50 

% 

2.057 

4.114 

6.171 

8.229 

10.29 

12.34 

14.40 

16.46 

20.58 

24.68 

28.80 

32.92 

10. 

2.083 

4.166 

6.250 

8.333 

10.41 

12.50 

14.58 

16.66 

20.82 

25.00 

29.16 

33.33 

% 

2.109 

4.219 

6.327 

8.438 

10.55 

12.65 

14.76 

16.87 

21.10 

25.30 

29.52 

33.75 

% 

2.135 

4.270 

6.405 

8.541 

10.67 

12.81 

14.94 

17.08 

21.34 

25.62 

29.88 

34.17 

mON    AND    STEEL. 


879 


Weig^bt  of  1  ft  in  lengrth  of  FI.AT  ]ROIiI.£D  IRON,  at  480  Ibf 
per  cubic  foot — (Continued.) 


11 

^.2 

1-16 

H, 

3-16 

T 

HICK 

5-16 

N£SS 

% 

IN  INCHES 

7-16          ^ 

% 

y* 

% 

1 

10^ 

2.162 

4.323 

6.486 

8.646 

10.81 

12.97 

15.13 

17.29 

21.62 

25.94 

30.26 

34.5S 

34 

2.188 

4.375 

6.564 

8.750 

10.94 

13.13 

15.31 

17.50 

21.88 

26.26 

30.62 

35.00 

2.2U 

4.427 

6.642 

8.854 

11.07 

13.28 

15.50 

17.71 

22.14 

26.56 

31.00 

35.42 

% 

8.239 

4.479 

6.717 

8.958 

11.20 

13.43 

15.67 

17.92 

22.40 

26.86 

31.34 

35.8S 

H 

2.266 

4.531 

6.798 

9.062 

11.33 

13.59 

15.86 

18.12 

22.66 

27,18 

31.72 

36.25 

u. 

2.291 

4.583 

6.873 

9.166 

11.46 

13.75 

16.04 

18.33 

22.90 

27.50 

32.08 

36.6^ 

H 

2.318 

4.636 

6.954 

9.271 

11.5^9 

13.91 

16.22 

18.54 

23.18 

27.82 

32.44 

37.08 

H 

2.3W 

4.688 

7.032 

9.375 

11.72 

14.06 

16.40 

18.75 

23.44 

28.12 

32.80 

37.50 

2.370 

4.740 

7.110 

9.479 

11.85 

14.22 

16.59 

18.96 

23.70 

28.44 

33.18 

37.92 

LZ 

2.395 

4.791 

7.185 

9.582 

11.97 

14.37 

16.76 

19.16 

23.94 

28.74 

33.52 

38.33 

64 

2.422 

4.844 

7.266 

9.688 

12.11 

14.53 

16.95 

19.37 

24.22 

29.06 

33.90 

38.7& 

H 

2.448 

4.896 

7.344 

9.792 

12.24 

14.68 

17.13 

19.58 

24.48 

29.36 

34.26 

39.16 

Vt 

2.474 

4.948 

7.422 

9.896 

12.37 

14.84 

17.32 

19.79 

24.74 

29.68 

34.64 

39.58 

12. 

2.500 

5.000 

7.500 

10.00 

12.50 

15.00 

17.50 

20.00 

25.00 

30.00 

36.00 

4o.oe 

IVeig-lit  of  Wrought  Iron  and  ISteel. 

Assuming*  485  lbs.  per  cub  ft,*  specific  gravity,  7.76;  a  cubic  incti 
weighs  0.28067  fi>s ;  and  a  pound  contains  3.5629  cubic  inches. 


Table,  page  875 

Wt.  of  plate,  1  ft  square,  in  lbs,  =  40.4167  D  ; 
'*     "    sq  bar,  1  ft  long,  in  lbs,  =    3.36K06  D2  ; 
"     *'    rd  bar,  1  ft  long,  in  lbs,  =    2.64527  D2  ; 
"     •♦    ball,  infts,  =    0.146959  D^ 


I>  =  thickness  or  diameter,  in  inches. 

Log  W  =  1.606  5605  +  Log  D 
Log  W  =  0.527  3792  -f-  2  Log  D 
Log  W  =  0^422  4693  +  2  Log  D 
Log  W  =  ri67  1966  +  3  Log  D 


Weight  of  a      _   ("weight  of  ball  having)   _   (weight  of   ball  having 
spherical  shell        (  outer  diameter  of  shell  J         \  inner  diameter  of  shell. 


Weights  of  equal  masses. 
For  lead,        at  700  lbs  per  cub  ft;  weight  =  1.44 


For  copper. 
For  brass. 
For  tin. 
For  zinc  or   , 
cast  iron. 


550 
500 
460 


=  1.44    X  1 

=  1.13    X 

weight 

=  1.03    X 

of 

=  0.948  X 

wrought 

-  0.928  X 

iron 

*  Very  pure  soft  wrought  iron  weighs  from  488  to  492  ibs  per  cubic  foot ;  average 
rolled  iron  about  480.  At  480  lbs,  a  bar  1  inch  square  weighs  exactly  10  Bbs  per 
yard  =  Z%  lbs  per  foot. 


880 


SHEET-IRON. 


Weig^lits  per  square  foot  of  galvanized  slieet  iron.     Standard  liat 
adopted  by  the  American  Galvanized  Iron  Ass'n,  at  Pittsburgh,  April,  1884. 


Ko. 

Ounces 

Sqft 

No. 

Ounces 

Sqft 

No. 

Ounces 

Sqft 

avoir 

per 

avoir 

per 

avoir 

per 

per  sq  ft 

2240  2)8. 

per  sq  ft. 

2240  fi>s. 

per  sq  ft. 

2210  Ibi. 

29 

12 

2987 

24 

17 

2108 

19 

33 

1086 

28 

13 

2757 

23 

19 

1886 

18 

38 

943 

27 

14 

2560 

22 

21 

1706 

17 

43 

833 

26 

15 

2389 

21 

24 

1493 

16 

48 

746      _ 

25 

16 

2240 

20 

28 

1280 

14 

60 

597      I 

i 


Tlie  g-alvanizingr  is  simply  a  tbin  film  of  zinc  on  both  sides  of  th< 

sheet,  as  in  what  is  known  as  "  tinned  plates,"  or  "  tin  ;  "  which  are  in  reality  sheet  iron  similarly 
«oated  with  tin.  Zinc,  like  tin,  resists  corrosion  from  ordinary  atmospheric  influences,  much  bettei 
than  iron  ;  and  hence  the  use  of  these  metals  as  a  protection  to  the  iron.  ,  A  well  galvanized  roof,j_ 
of  a  good  pitch,  will  sufiFer  but  little  from  5  to  6  years'  exposure  without  being  painted.  It  will  thea! 
take  paint  readily,  and  should  be  painted.    It  is  better,  however,  always  to  paint  tin  ones  at  once. 

Paint  does  not  adhere  well  to  neiv  zinc,  and  this  is  the  principal 

reason  why  new  galvanized  roofs  are  not  painted ;  but  this  may  be  remedied  by  first  brushing  the 
zinc  over  with  the  following:  One  part  of  chloride  of  copper,  1  part  nitrate  of  copper,  1  part  of  sal- 
ammoniac.  Dissolve  in  64  parts  of  water.  Then  add  1  part  of  commercial  hydrochloric  acid.  When 
brushed  jcith  this  solution,  the  zinc  turns  black  ;  dries  within  12  to  24  hours,  and  may  then  be  painted. 
Paint  of  some  mineral  oxide  of  a  brown  color  is  generally  used ;  one  coat  being  applied  to  both 
sides  in  the  shop  ;  and  the  other  after  being  put  on  the  roof.  Repainting  every  3  or  4  y«ars  will  sufQce 
afterward.  Ungalvanized  iron  (called  black  iron,  for  distinction)  is  also  very  enduring  for  roofs,  if 
well  painted  every  1  or  2  years.  The  chief  advantage  of  galvanized  roofing  is  that  it  does  not  require 
painting  so  often  as  the  black.  The  galvanizing  adds  about  ^  of  a  Bt»  per  square  foot  of  surface,  or 
about  Ji  lb  per  sq  ft  of  sheet  as  coated  on  both  sides;  without  regard  to  the  thickness  of  the  sheet. 
Paint  for  roofs  should  not  have  much  dryer.    See  Painting. 

The  sulphurous  fumes  from   coal  are  very  corrosive  of 

EITHER  GALVANIZED  OR  BLACK  IRON  ;  as  may  bc  secn  in  shops,  railroad  bridges,  or  engine  bouses, 
roofed  with  either ;  if  efficient  means  are  not  provided  for  carrying  ofiF  the  smoke  ;  and  the  same  with 
other  metals.     The  acid  of  oak  timber  is  said  to  destroy  the  zinc  of  galvanized  iron. 

Flat  iron  is  usually  nailed  upon  a  sheeting  of  boards ;  but  the  strength  of  corrugated  iron 
obviates  the  necessity  for  this,  and  enables  it  to  stretch  5  or  6  ft  from  purlin  to  purlin,  without  inter* 
mediate  support.  The  corrugated  sheets  are  riveted  together  on  the  roof,  by  rivets  of  galvanized 
wire  about  one-eighth  inch  thick,  300  to  a  pound,  well  driven  (so  as  to  exclude  rain)  3  or  4  inch«B 
apart,  all  around  the  edges.  The  rivet-holes  are  first  punched  by  machinery,  so  as  to  insure  coinei- 
dence  in  the  several  sheets ;  and  the  rivets  are  driven  by  two  men,  one  above,  and  one  beneath  the 
roof.  For  black  iron,  ungalvanized  nails,  boiled  in  linseed  oil  as  a  partial  preservative  from  rust,  are 
commonly  used;  as  also  in  shingling  or  slating.  Galvanized  ones,  however,  would  be  better  in  all 
these  cases ;  or  even  copper  ones  for  slating  because  geod  slate  endures  much  longer  than  either 
shingles  or  iron,  and  therefore  it  becomes  true  economy  to  use  durable  metals  for  fastening  it.  In 
none  of  these  cases,  however,  are  the  nails  fully  exposed  to  the  weather. 

The  sheets  of  flat  iron  are  put  tog^ether  toy  overlapping  and 

folding  the  edges,  much  the  same  as  shown  by  the  fig  page  916,  head  Tin  ;  the  joints  which  run 
up  and  down  the  roof  being  the  same  as  at  a  a,  and  the  horizontal  ones  as  at  t  t ; 
except  that  inasmuch  as  these  are  not  soldered  in  the  iron  sheets,  the  joint  is  made 
about  ?i  to  1  inch  wide,  instead  of  }i  inch,  the  better  to  provide  against  leaking. 
Cleats  are  used  asin  tin,  with  2  nails  to  a  cleat.  The  iron  plates  are  best  laid  on 
sheeting  boards  ;  but  in  sheds,  &c,  are  sometimes  laid  directly  on  rafters,  not  more 
than  about  18  ins  apart  in  the  clear ;  the  plates  being  allowed  to  sag  a  little  between 

the  rafters,  so  as  to  form  shallow  gutters.     In  such  cases  it  is  well  to  bevel  off  the  tops  of  the  rafters 

■lightly,  as  in  this  fig. 

A  serious  otojection  to  iron  as  a  roof  covering,  is  its  rapid  con- 
densation of  atmospheric  moisture;  which  falls  from  the  iron  in  drops  like  rain,  and  may  do  injury 
to  ceilings,  floors,  or  articles  in  the  apartments  immediately  beneath  the  roof.  Painting  does  not 
appreciably  diminish  this ;  it  may,  however,  be  obviated  by  plastering. 

Corrugated  sheet  iron.  The  size  of  sheets  generally  used  for  corrugating, 
is  30  inches  wide  by  96  inches  long.  Corrugation  reduces  the  width  to  27>^  inches.  When  the  cor- 
rugated sheets  are  laid  upon  the  roof,  the  overlapping  of  about  2^  inches  along  the  sides,  and  of  4 
inches  along  their  ends,  diminishes  the  area  of  roof  covered  by  a  sheet,  to  about  seven-eighths  of  that 
of  the  entire  corrugated  sheet  itself;  or,  the  weight  per  square  foot  of  roof  covered,  will  be  about 
one-seventh  greater  than  that  per  square  foot  of  the  corrugated  sheet ;  or,  the  weight  of  corrugated 
iron  per  square  foot  of  roof  covered  is  about  one-fifth  greater  than  that  of  the  flat  sheets  from  which 
it  is  made. 

About  6  inches  are  usually  allowed  for  the  extension  over  the  eaves. 

The  weights  per  square  foot  corresponding  to  the  different  numbers  of  the  Birmingham  wire  gauge, 
vary  somewhat  with  the  different  makers.  The  two  styles  of  corrugation  given  in  the  table  below, 
bxiyi  and  2}4  x  Y^,  are  those  most  frequently  used. 


CORRUGATED  SHEET   IRON. 


881 


No. 
Bmghm 
wire  ga. 

Thick- 
ness 

in  ins. 

Wt  in  !bs  per 
sq  ft  of  sheets. 

Wt  in  S)s  per 

sq  ft  of  roof. 

Black 
20 
22 

24 

26 

Black 
.035 
.028 
.022 
.018 

Black 
1.84 
1.50 
1.20 
1.00 

Lead  cotid 
or  galv'd 

2. 

1.6 

1.25 

1.12 

Black 
2.12 
1.73 
1.38 
1.15 

Lead  cot'd 
or  galv'd 

2.3 

1.84 

1.44 

1.29 

^ 


s 


m\mm\m\mW^ 


8treiigtb  of  Corrugrated  Iron.    Experiments  by  the  antlior. 

First.     A  sheet  d  cf,  of  No.  16  iron, 

(about  Y3-  inch  thick,)  27  ins  wide,  by  4  ft  long, 
with  five  complete  corrugations  of  5  ins  by  1  inch, 
was  laid  on  supports  3  ft  9  ins  apart.  A  block  of 
wood  c,  9  ins  wide,  by  7  ins  thick,  and  30  ins  long, 
was  placed  across  the  center,  and  gradually  loaded 
with  castings  weighing  1600  fts. 

This  caused  a  deflection  at  the  center  of  precisely  3^  an 
inch.  On  the  removal  of  the  load  after  an  hour,  no  perma- 
nent set  was  appreciable.  The  seyerity  of  the  test  was  pur- 
posely increased  by  applying  the  sereral  eastings  very 
roughly,  joltiag  the  whole  as  mueh  as  possible.*  The  sus- 
pended area  of  the  sheet  wag  8.44  sq  ft ;  and  since  the  actual  center  load  of  1600  lbs  is  about  equira- 

lent  to  3000  fts  equally  distributed,  it  amounts  to       -.=355  fts  per  sq  ft  distributed.    But  3000  lbs 

distributed  would  produce  a  deflection  of  but  about  full  }4  of  an  inch.  Again,  355  lbs  perlsq  ft 
is  about  4  times  the  weight  of  the  greatest  crowd  that  could  well  congregate  upon  a  floor.  Conse- 
quently this  iron,  at  3'  9"  span,  is  safe  in  practice  for  any  ordinary  crowd.  Moreover,  such  a  crowd 
would  produce  a  center  deflection  of  only  the  J^th  part  of  }4  of  an  inch ;  or  ^3^  of  an  inch ; 


•^T^" 


'j-zn 


of  tb«  clear  spaa ;  which  is  bat  two-thirds  of  Tredgold's  limit  of  -t^tj  °f  ^^^  span. 

In  one  experiment  the  ends  of  the  sheets  rested  upon  supports  dressed  so  as  to  present  undulations 
corresponding  tolerably  closely  with  the  shape  of  the  corrugations ;  but  in  the  other  the  supports 
were  flat,  and  each  end  of  the' sheet  rested  only  upon  the  lower  points  of  the  corrugations.  Wo  ap- 
preciable difiference  was  observed  in  the  results. 

(Second.    An  arch  of  ^o.  18  (-^jj  

inch)  iron,  corrugated  like  the  foregoing, 
but  the  depth  of  corrugation  increased  to 
IV^ins  by  the  process  of  arching  the  sheet; 
clear  span  6  ft  1  inch  ;  rise  10  ins ;  breadth  27 
ins,  (of  which,  however,  only  25  ins  bore 
against  the  abutments.) 

Each  foot  0  of  the  arch  abutted  upon  a  casting  j, 
the  inner  portion  t  of  which  was  undulated  on  top,  to 
correspond  with  the  corrugations  of  the  arch,  which 
rested  upon  it.  At  y,  (one-fourth  of  the  span,)  two 
vooden  blacks  were  placed,  occupying  a  width  of  9 
Inches,  and  extendjog  across  the  arch  ;  on  them  was 
piled  a  load,  I,  of  castings,  to  the  extent  of  4480  lbs, 
or  2  tons.  Under  this  load  the  arch  descended  about 
half  an  inch  at  y,  becoming  flatter  on  that  side  and 
slightly  more  curved  upward  along  the  unloaded  side  n.  Two  similar  blocks  were  then  placed  at  n, 
and  two  tons  of  load,  s,  were  piled  upon  them,  in  addition  to  the  2  tons  at  I;  making  a  total  of  8960 
fts,  or  4  tons.  This  brought  the  arch  more  nearly  back  to  its  original  shape ;  but  still  slightly 
straightened  at  both  n  and  y,  and  a  little  more  curved  in  the  center.  The  load  was  then  increased  to 
10000  B)s,  and  left  standing  for  several  days.  Two  iron  ties,  each  14  by  1%,  which  were  used  for  pre- 
▼snting  the  abu-tment  castings  j  from  spreading,  were  found  to  have  stretched  nearly  }4  of  an  inch. 
Additional  ones  were  inserted,  and  the  load  increased  to  a  total  of  6  tons,  or  13440  ftis:  parts  of  it  on 
8  and  I,  and  part  in  the  shape  of  long  broad  bars  of  iron  at  the  center  of  the  arch,  below  the  loads  a 
and  I,  and  between  n  and  y.  So  far  as  conld  be  judged  by  eye,  the  shape  of  the  arch  was  now  almost 
perfect.  Tfte  loads  s  and  1  did  not  touch  each  other.  After  standing  more  than  a  week,  the  load 
was  accidentally  overturned,  crippling  the  arch.     The  load  was  equal  to  about  1000  fts  per  sq  ft  of 

the  arch.    Such  arches  have  since  come  into  common  use  Instead  of  brick,  for 
fireproof  floors. 

Curved  roofs  of  25  to  30  ft  span,  rising  about  ^^  span,  may  be  made 
of  ordinary  corrugated  iron  of  Nos  16  to  13,  riveted  as  usual ;  and  having  no  acces- 
sories except  tie-rods  a  few  feet  apart ;  continuous  angle-iron  skewbacks ;  and  thin 
rerticai  rods  to  prevent  the  ties  from  sagging. 

*  Without  letting  the  deflection  exceed  H  iQch ;  which  was  prevented  by  a  stop  under  the  sheet. 

56 


882 


IRON   PIPES,   TUBES  AND   FITTINGS. 


Welded 

wron 

fftit-iron  pipes. 

for  steam,  gas.  and  water 

Usually  in 

lengths  of  about  18 

feet.    Standard  sizes. 

Inner  Diam. 

. 

^  ^ 

•1- 

Inner  Diam. 

. 

s-  ^ 

-1— 

,3 

^  o 

^1 

si 

ti 
"1 

1 

it 

2  a 

* 

a 

p 

a 
1 

*. 

a 
3 
1 

o 

< 

Ins. 

Ins. 

Ins. 

lbs. 

s 

Ins. 

Ins. 

Ins. 

»)S. 

s 

yi 

0.270 

0.068 

0.24 

27 

0.055 

81/^ 

3.548 

0.226 

9.00 

.    8 

0.95 

0.3641  0.088 

0.42 

18 

(( 

4 

4.026 

0.237 

10.66 

1.08 

/x 

0.494 

0.091 

0.56 

'< 

<' 

43^ 

4.508 

0.246 

12.49 

1.30 

0.623 

0.109 

0.84 

14 

0.085 

5 

5.045 

0.259 

14.50 

1.45 

0.824 

0.113 

1.12 

0.115 

6. 

6.065 

0.280 

18.76 

1.88 

1 

1.048 

0.134 

1.67 

11.5 

0.165 

7 

7.023 

0.301 

23.27 

2.35 

IM 

1.380 

0.140 

2.24 

0.225 

8 

7.982 

0.322 

28.18 

2.82 

1^ 

1.611 

0.145 

2.68 

*< 

0.27 

9 

9.001 

0.344 

33.70 

3.40 

2 

2.067 

0.154 

3.61 

<< 

0.36 

10 

10.019 

0.366 

40.00 

4.25 

2^ 

2.468 

0.204 

5.74 

8 

0.575 

11 

11.000 

0.375 

45.00 

4.75 

3 

3.067 

0.217 

7.54 

0.755 

12 

12.000 

0.375 

49.00 

5.20 

Fittingrs  for  Wrou^ht-iron  Pipes.  1,  Elbow.  2,  Service  Elbow. 
3,  Elbow  with  side  outlet.  4,  Keducing  T.  5,  T.  6,  Reducing  Cross.  7,  Reduc- 
ing Coupling  or  Socket.  8,  Return  Bend  with  side  outlet.  9,  Return  Bend  with 
back  outlet.    10,  Cross.    11,  Flange  Union.    12,  Oval  Flange.    13,  Plug. 


i5i^V^ 


liap- welded  cliar coal-iron  boiler  tubes,  in  lengths  up  to  20  ft. 

* 

Nom 

n^-^ 

•*_ 

Nom 

.*r 

* 

Nora 

n^-*T 

hi 

Thick- 

Wt. 

0^ 

a;  S 

Thick- 

Wt. 

,oi^ 

S  'a 

Thick- 

Wt. 

^'^ 

SS 

ness. 

per 
ft. 

^  c. 

Oft 

ness. 

■rr 

^1 

6S 

ness. 

r 

B 

B 

B 

Ins. 

Ins. 

WG 

ft>s. 

$ 

Ins. 

Ins. 

WG 

ft)S. 

S 

Ins. 

Ins. 

WG 

fibs. 

$ 

1 

0.95 

13 

0.90 

0.30 

3 

0.109 

12 

3.33 

0.35 

8 

0.165 

8 

13.65 

1.5(^ 

11^ 

(( 

1.15 

0.28 

3?| 

0.120 

11 

3.96 

0.40 

9 

0.180 

7 

16.76 

1.70 

1^ 

** 

1.40 

0.27 

*' 

" 

4.28 

0.44 

10 

0.203 

B 

21.00 

2.10 

m 

" 

1.66 

0.22 

" 

** 

4.60 

0.50 

11 

0.220 

5 

25.00 

2.50 

<i 

1.91 

0.20 

4 

0.134 

10 

5.47 

0.55 

12 

0.229 

4I/; 

28.50 

2.90 

214 

(( 

2.16 

0.24 

'iVo 

*' 

6.17 

0.62 

13 

0.238 

4 

32.06 

3.2a 

2^ 

0.109 

12 

2.75 

0.28 

5 

0.148 

9 

7.58 

0.75 

14 

0.248 

31^ 

36.00 

3.65 

2«^ 

« 

it 

3.04 

0.34 

6 

0.165 

8 

10.16 

1.00 

15 

0.259 

3 

40.60 

4.10 

7 

'* 

11.90 

1.20 

16 

0.270 

2>^ 

45.20 

4.6d 

*  For  pipes  give  the  "  nominal "  inner,  for  boiler  tubes  the  outer  diam.  See  p  526. 
ir  For  discounts,  see  price  list. 


BOLTS,     NUTS,    WASHERS. 


883 


Screw  Threads,  Bolts,  Nuts,  and  Washers. 

Serei¥  threads,  a  =  angle  between  two  sides  of  a 
thread ;  P  —  pitch ;  w  =^  width  of  flat  top  or  bottom  of 
each  thread;  all  measured  in  a  plane  containing  the 
axis  of  the  screw ;  N  =  number  of  threads  per  inch,  == 
1/P.  In  the  Sellers  or  Franklin  Institute 
Standard,  proposed  by  Mr.  William  Sellers  and 
adopted  by  the  Institute  in  1864,  a  =  60° ;  S  =  P ;  w  =^  c 
=  P/8  ;  F  =  0.75  P ;  M  =-  P  cos  a/2  =  0.8660  P  ;  D  (diam- 
eter) =  d  -f  2  X  0.866  X  0.75  P  =  d  +  1.299  P.  Under  the 
name  of  United  States  Standard,  the  U.  S.  Navy 
Department  in  1868  adopted  the  Sellers  system,  except 
for  finished  heads  and  nuts,  which  it  made"  the  same  as 
for  rough  heads  and  nuts. 


B 

d 

w 

X 

D 

a 

W 

N 

I> 

d    1   w 

IV 

B 

d 

w 

IV 

las 

ins 

ins 

ins 

ins 

ins 
^156 

s"" 

ins 
2 

ins 
1.712 

ins 
.0277 

4U 

ins 
4 

ins 
3.567 

ins 
.0413 

H 

.185 

.0062 

20 

.837 

3 

5-16  .240 

.0074 

18 

.940 

.0178 

7 

2^/1 

1.962 

.0277 

4g 

3.798 

.0435 

2;^ 
23I 

%    .294 

.0078 

16 

1.065 

.0178 

7 

m 

2.176 

.0312 

4 

4.028 

.0454 

7-16  .344 

.0089 

14 

ii 

1.160 

.0208 

6 

2.426 

.0312 

4 

4.256 

.0476 

}4     .400 

.0096 

13 

1.284 

.0208 

6 

3 

2.629 

.0357 

3^ 
31^ 
314 

5 

4.480 

.0500 

9-16 

.454 

.0104 

12 

1.389 

.0227 

51^ 

SV, 

2.879 

.0357 

514 

4.730 

.0500 

5/ 

.5071. 01  U{ 

11 

1.491 

.0250 

5 

6 

"?4 

3.100 

.0384 

5^ 
5/^ 

4.953 

.0526 

% 

.620  .0125 

10 

1% 

1.616 

.0250 

5 

3.317 

.0413 

3 

5.203 

.0526  2^ 

.731  .0138 

' 

6 

5.423 

.0555  234 

Dimensions  of  Heads  and  Nuts. 
Rou^h.       I      Finished. 


X 

H  Hn  head) 
H  (in  nxii) 


y^  inch. 


13^D  + 1-16  inch. 
D  — 1-16  inch. 


In  the  "Whit^vorth  (English)  standard  thread,  the  angle  a,  Fig  1,  is  55°. 
The  tops  and  bottoms  of  the  threads  are  rounded,  instead  of  flat  as  in  tne  Ameri- 
can standards.  The  number  (N)  of  threads  per  inch  is  the  same  as  above  for 
diams  of  bolt  up  to  three  ins,  except  for  D  —  %  inch  ;  where  N  ~  12. 

In  the  International  metric  screw  thread,  adopted  at  Zurich, 
October,  1898,  the  Sellers  thread  profile  is  used.  The  dimensions  are  as  follows, 
all  in  millimeters : 


Diam. 

6 

7 

' 

9 

lolll    12 

14  lellS  20 

22 

24  27  30  33  36  39142 

45  4852|56|60 

64  68  72 

76 

80 

Pitch  l.Ojl.25  1.5    1.75 

2.0  1     2.5     1  3.0 

3.5     4.0 

4.5 

5.0     5.5     6.0 

6.5 

7.0 

Intermediate  diameters  are  to  be  of  an  integral  number  of  millimeters,  and 
of  the  same  pitch  as  the  next  smaller  diameter  in  the  table.  Thus,  for  diam  65 
or  69  mm  ;  pitch  =  6.0  mm. 

Plate-iron  washers.  Standard  sizes.  Diameters  of  washers  and  bolt- 
holes  in  inches.  Approximate  thickness  by  Birmingham  wire  gauge.  Approxi- 
mate number  in  one  ft). 


Diams. 

Ths. 

No. 

Diams. 

Ths. 

No. 

Diams. 

Ths. 

No. 

1^ 

V, 

18 

450 

1^4 

9-16 

14 

43 

214 

15-16 

9 

8.6 

% 

5-16 

16 

210 

1V« 

12 

26 

2% 

234 

1  1-16 

9 

6.2 

% 

5-16 

16 

139 

% 

12 

22.5 

IM 

9 

5.2 

% 

Vs 

16 

112 

11-16 

10 

13.1 

3 

i4 

9 

4. 

1 

7-16 

14 

68 

2 

13-16 

10 

10.1 

3K|i^ 

9 

2.8 

884 


BOLTS,  NUTS,   WASHERS. 


A  square  head  and  nut  together,  weigh  about  as  much  as  a  length  of  the  bolt  equal  to  7  or  8  titxMSl 
D.    Hexagon,  6  or  7. 

With  the  above  dimensions  a  bolt  will  generally  fail  by  breakina 
ofiF  between  the  head  and  the  nut,  where  the  diameter  is  decreasea 
by  cutting     the  thread,  rather    than    by    stripping    ofiF    its    threads. 

The  diani  I>  of  the  thread  must  of  course  be  greater 

than  that  required  to  bear  safely  the  proposed  tensile  strain,  by  an  amount 
equal  to  twice  the  depth  of  the  thread.  The  waste  of  iron,  which  would 
result  from  making  the  entire  bolt  of  this  greater  diam,  is  frequently 
avoided  by  making  the  bolt  from  a  bar  of  only  sufficient  dimensions  to  bear 
the  Strain  safely,  and  upsetting  its  ends  aa  in  Fig  3, 
thus  increasing  their  diam  sufBciently  to  allow  for  the  cutting  of  tha 
threads. 

In  carpentry,  as  well  as  in  ties  for  masonry,  washers,  w  w,  of  either  cast 
or  wrought  iron,  are  placed  between  the  timber,  or  stone,  and  the  head 
and  nut;  in  order  to  distribute  the  pressure  over  a  greater  surface,  and 
thus  prevent  crushing;  especially  in  timber. 

When  much  strained  ag-a^inst  irood,  the  side 

of  a  square  wrought-iron  washer ;  or  the  diam  w  m;  of  a  circular  one,  should  not  be  less  than  4  diams 

of  the  screw,  as  in  the  fig ;  and  its  thickness,  tw,  hi  diam  at  least.  

Two  such  square  washers  will  together  weigh  as  much  as  18  diams  in 
length  of  a  round  rod  of  the  same  diam  as  the  screw.  Two  round 
washers  will  weigh  together  as  much  as  14  diams  of  rod  of  same  diam 
as  screw.  In  either  case,  a  square  head  and  nut  will  weigh  as  much 
as  6  diameters.  Cast-iron  washers,  being  more  apt  to  split  under 
heavy  strains,  may  be  made  about  twice  as  thick  as  wrought  ones. 
When  the  strain  is  very  great,  the  diam  of  the  washer  may  be  5  or 
6  times  that  of  the  screw ;  and  its  thickness  equal  to  diam ;  but  4 
diams  will  suffice  for  most  practical  purposes,  or  even  2.5  When  there 
is  but  little  strain,  and  the  thickness  may  then  be  but  .1  or  .2  diam  of 
bolt. 

Table   of    machine    and    car   bolts,  with 

square  and  hexagon  heads  and  nuts.  Figs  4  and  5 ;    made  by  Hoopes 

&  Townsend,  1330  Buttonwood  St,  Phila.    All  their  bolts  ^.      ,  ^.      ^ 

are  cut    with  U.   S.  iStandard   threads,   as         Fig.  4.       Fig.  5s 
per  first  table  on  p  883,  unless  otherwise  ordered.    Discounts,  see  price  list. 


fio§ 

Length, ins 

Weight,  ft)s  of 

exclusive  of  head. 

100  bolts. 

List  price 

,  8  per  100. 

S'^  - 

SS;3 

Min. 

Max. 

Min. 

Max. 

Min. 

Max. 

% 

IK 

8 

3.9 

13.2 

1.70 

2.74 

5-16 

** 

6.2 

20.3 

2.00 

3.56 

% 

<* 

12 

9.7 

43.5 

2.40 

5.76 

7-16 

<( 

" 

14.7 

58.3 

2.80 

7.00 

^ 

• 

20 

20.4 

122.0 

3.60 

13.22 

9--16 

K 

" 

26 

151.0 

5.20 

19.26      ■ 

% 

** 

24 

37 

224.0 

" 

22.30 

1 

'< 

♦* 

58 

330.0 

7.20 

29.70 

2 

<< 

97.7 

470.0 

11.20 

42.00 

1 

<( 

" 

145.0 

625.0 

16.00 

55.60 

Expansion  bolts,  for  fastening  plates,  timbers, 
etc.,  to  walls  of  brick  or  masonry.  The  wedge-shaped 
nut,  traveling  up  the  bolt,  as  the  latter  is  turned, 
presses  the  wings  against  the  sides  of  the  hole,  which, 
in  practice,  is  drilled  just  large  enough  to  admit  the 
nut  and  wings,  so  as  to  prevent  the  former  from  turn- 
ing with  the  bolt.  If  the  hole  is  made  larger,  as 
shown,  the  nut  must  be  held  by  a  small  wedge. 


BOLTS,   NUTS,   WASHERS.  885 

liOCk-nnt  washers.  When  bolts  are  subjected  to  much 
rough  jolting,  as  at  rail-joints,  &c,  the  nuts  are  liable  to  wear  loose, 
and  U7iscrew  themselves.  On  railroads  this  is  a  source  of  great 
annoyance,  and  innumerable  devices  for  preventing  it  have  been 
tried.  The  Verona  lock-nut  washer  *  is  a  simple  circular  washer 
made  of  steel ;  with  a  slit  s  s  cut  through  it,  leaving  sharp  edges. 
On  one  side,  a,  of  the  slit,  the  metal  is  pressed  upward  about  "% 
inch  ;  and  that  on  the  other  side,  c,  downward,  the  same  distance ; 
80  that  a  perspective  view  would  be  somewhat  as  at  t.  Now,  when 
the  nut  is  screwed  down  over  the  washer,  in  the  direction  of  the 
arrow,  the  slit  offers  no  obstruction;  but  if  the  nut  afterward 
tends  to  unscrew  itself,  the  sharp  upper  edge  of  the  slit,  along  a,  presents  friction 
against  the  bottom  of  the  nut,  which  tends  to  hold  it  in  place.  Besides,  the 
washer,  by  its  elasticity,  tends  to  resume  its  original  shape,  and  thus  presses  the 
threads  of  the  nut  against  those  of  the  bolt ;  and  the  additional  friction  thus 
produced  also  aids  in  holding  the  nut. 

Another  lock-nut  washer  consists  of  a  long  strip  of  steel,  with  two  holes,  each 
of  which  has  its  edges  formed  like  those  of  a  Verona  washer,  and  through  each 
of  which  passes  one  of  the  bolts  of  the  rail-joint. 

Another  device  is  to  cut,  at  the  end  of  the  screw,  a  few  threads  of  a  screw  of 
less  diameter  than  the  main  one,  and  in  the  opposite  direction.  The  nut  is  then 
•crewed  upon  the  larger  diameter;  and  after  it  the  lock-nut  is  screwed  in  the 
other  direction  upon  the  smaller  diam,  until  it  comes  into  contact  with  the  main 
nut.  In  the  Smith  lock-nut  bolt,  this  second  nut  is  only  about "%  inch  thick; 
and  after  being  driven  home,  one  of  its  corners  is  bent  over  the  edge  of  the 
main  nut. 

The  Atw^ood  lock-nuts  take  advantage  of  elasticity  in  the  nut  itself,  which 
is  obtained  either  by  slitting  the  nut,  or  by  reducing  its  thickness  near  the 
bolt  hole. 

It  is  claimed  that  if  the  threads  of  an  ordinary  bolt  and  nut  are  carefully  cut, 
so  as  to  be  in  contact  with  each  other  throughout,  no  lock-nut  contrivance  is 
necessary,  because  the  friction  between  the  two  threads  is  distributed  over  a 
larger  surface,  and  abrasion  does  not  take  place  so  readily  as  if  the  threads 
touched  each  other  at  only  a  few  points.  The  nuts  are  therefore  less  apt  to  wear 
loose  under  repeated  jarring. 

Owing  to  the  diflaculty  of  obtaining  such  perfect  fitting  bolts  and  nuts,  due  to 
the  wear  of  the  cutting  tools  used  in  their  manufacture,  bolts  and  nuts  have  been 
made  in  which  the  thread  on  the  bolt  differs  slightly  in  shape  from  that  in  the 
nut.  They  also  furnish  nuts  in  which  the  thread,  instead  of  being  of  uniform 
shape  throughout,  gradually  becomes  deeper  and  thicker,  by  having  its  side  angle 
made  more  acute,  and  its  top  truncated.  These  nuts  are  used  with  bolts  having 
the  usual  uniform  thread.  The  bolt  enters  the  nut  upon  the  side  where  the 
thread  is  of  the  same  shape  as  its  own ;  but  its  thread  encounters,  and  is  forced 
into,  the  gradually  narrowing  and  deepening  path  between  the  threads  of  the 
nut.  In  both  devices,  the  enforced  conformity  between  the  two  threads  is  relied 
upon  to  give  the  desired  completeness  of  contact  between  them.  The  greater  force 
required  in  screwing  on  the  nut  also  increases  the  friction  between  the  threads, 

BUCKI.ED   PlJkTES. 

Buckled  plates  are  usually  of  steel,  34  *<>  3^  iii  thick  and  3  to  4  ft  sq ;  some- 
times in  long  plates  having  several  buckles  each.  Buckle  2  to  3  ins.  Flat  rim 
or  fillet,  2  to  4  ins.  They  are  used  for  the  floors  of  buildings  and  of  highway 
bridges. 

Total  permissible  load,  lbs.  on  a  single  square  buckled  plate  of  any  size  and 
thickness.!  Load  =  4k  t  h;  where  k  =  permissible  unit  stress  in  metal,  fcs  per 
sq  in,  say  6000 ;  t  =  thickness  of  metal,  ins,  and  h  =  depth  of  buckle,  ins. 

Buckled  plates  are  stronger,  and  require  less  concrete,  etc,  for  filling,  when  laid 
with  convex  side  down.  They  weigh  but  little  more  than  flat  plates,  or  about  10 
ft)s  per  sq  ft  per  3^  in  of  thickness. 

*  Invented  by  Mr.  Thomas  Shaw,  M.  E.,  of  Philadelphia. 

t"  Steel  in  Construction,"  by  Pencoyd  Iron  Works,  Philadelphia,  1900,  p  147, 


WEIGHT  OF  METALS. 


WEIGHT  AND  STREXOTH  OF  IROX  BOL.TS.     (Original.) 

Diameters,  weights,  and  approximate  breaking  strains,  for  rouiKl  bolts; 

breaking  strain  per  square  inch  assumed  as  follows :  Up  to  1  inch  square,  or  1 
inch  diam,  20  tons,  or  44800  lbs;  from  1  to  2  ins  sq  or  diam,  19  tons;  2  to  3  ins, 
18  tons  ;  3  to  4  ins,  17  tons ;  4  to  5  ins,  16  tons  ;  5  to  6  ins,  15  tons. 

A  long  upset  rod  is  no  stronger  than  one  not  upset,  against  slowly  applied  loads 
or  strains.  Both  will  then  break  at  about  midlength,  under  equal  pulls.  In  such 
cases  use  columns  5  and  6. 

Square  bars.     Strength  or  wt  =  1.273  X  strength  or  weight  of  round  bar. 

<r<A«««%A>.  Ka.«a      /strength  =  0.8      X  strength  of  similar  iron  bar. 

^.opper  Dars.     |;^eight  =1.14   X  weight     *'        "         "       ** 


Ends  enlarged,  or 

upset. 

Ends  not 
enlarged. 

Ends  enlarged,  or  upset. 

Ends  not 
enlarged. 

Diam.  Weight 

Break. 

Break- 

Diam. 

Weight 

Diam. 

Weight 

Break- 

Break- 

Diam. 

Weigh! 

of 

per  foot 

iiig 

ing 

of 

per  foot 

of 

per  foot 

ing 

iug 

of 

per  foot 

shank 

run. 

strain. 

strain. 

shank 

run. 

shank 

run. 

strain. 

strain. 

shank 

run. 

las. 

Pds. 

Tons. 

Pds. 

Ins. 

Pds. 

Ins. 

Pds. 

Tons. 

Pds. 

Ins. 

Pds. 

i 

.0414 

.245 

549 

1| 

8.10 

45.7 

102368 

2.14 

12.0 

A 

.093 

.553 

1239 

HI 

8.69 

49.0 

109760 

2.22 

12.9 

? 

.165 

.983 

2202 

.35 

.321 

n 

9.30 

52.5 

117600 

2.30 

13.8 

tI 

.258 

1.53 

34-27 

.43 

.452 

^H 

9.93 

56.0 

125440 

2.38 

14.7 

1 

.372 

2.21 

4950 

.50 

.654 

2. 

10.6 

59.7 

133728 

2.45 

15.7 

.506 

3.00 

6720 

.58 

.897 

2^ 

12.0 

63.8 

142912 

2.59 

17.5 

.661 

3.93 

8803 

.66 

1.14 

2i 

13.4 

71.6 

160384 

2  73 

19.5 

f 

.837 

4.97 

11133 

.73 

1.41 

2g 

14.9 

79.7 

178528 

2.88 

21.6 

1.03 

6.14 

13754 

.80 

1.67 

^ 

16.5 

88.4 

198016 

3.02 

23.9 

1.25 

7.42 

16621 

.88 

2.03 

25 

18.2 

97.4 

218176 

3.16 

26.1 

1.49 

8.83 

19779 

.96 

2.41 

2| 

20.0 

106.9 

239456 

3.30 

28.5 

H 

1.75 

10.4 

23296 

1.04 

2.81 

2| 

21.9 

116.8 

261632 

3.45 

31.1 

I 

2.03 

12.0 

26880 

1.12 

3.26 

3. 

23.8 

127.2 

284928 

3.60 

33.9 

2.33 

13.8 

30912 

1.20 

3.77 

3i 

27.9 

141.0 

315840 

3.86 

39.1 

lin. 

2  65 

15.7 

3516S 

1.27 

4.27 

H 

32.4 

163.6 

366464 

4.12 

44.4 

2.99 

16.8 

37632 

1.35 

4.77 

Sf 

37.2 

187.7 

420448 

4.41 

51.0 

1 

3.35 

18.9 

42336 

1.42 

6.28 

4. 

42.3 

213.6 

478464 

4.70 

57.8 

3.73 

21.1 

47264 

1.49 

5.81 

47.8 

227.0 

508480 

4.98 

65.2 

4.13 

23.3 

52192 

1.55 

6.39 

44 

53.6 

254.5 

570080 

5.25 

72.9 

4.56 

25.7 

57568 

1.64 

7.04 

41 

59.7 

283.5 

635040 

5.53 

80.5 

11 

5.00 

28.2 

63168 

1.72 

7.74 

5. 

66.1 

314.2 

703808 

5.80 

88.1 

5.47 

30.8 

68992 

1.80 

8.48 

H 

72.9 

324.7 

72732S 

6.08 

97.0 

11 

5.95 

33.6 

75264 

1.87 

9.20 

H 

80.0 

356.4 

798336 

6.36 

106. 

6.46 

36.4 

81536 

1.94 

9  88 

5| 

87.5 

389.5 

872480 

6.63 

116. 

6.99 

39.4 

88256 

2.00 

10.6 

6 

95.2 

424.1 

949984 

6.90 

126. 

7.53 

42.5 

95200 

2.07   ;U.3      1 

WIRE   GAUGES. 


The  Birmingham  -wire  ^S'aug'e  is  the  one  in  most  general  use  for  Iron.  The 
new  British  w  g  went  into  effect  March  Ist  1884,  In  the  "American  "  w  g  of  Dar- 
ling, Brown  &  Sharpe,  Providence  R.  I.,  each  diam  or  thick  is  =  the  next  smaller 
one  X  1.122932.  We  take  the  wt  of  wrot  iron  per  cub  ft  at  485  lbs  in  the  first  two  ; 
and  at  486  in  the  last.  For  the  wt  of  steel,  mult  that  of  iron  by  1.01.  For 
lead,  mult  iron  by  1.46.  For  zinc,  mult  iron  by  .9.  For  brass  (approx),  mult 
iron  by  1.06.    For  copper,  mult  iron  by  1.134. 


Birmingham  W.  Ga. 

New  British  W.  Ga. 

American  W.  Ga. 

Diam  of 

Wtof 

Diam  of 

Wtof 

Diam  of 

Wt  of 

No. 

wire,  or 

Wt  of, 

iron 

wire,  or 

Wt  of 

iron 

wire,  or 

Wtof 

iron 

thickness 

iron  wire, 

sheets, 

thickness 

iron  wire, 

sheets, 

thickness 

iron  wire, 

sheets, 

of  sheet, 

in  fts  per 

in  lbs  per 

of  sheet. 

in  fts  per 

in  lbs  per 

of  sheet, 

in  3)s  per 

in  fts 

ins. 

lin  ft. 

sq  ft. 

ins. 

lin  ft. 

sq  ft. 

ins. 

lin  ft. 

persqft 

7-0 

.500 
.464 
.432 

.661 
.569 
.494 

20.21 
18.75 
17.46 

6-0 

.„„ 

6-0 

!!!!!!!!!!!! 

A-0 

■".454"* 

'.546 

18.35**" 

.400 

.423 

16.17 

.460000 

.561 

18.63 

Ho 

.425 

.479 

17.18 

.372 

.366 

15.03 

.409642 

.445 

16.58 

a-0 

.380 

.383 

15.36 

.348 

.320 

14.06 

.364796 

.353 

14.77 

sO 

.340 

.306 

13.74 

.324 

.278 

13.09   J. 
12.13  ^ 

.324861 

.280 

13.15 

^1 

.300 

.238 

12.13 

.300 

.238 

.289297 

.222 

11.70 

2 

.284 

.214 

11.48 

.276 

.202 

11.15 

.257627 

.176 

10.43 

3 

.259 

.178 

10.47 

.252 

.168 

10.19 

.229423 

.139 

9.291 

4 

.238 

.150 

9.619 

.232 

.142 

9.377 

.204307 

.111 

8.273 

5 

.220 

.128 

8.892 

.212 

.119 

8.568 

.181940 

.0877 

7.366 

6 

.203 

.109 

8.205 

.192 

.0976    -y 

7.760^ 

.162023 

.0696 

6.561 

7 

.180 

.0859 

7.275 

.176 

.0820 

7.113 

.144285 

.0552 

5.842 

8 

.165 

.0721 

6.669 

.160 

.0677 

6.466 

.128490 

.0438 

5.203 

9 

.148 

.0580 

5.981 

.144 

.0548 

6.820 

.114423 

.0347 

4.633 

10 

.134 

.0476 

6.416 

.128 

.0434 

6.173 

.101897 

.0275 

4.125 

11 

.120 

.0382 

4.850 

.116 

.0357 

4.688 

.090742 

.0218 

3.674 

12 

.109 

.0315 

4.405 

.104 

.0286    ; 

'4.203 

.080808 

.0173 

3.272 

13 

.095 

.0239 

3.840 

.092 

.0224    -"^ 

3.718 

.071962 

.0137 

2.914 

14 

.083 

.0183 

3.355 

.080 

.0169 

3.233 

.064084 

.0109 

2.595 

15 

.072 

.oic/r 

2.910 

.072 

.0137 

2.910 

.057068 

.00863 

2.310 

16 

.065 

.0112 

2.627 

.064 

.0108 

2.587 

.050821 

.00684 

2.053 

17 

.058 

.00891 

2.344 

.056 

.00832 

2.263 

.045257 

.00543 

1.832 

>18 

.049 

.00636 

1.980 

.048 

.00610 

,.-1.940 

.040303 

.00430 

1.631 

19 

.042 

.00467 

1.697 

.040 

.00423 

1.617 

.035890 

.00341 

1,453 

20 

.035 

.00325 

1.415 

.036 

.00344 

1.455 

.031961 

.00271 

1.293 

21 

.032 

.00271 

1.293 

.032 

.00269 

1.293 

.028462 

.00215 

1.152 

22 

.028 

.00208 

1.132 

.028 

.00207 

1.132 

.025346 

.00170 

1.026 

23 

.025 

.00166 

1.010 

.024 

.00152 

.9700 

.022572 

.00135 

.913 

24 

.022 

.00128 

.8892 

.022 

.00128 

>  .8891 

.020101 

.00107 

.814 

25 

.020 

.00106 

.8083 

.020 

.00106 

.8083 

.017900 

.000849 

.724 

26 

.018 

.000859 

.7225 

.018 

.000857 

.7275 

.015941 

.000673 

.644 

27 

.016 

.000678 

.6467 

.0164 

.000712 

.6628 

.014195 

.000534 

.574 

28 

.014 

.000519 

.5658 

.0148 

.000579 

.5982 

.012641 

.000423 

.511 

29 

.013 

.000448 

..5254 

.0136 

.000489 

.5497 

.011257 

.000336 

.455 

30 

.012 

.000382 

.4850 

.0124 

.000408 

''  .5012 

.010025 

.000266 

.405 

SI 

.010 

.000265 

.4042 

.0116 

.000357 

.4688 

.008928 

.000211 

.360 

22 

.009 

.000215 

.3638 

.0108 

.000309 

.4365 

.007950 

.000167 

.321 

33 

.008 

.000170 

.3233 

>.0100 

.000265 

.4042 

.007080 

.000133 

.286 

34 

.007 

.000130 

.2829 

.0092 

.000224 

.3718 

.006305 

.000105 

.254 

35 

.005 

.0000662 

.2021 

.0084 

.000187 

.3395 

.005615 

.0000837 

.226 

36 

.004 

.0000424 

.1617 

.0076 

.000153 

>  .3072 

.005000 

.0000662 

.202 

37 

.0068 
.0060 
.0052 
.0048 
.0044 
.0040 
.0036 
.0032 
.0028 
.0024 
i)020 

.000122 

.0000952 

.0000714 

.0000608 

.0000513 

.0000423 

.0000344 

.0000271 

.0000207 

.0000152 

.0000106 

.2748 
.2425 
.2102 
.1940 
.1778 
.1617 
.1455 
.1293 
.1132 
.0970 
.0808 

.004453 
.003965 
.003531 
.003144 

.0000525 
.0000417 
.0000330 
.0000262 

.180 

38 

.159 

39 



.142 

40 

.127 

41 



42 

43 





44 

!!!!!!!!*!!! 

45 

46 

47 

48 



.0016 

.0000068 

.0647 

888 


WIRE   GAUGES. 


American  g'an^e  for  sheet  and  plate  iron  and  steel  (1893).  We  omit 
the  columns  of  weight  in  kilograms  per  square  foot  and  in  pounds  per  square 
meter,  and  simplify  the  headings  of  the  remaining  columns. 

An  Act  establishing  a  standard  gauge  for  sheet  and  plate  iron  and  steel. 

Be  it  enacted  by  the  Senate  and  House  of  Representatives  of  the  United  States  of 
America  in  Congress  assembled^  Tliat  for  the  purpose  of  securing  uniformity  the 
following  is  established  as  the  only  standard  gauge  for  sheet  and  plate  iron  and 
steel  in  the  United  States  of  America,  namely : 


Approximate  thickness. 

Weight. 

No. 

Inches. 

Millimeters. 

Per  sq.  foot, 

in  avoirdupois 

Per  sq. 
meter,  in 

ounces. 

pounds. 

kilograms. 

7-0 

1-2      =.5 

12.7 

320 

20.00 

97.65 

6-0 

15-32    =.46875 

1  J. 90625 

300 

18.75 

91.55 

5- 

-0 

7-16    =.4375 

11.1125      . 

280 

17.50 

85.44 

4- 

-0 

13-32    =.40625 

10.31875 

260 

16.25 

79.33 

3- 

-0 

3-8      =.375 

9.525 

240 

15. 

73.24 

2- 

-0 

11-32    =.34375 

8.73125 

220 

13.75 

67.13 

0 

5-16    =.3125 

7.9375 

200 

12.50 

61.03 

1 

9-32     =.28125 

7.14375 

180 

11.25 

54.93 

2 

17-64    =.265625 

6.746875 

170 

10.625 

51.88 

3 

1-4      =.25 

6.35 

160 

10. 

48.82 

4 

15-64    =.234375 

5.953125 

150 

9.375 

45.77 

5 

7-32    =.21875 

5.55625 

140 

8.75 

42.72 

6 

13-64    =.203125 

5.159375 

130 

8.125 

39.67 

7 

3-16     =.1875 

4.7625 

120 

7.5 

36.62 

8 

11-64    =.171875 

4.365625 

110 

6.875 

33.57 

9 

5-32    =.15625 

3.96875 

100  ■ 

6.25 

30.52    . 

0 

9-64     =.140625 

3.571875 

90 

5.625 

27.46 

1 

1-8      =.125 

3.175 

80 

5. 

24.41 

2 

7-64    =.109375 

2.778125 

70 

4.375 

21.36 

3 

3-32     =.09375 

2.38125 

60 

3.75 

18.31 

4 

5-64    =.078125 

1.984375 

50 

3.125 

15.26 

5 

9-128  =.0703125 

1.7859375 

45 

2.8125 

13.73 

6 

1-16    =.0625 

1.5875 

40 

2.5 

12.21 

7 

9-160  =.05625 

1.42875 

36 

2.25 

10.99 

8 

1-20    =.05 

1.27 

32 

2. 

9.765 

9 

7-160  =.04375 

1.11125 

28 

1.75 

8.544 

20 

3-80    =.0375 

.9525 

24 

1.50 

7.324 

21 

11-320  =.034375 

.873125 

22 

1.375 

6.713 

22 

1-32    =.03125 

.793750 

20 

1.25 

6.103 

23 

9-320  =.028125 

.714375 

18 

1.125 

5.493 

24 

1-40    =.025 

.635 

16 

1. 

4.882 

25 

7-320  =.021875 

.555625 

14 

.875 

4.272 

26 

3-160  =.01875 

.47625 

12 

.75 

3.662 

27 

11-640  =.0171875 

.4365625 

11 

.6875 

3.357 

28 

1-64    =.015625 

,396875 

10 

.625 

3.052 

29 

9-640  =.0140625 

.3571875 

9 

.5625 

2.746 

30 

1-80    =.0125 

.3175 

8 

.5 

2.441 

31 

7-640  =.0109375 

.2778125 

7 

.4375 

2.136 

32 

13-1280=.01015625 

.25796875 

6i 

.40625 

1.983 

33 

3-320  =.009375 

.238125 

6 

.375 

1.831 

34 

11-1280=.00859375 

.21828125 

5i 

.34375 

1.678 

35 

5-640  =.0078125 

..1984375 

5 

.3125 

1.526 

36 

9-1280=.00703125 

.17859375 

4^ 

4i 

.28125 

1.373 

87 

17-2660=.006640625 

.168671875 

.265625 

1.297 

38 

1-160  =.00625 

.15875 

4 

.25 

1.221 

And  on  and  after  July  first,  eighteen  hundred  and  ninety-three,  the  same  and 
no  other  shall  be  used, in  determining  duties  and  taxes  levied  by  the  United 
States  of  America  on  sheet  and  plate  iron  and  steel.  But  this  act  shall  not  be 
construed  to  increase  duties  upon  any  articles  which  may  be  imported. 

Sec.  2.  That  the  Secretary  of  the  Treasury  is  authorized  and  required  to  pre- 
pare suitable  standards  in  accordance  herewith. 

Sec.  3.  That  in  the  practical  use  and  application  of  the  standard  gauge  hereby 
established  a  variation  of  two  and  one-half  per  cent,  either  way  may  be  allowea. 

Approved  March  3, 1893. 


CIRCULAR   MEASURE. 


889 


CIRCUI.AB  MEASURE. 

Used  in  comparing  cross  sections  of  wires,  etc. 

A  circular  unit  is  the  area  of  a  circle  whose  diameter  is  one  linear  unit 
Thus,  a  circular  inch  is  the  area  (=0.7854  square  inch)  of  a  circle  whose 
diameter  is  one  inch. 

The  following  table  is  adapted,  by  permission,  from  Mr.  Carl  Bering's  valu- 
able Tables  of  Equivalents  of  Units  of  Measurement,  New  York,  1888.  Inas- 
much as  we  take  1  meter  ^  39.37  inches,  instead  of  39.37079  inches,  our  values 
differ  slightly  from  his. 

Logarithm. 

1  O  mil* =     •  0.78540  D  mil* 1.895  0899 

=       0.00064516  O  millimeter 1.809  6692 

=       0.00050671  D  millimeter ^ "4.704  7591 

1  D  mil* =       1.2732  O  mils* 0.104  9101 

=       0.00082145  O  millimeter 1.914  5793 

1  O  millimeter  =  1550.0  0  mils* 3.190  3308 

=  1217.4  D  mils* 3.085  4207 

==       0.78540  D  millimeter 1.895  0899 

1  n  millimeter  =  1973.5  O  mils* , 3.295  2409 

=       1.2732  O  millimeters 0.104  9101 

EDISON  STANDARD   WIRE   OAIJOE. 

Adopted  by  the  Associated  Edison  Illuminating  Companies. 
In  this  table  the  gauge  number  is  approximately  equal  to 

Tijtjo  X  area  of  cross  section  in  circular  mils  * 
=  Ys^  X  square  of  diameter  in  mils.  * 


No. 

Diameter, 

No. 

Diameter, 

No. 

Diameter, 

in  mils. 

in  mils. 

in  mils. 

3 

54.78 

65 

254.96 

.  160 

400.00 

5 

70.72 

70 

264.58 

170 

412.32 

8 

89.45 

75 

273.87 

180 

424.27 

12 

109.55 

80 

282.85 

190 

436.89 

15 

122.48 

85 

291.55 

200 

447.22 

20 

141.43 

90 

300.00 

220 

469.05 

25 

158.12 

95 

308.23 

240 

489.90 

30 

173.21 

100 

316.23 

260 

509.91 

35 

187.09 

110 

831.67 

280 

529.16 

40 

200.00 

120 

346.42 

300 

547.73 

45 

212.14 

130 

360.56 

320 

565.69 

50 

223.61 

140 

374.17 

340 

583.10 

55 

234.53 

150 

387.30 

360 

600.00 

60 

244.95 

*  1  mil  =  ^ 


r  inch. 


890 


WIRE    GATTGES. 


"So  trade  Stupidity  is  more  thoroughly  senseless  than  the  adherence  to 
the  various  Birmingham,  Lancashire,  &c,  gauges ;  instead  of  at  once  denoting  the 
thickness  and  diameter  of  sheets,  wire,  &c,  by  the  parts  of  an  inch ;  as  has  long 
been  suggested.  Thus,  No.  J/^,  or  No.  ^^^  wire,  or  sheet-metal  of  any  kind,  should 
be  understood  to  mean  }/^  or  -^^  of  an  inch  diam,  or  thickness.  To  avoid  mistakes, 
whicli  are  very  apt  to  occur  from  the  number  of  gauges  in  use;  and  from  the  absurd 
practice  of  applying  the  same  No.  to  different  thicknesses  of  different  metals,  in  dif- 
ferent towns,  it  is  best  to  ignore  them  all ;  and  in  giving  orders,  to  define  the  diam- 
eter of  wire,  and  the  thickness  of  sheet-metal,  by  parts  of  an  inch.  Or  the  weight 
per  hundred  ft  for  wire ;  or  per  sq  ft  for  sheets,  may  be  employed.  We  believe  that 
the  foregoing  Birmingham  gauge  applies  to  zinc,  copper,  brass,  and  lead;  although 
it  is  generally  stated  to  be  for  iron  and  steel  only.  Another  Birmingham  gauge  is 
used  for  sheet-brass,  gold,  silver,  and  some  other  metals;  but  we  have  never  seen  it 
stated  what  tliose  others  are.  Jhere  are  different  gauges  even  for  wire  to  be  used 
for  different  purposes  ;  and  various  firms  have  gauges  of  their  own ;  not  even  accord* 
ing  among  themselves. 

As  Mr.  Stubs  makes  various  English  gauges,  the  term  "  Stubs  g'auge  "  by 
itself  means  nothing.  Generally,  however,  in  our  machine  shops,  it  applies  to  the 
Binningham  gauge  of  the  preceding  table. 

Birmingham  g;aug;e  for  sheet  Brass,  SilTer,  Oold,  and  all  metals 

except  iron  and  steel  ? 


No. 

Thiokn's. 

No. 

Thiokn's. 

No. 

Thickn'8. 

No. 

Thickn'8. 

No. 

Thiokn's. 

No. 

Thickn'a. 

Inch 

Inch 

Inch 

Inch 

Inch 

Inch 

1 

.004 

7 

.015 

13 

.036 

19 

.064 

25 

.095 

31 

.13S 

2 

.005 

8 

.016 

14 

.041 

20 

.067 

26 

.103 

32 

.143 

3 

.008 

9 

.019 

15 

.047 

21 

.072 

27 

.113 

33 

.145 

4 

.010 

10 

.024 

16 

.051 

22 

.074 

28 

.120 

34 

.148 

5 

.012 

11 

.029 

17 

.057 

23 

.077 

29 

.124 

35 

.158 

6 

.013 

12 

.034 

18 

.061 

24 

.082 

SO 

.126 

36 

.167 

The  mills  rolling  sheet  iron  in  the  United  States  generally 
use  the  following,  which  varies  slightly  from  the  Birmingham  gauge : 


No. 

lbs  per 
sqft 

No. 

liis  per 
sqft 

No. 

lbs  per 
sqft 

No. 

lbs  per 
sqft 

1 

12.50 

8 

6.86 

15 

2.81 

22 

1.25 

2 

12.00 

9 

6.24 

16 

2.50 

23 

1.12 

3 

11.00 

10 

5.62 

17 

2.18 

24 

1.00 

4 

10.00 

11 

5.00 

18 

1.86 

25 

.90 

5 

8.75 

12 

4.38 

19 

1.70 

26 

.80 

6 

8.12 

13 

3.75 

20 

1.54 

27 

.72 

7 

7.50 

14 

3.12 

21 

1.40 

28 

.64 

When  -vrire,  sheet-metal)  &«.,  are  ordered  by  gauge  number,  and  it  is 
not  specified  what  gauge  is  intended ;  dealers  in  the  United  States  fill  the  order  as 
follows : 

Brass,  bronze  or  German  Silver  in  sheets,  German  Silver  wire,  brazed  brass,  bronze, 
zinc  or  copper  tubing,  by  Brown  &  Sharpe's  (or  "  American  ")  gauge. 

Copper  in  sheets ;  brass  and  copper  wire ;  seamless  brass,  bronze  or  copper  tubing; 
and  small  brass  rods;  by  Stubs'  (or  Birmingham)  gauge. 


UnanneaJed  or  hard  hrass  wire  has  about  %ths  the  strengths  of  the  table  p.  891, 
and  about  1  more  weight.    If  annealed,  only  full  half  the  strength. 

Hard  copper  virire  may  be  token  at  %  of  the  tabular  strengths,  and  foJt 
1  more  weight 


IRON   WIRE. 


891 


Table  of  Charcoal  Iron  l¥ire  made  by  Trenton  Iron  Co., 

Trenton,  N.  J.  The  numbers  in  the  first  column  are  those  of  the  Trenton  Iron 
Co's  g-aug-e.  The  corresponding  diameters  in  the  second  column  will  be  seen  to 
be  somewhat  less  than  those  of  the  Birmingham  gauge. 


No. 

Diam. 
ins. 

Lineal 

feet  to  the 
Pound. 

Tensile 

Str'gtli 

Approx 

lbs. 

No. 

Diam. 
ins. 

Lineal 

leet  tc  the 

Pound. 

Tensile 

Str'gth 

Approx 

lbs. 

No. 

Diam. 
ins. 

Lineal 

feet  to  the 

Pound. 

00000 

.450 

1.863 

12598 

11 

.1175 

27.340 

1010 

26 

.018 

1164.689 

0000 

.400 

2.358 

9955 

12 

.105 

34.219 

810 

27 

.017 

1305.670 

000 

.360 

2.911 

8124 

13 

.0925 

44.092 

631 

28 

.016 

1476.869 

00 

.330 

3.465 

6880 

14 

.080- 

58.916 

474 

29 

.015 

1676.989 

0 

.305 

4.057 

5926 

15 

.070 

76.984 

372 

30 

.014 

1925.321 

1 

.285 

4.645 

5226 

16 

.061 

101.488 

292 

31 

.013 

2232.653 

2 

.265 

5.374 

4570 

17 

.0525 

137.174 

222 

32 

.012 

2620.607 

3 

.245 

6.286 

3948 

18 

.045 

186.335 

169 

33 

.011 

3119.092 

4 

.225 

7.454 

3374 

19 

.040 

235.084 

137 

34 

.010 

3773.584 

5 

.205 

8.976 

2839 

20 

.035 

308.079 

107 

35 

.0095 

4182.508 

6 

.190 

10.453 

2476 

21 

.031 

392.772 



36 

.009 

4657.728 

7; 

.175 

12.322 

2136 

22 

.028 

481.234 

37 

.0085 

5222.035 

8 

.160 

14.736 

1813 

23 

.025 

603.863 

38 

.008 

5896.147 

9 

.145 

17.950 

ir>07 

24 

.0225 

745.710 

39 

.0075 

6724.291 

10 

.130 

22.333 

1233 

25 

.020 

943.396 



40 

.007 

7698.253 

Tbe  wire  in  tbis  table  is  supposed  to  b«  hard,  bright, 
The  figures  in  the  column  of  tensile  strength  are  based  upon  tests 
charcoal  iron  wire  from  Trenton  blooms. 

The  tensile  strength  of  wire  made  of  is  about 

Good  refined  iron 15  percent,  less"] 

Swedish  charcoal  iron 10       "        " 

Mild  Bessemer  steel 10       "      morel- 

Ordinary  crucible  steel 25       "         '• 

Special  crucible  steel 30  to  120       "         "     J 

Annealing  renders  wire  more  pliable  and  ductile,  but  less  elastic ;  and  reduces  the 
tensile  strength  by  from  20  to  25. per  cent. 


or  unannealei 
made  with  good 


than  that  of 
bright  charcoal 
wire,  given  in 
the  above  table. 


To  find  approximately  the  nnmber  of  straig^ht  wires  tbat 
can  be  got  into  a  cable  of  g-iven  diameter. 

Divide  the  diameter  of  the  cable  in  inches,  by  the  diameter  of  a  wire  in  inches. 
Square  the  quotient.  Multiply  said  square  by  the  decimal  .77.  The  result  will  be 
correct  within  about  4  or  5  per  cent  at  most,  in  a  cylindrical  cable. 

Tbe  solidity,  or  metal  area  of  all  tbe  wires  in  a  cable,  will  be 
to  the  area  of  the  cable  itself,  about  as  1  to  1.3.  In  other  words,  the  area  of  the 
voids  is  nearly  i^  that  of  the  cable ;  while  that  of  the  wires  is  fully  %  that  of  th« 
cable.    All  approximate.  ' 


892 


I-BEAMS. 


li  =  span  in  ft 

W  =  uniformly  distributed  safe  load  in  fl)s  = 

WL^C 

8     ~  8^ 


M  =  moment  of  load,  in  ft-fi)s  : 


\  =  stress  in  extreme  fibres,  in  fts  per  sq  in  = 


C 

L 

12 
M 


I,  i  =  moment  of  inertia ;  I,  about  XY  ;  i,  about  A  B 
R,  r  =  radius  of  gyration ;  R,     "       "      r,     "       " 

X      =**  section  modulus "        •  '*       "      X= — 5— 


CARXEOIE 


C       =  coefficient  for  uniformly  distributed  safe  load  =  WL  =  8  M  = 

12 
Cg,  for  static  loads;  S  =  16,000  lbs.    C^,  for  moving  loads ;  S  =  12,500  lbs. 
I>      =  distance  required  to  make  r  =  R 


Section 
index. 

H. 

Depth 
ins. 

Weight 
per  ft 

flt)S. 

Area  of 
section 
sq  in. 

Web 

thickness 

ins. 

Flange 

width 

ins. 

B    1 

24 

100.00 
80.00 

29.41 
23.32 

0.754 
0.500 

7.254 
7.000 

B    2 

20 

100.00 

moo 

29.41 
23.73 

0.884 
0.600 

7.284 
7.000 

B   3 

20 

75.00 
65.00 

22.06 
19.08 

0.649 
0.500 

6.399 
6.250 

B80 

18 

70.00 
55.00 

20.59 
15.93 

0.719 
0.460 

6.259 
6.000 

B    4 

15 

100.00 
80.00 

29.41 
23.81 

1.184 
0.810 

6.774 
6.400 

B    5 

15 

75.00 
60.00 

22.06 
17.67 

0.882 
0.590 

6.292 
6.000 

B^  7 

15 

55.00 
42.00 

16.18 
12.48 

0.656 
0.410 

5.746 
5.500 

B    8 

12 

55.00 
40.00 

16.18 
11.84 

0.822 
0.460 

5.612 
5.250 

B    9 

12 

35.00 
31.50 

10.29 
9.26 

0.436 
0.350 

5.086 
5.000 

Bll 

10 

40.00 
25.00 

11.76 
7.37 

0.749 
0.310 

.      5.099 
4.660 

B13 

9 

35.00 
21.00 

10.29 
6.31 

0.732 
0.290 

4.772 
4.330 

B15 

8 

25.50 
18.00 

7.50 
5.33 

0.541 
0.270 

4.271 
4.000 

B17 

7 

20.00 
15.00 

5.88 
4.42 

0.458 
0.250 

3.868 
3.660 

B19 

6 

17.25 
12.25 

5.07 
3.61 

0.475 
0.230 

3.575 
3.330 

B21 

5 

14.75 
9.75 

4.34 

2.87 

0.504 
0.210 

3.294 
3.000 

B23 

4 

10.50 
7.50 

3.09 
2.21 

0.410 
0.190 

2.880 
2.660 

B77 

3 

7.50 
5.50 

2.21 
1.63 

0.361 
0.170 

2.521 
2.330 

I-BEAMS. 


893 


I-BEAMS. 


-Y 


e^ 


The  table  gives  the  maximum 
and  the  minimum  weight  of  each  section. 
The  minimum  weights  are  standard. 
Others  are  special. 

Caution. — With  very  short  spans, 
the  loads  found  by  means  of  columns 
Cg  and  Cm,  although  safe  against  hend- 
i7ig,  may  be  so  great  as  to  endanger  a 
crmhing  of  the  ends  of  the  beam,  or  of 
the  walls,  etc.,  under  them,  unless  the 
beam  has,  at  its  ends,  a  greater  length  of 
bearing  than  would  otherwise  be  needed. 


I 

i 

R 

ins. 

r 

ins. 

X 

lbs. 

lbs. 

ins. 

Section 
index. 

2380.3 
2087.9 

48.56 
42.86 

9.00 
9.46 

1.28 
1.36 

198.4 
174.0 

2,115,800 
1,855,900 

1,653,000 
1,449,900 

17.82 
18.72 

B^^l 

1655.8 
1466.5 

52.65 
45.81 

7.50 
7.86 

1.34 
1.39 

165.6 
146.7 

1,766,100 
1,564,300 

1,379,800 
1,222,100 

14.76 
15.47 

B   2 

1268.9 
1169,6 

30.25 
27.86 

7.58 
7.83 

1.17 
1.21 

126.9 
117.0 

1,353,500 
1,247,600 

1,057,400 
974,700 

14.98 
15.47 

B   8 

921.3 
795.6 

24.62 
21.19 

6.69 
7.07 

1.09 
1.15 

102.4 
88.4 

1,091,900 
943,000 

853,000 
736,700 

13.20 
13.95 

B80 

900.5 
795.5 

50.98 
41.76 

5.53 
5.78 

1.31 
1.32 

120.1 
106.1 

1,280,700 
1,131,300 

1,000,600 
883,900 

10.75 
11.25 

B   4 

691.2 
.    609.0 

30.68 
25.96 

5.60 
5.87 

1.18 
1.21 

92.2 
81.2 

983,000 
866,100 

768,000 
676,600 

10.95 
11.49 

B    5 

511.0 
441.7 

17.06 
14.62 

5.62 
5.95 

1.02 
1.08 

68.1 
58.9 

726,800 
628,300 

567,800 
490,800 

11.05 
11.70 

B    7 

321.0 
268.9 

17.46 
13.81 

4.45 

4.77 

1.04 
1.08 

53.5 
44.8 

570.600 
478,100 

445,800 
373,500 

8.65 
9.29 

B   8 

228.3 
215.8 

10.07 
9.50 

4.71 
4.83 

0.99 
1.01 

38.0 
36.0 

405.800 
383,700 

317,000 
299,700 

9.21 
9.45 

B   9 

158.7 
122.1 

9.50 
6.89 

3.67 
4.07 

0.90 
0.97  ■ 

31.7 
24.4 

333,500 
260,500 

264,500 
203,500 

7.12 
7.91 

Bll 

111.8 
84.9 

7.31 
5.16 

3.29 
3.67 

0.84 
0.90 

24.8 
18.9 

265,000 
201,300 

207,000 
157,300 

6.36 

7.12 

B13 

68.4 
56.9 

4.75 
3.78 

3.02 
3.27 

0.80 
0.84 

17.1 
14.2 

182,500  « 
151,700 

142,600 
118,500 

5.82 
6.32 

^}^ 

42.2 
36.2 

3.24 

2.67 

2.68 
2.86 

0.74 
0.78 

12.1 
10.4 

128,600 
110,400 

100,400 
86,300 

5.15 
5.50 

BIT 

26.2 
21.8 

2.36 
1.85 

2.27 
2.46 

0.68 
0.72 

8.7 
7.3 

93,100 
77,500 

72,800 
60,500 

4.33 
4.70 

B19 

15.2 
12.1 

1.70 
1  23 

1.87 
2  05 

0.63 
0  65 

6.1 

4.8 

64,600 
51,600 

38,100 
31,800 

20,700 
17,600 

50,500 
40,300 

29,800 
24,900 

16,200 
13,800 

B21 

7.1 
6  0 

1.01 

0  77 

1.52 
1  64 

0.57 
0.59 

3.6 
3.0 

B23 

2.9 

2.5 

0.60 
0.46 

1.15 
1.23 

0.52 
0.53 

1.9 
1.7 

B77 

894 


CHANNELS. 


CARNIBOIB 


Ij  ^  span  in  ft 

"W  =  uniformly  distributed  safe  load  in  B)s  =  - 

L 


M  =  moment  of  load,  in  ft-S)s  = 


WL 


SX 

''  12 

M 


S  =  stress  in  extreme  fibres,  in  fl>s  per  sq  in  =  ^ 

I,  i  =  moment  of  inertia ;  I,  about  XY  ;  i,  about  A  B 
R,  r  =  radius  of  gyration ;  R,     "       "      r,     '*       '* 

X      ="  section  modulus "  **       "      X  =  — ^ 

C       =  coefficient  for  uniformly  distributed  safe  load  =  WL  =  8  M  = 

Cg,  for  static  loads ;  S  =  16,000  fi)s.    Cjjj,  for  moving  loads ;  S  =  12,500  fts. 

I>      =  distance  required  to  make  r  =  B, 


8SX 


Section 
index. 

H. 

Depth 

ins. 

Weight 
per  ft 
lbs. 

Area  of 
section 
sq  in. 

Web 

thickness 

ins. 

E. 

Flange 

width 

ins. 

CI 

15 

55.00 
33.00 

16.18 
9.90 

0.818 
0.400 

3.818 
3.400 

C    2 

12 

40.00 
20.50 

11.76 
6.03 

0.758, 
0.280 

3.418 
2.940 

C   3 

10 

35.00 
15.00 

10.29 
4.46 

0.823 
0.240 

3.183 
2.600 

C   4 

9 

25.00 
13.25 

7.35 

3.89 

0.615 
'       0.230 

2.815 
2.430 

C   5 

8 

21.25 
11.25 

6.25 
3.35 

0.582 
0.220 

2.622 
2.260 

C^6 

7 

19.75 
9.75 

5.81 
2.85 

0.633 
0.210 

2.513 
2.090 

^u 

6 

15.50 
8.00 

4.56 
2.38 

0.563 
0.200 

2.283 
1.920 

C    8 

5 

11.50 
6.50 

3.38 
1.95 

0.477 
0.190 

2.037 
1.750 

^J 

4 

7.25 
5.25 

2.13 
1.55 

0.325 
0.180 

1.725 
1.580 

C72 

3 

6.00 
4.00 

1.76 
1.19 

0.362 
0.170 

1.602 
1.410 

Fire-proof  floors  of  I  beams  and  briek-arelies. 

The  arches  are  usually  "  four-inch"— or  *'  half  a  brick"  deep;  span,  s,  from  4 
to  6  feet;  rise  about  one-twelfth  to  one-sixteenth  of  the  span.  Tie-rods,  T,  % 
inch  to  1  inch  diameter,  from  4  to  6  or  8  feet  apart,  and  anchored  into  each  wall 


irith  a  stout  washer,  W.  At  each  wall  an  angle  iron,  a,  or  a  tee  iron  is  generally 
used  instead  of  a  beam.  The  spandrels  are  leveled  up  with  concrete,  enclosing 
wooden  strips,  m  m,  about  1  inch  X  2  inches,  two  over  each  arch.  To  these  strips 
the  flooring  is  nailed. 


CHANNELS. 


895 


X-; 


^ — -D- 

|.-Y 


Tlie  table  gives  the  maximum 
and  the  minimum  weight  of  each  section. 
The  minimum  weights  are  standard. 
Others  are  special. 

Caution. — With  very  short  spans, 
the  loads  found  by  means  of  columns 
Cs  and  Cm,  although  safe  against  bend- 
ing, may  be  so  great  as  to  endanger  a 
crushing  of  the  ends  of  the  beam,  or  of 
the  walls,  etc.,  under  them,  unless  the 
beam  has,  at  its  ends,  a  greater  length  of 
bearing  than  would  otherwise  be  needed. 


I 

i 

R 

ins. 

r 

ins. 

X 

€s 

ft)S. 

fts. 

ins. 

Section 
index. 

430.2 
312.6 

12.19 
8.23 

5.16 
5.62 

0.868 
0.912 

57.4 
41.7 

611,900 
444,500 

478,000 
347,300 

8.53 
9.50 

CI 

197.0 
128.1 

6.63 
3.91 

4.09 
4.61 

0.751 

0.805 

32.8  * 
21.4 

350,200 
227,800 

273,600 
178,000 

6.60 

7.67 

C^^2 

115.5 
66.9 

4.66 
2.30 

3.35 
3.87 

0.672 
0.718 

23.1 
13.4 

246,400 
142,700 

192,500 
111,500 

5.17 
6.33 

€3 

70.7 
47.3 

2.98 
1.77 

3.10 
3.49 

0.637 
0.674 

15.7 
10.5 

167,600 
112,200 

130,900 

87,600 

4.84 
5.63 

C    4 

47.8 
32.3 

2.25 
1.33 

2.77 
3.11 

0.600 
0.630 

11.9 
8.1 

127,400 
86,100 

99,500 
67,300 

4.23 
4.94 

Co 

33.2 
21.1 

1.85 
0.98 

2.39 
2.72 

0.565 
0.586 

9.5 
6.0 

101,100 
66,800 

79,000 
52,200 

3.48 
4.22 

C^^6 

19.5 
13.0 

1.28 
0.70 

2.07 
2.34 

0.529 
0.542 

6.5 
4.3 

69,500 
46,200 

54,300 
36,100 

2.91 
3.52 

QJ 

10.4 
7.4 

0.82 
0.48 

1.75 
1.95 

0.493 
0.498 

4.2 
3.0 

44,400 
31,600 

34,700 
24,700 

2.34 
2.79 

C   8 

4.6 
3.8 

0.44 
0.32 

1.46 
1.56 

0.455 
0.453 

2.3 
1.9 

24,400 
20,200 

19,000 
15,800 

1.85 
2.06 

C   9 

2.1- 
1.6 

0.31 
0.20 

1.08 
1.17 

0.421 
0.409 

1.4 

1.1 

14,700 
11,600 

11,500 
9,100 

1.07 
1.31 

C72 

The  weight  of  a 

exclusive  of  the  li 

A  dense  crowd 

1 

4-inch  arch, 
eams,  is  abou 
of  persons  wi 

with 

t70  1 
Ihar 

T1 

its 
bs. 
dly 

CO 

pel 
w< 

ncrete  filli 
"  square  fo 
3igh  more 

Qg  and  wood 
ot  of  floor, 
than  80  lbs.  ] 

en  flooi 
3er  squ 

'ing,  but 
are  foot. 

Lt|^ 

Siip^^^ 

:^^^^ 

^ 

^ 

^ 

P 

has  fas 
e  cente 

=^^^^ 

^^^^^^^==:z 

=55MU 

Each  c 
hook  by 

enter,  C, 
which  th 

tened 
r  is  su. 

to  it 
spend 

at  ( 
edi 

mc 
fro 

h  end  a  b€ 
m  the  low( 

nt  iron  stra 
ir  flanges  of 

p,  h,  fo 
the  bea 

rming  a 
ms. 

896 


ANGLES   AND   T  SHAPES. 


CARNEGIE  ANGI.es 


di  =  distance  between  center  of  gravity  and  back  of  flange  W 

s  =         "  '*  *'        "        "  *'  "       "        H 

I,  i      =  moment  of  inertia ;  I,  about  XY ;    i,  about  A  B 


=  least  " section  modulus' 


X, 


12  M. 

S 


t 


R,  R'  =  radius  of  gyration ;  R,    "        **       R%    " 

r  —  least  radius  of  gyration,  about  neutral  axis  forming  acute  angle  a 

with  each  flange.    In  angles  with  equal  legs,  a  =  45° 


=  coefficient  for  uniformly  distributed  safe  load :  "] 
Cg  for  static  loads  ;  fibre  stress  =  16,000  fibs.  >■ 
Cj^,  for  moving  loads  ;  fibre  stress  =  12,000  fos.  J 


For  T  shapes  only. 


Size 
H     W 
ins.  ins. 


Thick- 

nesss 
ins. 


Weight 
per  ft 


Area  of 
section 
sq  ins. 


ins. 


Angeles  \¥itli  Unequal  lieges. 


X3> 

X4 
X4 

X3K 
X3i| 

X4 

X4 

X3K 
X3i^ 

X3 
X3 


4KX3 
4^X3 


X33^ 
X3^ 

X3 
X3 


3KX5 

33^  x: 

3>^X2K 
3>|X23.' 

334X2 
3^X2 

3      X23^ 
3      X2>| 

3X2 
3X2 

2^X2 
2>^X2 

2%  X  13^ 
2H  X  i>| 

2      Xlfl 
2      Xl^ 

1%X1 
1%X1 


1 

32.3 
15.0 

1 

30.6 
12.3 

1 

Vs 

28.9 
11.7 

g 

24.2 
11.0 

1 

22.7 
8.7 

^ 

19.9 

8.2 

X 

18.5 

7.7 

i 

18.5 

7.7 

TB 

17.1 
7.1 

^ 

15.7 
6.6 

II 

12.4 
4.9 

li 

9.0 
4.3 

§ 

9.5  • 
4.5 

A: 

7.7 
4.0 

^ 

6.8 
2.8 

I 

5.5 
2.3 

i 

2.7 
2.1 

A 

1.8 
1.0 

9.50 
4.40 

9.00 
3.61 

8.50 
3.42 

7.11 
3.23 

6.67 
2.56 

5.84 
2.40 

5.43 
2.25 

5.43 
2.25 

5.03 
2.09 

4.62 
1.93 

3.65 
1.44 

2.64 
1.25 

2.78 
1.31 

2.25 
1.19 

2.00 
0.81 

1.63 
0.67 

0.78 
0.60 

0.53 
0.28 


2.71 
2.50 

2.17 
1.94 

2.26 
2.04 

1.71 
1.53 

1.79 
1.59 

1.86 
1.68 

1.65 
1.47 

1.36 
1.18 

1.44 
1.26 

1.23 
1.06 

1.27 
1.11 

1.21 
1.09 

1.02 
0.91 

1.08 
0.99 

0.88 
0.76 

0.86 
0.75 

0.69 
0.66 

0.48 
0.44 


0.96 
0.75 

1.17 
0.94 

1.01 
0.79 

1.21 
1.03 

1.04 
0.84 

0.86 
0.68 

0.90 
0.72 

1.11 
0.93 

0.94 
0.76 

0.98 
0.81 

0.77 
0.61 

0.59 
0.48 

0.77 
0.66 

0.58 
0.49 

0.63 
0.51 

0.48 
0.37 

0.37 
0.35 

0.29 
0.26 


45.37 
22.56 

30.75 
13.47 

29.24 
12.86 

16.42 
8.14 

15.67 
6.60 

13.98 
6.26 

10.33 
4.69 

7.77 
3.56 

7.34 
3.38 

4.98 
2.33 

4.13 
1.80 

2.64 
1.36 

2.28 
1.17 

1.92 
1.09 

1.14 

0.51 

0.82 
0.34 

0.37 
0.24 

0.09 
0.05 


7.53 
3.95 

10.75 
4.90 

7.21 
3.34 

9.23 
4.67 

6.21 
2.72 

3.71 
1.75 

8.60 
1.73 

5.49 
2.59 

3.47 
1.65 

3.33 
1.58 

1.72 
0.78 

0.75 
0,40 

1.42 
0.74 

0.S7 
0.39 

0.64 
0.29 

0.26 
0.12 

0.12 
0.09 

0.04 
0.02 


♦  Special  sections,    f  For  M  and  S  see  p.  892  or  p.  894. 


ANGLES   AND   T   SHAPES. 


AND  T  SCEAPES. 


-X -Y- 


^^ 


W- 


X  ---Y- 


-W^ 


/f  -X- 


I 
I 


^ 


W- 


H 


^& 


-X 


-r^ 


-w- 


R 


R' 


Maximum  and  Minimum  "Weight  of  each  Section. 


2.96 
1.47 

2.19 
2.26 

0.89 
0.95 

0.88 
0.89 

3.79 
1.60 

1.85 
1.93 

1.09 
1.17 

0.85 
0.88 

2.90 
1.23 

1.85 
1.94 

0.92 
0.99 

0.74 
0.77 

3.31 
1.57 

1.52 
1.59 

1.14 
1.20 

0.84 
0.86 

2.52 
1.02 

1.53 
1.61 

0.96 
1.03 

0.75 
0.76 

1.74 
0.75 

1.55 
1.61 

0.80 
0.85 

0.64 
0.66 

1.71 
0.76 

1.38 
1.44 

0.81 
0.88 

0.64 
0.66 

2.30 
1.01 

1.19 
1.26 

1.01 
1.07 

0.72 
0.73 

1.68 
0.74 

1.21 
1.27 

0.83 
0.89 

0.64 
0.65 

1.65 
0.72 

1.04 
1.10 

0.85 
0.90 

0.62 
0.63 

0.99 
0.41 

1.06 
1.12 

•    0.67 
0.74 

0.53 
0.54 

0.53 
0.26 

1.00 
1.04 

0.53 
0.57 

0.44 
0.45 

0.82 
0.40 

0.91 
0.95 

0.72 
0.75 

0.52 
0.53 

0.47 
0.25 

0.92 
0.95 

0.55 
0.57 

0.43 
0.43 

0.46 
0.20 

0.75 
0.79 

0.56 
0.60 

0.42 
0.43 

0.26 
0.11 

0.71 
0.72 

0.40 
0.43 

0.39 
0.40 

0.12 
0.09 

0.63 
0.63 

0.39 
0.40 

0.30 
0.31 

0.05 
0.03 

0.41 
0.44 

0.27 
0.29 

0.22 
0.22 

57 


*  Special  sections. 


898 


ANGLES   AND   T   SHAPES. 


CARNEOIE  AXOIiES 

Cl  =»  distance  between  center  of  gravity  and  back  of  flange  W 

•  =         "  "  "        "        "  *'  "       ''        H 

I,  i      =  moment  of  inertia ;  I,  about  XY  ;    i,  about  A  B 

X,  X  =  least  "  section  modulus " ;    X,    "        "        x,     "        "    =  ~^1* 

B,  R' =  radius  of  gyration ;  R,    "        "       R%    "        " 

r  =  least  radius  of  gyration,  about  neutral  axis  forming  acute  angle  a 

with  each  flange.    In  angles  with  equal  legs,  a  =  45° 

C  ==  coefficient  for  uniformly  distributed  safe  load :  ^ 

Cg  for  static  loads ;  fibre  stress  =  16,000  ft>s.        >For  T  shapes  only. 
Cj^,  for  moving  loads  ;  fibre  stress  =  12,000  lbs.  j 


Size 
H     W 

ins.  ins. 


Thick- 
nesss 
ins. 


Weight 

per  ft 

fi)s. 


Area  of 
section 
sq  ins. 


d 

ins. 


Angles  witli  Equal  Jjeg9m 


X8 
X8 
X6 
X6 
X5 
X5 
X4 
X4 
3KX3K 
3>|X3>| 
3  X3 
3X3 
2^X2|^ 
2^  X  "'^■ 


X2K 


2^X21^ 
2^X21^ 


i^/lxl^/l 
i>|xi>l 

134  XiM 
1^X1^ 


56.9 
26.4 
37.4 
14.8 
30.6 
12.3 
19.9 
8.2 
17.1 
7.1 
11.4 
4.9 
8,5 
4.5 
7.7 
3.1 
6.8 
2.8 
5.3 
2.5 
4.6 
2.1 
3.4 
1.2 
2.4 
1.0 
1.5 
0.8 
1.0 
0.7 
0.8 
0.6 


16.73 
7.75 

11.00 
4.36 
9.00 
3.61 
5.84 
2.40 
5.03 
2.09 
3.36 
1.44 
2.50 
1.31 
2.25 
0.90 
2.00 
0.81 
1.56 
0.72 
1.30 
0.62 
0.99 
0.36 
0.69 
0.30 
0.44 
0.24 
0.29 
0.21 
0.25 
0.17 


2.41 
2.19 
1.86 
1.64 
1.61 
1.39 
1.29 
1.12 
1.17 
0.99 
0.98 
0.84 
0.87 
0.78 
0.81 
0.69 
0.74 
0.63 
0.66 
0.57 
0.59 
0.51 
0.51 
0.42 
0.42 
0.35 
0.34 
0.30 
0.29 
0.26 
0.26 
0.23 


97.97 
48.63 
35.46 
15.39 
19.64 
8.74 
8.14 
3.71 
5.25 
2.45 
2.62 
1.24 
1.67 
0.93 
1.23 
0.55 
0.87 
0.39 
0.54 
0.28 
0.35 
0.18 
0.19 
0.08 
0.09 
0.044 
0.037 
0.022 
0.019 
0.014 
0.012 
0.009 


T  Shapes. 


13.6 
15.6 
9.3 
11.7 
11.8 
8.5 
7.2 
4.9 
3.1 

3.99 
4.56 
2.73 
3.45 
3.48 
2.49 
2.10 
1.44 
0.90 

0.75 
1.56 
0.78 
1.06 
1.32 
1.09 
0.97 
0.69 
0.54 

2.6 
10.7 
2.0 
3.7 
5.2 
2.9 
1.8 
0.66 
0.23 

5.6 

2.8 

2.1 

1.89 

1.21 

0.93 

0.54 

0.33 

0.12 


*  Special  sections,    f  For  M  and  S  see  p.  892  or  p.  894. 


ANGLES   AND   T   SHAPES. 


ANI>  T  SHAPES.— Continued. 


^ 


-W- 


-W- 


X 

X 

R 

B' 

r 

^s 

Cm 

Section 
Index. 

Maximum  and  Minimum  Weight  of  each  Section. 

17.53 
8.37 
8.57 
3.53 
5.80 
2.42 
3.01 
1.29 
2.25 
0.98 
1.30 
0.58 
0.89 
0.48 
0.73 
0.30 
0.58 
0.24 
0.40 
0.19 
0.30 
0.14 
0.19 
0.070 
0.109 
0.049 
0.056 
0.031 
0.033 
0.023 
0.024 
0.017 


2.42 
2.50 
1.80 
1.88 
1.48 
1.56 
1.18 
1.24 
1.02 
1.08 
0.88 
0.93 
0.82 
0.85 
0.74 
0.78 
0.66 
0.70 
0.59 
0.62 
0.51 
0.54 
0.44 
0.46 
0.36 
0.38 
0.29 
0.31 
0.26 
0.26 
0.22 
0.23 


55 
58 
16 
19 
,96 
99 
77 
79 
67 
69 
1.57 
59 
52 
55 
,47 
49 
,43 
44 
39 
40 


Selected  Sections. 


1.18 
3.10 
0.88 
1.52 
1.94 
1.21 
0.87 
0.42 
0.19 


2.22 
1.41 
1.05 
1.08 
0.81 
0.62 
0.43 
0.30 
0.14 


0.82 
1.54 
0.86 
1.04 
1.23 
1.09 
0.92 
0.68 
0.51 


1.19 
0.79 
0.88 
0.74 
0.59 
0.61 
0.51 
0.48 
0.37 


12,550 

33,070 

9,430 

16,210 

20,650 

12,910 

9,280 

4,480 

2,050 


9,410 

24,800 

7,070 

12,160 

15,480 

9,680 

6,960 

8,360 

1,540 


*  Special  sections. 


900 


SEPARATORS    FOR    I    BEAMS. 


€ARN£OI£   STANDARD   CAfST   IRON   SEPARATORS   FOR 
I  BEAMS. 


Separators  for  18^',  2(y'  and  24"  beams  are  made  of  %''  metal. 
"  "     6''  to  15''  beams  are  made  of  3^''  metal. 

"  "     5''  beams  and  under  are  made  of  %"  metal. 


Designation 
OP  Beam. 

Distances. 

Bolts. 

Weights. 

c 

c 
o 

1 

4 

1 

"Si 

SB 

i 

il 

8 

1 

u 

^  '-IS 

4^  w  52 

iJJ 

^  1 

o 

■i   i 

!J 

III 

1  i 

Ins. 

Lbs. 

Ins. 

Ins. 

In. 

Ins. 

Ins. 

Lbs. 

Lbs. 

Lbs 

Lbs. 

Separators  witb  Two  Bolts. 


B  1 
B  2 
B  3 

B  80 
B  4 


24 

20 
20 
18 
15 
15 
15 
12 
12 


80. 
65. 
55. 
80. 
60. 
42. 
40. 
31.5 


7 
6% 

6% 


12 
10 
10 
9 

6 


9M 

8M 

9 

8K 

7>| 

7>| 

7H 


3.41 

.250 

32. 

3.41 

28. 

3.23 

25. 

3.16 

16. 

3.35 

15. 

3.23 

15. 

2.98 

15. 

2.98 

11. 

2.92 

11. 

Separators  with  One  Bolt. 


B  8 
B  9 
B  11 
B  13 
B  15 

B  17 
B  19 
B  21 
B  23 

B  77 


12 

12 
10 
9 

8 

7 
6 
5 
4 


40.0 
31.5 
25.0 
21.0 
18.0 

15.0 
12.25 
9.75 
7.50 
5.50 


11^ 

lOM 

i 

5^ 


6 

5^ 

r^ 

3M 


7H 

1.49 

'^. 

1.46 

1.40 

1 

1.34 
1.28 

5y^ 

1.25 

'4 

Wo 

1.22 

1.16 

1.13 

4^ 

0.70 

.125 


Z  BAR    COLUMNS. 


CARNEOIE  STEEIi  Z-BAR  COIiUMXS. 
Table  of  dimensions,  in  inches. 

U- -H >\ 


Diameter  of  bolt  or     By'^^ 
rivet  =  %  inch.  i 

k 


For  area  of  section- 
weight  per  yard,  least 
radius  of  gyration  and 
safe  load,  see  tables,  pp? 
902  and  903. 


Thickness 

See  figure  above. 

s 

of 
Metal. 

A 

B 

0 

D 

E 

F 

0 

H 

I 

S 

9 

i 

12A 

3J 

5A 

2J. 

2i 

1 

2-i 

8i 

3? 

I 

A 

12f 

3A 

5A 

2J 

2i 

2f 

8f 

3 

« 

^ 

1 

12A 

3A 

5A 

2* 

2^ 

1- 

2Ti 

8:- 

3 

i^ 

12i 

3/i 

5A 

2-- 

2:, 

1 

21 

8-- 

3 

8 

i 

12 

3i 

5A 

2-. 

2^ 

1 

2-i 

8 

3^ 

« 

A 

12A 

3ii 

5A 

2f 

2^ 

1 

2j 

7f 

3f 

m 

i 

14Ti 

H 

6A 

3 

3 

IJ 

3A 

^^ 

4J 

8 

A 

14t 

4A 

6A 

3 

■ 

3 

1| 

3* 

9^ 

4^ 

s 

f 

14il 

4i^K 

6A 

3 

3 

If 

3A 

9 

45- 

s 

A 

14^ 

4/j 

of 

3 

3 

If 

3A 

9 

4tV 

i 

14?ff 

4A 

5f 

3, 

3 

1^ 

H 

9 

4t% 

A 

A 

14| 

4M 

5J 

3. 

■ 

3 

IJ 

3A 

8f 

4H 

1 

14i 

4A 

5^ 

-J 

3- 

3 

1} 

3A 

8- 

4f 

• 

H 

14A 

4|' 

5 

1 

h 

3 

If 

H 

a 

4f 

« 

f 

14f 

5 

i 

3: 

3 

1- 

3A 

8^ 

4f 

OB 

A 

16J 

5A 

6i^^ 

3^ 

- 

h 

If 

3J 

lOf 

5A 

8 

s 

s 

i 

16A 

5? 

6t^^ 

3, 

■ 

H 

If 

3A 

10 

5A 

A 

16f 

5H. 

6t^^ 

3. 

■ 

3? 

If 

3f 

n 

5tV 

i 

16f 

5? 

6? 

3. 

r 

3? 

If 

3i 

9f 

5} 

9 

A 

16A 

54 

6i 

3, 

■ 

3i 

If 

3A 

9f 

5f 

•g 

1 

16i 

5A 

ej 

3. 

■ 

3i 

If 

3f 

9^ 

5  ■ 

6 

H 

16A 

5ii 

6tV 

3. 

: 

31 

If 

H 

9f 

5H 

•P4 

i 

16i 

5A 

6tV 

3^ 

r 

3i 

If 

3A 

9?     StI 

M 

16A 

5H 

6tV 

3i 

': 

3i 

If 

3| 

9^     5t| 

7 

f 

18| 

6A 

7-8- 

4 

4 

2 

3i 

Ht 

6f 

a 

A 

IStI 

6A 

7^ 

4 

4 

2 

3A 

Hi 

6f 

S 

i 

19 

6| 

7i 

4 

4 

2 

3f 

11 

6f 

8 

A 

18H 

6A 

6if 

4 

4 

2 

4 

lOf 

6^ 

0 

1 

18i 

6f 

^H 

4 

4 

2 

3A 

lOf 

6  4 

a 

a 

18i| 

6M 

m 

4 

4 

2 

3f 

lOf 

^rl 

8 

i 

ISA 

6| 

6f 

4 

4 

2 

3i 

m 

6? 
6| 

•P4 

a 

18| 
18H 

6il 

6f 

4 

4 

2 

3A 

lOf 

i 

6A 

6| 

4 

4 

2 

3f 

lOJ 

7 

902 


Z-BAR   COLUMNS. 


CARNEOIi:  STEEI.  Z-BAR  COIiUMNS. 

Table  of  ISafe  Lioads  as  given  by  Carnegie  Steel  Co.,  for  columns  with 
square  ends.  Safety  factor  =  4.  The  loads  given  are  based  upon  the  following 
allowed  strains  in  pounds  per  square  incli  : 

Each  Z-bar  column  is  made  up  of  four  Z-bars  and  one  web-plate  (all  of  uni- 
form thickness)  bolted  or  riveted  together,  as  shown  in  the  figure  on  page  901, 

6-incli  Steel  Z-bar  Colamns. 

Composed  of  four  Z-bars  about  3  inches  deep  and  one  web-plate  5%  inches  wide. 


Thickness  of  metal, ) 
inch.                       J 

i 

A 

f 

tV 

i 

A 

Area  of  section,  sq. ) 
ins.                         i 

9.31 

11.7 

13.6 

16.0 

17.6 

20.0 

Weight  per    yard,) 
pounds.                  ) 

95.1 

119.4 

138.6 

162.9 

179.7 

203.7 

Least  rad  of  gyr,     ) 
inches.                   / 

1.86 

1.90 

1.88 

1.93 

1.90 

1.9o 

Length  of  column. 
Feet. 

Safe  load  of  column,  in  pounds. 

12  or  less. 
14 
16 
18 
20 
22 
24 
26 
28 
30 

111800 
111400 
104600 
07600 
90800 
84000 
77200 
70400 
63400 
56600 

140600 
140600 
133000 
124600 
116200 
107800 
99400 
91000 
82600 
74200 

163200 
163200 
153200 
143400 
133400 
123600 
113800 
103800 
94000 
84000 

191600 
191600 
182600 
171200 
159800 
148600 
137200 
126000 
114600 
103400 

211400 
211400 
199800 
187200 
174400 
161800 
149200 
136400 
123800 
111000 

239600 
239600 
229600 
215600 
201600 
187600 
173600 
159600 
145600 
131600 

8-ineli  Z-bar  Columns. 

Composed  of  four  Z-bars  about  4  inches  deep  and  one  web-plate  6^^  inches  wide. 


Thickness  of  metal, ) 
inch.                      j 

i 

A 

f 

A 

i 

A 

1 

tt 

f 

Area  of  section,  sq.  \ 
ins.                         j 

11.3 

14.1 

17.1 

19.0 

21.9 

24.8 

2r..3 

29.0 

31.9 

Weight  per    yard,| 
pounds.                 j 

114.9 

144.3 

174.0 

194.1 

221.1 

252.3 

267.6 

296.4 

325.2 

Least  rad  of  gyr,) 
inches.                  j" 

2.47 

2.52 

2.57 

2.49 

2.55 

2.60 

2.52 

2.58 

2.63 

Length  of  column. 
Feet. 

Safe  load  of  column,  in  pounds. 

18  or  less. 
20 
22 
24 
26 
28 
30 
32 
34 
36 
38 
40 


135000 
130000 
123800 
117600 
111400 
105200 
98800 
92600l 
86400| 
802001 
740001 
67800 


169600 
165000 
157400 
149600 
142000 
134200 
126600 
119000 
111200 
103600 
96000 
88200 


204800  228400 


201000 
191800 
182600 


221000 
210600 
200200 


173600!  189600 
164600  179200 
155400!  168800 
1464001158400 
1374001148000 
128200' 137400 
119200127000 
110000  116600 


262400 
256400 
244800 
233000 
221200 
209400 
197600 
186000 
174200 
162400 


2970001315000 
2928001306600 
279800i292400 
266800i278200 
2538001264000 


240600 
227600 
214600 
201600 
188600 


150600  175600 
139000  162600 


249600 
235400 
221200 
207000 
192800 
178800 
164400 


348600 
342600 
327000 
311600 
296200 
280800 
265400 
250000 
234600 
219200 
203800 
188400 


382400 
379200 
362600 
346000 
329400 
312800 
296400 
279800 
263200 
246600 
230000 
213400 


Z-BAR    COLUMNS. 


903 


CARNEOIE  ST££Ii  Z-BAR  COIiUMNS. 

Table  of  Safe  liOads  (continued). 

lO-inch  Steel  Z-bar  Columns. 

Composed  of  four  Z-bars  about  5  inches  deep  and  one  web-plate  7  inches  wide. 


Thickness  of  metal, ) 
inch.         i 

A 

f 

tV 

i 

A 

i 

H 

f 

« 

Area  of  section,  sq. ) 
ins.          i 

15.8 

19.0 

22.3 

24.5 

27.7 

30.9 

32.7 

35.8 

39.0 

Weight  per  yard, ) 
pounds.       j 

161.1 

194.1 

227.4 

249.9 

282.6 

316.6 

330.0 

368.4 

397.8 

Least  rad  of  gyr,  "1 
inches.        j 

3.08 

3.13 

3.18 

3.10 

3.15 

3.21 

3.13 

3.18 

3.25 

Length  of  column. 
Feet. 

Safe  load  of  column,  in  pounds. 

22  or  less. 
24 
26 
28 
30 
32 
34 
36 
38 
40 
42 
44 
46 
48 
•50 

189400  228400 
185600  225200 
178600  217200 
171600  208800 
164600  200400 
157600  192200 
150600183800 
143600  175600 
136600167200 
129600158800 
122600:150600 
115400142200 
1084001134000- 
101400!125600| 
944001172001 

267800 
266200 
256600 
247000 
237400 
227600 
218200 
208600 
199000 
189400 
179800 
170200 
160600 
151000 
141400 

294000 
289200 
278400 
267600 
256800 
246000 
235200 
224400 
213600 
2028001 
192000: 
1812001 
170400 
159600, 
148800 

332400 
329600 
317400 
305400 
293400 
281400 
269400 
257400 
245400 
233400 
221200 
209200 
197200 
185200 
173200 

371200 
370600 
357400 
844200 
331000 
317800 
304600 
291400 
278200 
265000 
251800 
238600 
225400 
212200 
199000 

392000 
387200 
373000 
358600 
344400 
330000 
315800 
301400 
287200 
27.3000 
258800 
244400 
230200 
215800 
201600 

429800 
427800 
412400 
397000 
381600 
366200 
350800 
335600 
320000 
304600 
289200 
273800 
258400 
243000 
227600 

468000 
468000 
453200 
436800 
420400 
404000 
387600 
371200 
354800 
338200 
321800 
305400 
289000 
272600 
256200 

12-iiicli  Steel  Z-bar  Columns. 

Composed  of  four  Z-bars  about  6  inches  deep  and  one  web-plate  8  inches  wide. 

Thickness  of  metal, ) 
inch.         / 

i 

A 

i 

A 

1 

a 

f 

If 

i 

Area  of  section,  sq.  \ 
ins.          J 

21.4 

25.0 

28.8 

31.2 

34.8 

38.5 

40.5 

44.1 

47.7 

Weight  per  yard,) 
pounds.       J 

218.1 

255.6 

293.4 

318.6 

355.5 

392.7 

413.4 

449.7 

486.3 

Least  rad  of  gyr,! 
inches.        j 

3.67 

3.72 

3.77 

3.70 

3.75 

3.73 

3.68 

3.66 

3.64 

Length  of  column. 
Feet. 

Safe  load  of  column,  in  pounds. 

26  or  less. 
28 
30 
32 
34 
36 
38 
40 
42 
44 
46 
48 
50 

256600|300600 
254000  299400 
246000 1290200 
238000,281000 
230200,271800 
222200 1262600 
2142001253400 
2062001244200 
1982001235000 
1902001225800 
182400216600 
174400  i  207200 
1664001 198200 

345200 
345000 
835200 
324800 
314400 
304000 
293600 
283000 
272600 
262200 
252400 
241400 
231000 

374600  418200 
372000  417800 
360400  405000 
349000  392200 
337400379600 
325800366800 
314200354000 
302800i341400 
291000)328800 
2796001316000 
2680001303200 
256400290600 
244800:277800 

462000i486000 
460600:481600 
4466001466400 
432600451400 
4184001436400 
4042001421200 
390200!406200 
376000  391200 
361800I376000 
347800i361000 
383600I345800 
3196001330800 
305400J315800 

529000 

522800 
506400 
490000 
473400 
456800 
440400 
423800 
407400 
391000 
3744001 
358000: 
341400 

572200 
564200 
546400 
528400 
510400 
492600 
474600 
456600 
438800 
420800 
402800 
384800 
367000 

For  loads  g-reater  tban  tbose  griven  in  tbe  tables,  the  Z-bar 

columns  may  be  re-enforced  by  additional  plates,  riveted  to  the  flanges.  The 
addition  of  such  plates  does  not  in  any  case  diminish  the  least  radius  of  gym- 
tion.    Hence  the  same  load  per  square  inch  of  cross-section  may  be  used. 


904 


PHCENIX    COLUMNS. 


Table  of   rolled-iroo    seg^ment-colamns    of   tbe   Phoenix 
Iron  Co,  410    Walnut    St,   Philada. 


The  dimensions  givea 

are  subject  to  slight  variations  which  are  unavoidabU 

in  rolling  iron  shapes.   The  iveig^hts  of  columns  given 

are  those  of  the  4,  6,  or  8  segments,  of  which  they  ar« 

composed.    The  shanks  of  the  rivets  used  in  joining  them 

together,  of  course,  merely  make  up  the  quantity  of  metal 

j  punched  or  drilled  out,  in  making  the  holes ;  but  the  rivet- 

{  heads  add  from  2  to  5  per  cent  to  the  weights  given.  The 

I  rivets  are  spaced  3,  4,  or  6  ins  apart  from  cen  to 

'  cen. 

Any  desired  thickness  between  the  minimum 
and  maximum  for  any  given  size,  can  be  furnished. 
We  give  the  dimensions,  weights,  &c,  corresponding  t% 
-it  the  principal  thicknesses.  G  columns  have  8  segment^ 
E,  6  segments.    All  others,  4  segments. 


Bi 


S 


Diameters,  ins. 


»/ 


One  column. 


Area 

of  cross 

sec, 
sq  ins. 


Wtper 

ft  run, 

lbs. 


Lieast 
radof 

ins. 


Size  of 
Biveti. 


3% 

4H 


14^ 


r^ 


tk 
^ 


4 

12A 
I2i| 

20% 
21 


3.8 
4.8 
5.8 
6.8 
6.4 
9.2 
12. 
14.8 
7.4 
10.6 
13.8 
17. 
10. 
18. 
25.2 
33.2 
41.2 
16.8 
26.4 
37.8 
49.8 
61.8 
24. 
36. 
52. 
68. 
92. 


12.6 

16. 

19.3 

22.6 

21.3 

30.6 

40. 

49.3 

24.6 

36.3 

46. 

56.6 

33.3 

60. 

84. 

110.6 

137.3 

56. 

88. 

126. 

166. 

206. 

80. 

120. 

173.3 

226.6 

306.6 


1.45 
1.50 
1.56 
1.59 
1.92 
2.02 
2.11 
2.20 
2.34 
2.43 
2.52 
2.61 
2.80 
2.98 
3.16 
3.34 
3.52 
4.18 
4.36 
4.55 
4.73 
4.91 
5.46 
5.59 
5.77 
5.95 
6.23 


THE   GRAY   COLUMN. 


905 


THE  GRAY  COIilJMIir. 

The  Gray  Coliimn,  designed  and  patented  by  Mr.  J.  H.  Gray,  consists, 
in  its  original  form,  of  angles,  connected  at  intervals  D  (generally  of  2  ft  6  ins) 
by  transverse  bent  tie-plates  T,  usually  9  X  %  ins.  This  construction  renders 
the  parts  of  the  column  easily  accessible,  for  painting,  etc.,  but  under  trans- 
verse or  buckling  stresses  the  column  must  act  somewhat  like  a  rectangular 
frame  without  diagonals.  To  remedy  this,  a  later  form,  the  "twelve-angle" 
column.  Fig  5,  has  been  designed,  having,  in  the  square  column.  Figs  1  and  2, 
instead  of  the  bent  tie-plates  T,  four  additional  angles,  running  longitudinally 
like  the  others,  and  placed  centrally,  as  shown.  These  angles  supply  the  column 
with  two  webs,  intersecting  at  right  angles. 

Figs  1  and  2  show  the  square  column,  used  in  the  interior  of  buildings.  It  is 
used  chiefly  in  the  14,  15,  and  16  inch  sizes.  Fig  2  shows  plates  riveted  to  the 
outer  flanges,  as  is  done  in  some  of  the  heaviest  columns.  Figs  3  and  4  show  the 
wall  and  corner  columns  respectively.  Fig  5  shows  the  '*  twelve-angle"  column, 
and  Fig  6  one  of  many  forms  of  fireproofing,  with  plaster  laid  on  terra  cotta 
blocks.    The  rivets  are  usually  %  inch  in  diameter. 

The  safe  load,  in  pounds  per  sq  inch,  of  the  ordinary  column,  is  stated  as 

17,100  —  57  — ,  where  L  =  length  of  column  and  r  =  its  least  rad  of  gyration, 
r 
The  cost  of  the  Gray  column,  at  shop,  is  from  1  to  1.5  cents  per  S)  plus  tha 
eost  of  the  angles. 


rig.  6. 


FifiT.  1. 


Fig.  4. 


906 


THE   GRAY   COLUMN. 


Oray  Colninn.    liist  of  Selected  Sizes. 

Size 

S 

ins. 

Angles 
ins. 

Area 
sq  in. 

I 

Mom 
of  in. 

r 

Least 

rad. 

of  gyr. 

ius. 

r2 

Safe  loads*  for 

column  lengths 

of 

12  ft.       30  ft. 

Square  Columns.    Figs.  1  and  2. 


8.48 

64 

2.7 

7.3 

115 

18.00 

119 

2.6 

6.8 

250 

9.52 

95 

3.1 

9.6 

135 

20.00 

179 

3.0 

9.0 

285 

16.88 

241 

3.8 

14.4 

250 

30.00 

327 

3.3 

10.9 

435 

16.88 

285 

4.1 

16.8 

255 

43.52 

552 

3.6 

13.0 

645 

16.88 

336 

4.5 

20.2 

255 

29.36 

526 

4.3 

18.5 

445 

26.00 

444 

4.2 

17.6 

390 

30.00 

463 

4.0 

16.0 

450 

31.92 

597 

4.4 

19.4 

485 

39.36 

624 

4.0 

16.0 

590 

36.00 

526 

3.8 

14.4 

535 

22.00 

496 

4.8 

23.0 

335 

31.92 

693 

4.7 

22.1 

490 

39.36 

731 

4.4 

19  .*4 

600 

30.00 

653 

4.7 

22.1 

440 

41.84 

817 

4.4 

19.4 

635 

46.88 

828 

4.2 

17.6 

710 

22.00 

570 

5.1 

26.0 

340 

36.88 

912 

5.0 

25.0 

570 

55.52 

1134 

4.6 

21.2 

850 

22.00 

746 

5.8 

33.6 

345 

36.88 

1182 

5.7 

32.5 

575 

46.88 

1465 

5.6 

31.4 

730 

36.88 

1485 

6.4 

41.0 

580 

67.52 

2588 

6.2 

38.4 

1065 

41.80 

4147 

9.9 

98.0 

680 

264.40 

22688 

9.2 

84.6 

4285 

Wall  Columns.    Fig.  3. 


12 

3      X4 

X    Ys 
zX    K 

14.88 

94 

2.5 

6.2 

205 

130 

14 

3      X3> 

18.00 

160 

3.0 

9.0 

255 

185 

*♦ 

33^X5 
33^X4 

29.52 

241 

2.9 

8.4 

420 

295 

15 

X  >! 

21.00 

'111 

3.3 

10.9 

305 

225 

" 

4X5 

31.38 

317 

3.2 

10.2 

455 

335 

16 

3      X3 

16.50 

200 

3.5 

12.2 

240 

185 

<< 

4X6 

X  Y^ 

35.16 

375 

3.3 

10.9 

510 

380 

18 

4      X4 

X  % 

27.66 

434 

4.0 

16.0 

415 

330 

20 

4X5 

X  % 

31.38 

562 

4.2 

17.6 

475 

380 

30 

5      X4 

X  % 

31.38 

1408 

6.8 

46.2 

495 

440 

" 

6      X6 

xif 

198.30 

8937 

6.7 

44.9 

3150 

2785 

Corner  Columns.    Fig.  4. 


10^ 

3KX3i^X    \% 

23.48 

272 

3.4 

11.6 

345 

260 

13     1 

3Kx3i^x  y^ 

15.75 

288 

4.3 

18.5 

240 

195 

13  1 

14  1 

4X5X5^ 

24.91 

425 

4.2 

17.6 

375 

305 

*  In  thousands  of  lbs,  by  formula :    Safe  load  =  17100  —  57  -  in  Bbs  per  sq  in. 
t  Three  l-in  plates  riveted  to  each  pair  of  angles.    Fig.  2. 


STRENGTH    OF   IRON    PILLARS. 


907 


TABLE  OF   BR£AKIXO   I.OABS  OF  IRON  PIL.I.ARS, 

in  tons  per  square  inch  of  metal  area.  Deduced  from  Gordon.  The  ends  ftra 
•upposed  to  be  planed  to  form  perfectly  true  bearings ;  and  all  parts  to  be  equally 
pressed.  The  last  is  rarely  the  case  in  practice.  If  tlie  pillar  is  rectangu- 
lar instead  of  square,  use  the  least  side  for  a  measure  of  length.  (Original.) 


Hollow 

Hollow 

Solid 

Solid 

Round. 

Square. 

Round « 

Square. 

Length 
measd 



Length 
measd 

Breakg  loads  per 

in  sides 

Breakg  loads  per 

Breakg  loads  per 

in  sides 

Breakg  loads  per 

sq  inch  of  metal 

or 

sq  inch  of  metal 

sq  inch  of  metal 

sq  inch  of  metal 

area  of 

diams. 

area  of 

area  of 

diams. 

area  of 

trans  vers 

e  section. 

transverse  section. 

transverse  section. 

trans  vers 

e  section. 

Cast. 

Wrt. 

Cast. 

Wrt. 

Tons. 

Cast. 

Wrt. 

Cast. 

Wrt. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

S5.7 

16.1 

1 

35.7 

16.0 

.35.6 

16.1 

1 

35.6 

16.1 

35.5 

16.1 

2 

35.5 

16.0 

35.2 

16.1 

2 

35.4 

16.0 

35.2 

16.1 

3 

35.3 

16.0 

34.6 

16.0 

3 

34.9 

16.0 

34.8 

16.0 

4 

35.0 

16.0 

33.9 

15.9 

4 

34.3 

16.0 

34.3 

16.0 

5 

34.6 

16.0 

33.0 

15.9 

5 

33.6 

16.0 

33.7 

16.0 

6 

34.2 

16.0 

'    31.9 

15.8 

6 

32.8 

15.9 

33.0 

15.9 

7 

33.7 

16.0 

30.6 

15.7 

7 

31.8 

15.8 

32.3 

15.8 

8 

33.1 

15.9 

29.3 

15.6 

8 

30.8 

15.7 

31.5 

15.8 

9 

32.4 

15.9 

28.0 

15.5 

9 

29.7 

15.7 

30.6 

15.7 

10 

31.7 

15.8 

26.7 

15.4 

10 

28.6 

15.6 

29.7 

15.6 

11 

31.0 

15.8 

25.4 

15.2 

11 

27.4 

15.5 

28.8 

15.5 

12 

30.3 

15.7 

24.1 

15.1 

12 

26.3 

16.3 

27.9 

15.5 

13 

29.5 

15.7 

22.8 

14.9 

13 

25.1 

15.2 

27.0 

15.4 

14 

28.7 

15.6 

21.6 

14.8 

14 

24.0 

15.1 

26.0 

15.3 

15 

27.9 

15.5 

20.5 

14.6 

15 

22.9 

15.0 

25.1 

15.2 

16 

27.1 

15.5 

19.4 

14.4 

16 

21.8 

14.8 

24.2 

15.1 

17 

26.3 

15.4 

18,3 

14.2 

17 

20.7 

14.7 

33.3 

15.0 

18 

25.4 

15.3 

17.2 

14.0 

18 

19.7 

14.5 

22.3 

14.9 

19 

.  24.6 

15.2 

16.2 

13.8 

19 

18.8 

14.3 

21.4 

14.8 

20 

23.8 

15.1 

15.3 

13.6 

20 

17.9 

14.2 

20.6 

14.7 

21 

23.0 

15.0 

14.5 

13.4 

21 

17.0 

14.0 

19.8 

14.6 

22 

22.2 

14.9 

13.7 

13.2 

22 

16.2 

13.8 

19.0 

14.4 

23 

21.5 

14.8 

12.9 

13.0 

23 

15.4 

13.6 

18.3 

14.3 

24 

20.8 

14.7 

12.2 

12.8 

24 

14.6 

13.5 

17.5 

14.2 

25 

20.1 

14.6 

11.6 

12.6 

25 

13.9 

13.3 

16.8 

14.0 

26 

19.4 

14.5 

11.0 

12.4 

26 

13.3 

13.1 

16.2 

13.9 

27 

18.7 

14.4 

10.4 

12.1 

27 

12.7 

12.9 

15.5 

13.7 

28 

18.0 

14.3 

9.90 

11.9 

28 

12.1 

12.7 

14.9 

13.5 

29 

17.4 

14.2 

9.41 

11.7 

29 

11.5 

12.6 

14.3 

13.4 

80 

16.8 

14.0 

8.93 

11.5 

30 

11.0 

12.4 

13.8 

13.2 

31 

16.3 

13.9 

8.50 

11.3 

31 

10.5 

12.2 

13.2 

13.1 

32 

15.7 

13.8 

8.10 

11.1 

32 

10.0 

12.0 

12.7 

12.9 

33 

15.2 

13.6 

7.70 

10.8 

33 

9.59 

11.8 

12.3 

12.8 

34 

14.6 

13.5 

7.36 

10.6 

34 

9.18 

11.6 

11.7 

12.6 

35 

14.1 

13.3 

7.04 

10.4 

35 

8.79 

11.4 

11.3 

12.5 

36 

13.6 

13.2 

6.71 

10.2 

36 

8.42 

11.2 

10.9 

12.3 

37 

13.2 

13.1 

6.41 

10.0 

37 

8.07 

11.0 

10.5 

12.2 

38 

12.7- 

13.0 

6.11 

9.79 

38 

7.75 

10.8 

10.1 

12.0 

39 

12.3 

12.9 

5.85 

9.59 

39 

7.44 

lO.T 

9.75 

11.9 

40 

11.9 

12.7 

5.62 

9.39 

40 

7.14 

lO.S 

9.40 

11.7 

41 

11.5 

12.6 

5.40 

9.20 

41 

6.86 

10.1 

9.07 

11.6 

42 

11.1 

12.4 

5.19 

9.00 

42 

6.60 

10.1 

-8.76 

11.4 

43 

10.8 

12.3 

4.99 

8.81 

43 

6.35 

9M 

8.46 

11.3 

44 

10.4 

12.2 

4.79 

8.64 

44 

6.11 

9.7T 

8.16 

11.1 

45 

10.1 

12.0 

4.59 

8.46 

45 

5.89 

9.69 

7.88 

10.9 

46 

9.80 

11.9 

4.42 

8.27 

46 

5.68 

9.42 

7.61 

10.8 

47 

9.50 

11.7 

4.26 

8.10 

47 

5.48 

9.25 

7.36 

10.6 

48 

9.20 

11.6 

4.11 

7.94 

48 

5.28 

9.09 

7.13 

10.4 

J9 

8.94 

11.4 

3.97 

7.76 

49 

5.10 

8.92 

6.91 

10.3 

50 

8.66 

11.3 

3.84 

7.60 

50 

4.92 

8.77 

6.90 

9.61 

55 

7.47 

10.7 

3.22 

6.86 

55 

4.16 

8.00 

6.10 

8.92 

60 

6.49 

10.0 

2.75 

6.18 

60 

3.57 

7.30 

4.44 

8.29 

65 

5.68 

9.43 

2.37 

5.59 

65 

3.09 

6.67 

3.90 

7.69 

70 

5.01 

8.84 

2.06 

5.06 

70 

2.70 

610 

3.44 

7.14 

75 

4.44 

8.30 

1.80 

4.59 

75 

2.37 

5.59 

8.08 

6.64 

80 

3.97 

7.77 

1.59 

4.18 

80 

2.10 

5.13 

2.46 

5.73 

90 

3.21 

6.88 

1.27 

3.49 

90 

1.68 

4.34 

2.02 

4.99 

100 

2.65 

6.02 

1.04 

2.95 

100 

1.37 

3.71 

908 


STRENGTH   OF   IRON   PILLARS. 


Table  1.     B[OIiI.OW  CTUNB  CAST  IRON  PIIiliARS. 

Breaking:  loa<l»«,  flat  ends,  perfectly  true, and  firmly  fixed; 
and  the  loads  pressing;  equally  on  every  part  of  tbe  top. 

By  Gordon's  formula. 
For  dianDS  or  leng^ttis  intermediate  of  tbose  in  tbe  table,  the 

locds  may  be  found  near  enough  by  simple  proportion. 

For  tbicbnesses  less  tban  those  in  the  table,  the  breaking  loadB 

may  safely  be  assumed  to  diminish  in  the  same  proportion  as  the  thickness,  while  the  outer  diam 
remains  the  same.  But  for  greater  thicknesses  than  those  in  the  table,  the  loads  do  not  increase 
as  rapidly  as  the  new  thickness.     Still,  in  practice,  they  may  be  assumed  to  do  so  approximately, 

if  the  new  thickness  does  not  exceed  aboat  3^  part  of  th« 
outer  diam. 


CAST  IKON. 

THICKNESS  H  INCH.    (OrigiMl.) 

1 

Outer  Diameter  in  inches. 

2 

2M 

2H 

2H 

3 

3^ 

4            iH     , 

5 

iH 

6 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

45.2 

52.5 

59.8 

66.8 

74.1 

88.5 

103.0 

117.1 

131.3 

145.5 

159.8 

36.2 

43.4 

51.0 

58.6 

66.5 

81.5 

96.7 

111.3 

126.9 

140.5 

155.2 

27.2 

34.1 

41.4 

48.8 

56.7 

71.9 

87.6 

102.7 

117.8 

132.9 

148.1 

20.2 

26.3 

32.9 

39.7 

47.0 

61.9 

77.5 

92.9 

108.3 

123.7 

139.1 

15.1 

20.2 

25.9 

32.0 

38.6 

52.6 

67.4 

82.6 

97.9 

113.4 

129.1 

11.6 

15.8 

20.7 

25.9 

31.6 

44.3 

58.2 

72.7 

87.7 

lOS.O 

118.7 

9.1 

12.5 

16.6 

21.0 

26.1 

37.4 

50.1 

6.3.7 

78.1 

92.9 

108.3 

7.3 

10.1 

13.5 

17.3 

21.7 

31.6 

48.2 

55.8 

69.3 

83.4 

98.4 

5.9 

8.2 

11.1 

14.3 

18.2 

26.9 

87.3 

48.8 

61.5 

74.9 

89.2 

4.9 

6.9 

9.3 

12.0 

15.4 

23.1 

32.4 

42.9 

54.6 

67.1 

80.7 

4.1 

5.8 

7.9 

10.3 

13.2 

20.0 

28.3 

37.8 

48.6 

60.3 

73.0 

3.5 

5.0 

6.8 

8.9 

11.4 

17.4 

24.9 

33,5 

43.3 

54.2 

66.1 

3.0 

4.2 

5.8 

7.6 

9.9 

15.2 

21.9 

29.7 

38.8 

48.8 

60.0 

2.6 

3.6 

5.1 

6.7 

8.7 

13.5 

19.5 

26.6 

34.9 

44.1 

54.5 

2.3 

8.2 

4.5 

6.0 

7.7 

11.9 

17.4 

23.9 

31.4 

39.9 

49.7 

2.0 

2.8 

4.0 

5.3 

6.9 

10  7 

15.6 

21.5 

28.4 

36.4 

45.6 

1.6 

2.8 

3.2 

4.2 

5.5 

8.6 

12.7 

17.5 

23.4 

30.2 

88.0 

1.3 

1.8 

2.6 

3.4 

4.5 

7.1 

10.5 

14.7 

19.7 

25.6 

32.3 

1.7 

2.8 

3.0 

4.7 

7.0 

9.8 

13.8 

17.6 

22.5 

1.2 

1.6 

2.1 

3.2 

50 

7.0 

9.4 

12.6 

16.1 

1.5 

2.4 

8.7 

5.2 

7.1 

9.4 

12.2 

1.2 

1.9 

2.8 

4.0 

5.4 

7.2 

9.5 

4.31  I 

1.38  1 


SI 
2 


W^eight  of  1  foot  of  length  of  pillar  in  pounds. 
4.91   I      5.53  I    6.13    |      6.75  |      7.97  |      9.22  |    10.4     |    11.7     |    12.9    |    14.1 


Area  of  ring  of  solid  metal  in  square  inches. 
1.57  I      1.77   I    1.96    I      2.16  1      2.55  |      2.95  |      3.34  |      3.73  |      4.12  |      4.52 


a 

CAST  IBON 

.     THICKNESS  H  INCH.    (Original.) 

B 

y 

tl 

a 

Outer  Diameter  in  inches. 

5 

5>^ 

6 

6H 

7 

7^ 

8 

8^ 

9 

10 

11 

12 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tom. 

Tons. 

2 

239 

268 

297 

325 

354 

383 

412 

440 

469 

526 

583 

640 

2 

4 

205 

235 

266 

296 

326 

356 

387 

416 

445 

504 

563 

622 

4 

6 

166 

196 

227 

257 

288 

319 

350 

380 

410 

471 

532 

593 

6 

8 

131 

159 

188 

218 

248 

279 

310 

340 

371 

432 

494 

557 

8 

10 

103 

127 

154 

182 

210 

240 

270 

300 

330 

391 

454 

517 

10 

12 

82 

103 

126 

151 

177 

205 

233 

262 

292 

351 

418 

475 

12 

14 

66 

84 

104 

126 

149 

174 

200 

227 

255 

313 

372 

434 

14 

16 

54 

69 

87 

106 

126 

149 

173 

198 

224 

277 

334 

394 

16 

18 

45 

58 

73 

90 

108 

128 

148 

172 

196 

246 

300 

357 

18 

20 

37 

48 

62 

77 

93 

111 

130 

151 

173 

219 

270 

323 

20 

22 

32 

41 

53 

66 

80 

96 

113 

132 

151 

194 

241 

292 

22 

25 

25 

34 

43 

54 

65 

79 

93 

109 

126 

164 

206 

252 

26 

SO 

18 

24 

31 

39 

48 

59 

70 

83 

96 

126 

160 

199 

30 

35 

13 

18 

24 

30 

36 

44 

53 

63 

73 

98 

126 

W9 

35 

40 

10 

14 

18 

23 

28 

35 

42 

50 

59 

78 

102 

129 

40 

46 

8 

11 

15 

19 

23 

28 

34 

41 

48 

64 

84 

107 

46 

60 

6 

9 

12 

15 

19 

23 

28 

33 

39 

53 

70 

86 

50 

60 

4 

6 

9 

11 

14 

17 

20 

24 

28 

38 

50 

65 

«0 

70 

3 

4 

6 

8 

10 

12 

16 

18 

21 

29 

38 

50 

TO 

80 

3 

4 

5 

6 

8 

13 

16 

22 

80 

88 

80 

Weight  of  1  foot  of  length  of  pillar,  in  pounds. 
22.1  I    24.5    I   27.0    |   29.4    |    31.9   |    34.4    |    36.9    |    39.4   |    41.9  |    46.6  |   51.6  |    56.6 

Area  of  ring  of  solid  metal,  in  square  inches. 

7.07  (     7.85    .    8.64   |    9.43  f  10-2   I    H-O    |    11.8    |   12.6   |   13.4   |    14.9   |   ^6.5  |    18.1 


STRENGTH    OF   IRON   PILLARS. 


909 


TaMe  1.    HOIil^OW  CYI^IBTD  CAST  IRON  PII^IiARS. 

BREAKING  LOADS.— (Continued.)  By  Gordon's  Rule. 


CAST  IKON.  THICKNESS  1  INCH.  (Original.) 

1 

Outer  Diameter  in  Inches. 

12 

13 

14 

15 

16 

18 

20 

22 

24 

27 

30 

36 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tona. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

1188 

1301 

1415 

1530 

1645 

1874 

2103 

2830 

2557 

2896 

3236 

3915 

1138 

1253 

1368 

1484 

1601 

1833 

2066 

2295 

2625 

2866 

3208 

3890 

1065 

1184 

1303 

1423 

1543 

1779 

2015 

2247 

2479 

2824 

3170 

3860 

989 

1110 

1231 

1355 

1475 

1716 

1957 

2193 

2430 

2780 

3128 

3823 

909 

1030 

1152 

1275 

1399 

1644 

1889 

2129 

2369 

2723 

8076 

3778 

829 

949 

1071 

1195 

1320 

1566 

1813 

2056 

2300 

2659 

3016 

3726 

756 

873 

992 

1114 

1237 

1484 

1733 

1979 

2226 

2589 

2951 

3668 

683 

796 

913 

1034 

1155 

1401 

1651 

1899 

2147 

2515 

2879 

3604 

618 

727 

840 

958 

1077 

1320 

1568 

1817 

2066 

2437 

2805 

3536 

559 

663 

772 

88T 

1002 

1241 

1486 

1734 

1982 

2356 

2726 

3464 

508 

606 

709 

818 

929 

1163 

1404 

1650 

1899 

2272 

2644 

838T 

459 

553 

651 

75« 

863 

1090 

1326 

1570 

1816 

2188 

2560 

3308 

il8 

50() 

598 

697 

800 

1020 

1250 

1489 

1733 

2103 

2475 

3226 

380 

462 

549 

644 

743 

954 

1178 

1412 

1653 

2020 

2392 

3143 

347 

424 

506 

595 

689 

893 

1110 

1338 

1575 

1938 

2308 

3059 

318 

390 

467 

552 

641 

836 

1046 

1268 

1500 

1857 

2225 

2974 

292 

359 

432 

511 

596 

783 

984 

1199 

1427 

1779 

2143 

2889 

268 

331 

400 

475 

556 

734 

928 

1136 

1361 

1704 

2063 

2804 

247 

305 

370 

441 

518 

687 

874 

1076 

1296 

1630 

1984 

2720 

229 

283 

344 

411 

U84 

645 

825 

1019 

1232 

1560 

1909 

2636 

212 

263 

321 

363 

i52 

605 

776 

963 

1169 

1491 

1834 

2552 

197 

246 

299 

358 

423 

569 

734 

915 

1114 

1428 

1765 

2474 

183 

229 

280 

335 

397 

536 

694 

869 

1060 

1367 

1696 

2396 

170 

213 

261 

314 

373 

505 

656 

824 

1008 

1,306 

1627 

2319 

144 

181 

222 

269 

320 

438 

573 

725 

893 

1172 

1474 

2135 

124 

157 

192 

233 

278 

383 

503 

641 

795 

1054 

1333 

1964 

105 

134 

165 

202 

242 

335 

443 

568 

709 

951 

1211 

1808 

93 

118 

146 

178 

213 

297 

394 

507 

636 

853 

1099 

1664 

73 

92 

113 

139 

168 

235 

315 

408 

516 

705 

914 

1414 

68 

73 

91 

112 

136 

191 

257 

335 

426 

586 

768 

1209 

48 

60 

75 

92 

112 

157 

213 

279 

356 

491 

651 

1040 

Weight  of  one  foot  of  length  of  pillar,  in  pounds. 

108  1  118  I  128  I  138  I  147   [  167  |  187  |  206  |  226  |  255  |  285  |  344 

Area  of  ring  of  solid  metal,  in  square  inches. 
34.6  I    37.7  I     40.8   |     44.0  ]       -       ■      -       "  —-._-- 


53.4   I     59.7  I    66.0  |   72.2   |    81.7   |    91.1  |  110.0 


c 

CAST  IRON.  THICKNESS  3  INCHES.  (Original.) 

a 

t'i 

a 

Outer  Diameter  in  Feet. 

V 

2 

3 

3M 

4 

i^     i   5 

5^ 

6 

7 

8 

10 

12 

Tons. 

Tons. 

Tons. 

Tons,  j  Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

10 

10806 

12878 

14906 

16967 

18986 

21038 

23052 

27143 

31196 

39254 

47375 

10 

15 

10453 

12564 

14628 

16712 

18754 

20825 

22855 

26972 

31045 

39133 

47273 

15 

20 

9996 

12150 

14250 

16369 

18440 

20533 

22585 

26737 

30837 

38962 

47131 

20 

30 

8884 

11102 

13275 

15459 

17593 

19743 

21847 

26084 

30257 

38486 

46729 

30 

40 

7688 

9907 

12113 

14344 

16531 

18735 

20891 

25224 

29476 

37838 

46175 

40 

50 

6554 

8702 

10888 

13126 

15341 

17580 

19780 

24107 

28532 

37037 

45484 

50 

60 

5553 

7575 

9690 

11892 

14100 

16349 

18570 

23049 

27460 

36103 

44666 

60 

70 

4704 

6570 

8576 

10702 

12870 

15097 

17319 

21824 

26287 

35058 

43737 

70 

80 

3998 

5699 

7570 

9595 

11692 

1 3873 

16070 

20566 

25048 

33924 

42712 

80 

90 

3417 

4953 

6683 

8588 

10594 

12706 

14856 

19304 

23791 

32725 

41670 

90 

100 

2940 

4321 

5909 

7665 

9588 

11614 

13700 

18065 

22521 

31481 

40438 

lOO 

110 

2547 

3788 

5238 

6888 

8677 

10606 

12613 

16869 

21270 

30213 

39220 

110 

125 

1950 

2954 

4400 

6864 

7483 

9257 

11132 

14650 

19448 

27664 

36692 

125 

liO 

1532 

2350 

3353 

4547 

5900 

7417 

9058 

12701 

16668 

25182 

34127 

150 

Weight  of  one  foot  of  length  of  pillar,  in  pounds. 
972    I    1150    I    1325    |    1503    |    1678    |    1856    |    2031    |    2388    |    2740     |    3444     |    4153 

Coeffieients  of  safety  for  bollow  cast  iron  i»illars. 

Mr.  James  B.  Francis,  of  Lowell.  Mass.,  a  high  authority,  in  his  "Tables  of  Cast  Iron  Pillars." 
recommends  that  in  order  to  allow  for  unequal  loading,  imperfect  casting,  bad  end  bearings,  side 
tolows.  &c.,  we  should  not  take  the  safe  load  at  more  than  one-fifteenth  of  Hodgkinson's  breaking 
load,  if  the  pillars  are  rouelily  cast,  and  the  ends  not  perfectly  planed  and  adjusted;  and  one-fiftb 
when  they  are  so,  and  the  loads  about  equally  distributed. 


910 


STKENGTH   OF   IRON   PILLARS. 


HOIiliOW  CTI^IXDRICAIi  WROUGHT  IROX  PII.I.ARS. 

Table  4,  of  breaking:  loads  in  tons  of  bollow  cylindrical 
wrougrbt  iron  pillars,  witb  Hat  ends,  perfectly  true,  and 
firmly  fixed,  and  tbe  loads  pressing:  equally  on  every  part 
of  tbe  top.    Calculated  by  Gordon's  formula. 


(Original.) 


WROUGHT  IRON.    THICKNESS  H  INCH. 


Outer  diameter  in  inclies. 


H      \      I      \     IH     \     l}4    \     IH    \      2      ]     2H    I    -21^     \   2H     \     S 


Tons. 
18.0 
17.8 
17.3 
16.7 
16.0 
15.2 
14.4 
13.5 
12.6 
11.8 
11.0 
10.2 
9.5 
8.9 


BREAKING 

LOAD 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

3.64 

5.27 

6.88 

8.50 

10.1 

11.7 

13.2 

14.8 

16.4 

2.94 

4.64 

6.32 

8.00 

9.6 

11.2 

12.8 

14.5 

16.1 

2.30 

3.86 

5.57 

7.28 

8.9 

10.6 

12.2 

13.9 

15.6 

1.77 

3.13 

4.74 

6.36 

8.1 

9.9 

11.6 

13.3 

15.0 

1.36 

2.51 

4.07 

5.66 

7.3 

9.1 

10.8 

12.5 

14.2 

1.04 

2.03 

3.46 

4.91 

6.6 

8.3 

9.9 

11.6 

13.4 

.81 

1.65 

2.91 

4.24 

5.7 

7.4 

9.1 

10.8 

12.6 

.61 

1.36 

2.46 

3.67 

5.1 

6.7 

8.3 

9.9 

11.7 

.50 

1.05 

2.03 

3.18 

4.5 

6.0 

7.5 

9.1 

10.8 

.41 

.95 

1.75 

2.77 

4.0 

5.4 

6.9 

8.4 

10.1 

.34 

.81 

1.52 

2.41 

3.6 

4.8 

6.2 

7.7 

9.3 

.29 

.70 

1.34 

2.14 

3.2 

4.3 

5.6 

7.0 

8.6 

.24 

.60 

1.16 

1.88 

2.8 

8.9 

5.2 

6.5 

8.0 

.21 

.53 

1.03 

1.69 

2.5 

8.5 

4.7 

6.0 

7.4 

.19 

.47 

.91 

1.50 

2.3 

8.2 

4.3 

5.5 

6.9 

.18 

.42 

.84 

1.38 

2.1 

2.9 

4.0 

5.1 

6.4 

.14 

.33 

.67 

1.11 

1.7 

2.4 

3.4 

4.4 

5.6 

.27 

.55 

.91 

1.4 

2.0 

2.8 

3.7 

4.7 

.9 

1.4 

2.0 

2.6 

3.4 

Weight  of  one  foot  of  length  of  pillar,  in  pounds. 
I     1.15     I     1.47     I     1.80     I    2.13     I     2.45    |     2.78     |     3.11     |    3.43     |  3.77 


Area  of  ring  of  solid  metal,  in  square  inches. 
.246     i    .344      I     .442     |     .540    |     .638     |     .736    |    .835     |     .933     |    l.C 


I    1.13 


a 

WROUGHT  IRON. 

THICKNESS  }4  INCH. 

h 

Outer  diameter  in  inches. 

^ 

a 

^H      1 

2H 

■2H 

3 

3H    1      4     1    4«      1     5 

5H 

1      6 

^ 

BREAKING    LOAD. 

Tons. 

Tons. 

Tons. 

Tom. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

1 

21.9 

25.4 

28.3 

31.4 

34.5 

40 

47 

53 

60 

66 

72 

1 

2 

31.1 

24.3 

27.6 

30.7 

33.9 

40 

47 

53 

60 

66 

72 

2 

3 

19.9 

23.1 

26.4 

29.7 

33.0 

?9 

46 

52 

59 

65 

71 

3 

4 

18.6 

21.8 

25.3 

28.5 

31.9 

38 

46 

51 

58 

64 

71 

4 

5 

17.0 

20.4 

23.5 

27.3 

30.7 

37 

44 

50 

57 

63 

70 

5 

6 

15.4 

18.8 

22.1 

25.7 

29.2 

36 

43 

49 

56 

62 

69 

6 

7 

13.9 

17.3 

20.5 

23.8 

27.8 

34 

41 

47 

54 

61 

68 

7 

8 

12.5 

15.6 

19.1 

22.3 

25.9 

32 

40 

46 

53 

60 

67 

8 

9 

11.2 

14.2 

17.5 

20.6 

24.3 

80 

38 

44 

51 

58 

65 

9 

10 

lo.a 

13.0 

16.1 

19.1 

22.7 

29 

37 

43 

50 

57 

64 

10 

11 

9.0 

10.7 

15.7 

17.6 

21.1 

27 

35 

41 

48 

55 

62 

11 

12 

8.1. 

10.6 

13.5 

16.4 

19.6 

26 

33 

40 

46 

54 

61 

12 

13 

7.3 

9.6 

12.4 

15.1 

18.2 

24 

31 

38 

44 

52 

59 

13 

14 

6.6 

8.8 

11.3 

14.0 

17.0 

23 

30 

36 

43 

51 

57 

14 

15 

6.0 

8.0 

10.4 

12.9 

15.8 

21 

28 

34 

41 

49 

55 

15 

16 

5.5 

7.3 

9.5 

12.0 

14.6 

20 

27 

33 

40 

47 

54 

16 

18 

4.6 

6.0 

8.0 

10.3 

12.7 

18 

24 

30 

37 

43 

50 

18 

20 

3.8 

5.1 

.    6.8 

8.7 

11.0 

16 

21 

27 

34 

40 

47 

20 

25 

7.9 

12 

16 
13 

21 
17 

27 
22 

33 
27 

39 
32 

25 
30 

30 

86 

10 

14 

18 

22 

27 

35 

40 

14 
11 

8 

18 
15 
12 

23 
19 
16 

40 
45 

45 

50 

50 

Weight  of  one  foot  of  length  of  pillar,  /b  pounds. 
4.60    I    5.23      I    6.90     |    6.53    |    7.20    |    8.50     |    9.83    j    11.1     |    12.4    |    13.7    |    15.9 

Area  of  ring  of  solid  metal,  in  square  inches. 
1.38    I    1.57      i    1.77     I    1.96    |    2.16    |    2.55    |    2.95    |    3.34    |    3.73    !    4.12    |    4.51 


STRENGTH   OF   IRON   PILLARS.  911 

MOIiliOW  CTIillTDRICAIi  WROUOHT  IBOHr   PIIil^ARS. 


Table  4, 

(Con 

tinued.)     (Original.) 

3 

WROUGHT  IRON.     THICKNESS  H  INCH. 

a 

%i 

Outer  diameter  in  inches. 

II 

.2 

5 

5% 

1      6 

1  en 

1      7 

1    7>^    1     8      1    8>^     1     9      1 

10 

11 

12 

^ 

BREAKING   LOAD. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

2 

112 

125 

139 

152 

166 

177 

189 

201 

214 

238 

265 

290 

2 

4: 

110 

123 

136 

149 

163 

174 

186 

199 

212 

237 

262 

289 

4 

6 

106 

119 

132 

145 

158 

171 

184 

197 

210 

235 

261 

288 

6 

8 

101 

114 

127 

140 

154 

167 

181 

194 

207 

232 

258 

284 

8 

10 

95 

108 

123 

136 

149 

162 

176 

189 

203 

228 

254 

280 

10 

Vi 

89 

102 

116 

129 

143 

157 

171 

185 

199 

224 

250 

276 

12 

U 

83 

95 

108 

122 

137 

151 

165 

179 

194 

219 

245 

272 

14 

16 

76 

89 

103 

117 

131 

145 

160 

173 

187 

213 

240 

268 

16 

18 

70 

83 

97 

110 

lU 

138 

153 

166 

180 

207 

235 

263 

18 

2C 

64 

77 

91 

104 

117 

131 

145 

159 

178 

201 

227 

257 

20 

22 

58 

70 

83 

96 

109 

123 

138 

151 

165 

192 

220 

250 

22 

25 

52 

64 

76 

89 

102 

115 

129 

143 

157 

183 

212 

241 

25 

30 

42 

52 

63 

74 

87 

100 

113 

127 

141 

167 

195 

224 

30 

35 

34 

43 

53 

64 

75 

87 

99 

112 

125 

151 

178 

207 

35 

40 

27 

36 

44 

53 

64 

75 

86 

98 

110 

135 

163 

190 

40 

45 

23 

30 

38 

46 

55 

65 

76 

87 

98 

128 

148 

174 

45 

50 

19 

24 

32 

38 

47 

56 

66 

76 

87 

109 

133 

158 

50 

60 

15 

19 

24 

29 

36 

43 

51 

60 

69 

88 

109 

132 

60 

70 

11 

u 

18 

23 

28 

S4 

40 

48 

56 

78 

91 

111 

70 

80 

9 

11 

14 

18 

22 

27 

82 

37 

44 

57 

74 

93 

80 

90 

7 

9 

11 

14 

18 

22 

26 

31 

86 

49 

63 

78 

90 

100 

6 

T 

9 

12 

15 

18 

22 

26 

SO 

41 

53 

66 

100 

Weight  of  one  foot  of  lengrth  of  pillar,  in  pounds. 
23.6    I    26.2  I    28.8  |    31.i  |    34.0  |    36.6  |    39.3    |    42.0    |    44.7    |    49.7    |    55.0   |    60.8 

Area  of  ring  of  solid  metal,  in  square  inches. 
7.07    I    7.85  I    8.64  |    9.43  |    10.2  |    11.0  |    11.8   |    12.6   |    13.4  |    14.9        16.5  |    18.1 


Table  4,  (Continued.)    (Original.) 


a 

WROUGHT  IRON.    THICKNESS  1  INCH. 

,fl 

ti 

Outer  diameter  in  inches. 

II 

2 

13 

14    1     15 

16    1      17    1      18    1      20    1      22    1      24     t      26    |      28    |    80 

J 

BB 

BAKING  LOAD. 

Tom. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

1 

60S 

653 

704 

758 

805 

854 

955 

1056 

1157 

1257 

1857 

1458 

1 

10 

588 

638 

691 

742 

795 

846 

949 

1049 

1149 

1248 

1354 

1457 

10 

20 

543 

595 

651 

702 

759 

810 

913 

1016 

1120 

1223 

1327 

1430 

?0 

30 

479 

538 

594 

645 

699 

758 

866 

973 

1077 

1186 

1289 

1394 

30 

40 

415 

470 

528 

584 

636 

691 

806 

912 

1027 

1130 

1237 

1348 

40 

50 

355 

405 

462 

516 

570 

627 

740 

848 

961 

1067 

1179 

1294 

50 

60 

300 

348 

400 

452 

505 

659 

669 

781 

891 

1005 

1115 

1228 

60 

70 

256 

300 

348 

398 

448 

499 

606. 

715 

824 

936 

1046 

1160 

70 

80 

215 

255 

298 

344 

392 

440 

543  • 

649 

757 

868 

978 

1092 

80 

90 

185 

222 

261 

303 

347 

892 

489 

590 

694 

800 

910 

1023 

90 

100 

157 

190 

225 

262 

303 

345 

436 

532 

631 

735 

843 

955 

100 

110 

134 

162 

193 

227 

264 

302 

886 

474 

568 

666 

770 

877 

110 

125 

111 

135 

162 

192 

225 

259 

336 

416 

505 

598 

697 

799 

125 

150 

82 

101 

122 

145 

171 

198 

262 

328 

405 

485 

574 

666 

150 

175 

62 

78 

95 

112 

133 

155 

208 

266 

331 

<00 

478 

560 

175 

200 

49 

«0 

74 

89 

106 

124 

168 

216 

269 

828 

895 

467 

200 

Weigh 

t  of  one  foot  of  length  of  pillar,  in  pounds. 

126 

!    136   1     147 

!    157     1    168    1    178     1  199     |    220    |    241     |    262     |    283     |    304 

Area 

of  ring  of  solid  metal,  in  square  inches. 

37.7 

1    40.8  1    44.0 

1    47.1    1    50.3  1    53.4   |   59.7    |    66.0  |    72.3  |    78.5    |    84.8    |    91.1 

The  bi 

reakin^ 

loads   for    less   tbicknesses  may  safely  be  aa«am*d  tt 

^min 

ish  at  t 

he  sam 

e  rate 

&a  the 

thiokne 

BS. 

912 


STRENGTH   OF   IRON   PILLARS. 


Table  of  approximate  averag^e  ultimate  loads  in  lbs  per 
square  incb,  as  found  by  experiment  with  carefully  prepared  specimens.  In 
practice,  allowance  must  be  made  for  the  rougher  character  of  actual  work,  for 
jarrings  etc  etc. 


00 

Pencoyd  Angl 

3s,  Tees, 

I  beams  and  Channels.* 

Phcenlx 
columns.f 

00 

s 

% 

1-.? 

Steel. 

IroD 

Iron. 

l§ 

Ji2 

Hard;  .36 
per  cent 

Mild ;  .12 
per  cent 

% 

oarbon 

oarboQ 

Flat 

Flat 

Fixed 

Flat 

Hinged 
ends 

Round 

Flat 

ends 

ends 

ends 

eudb 

ends 

ends 

>A 

3 

57200 

3 

17 



50400 

17 

20 

100000 

70000 

46000 

46000 

46000 

iiooo 

48000 

20 

30 

74000 

51000 

43000 

43000 

43000 

40250 

40000 

30 

40 

62000 

46000 

40000 

40000 

40000 

36500 

37000 

40 

50 

60000 

44000 

38000 

38000 

38000 

33500 

37000 

50 

60 

58000 

42000 

36000 

36000 

36000 

30500 

37000 

60 

70 

55500 

40000 

34000 

34000 

33750 

27750 

37000 

70 

80 

53000 

38000 

32000 

32000 

31500 

25000 

36000 

80 

90 

49700 

36000 

31000 

30900 

29750 

22750 

35000 

90 

100 

46500 

34000 

30000 

29800 

28000 

20500 

35000 

100 

120 

40000 

30000 

28000 

26300 

24300 

16500 

34500 

120 

140 

33500 

26000 

25500 

23500 

21000 

12800 

140 

160 

28000 

22000 

23000 

20000 

16500 

9500 



160 

200 

19000 

14800 

17500 

14500 

10800 

6000 



200 

300 

8500 

7200 

9000 

7200 

6000 

2800 



300 

The  following  simple  formula,  by  Mr.  D.  J.  Whittemore,  was  found  t* 
agree  rery  closely  with  the  results  of  the  experiments  on  Phoenix  columns  rj- 


Breaking  load  in  fts 
per  sq  inch  of  area 
of  cross  section  of  pillar 


:  [(1200  — H)  X  30]  +- 


H2 


where  H  = 


_  length  of  pillar 
'  diam  D,  fig  p  904 


both  in  the  same  unit. 


Mr.  Christie*  adopts  the  following  formulae  for  obtaining  the  proper  factor 
af  safety  for  pillars  of  wrought  iron  or  steel : 


For  flat  and  fixed  ends, 


Factor  of  safety  =  3  +  I  .01 


(■' 


length 


least  rad  of  gyr/ 
length 


least  rad  of  gyr/ 


For  hinged  and  round  ends,     Factor  of  safety  =  3  4-  (  015 

It  will  be  noticed  that  the  factor  of  safety,  as  found  by  these  formulse,  in- 
creases with  the  ratio  of  the  length  of  the  pillar  to  the  least  radius  of  gyration 
of  its  cross  section;  and  is  greater  for  round  and  hinged  ends  than  for  fiat  and 
fixed  ends. 


*  See  "  Wrought  Iron  and  Steel  in  Construction  ",  by  Pencoyd  Iron  Works  ;  published  by  John 
Wiley  &  Sons,  New  York,  1884. 
t  See  Transactions,  American  Society  of  Civil  Engineers ;  Jan,  Feb  and  March  1882. 


STRENGTH   OF   IRON   PILLARS. 


913 


Ultimate  crippling  streng^tlis  in  lbs  per  sq  iiicb  of  metal 
section  of  the  four  wroujjriit  iron  pillars  below.  These  formulas 
are  deduced  by  Chs.  Shaler  Smith,  from  many  tests  by  G.  Bousearen,  C.  E.,  of 
large  pillars  of  good  American  iron.  The  lower  Table  is  an  abridgment  of  the 
full  ones  by  C.  L.  Gates,  C.  E.,  in  the  Trans,  Am.  Soc.  C.  E.,  Oct.,  1880. 

^  length  between  end  heaHr^s  ^^^^  .^  ^^^  ^^^^  ^^^^^^    ^^^  ^  ^^  ^^  ^^^^^ 

least  diameter  d 
For  safety  take  from  3^  to  ]/q,  according  to  circumstances. 


Flat  ends.. 


One  pin  end- 
Two  pin  ends. 


Ultimate  and  safe  loads  in  lbs  per  sq  inch,  of  the  above  four  pillars,  with 
flat  ends,  and  equally  loaded.  Coef  of  Safety  =  4  +  .05  H.  By  C.  L.  Gates,  C.  E. 


H. 

A.  Square  Col. 

B.  Phoenix  Col. 

C.  American  Col. 

D.  Common  Col, 

Ult. 

Safe. 

Ult. 

Safe. 

Ult. 

Safe. 

Ult. 

Safe. 

15 

37067 

7822 

40476 

8521 

34434 

7249 

33693 

7093 

16 

36876 

7683 

40212 

8377 

34167 

7118 

33339 

6946 

18 

36470 

7443 

39645 

8091 

33597 

6856 

32589 

6651 

20 

36024 

7205 

39030 

7806 

32982 

6596 

31790 

6358 

22 

35544 

6970 

38373 

7524 

32327 

6338 

30952 

6069 

25 

34767 

6622 

37317 

7110 

31285 

5959 

29639 

5646 

30 

83344 

6063 

35424 

6440 

29435 

5352 

27375 

4977 

35 

31806 

5531 

33406 

5810 

27512 

4789 

25108 

4367 

40 

30198 

5033 

31352 

5226 

25584 

4264 

22919 

3820 

45 

28562 

4570 

29310 

4690 

23701 

3792 

20857 

3337 

50 

26932 

4143 

27321 

4203 

21900 

3369 

18952 

2916 

55 

25333 

3728 

25415 

3765 

20203 

3004 

17214 

2650 

§0 

23787 

3398 

23611 

3373 

18621 

2660 

15643 

2236 

58 


914 


FLOOR  SECTIONS. 


PENCOYD  FI.OOR  SECTIONS. 

L  =  span,  in  feet. 
C  =  coefficient. 
W  =  distributed  load,  in  ft)s,  per  foot  of  floor  width. 


w  =  o 


Corru^erated  flooring-,  for  bridges  and  buildings. 

W  =  load  producing  fiber  stress  of  15,000  fts  per  square  inch. 


SECTION  210  M. 
Dimensions  in  inches. 


Thickness, 

inches. 

Weight, 

Web. 

Flange. 

ibs  per  sq  ft. 

C 

^ 

i 

14.8 

44,000 

^1 

A 

18.4 

55,000 

^9 

1 

21.9 

66,000 

ii      - 

^ 

25.5 

77,400 

1 

I 

29.1 

88,800 

SECTION  260  M. 
Dimensions  in  inches. 


Thickness,  inches. 
Web.  Flange, 

i  ito| 

h  i  to  I 

I  I  to  I 


Weight, 
fl>s  per  sq  ft. 
20.0  to  30.7 
26.5  to  37.2 
29.4  to  40.1 


105,000  to  186,000 
143,000  to  224,000 
153,000  to  237,000 


Z  Bar  Flooring^. 

W  =  safe  load. 


Section 

Dimensions,       Thickness,  ins. 

Weight, 

Q 

No. 

in  inches.      Z  bars.      Plates. 

lbs  per  sq  ft. 

1 

A     B     C      I>(   i  ) 

15      6      9      4  ^  tJ  |-       \io\ 

f  25.9  to  36.1 
<  29.1  to  39.3 
(  32.3  to  42.5 

93,400  to  147,400 
104,000  to  157,000 
114,400  to  167,000 

2 

18      8    10      5  ^   f  V     T«B  to  ^^ 

(■  32.1  to  42.3 
<  35.2  to  45.4 
(  38.4  to  48.6 

143,000  to  209,400 
155,000  to  221,400 
166,400  to  233,000 

3 

21      9    12      &\.h  V       1  to  i 

i  39.3  to  49.5 
-I  42 A  to  52.6 
i  45.5  to  55.7 

203,400  to  281,000 
217,400  to  294,000 
231,000  to  307,200 

CHAINS. 


915 


WEIOHT  AND  STRENGTH  OF  lUON  CHAINS. 

Table  of  streng^th  of  chains. 

Chains  of  superior  iron  will  require  ^  to  3^  more  to  break  them.    (Original.) 


Diana  of  rod 
of  which 
the  links 
are  made. 


Weight 
of  chain 
per  ft  run. 


Ins. 
3-16 

fee 
% 

13-16 


Pds. 

.5 
.8 
1. 
1.7 
2. 
2.5 
3.2 
4.3 
5. 
5.8 
6.7 


Breaking  strain 
of  the  chain. 


1731 
3069 
4794 
6922 
9408 
12320 
15590 
19219 
23274 
27687 
32301 
37632 
43277 


Tons. 

.773 
1.37 
2.14 
3.09 
4.20 
5.50 
6.96 
8.58 
10.39 
12.36 
14.42 
16.80 
19.32 


I)iam  of  rod 
of  which 
the  links 
are  made. 


Weight 
of  chain, 
per  ft  run. 


Pds. 

10.7 

12.5 

16. 

18.3 

21.7 

26. 

28. 

32. 

38. 

54. 

71. 


Breaking  strain 
of  the  chain. 


Pds. 


49280 

22.00 

59226 

26.44 

73114 

32.64 

88301 

39.42 

105280 

47.00 

123514 

55.14 

143293 

63.97 

164505 

73.44 

187152 

83.55 

224448 

100.2 

277088 

123.7 

335328 

149.7 

398944 

178.1 

The  links  of  ordinary  iron  chains  are  usually  made  as  short  as  is  consistent 
with  easy  play,  in  order  that  they  may  not  become  bent  when  wound  around 
drums,  sheaves,  &c.;  and  that  they  may  be  more  easily  handled  in  slinging 
large  blocks  of  stone,  &c.  U.  S.  Government  experiments,  1878,  prove  that 
studs  weaken  the  links. 

When  so  made,  their  weight  per  foot  run  is  quite  approximately  Z}^  times  that 
of  a  single  bar  of  the  round  iron  of  which  they  are  composed.  Since  each  link 
consists  of  two  thicknesses  of  bar,  it  might  be  supposed   that  a  chain  would 

gossess  about  double  the  strength  of  a  single  bar;  but  the  strength  of  the  bar 
ecomes  reduced  about  30  per  cent,  by  being  formed  into  links ;  so  that'  the  chain 
has  but  about  70  per  cent  of  the  strength  of  two  bars.  As  a  thick  bar  will  not 
sustain  as  heavy  a  unit  stress  as  a  thinner  one,  so  of  course,  stout  chains  are 
proportionally  weaker  than  slighter  ones.  In  the  foregoing  table,  20  tons 
per  sqiiare  inch,  is  assumed  as  the  average  breaking  strain  o^a  single  straight 
bar  of  ordinary  rolled  iron,  1  inch  in  diameter  or  1  inch  square ;  19  tons,  from 
1  to  2  inches  ;  and  18  tons,  from  2  to  3  inches.  Deducting  30  per  cent  from  each, 
we  have  as  the  breaking  strain  of  the  two  bars  composing  each  link,  as  follows : 
14  tons^er  square  inch,  up  to  1  inch  diameter  ;  13.3  tons,  from  1  to  2  inches ;  and 
12.6  tons,  from  2  to  3  inches  diameter ;  and  upon  these  assumptions  the  table  is 
based.  The  weights  are  approximate  ;  depending  upon  the  exactness  of  diameter 
of  the  iron,  and  shape  of  link. 


916 


TIN   AND   ZINC. 


% 


Tinr  AXD  zinrc. 

The  pare  metal  Is  called  block  tin.  When  perfectly  pure,  (which  It 
rarely  is,  being  purposely  adulterated,  frequently  to  a  large  proportion,  with  the 
cheaper  metals  lead  or  zinc,)  its  sp  grav  is  i  .29 ;  and  its  weight  per  cub  ft  is  455  ft)s. 
It  is  sufficiently  malleable  to  be  beaten  into  tin  foil,  only  joVo"  °^  ^^  inch  thick. 
Its  tensile  strength  is  but  about  4600  Bbs  per  sq  inch ;  or  about  7000  fts  when  made 
into  wire.  It  melts  at  the  moderate  temperature  of  442°  Fah.  Pure  block  tin  is 
not  used  for  common  building  purposes  ;  but  thin  plates  of  sheet  iron,  covered  with 
it  on  both  sides,  constitute  the  tinned  plates,  or,  aa  they  are  called,  the  tin,  used  for 
covering  roofs,  rain  pipes,  and  many  domestic  utensils.    For  roofs  it  is  laid  on  boards. 

Tbe  sbeets 
of  tin  are  uni« 
ted  as  shown  in 
this  fig.  First,  sev- 
erar  sheets  are 
joined  together  in 
the  shop,  end  for 
end,  as  at  tt;  by 
being  first  bent 
over,  then  ham- 
mered flat,andthen 

soldered.  These  are 

then  formed  into  a 
roll  to  be  carried 
to  the  roof;  a  roll 

being  long  enough  to  reach  from  the  peak  to  the  eaves.  Different  rolls  being  spread 
up  and  down  the  roof,  are  then  united  along  their  sides  by  simply  being  bent  as  at  a 
and  s,  by  a  tool  for  that  purpose.  The  roofers  call  the  bending  at  5  a  double  groove,' 
or  double  lock  ;  and  the  more  simple  ones  at  t,  a  sir^le  groove,  or  lock. 

To  hold  the  tin  securely  to  the  sheeting  boards,  pieces  of  the  tin  3  or  4  ins  long, 
by  2  ins  wide,  called  cleats,  are  nailed  to  the  boards  at  about  every  18  ins  along  the 
joints  of  the  rolls  that  are  to  be  united,  and  are  bent  over  with  the  double  groove  s. 
This  will  be  understood  from  y,  where  the  middle  piece  is  the  cleat,  before  being 
bent  over.  The  nails  should  be  4-penny  slating  nails,  which  have  broader  heada 
than  common  ones.  As  they  are  not  exposed  to  the  weather,  they  may  be  of  plain  iron. 
Mnch  use  is  made  of  what  is  called  leaded  tin,  or  ternes,  for  rooftng.  It  is 
■imply  sheet-iron  coated  with  lead,  instead  of  the  more  costly  metal  tin.  It  is  not 
as  durable  as  the  tinned  sheets,  but  is  somewhat  cheaper. 

The  best  plates,  both  for  tinning  and  for  ternes,  are  made  of  charcoal  iron ;  which, 
being  tough,  bears  bending  better.  Coke  is  used  for  cheaper  plates,  but  inferior  as 
regards  bending.  In  giving  orders,  it  is  important  to  specify  whether  charcoal 
plates  or  coke  ones  are  required;    also  whether  tinned  plates,  or  ternes. 

Tinned  and  leaded  sheets  of  Bessemer  and  other  cheap  steel,  are  now  much  used. 
They  are  sold  at  about  the  price  of  charcoal  tin  and  terne  plates. 

There  are  also  in  use  for  roofing,  certain  compound  metals  which  resist  tarnish 
better  than  either  lead,  tin,  or  zinc ;  but  which  are  so  fusible  as  to  be  liable  to  be 
melted  by  large  burning  cinders  falling  on  the  roof  from  a  neighboring  conflagration. 
A  roof  covered  with  tin  or  other  metal  should,  if  possible,  slope  not  much  less  than 
five  degrees,  or  about  an  inch  to  a  foot ;  and  at  the  eaves  there  should  be  a  sudden 
fall  into  the  rain-gutter,  to  prevent  rain  from  backing  up  so  as  to  overtop  the  double- 
groov©  joint  s,  and  thus  cause  leaks.  Where  coal  is  used  for  fuel,  tin  roofs  should 
receive  two  coats  of  paint  when  first  put  up,  and  a  coat  at  every  2  or  3  years  after. 
Where  wood  only  is  used,  this  is  not  necessary ;  and  a  tin  roof,  with  a  good  pitch, 
will  last  20  or  30  years. 

Two  good  workmen  can  put  on,  and  paint  outside,  from  250  to  300  sq  ft  of  tin  roof, 
per  day  of  8  hours. 

Tinned  iron  plates  are  sold  by  the  box.  These  boxes,  unlike  glass,  have  not  equal 
areas  of  contents.  They  may  be  designated  or  ordered  either  by  their  names  or 
sizes.  Many  makers,  however,  have  their  private  brands  in  addition ;  and  some  of 
these  have  a  much  higher  reputation  than  others. 


TIN   AND   ZINC. 


917 


Table  of  Tinned  and  Terne  Plates. 
C!»atlon. — Boxes  often  contain  considerably  less  weight  of  tin  plate  than  Um 
table  requires;  the  plates  being  rolled  thin  and  plated  thin,  in  order  to  enablt 
mechanics  to  get  pay  for  more  material  than  they  furnish. 

The  marks  indicate  the  thicknesses,  approximately  as  follows : 


IC 

IX 

IXX 

IXXX 

IXXXX 


Number 
Birmingham 
wire  gauge. 


30 

28 
27 


.012 
.014 
.016 

.018 
.020 


Lbs 
per  sq  ft. 


.48 
.56 
.64 


DC 

DX 

DXX 

DXXX 

DXXXX 


Number 

Birmingham 

Ins. 

wire  gauge. 

27 

.018 

25 

.020 

23 

.025 

22 

.028 

21 

.032 

Lbs 
per  sq  ft. 

.64 

.80 
1.00 
1.13 
1.29 


inches. 

Mark. 

O  S   8S 

m 

Size, 
inches. 

Mark. 

i§5 

Size, 
inches. 

Mark. 

III 

®  U  3 

&5«  fl 

^s.a 

5z;«  fl 

^s.a 

^".2 

^S.S. 

y  X18 

IC 

225 

ISO 

13X13 

IXX 

225 

194 

16X16 

IC 

225 

205 

IX 

" 

162 

13X26 

IC 

112 

135 

IX 

256 

10  X  10 

IC 

«• 

80 

IX 

169 

'1 

IXX 

" 

294 

IX 

" 

100 

'• 

IXX 

<< 

194 

16X19 

IX 

112 

152 

10X14 

IC 

«« 

112 

14X14 

IC 

225 

156 

17X17 

IX 

141 

IX 

<< 

140 

IX 

196 

IXX 

« 

166 

«' 

IXX 

>i 

161 

" 

IXX 

t< 

225 

17X25 

DX 

100 

252 

" 

IXXX 

'» 

182 

tt 

IXXX 

tt 

254 

DXX 

294 

>' 

IXXXX 

it 

203 

14X20 

IC 

112 

112 

18X18 

IX 

112 

162 

10  X  20 

IC 

«< 

160 

IX 

140 

IXX 

180 

IX 

•« 

200 

•' 

IXX 

« 

161 

tt 

IXXXX 

<« 

235 

11  xii 

IC 

<« 

97 

tt 

IXXX 

<• 

182 

20X20 

IX 

«< 

200 

IX 

it 

121 

f 

IXXXX 

<i 

203 

IXX 

tt 

230 

11  X22 

IC 

112 

97 

14  X  22 

IX 

« 

154 

" 

IXXX 

<• 

260 

IX 

121 

IXX 

" 

177 

t' 

IXXXX 

«« 

290 

•  ' 

IXX 

139 

14  X  24 

IX 

tt 

168 

20X28 

IC 

tt 

224 

12  X  12 

IC 

225 

112 

IXX 

tt 

193 

IX 

't 

280 

IX 

140 

14X25 

IC 

(< 

140 

It 

IXX 

tt 

322 

<< 

IXX 

«< 

161 

IX 

ti 

175 

tt 

IXXX 

tt 

364 

12  X  24 

IC 

112 

115 

" 

IXX 

** 

201 

" 

IXXXX 

'* 

406 

IX 

144 

14X  26 

IXXX 

237 

<» 

IXX 

>t 

166 

14X28 

IC 

tt 

157 

«< 

IXXX 

<i 

187 

IX 

it 

196 

Terne  P 

lates 

1. 

li^i  X  17 

DC 

100 

98 

" 

LXX 

«< 

225 

DX 

•  t 

120 

14X30 

IXX 

" 

241 

10X20 

IC 

112 

80 

DXX 

f 

147 

14X31 

IX 

*t 

217 

*< 

IX 

tt 

100 

DXXX 

"^ 

168 

IXX 

tt 

249 

14X20 

IC 

'» 

112 

DXXXX 

<< 

189 

15X15 

IX 

225 

225 

IX 

*« 

140 

13  X  IS 

IC 

225 

135 

IXX 

«• 

253 

20X28 

IC 

tt 

224 

IX 

" 

169 

•' 

IXXX 

'• 

326 

IX 

'* 

280 

Sheets  of  larger  size  may  be  made  to  special  order;  those  of  tinned  iron,  in  England;  but  leaded 
ternes  are  made  in  Pbilada  also,  and  elsewhere. 

A  box  of  225  sheets  of  13?^  by  10,  contains  214.84  sq  ft ;  but,  allowing  for  overlapping,  it  will  oorer 
but  about  150  sq  ft  of  roof;  even  without  any  allowance  for  the  waste  which  occurs  in  cutting  away 
portions  in  order  to  fit  at  angles,-&c. 

To  find  the  area  of  roof  covered  by  any  sheet,  first  deduct  2  ins  from  its  width,  and  1  inch  from  iti  ' 
length. 

Zinc,  in  sbeets,  and  laid  in  the  same  manner  as  slates,  is  much  used  in  some 
parts  of  Europe  for  roofing.  By  exposure  to  the  weather,  it  soon  becomes  covered  by  a  thin  film  of 
white  oxide,  which  protects  it  from  further  injury,  and  renders  the  roof  very  durable.  Corrugated 
sheet  zinc  is  also  used.     See  Galvanized  Sheet  Iron. 

Zinc  sheets  are  usually  about  3  ft  bv  7  or  8  ft.  The  gauge  difiFers  from  that  of  iron  ;  thus  No  13  is 
.032  of  an  inch  thick,  or  1.22  fts  per  sq  ft ;  No  14  =  .035  inch,  and  1.35  fts ;  No  15  =  .042  inch,  and 
1.49  fts ;  No  16  =  .049  inch,  and  1.62  fts  per  sq  ft.  Any  of  these  numbers  may  be  used  on  roofs,  for 
which  purpose  it  should  be  very  pure. 

Water  kept  in  zinc  vessels  is  said  to  become  injurious  to  health;  and 

recently  an  outcry  has  on  that  account  arisen  against  galvanized-iron  service-pipes  in  dwellings. 
Tet  such  have  been  in  use  for  many  years  in  New  England,  Philada,  and  elsewhere,  without  as  yet 
any  deleterious  effects.  This  is  possibly  owing  to  the  fact  that  service-pipes  being  short,  the  water 
Is  usually  all  drawn  through  them  several  times  a  day ;  and  hence  does  not  remain  in  contact  with 
the  zinc  or  lead  long  enough  to  acquire  a  poisonous  character.  In  taking  possession  of  a  house  in 
Wbieb  the  water  has  remained  stagnant  in  the  service-pipes  for  some  considerable  time,  such  water 
•kouM  ail  be  run  to  waste ;  otherwise  sickness  may  ensue  from  its  use. 


918 


WEIGHT   OF   METALS. 


Roof  copper  is  usually  in  sheets  of  2>^  feet  X  5  feet ;  or  Vz%  square  feet, 

weighing  10  to  14  ibs  per  sheet ;  and  is  laid  on  boards.*  No  solder  is  used  in  the 
horizontal  joints  as  it  is  in  tin  roofs;  but  both  the  horizontal  and  the  sloping 
joints  are  formed  by  only  overlapping  and  bending  the  sheets,  much  as  shown 
by  the  figs  on  page  916 ;  except  that  the  horizontal  joints  are  bent  or  locked 
together,  and  then  flattened  down  close. 

Sheet  lead.    List  of  standard  weig^hts  in  lbs  per  square  foot.    Thick* 
nesses  in  decimals  of  an  inch. 


wt. 

Th. 

Wt. 

Th. 

Wt. 

.Th. 

Wt. 

Th. 

Wt.     1     Th. 

Wt. 

Th. 

2.5 
3 

.042 
.051 

4 
5 

.068 
.085 

6 

7 

.102 
.119 

8 
9 

.136 
.153 

10 
12 

.170 
.203 

14 
16 

.237 
.271 

^Veiglit  of  Metal  Balls. 

W  =  Weight  of  ball,  in  pounds^.     D  =  Diameter  of  ball,  in  inches. 
liead       ^  (700  fibs  per  cub  ft)  W  =  0.212106  D3 ;  log  W  =X326  5529  +  3  log  D 
Copper  =  (550  Bbs  per  cub  ft)  W  =  0.166655  D3 ;  log  W  =T221  8176  +  3  log  D 
Brass     =  (500  fts  per  cub  ft)  W  =  0.151504  D3 ;  log  W  =Tl80  4249  +  3  log  D 

Steel  and)        fA<iK^a  — 

Wrought  y  =  ^   ,?.r^„K  fn  W  =  0.146959  D3 ;  log  W  =  1.167  1966  +  3  log  D 
Iron  J  per  CUD  It; 

Iron  }  =  (450  ^s  Pe^  c^^  ^t)  W  =  0.136354  D3  ;  log  W  =T.134  6674  +  3  log  D 

For  steel,  wrought  iron  and  cast  iron  balls,  see  also  tables,  pp 
875  and  877. 


liead  Pipe. 
Standard  Sizes. 


«5 

• 

Wis  per  ft  (F) 

fcS 

WtS  per  ft  (F) 

1^ 

i| 

■ 
it 

i^ 

and  per  rod  (R) 

^5 

and  per  rod  (R) 

•-? 

Ss 

of  161^  ft. 

A^ 

of  163^  ft. 

A° 

ga 

JSa 

g^ 

^'C 

H 

»« 

H 

•8 

H 

« 

H 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

H 

.06 

7      ftsR 

H 

.08 

16      ftsR 

IK 

.14 

3.5 

3 

3-16 

9 

.08 

10      oz  P 

.10 

IH  ftsF 

.17 

4  25 

M 

12 

m 

.12 

1      lb   F 

" 

.12 

l^ftsF 

•« 

.19 

5. 

<< 

5-16 

16 

" 

.16 

134  B)s  F 

" 

.16 

2%  ftsF 

«« 

.23 

6,5 

»* 

'Kr 

20 

«« 

.19 

\-^  IbsP 

" 

.20 

3      ftsP 

•' 

.27 

8 

'm 

3-16 

9.5 

H 

.07 

9      IbsK 

" 

.23 

3>^ftsP 

i%r 

.13 

4 

H 

15 

.09 

%-Si    Y 

'« 

.30 

W,  fts  F 

•i 

.17 

5 

♦« 

5-16 

18.5 

M 

.11 

1      lb    F 

1 

.10 

24%  fts  R 

" 

.21 

6  5 

" 

'^ 

22 

« 

.13 

IK  ttsF 

" 

.11 

2      ftsP 

•« 

.27 

85 

4 

3-lfi 

125 

" 

.16 

\H  l>sP 

•' 

.14 

2^  IbsP 

2 

.15 

4.75 

" 

H       16 

M 

.19 

2      ft8F 

«• 

.17 

^\i  ttsP 

«« 

.18 

6 

<« 

5-16 

21 

<« 

.25 

3      IbsP 

♦« 

.21 

4^*  fts  P 

•  « 

.22 

7 

" 

"Hi 

2a 

X 

.08 

12      ttsR 

•« 

.24 

45ittsP 

«« 

.27 

9 

4K 

3-16 

14 

.09 

1      ft    F 

IV* 

.10 

2      ftsP 

2V5 

3-16 

8 

V, 

18 

♦« 

.13 

IK  ftsF 

.12 

2%  ftsP 

M 

11 

5 

H 

20 

.16 

2      ftsF 

.14 

3      ftsP 

5-16 

14 

% 

31 

•• 

.20 

2>i  ftsF 

♦• 

.16 

3?i  ft«P 

" 

% 

17 

«< 

.22 

2?i  ftsP 

" 

.19 

4Ji  ftsP 

" 

.25 

3>^  ttsF 

" 

.25 

6      ftsF 

liead  service  pipes  for  single  dwellings  in  Philadelphia  are  usually  of  from 
l^  inch  bore,  wt  1  to  2^  Tbs ;  to  %  inch  bore,  wt  1^  to  3  Bbs  per  ft  run,  according 
to  head.  They  rarely  burst  from  sudden  closing  of  stopcocks  ;  but  sometimes 
do  so  from  the  freezing  of  the  contained  water. 

*  To  which  it  is  held  by  copper  cleats ;  as  at  Fig  y,   page  916. 


METALS. 


919 


ROIiliED  l.EAI>,  COPPER 

,  and  BRASS:  Slieets  and  Bars. 

LEAD. 

BRASS 

Thickness 

COPPER. 

] 

ThicknesI 
or 

Diameter, 

Diameter, 

or  side, 

Sheets, 

Square 

Round 

Sheets, 

Square 

Bound 

Sheets, 

Square 

Round 

or  side, 

in 

per 

Bars ; 

Bars; 

per 

Bars; 

Bars; 

per 

Bars; 

Bars ; 

in 

Inches. 

Square 

1  Foot 

IFoot 

Square 

IFoot 

1  Foot 

Square 

IFoot 

IFoot 

Inches. 

Foot. 

long. 

long. 

Foot. 

long. 

long. 

Foot. 

loiig. 

long. 

Lbs. 

Lbs. 

Lba. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

LbB. 

Lbs. 

1-32 

1.86 

.005 

.004 

1.44 

.004 

.003 

1.36 

.004 

.003 

1-32 

1-J6 

3.72 

.019 

.015 

2.89 

.015 

.012 

2.71 

.014 

.011 

1-16 

3-32 

5.58 

.044 

.034 

4.33 

.034 

.027 

4.06 

.032 

.025 

3-32 

H 

7.44 

.078 

.061 

5.77 

.060 

.047 

5.42 

.056 

.044 

H 

6-32 

9.30 

.121 

.095 

7.20 

.094 

.074 

6.75 

.088 

.069 

6-32 

3-16 

11.2 

.174 

.137 

8.66 

.135 

.106 

8.13 

.127 

.100 

816 

7-32 

13.0 

.237 

.187 

10.1 

.184 

.144 

9.50 

.173 

.136 

7-32 

H 

14.9 

.310 

.244 

11.5 

.240 

.189 

10.8 

.226 

.177 

y* . 

5-16 

18.6 

.485 

.381 

14.4 

■    .376 

.295 

13.5 

.353 

.277 

616 

v$ 

22.3 

.698 

.548 

17.3 

.541 

.425 

16.3 

.508 

.399 

% 

7-16 

26.0 

.950 

.746 

20.2 

.736 

.578 

19.0 

.691 

.543 

7-16 

}4 

29.8 

1.24 

.974 

23.1 

.962 

.755 

21.7 

.903 

.709 

^ 

9-16 

33.5 

1.57 

1.23 

26.0 

1.22 

.955 

24.3 

1.14 

.900 

9-16 

H 

37.2 

1.94 

1.52 

28.9 

1.50 

1.18 

27.1 

1.41 

1.11 

% 

11-16 

40.9 

2.34 

1.84 

31.7 

1.82 

1.43 

29.8 

1.70 

1.34 

11-16 

H 

44.6 

2.79 

2.19 

34.6 

2.16 

1.70 

32.5 

2.03 

1.60 

% 

13-16 

48.3 

3.27 

2.57 

37.5 

2.55 

1.99 

35.2 

2.38 

1.87 

13-16 

Vs 

52.1 

3.80 

2.98 

40.4 

2.94 

2.31 

37.9 

2.76 

2.17 

% 

15-16 

56.0 

4.37 

3.42 

43.3 

3.38 

2.65 

40.6 

8.18 

2.49 

15-16 

1. 

59.5 

4.96 

3.90 

46.2 

3.85 

3.02 

43.3 

3.61 

2.84 

1. 

IH 

66.9 

6.27 

4.92 

52.0 

4.87 

3.82 

48.7 

4.57 

3.60 

V4 

74.4 

7.75 

.  6.09 

57.7 

6.01 

4.72 

54.2 

5.64 

4.43 

^% 

81.8 

9.37 

7.37 

63.5 

7.28 

5.72 

59.6 

6.82 

5.37 

^% 

l^ 

89.3 

11.2 

8.77 

69.3 

8.65 

6.80 

65.0 

8.12 

6.88 

I^ 

1% 

96.7 

13.1 

10.3 

75.1 

10.2 

7.98 

70.4 

9.53 

7.49 

^% 

m 

104. 

15.2 

11.9 

80.8 

11.8 

9.25 

75.9 

11.1 

8.08 

IH 

IK 

112. 

17.5 

13.7 

866 

1.H.5 

10.6 

81.3 

12.7 

9.97 

IK 

2. 

119. 

19.8 

15.6 

92.3 

15.4 

12.1 

86.7 

14.4 

11.3 

2. 

Seamless  brass  tnbes.    Principal  sizes.    Extras,  in  cents  per  pound, 
over  base  price.     For  base  price,  see  price  list. 
Copper  tubes,  3  cents  per  pound  extra. 


Thickness. 

Outer  Diameter,  inches. 

Stubs 
gage. 

Ins. 

u 

y2 

1 

IM 

2 

3 

4 

5 

6 

7 

v% 

4 
11 
16 
18 
20 
22 

0.238 

0.120 

0.065 

0.049 

0.035 

0.0295 

0.0230 

*40"* 
43 
50 
65 

12 
ti 

13 
15 
18 
21 
28 

6 

8 
9 
13 
16 
22 

3 

4 
6 
10 
15 
24 

1 

4 
6 
9 
13 

1 

4 

7 
11 
16 

2 
t« 

7 
11 
15 
.20 

5 

11 
15 
19 

9 

15 
19 
23 

13 

<( 

19 
23 

27 

24 
28 
32 

25 

920 


METALS. 


Average  ultimate  tensile  strengtli  of  Metals. 


The  ultimate  tensile  or  pulling  load  per  square  inch  of  any 
material  is  frequently  called  its  constant,  coefficient,  or  modulus  of 
tensioH,  or  of  tensile  strength. 


Pounds 

per 
sq.  inch. 


Tons 

per 

sq.  in. 


Antimony,  cast , 

Bismuth,  cast 

Brass,  cast  8  to  13  tons,  say  18000  to  29000  lbs 

"    wire,  unannealed  or  hard,  80000.    Annealed 

Bronze,  phosphor  wire,  hard,  150000.    Annealed .-.. 

Copper,  cast  18000  to  30000 , 

boits,*28oooTo'Sooo"V.V."*.V.*.!*.l.V.!!!.\\*".!!!.*!.\\*^^ 

"       wire  (annealed  16  tons) ;  unannealed 

Grold,  cast , 

"      wire,  25000  to  30000 

Gun  metal  of  copper  and  tin,  23000  to  55000 , 

"  cast  iron,  U.  S.  ordnance,  36000  to  40000 

Iron,  cast,  English 13400  to  22400 

"        "     ordinary  pig..l3000  to  16000 : 

American  cast  iron  averages  one-fourth  more  than  the  above. 

Average  cast  iron,  when  sound,  stretches  about  .00018 ;  or  1  part 
in  5555  of  its  length ;  or  ]/q  inch  in  67.9  ft.  for  every  ton  of  ten- 
sile strain  per  sq  inch,  up  to  its  elastic  limit,  which  is  at  about 
^  its  break-strain.  The  extent  of  stretching,  however,  varies 
much  with  the  quality  of  the  iron ;  as  in  wrought-iron. 

Cast,  malleable,  annealed  18  to  25  tons.    

Iron  and  Steel,  rolled.— See  Digests  of  Specifications. 

Lead,  cast,  1700  to  2400 by  author.. 

"      wire,  1200  to  1600.    Pipe  1600  to  1700 "        "      ., 

Platinum  wire,  annealed,  32000.     Unannealed 

Steel  and  Ifon,  rolled. — See  Digests  of  Specifications. 

Silver,  cast 

Tin,  English  block 

**    wire 

Zinc,  cast. ..3000  to  3700;  (the  last  by  author) 


1000 
3200 
23500 
49000 
63000 
24000 
30000 
33000 
60000 
20000 
27500 
39000 
38000 
17900 
14500 


48160 

2050 

1650 

56000 

41000 
4600 
7000 
3350 


.45 

1.4 
10.5 
22 
28.1 
10.7 
13.4 
14.7 
26.8 

8.9 
12.3 
17.4 
17 

8 

6.47 


ai.5 

0.92 
0.74 
25 

18.3 
2.0 
8.1 
1.5 


liargre  bars  of  metal  bear  less  per  sq  inch  than  small  ones. 
Iron  bars  re-rolled  cold  have  tensile  strength  increased  25  to  60  per 
ct,  with  no  increase  of  density.    They  are  said  to  lose  this  strength  if  reheated. 


METALS. 


921 


Sheet  lead  is  sometimes  placed  in  the  joints  of  stone  col- 
umns, with  a  view  to  equalize  the  pressure,  and  thus  increase  the  strength  of 
the  eokimn.  But  experiments  have  proved  that  the  effect  is  directly  the  reverse, 
and  that  the  column  is  materially  weakened  thereby. 

Average  crushing  load  for  Metals. 

It  must  be  remembered  that  these  are  the  loads  for  pieces  but  two  or  three 
times  their  least  side  in  height.  As  the  height  increases,  the  crushing  load 
diminishes.    See  "Strength  of  Pillars." 


Cast  Iron,  usually 

It  is  usually  assumed  at  100000  fts,  or  say  45  tons  per  sq  inch.  Its 
crushing  strength  is  usually  from  6  to  7  times  as  great  as  its  tensile. 
"Within  its  average  elastic  limit  of  about  15  tons  per  sq  inch,  average 
cast  iron  shortens  about  1  part  in  5555 ;  or  %  inch  in  58  ft  under  each 
ton  per  sq  inch  of  load ;  or  about  twice  as  much  as  average  wrought 
iron.  Hence  at  15  tons  per  sq  inch  it  wHl  shorten  about  1  part  in  370; 
or  full  %  inch  in  4  feet.  Different  cast  irons  may  however  vary  10  to 
15  per  ct  either  way  from  this. 

IJ.  S.  Ordnance,  or  gun  metal  ;  Some 

Wrought  iron,  within  elastic  limit 

Its  elastic  limit  under  pressure  averages  about  13  tons  per  sq  inch. 
It  begins  to  shorten  perceptibly  under  8  to  10  tons,  but  recovers  when 
the  load  is  removed.  With  from  18  to  20  tons,  it  shortens  permanently, 
about  ^^th  part  of  its  length ;  and  with  from  27  to  30  tons,  about  y^^*^ 
part,  as  averages.  The  crushing  weights  therefore  in  the  table  are 
not  those  which  absolutely  masti  wrought  iron  entirely  out  of  shape, 
but  merely  those  at  which  it  yields  too  much  for  most  practical  build- 
ing purposes.  About  4  tons  per  sq  inch  is  considered  its  average  safe 
load,  in  pieces  not  more  than  10  diams  long ;  and  will  shorten  it  %  inch 
in  30  ft.  average. 

Brass,  reduced  y^yth  part  in  length,  by  51000;  and  ^by 
Copper,  (cast,)  crumbles 

(wrought)  reduced  ^th  part  in  length,  by 

Tin,  (cast,)  reduced  jVth  in  length,  by  8800;  and  ^  by 
I^ead,  (cast,)  reduced  >^  of  its  length,  by  7000  to  7700.... 

"  By  writer.  A  piece  1  inch  sq,  2  ins  high,  at  1200 fts  the  com- 
pression was  1-200  of  the  ht  j  at  2000,  1-29;  at  3000,  1-8;  at 
5000,  1-3  ;  at  7000,  1-2  of  the  ht. 

Spelter  or  Zinc,  (cast.)    By  writer.    A  piece  1  inch 

square,  4  ins  high,  at  2000  fl)s  was  compressed  1-400  of  its  ht ;  at  4000, 

1-200 ;  at  6000,  1-100;  at  10000,  1-38  ;  at  20000,  1-15  ;  at  40000  yielded 

rapidly,  and  broke  into  pieces. 

Steel,  224000  ft)s  or  100  tons  shorten  it  from  .2  to  .4  part. 

"        American.     Black   Diamond   steel-works,   Pittsburg,   Penn, 

experiments  by  Lieut  W.  H.  Shock,  y.  S.  N.,  on  pieces  J^  in 

square;  and  3>^  ins,  or  7  sides  long. 

'*        Untempered.  100100  to  104000 

'•        Heated  to  light  cherrv  red,  then  plunged  into  oil  of  82^"  Fah, 

173200  to  199200. . . ." 

"        Heated  to  light  cherry  red,  then  plunged  into  water  of  79° 

Fah  ;  then  tempered  on  a  heated  plate,  325400  to  340800.... 

"        Heated  to  light  cherry  red,  then  plunged  into  water  of  79° 

Fah,  275600  to  400000 : 

"      Elastic  limit,  15  to  27  tons 

"      Compression,  within  elas  limit  averages  abt 

1  part  in  13300,  or  .  1  of  an  inch  in  1 1 1  ft  per  ton  per  sq  inch ; 
or  .1  of  an  inch  in  5.3  ft  under  21  tons  per  sq  inch. 

Best  Steel  knife  edges,  of  large  R  R  weigh  scales 

are  considered  safe  with  7000  fl)s  pres  per  lineal  inch  of  edge ;  and 

solid  cylindrical  steel  rollers  under  bridges,  and 

rolling  on  steel,  safe  with  /diam  in  ins  X  3  100  000,  in  lbs  per  lineal 
inch  of  roller  parallel  to  axis.     And  per  the  same*  for 


85000  to  125000 


Pounds  per 
sq.  inch. 


175000 

22400  to  35840 
29120 


..165000.. 
..117000.. 
..103000.. 
....15500.. 
7350.. 


..102050.. 

,.186200.. 

..333100.. 

..337800.. 
...47040.. 


Tons  per 
sq.  inch. 


38  to  56 


78.1 

10  to  IS 

13 


73.6 

52.2 
46.0 
6.92 
3.28 


46.S 

83.1 

148.7 

150.8 
21 


Solid  cast  Iron  wheels  rolling  on  wrought  iron,  l/Diam  ins  X  352  000. 

"  *'        »•  '•  *•  "  cast  iron, 

SoUd  steel 


y/Diam  ins  X  222  222. 
]/Diam  ins  X  1300000. 


'  "  "  steel, 

"  "  "  "  wrought  iron,  V'Diamins  X 1024000. 

"  "  "  "  cast  iron,         ]/Diam  ins  X  850  000. 

'  SpecificatioDS  for  Iron  Drawbridge  at  Milwaukee,"  by  Don  J.  Wbittemore,  C.  E. 


922 


STONE,    ETC. 


Averag^e  ultimate  tensile  streng-ttis  of  Stone,  etc. 

The  strengths  iu  ail   these 
tables  may  readily  be  one-third 

Pounds 

Tons 

Pounds 

Tons 

part  more  or    less    than   our 

per 

per 

per 

per 

averages. 

sq.  inch. 

sq.  ft. 

Marble,8trong,wh.Italy.* 
♦'        Champlaln,varie- 

sq.  inch. 

sq.  ft. 

Brick,  40  to  400 

220 

14.1 

1034 

66.5 

Caen  stone,  100  to  200 

150 

9.7 

gated* 

1666 

107.1 

"        Glenn's  F'lls,N.Y. 

blk,*  750tol034. 

892 

67.4 

"       Montg'y    CO,  Pa, 

gray  * 

1175 

75.6 

"             "       white*... 

734 

47.2 

"       Lee,Mas8,white.* 

875 

56.3 

Cement    and    concrete, 

"       Manchester,  Vt  * 

see    articles,    Cement 
and  Concrete. 

550  to  800.  .  .   . 

675 

43  4 

"       Tennessee,  varie- 

gated*   

1034 

66.5 

Oolites,  100  to  200 

150 

9.7 

Plaster  of  Paris,  well  set. 

70 

4.5 

Rope,  Manilla,  best 

12000 

771 

Glass,  2500  to  9000 

5750 

369.6 

"      hemp,  best 

15000 

966    . 

Glue  holds  wood  together 

Sandstone,  Ohio* 

105 

6.75 

with  from  300  to  800... 

550 

35 

Pictou,  N.  S.* 

434 

27.9 

Horn,  ox 

9000 

579 

"           Conn,  red.*.... 

590 

37.9 

Ivory 

16000 

1029 

Slate  Lehigh* 

2475 

159.1 

Leather    belts,    1500   to 

"      Peach  bot'm,*  3025 

5000.    Good 

3000 

193 

to  4600  

3812 
300 

2451 

Mortar,  common,  6  mos 

Stone,  Ransome's  artif,... 

19.3 

old,  10  to  20 

15 

.96 

Whalebone 

7600 

489 

*  By  tlie  autbor's  trials  with  one  of  Biehl6's  testing  machines.    £eetioai« 

broken  13^  sq  iaobes. 


STONE,    ETC. 


923 


Ultimate  averag'e  criisliiii^  loads  in  tons,  per  square 
foot,  for  stones,  «fec.  The  stones  are  supposed  to  be  on  bed,  and  the  heights 
of  all  to  be  from  1.5  to  2  times  the  least  side.  Stones  generally  begin  to  crack  or 
split  under  about  one-half  of  their  crushing  loads.  In  practice,  neither  stone  nor 
brickwork  should  be  trusted  with  more  than  %  to  y  gth  of  the  crushing  load,  ac- 
oordin^  to  circumstances.  When  tliorou;;'lily  wet  some  absorbent  sand- 
stones lose  fully  half  their  strength. 


Granites  and  Syenites. 

Basalt 

Limestones  and  Mar- 
bles*  

Oolites,  good 

Brownstone : 
Connecticut — 

"Building  " 

"Bridge" 


300  to  1200 


Brick* 

Brickwork,  ordinary, 
cracks  with* 

Brickwork,  good,  in  ce- 
ment*   

Brickwork,  first/-rate, 
in  cement 

Slate 

Caen  Stone 

"        "      to  crack 

Chalk,  hard, 

Plaster  of  Paris,  1  day 
old 


Tons  per 
sq.  ft. 


^ean. 
Tons. 


250  to  1000 
100  to  250 


570  to  970 
400  to  630 

40  to  300 

20  to  30 

30  to  40 

50  to  70 

400  to  800 

70  to  200 


750 
700 


625 
175 


775 
535 

170 

25 

35 

60 
600 
135 
70 
25 

40 


Cement,    Portland, 

neat,U,  S.  or  foreign, 

7  days  in  water 

Common  U.S.cements, 

neat,  7  days  in  water 
ConcreteofPort. 

cement,    sand,    and 

gravel  or  brok  stone 

in  theproper  propor- 

tions,rammed  1  m  old 

6  months  old 

12  months  old 

With  good  common 

hya      cements, 

abt  .2  to  .25  as  much 
€oig-net  be  ton,  3 

months  old 

Rubble      masonry, 

mortar,  rough 

Glass,  green,crowu  and| 

flint tl300to2300 

or  3  times  that  of  granita 
Ice,  firmf |   12  to  18   |    15 


Tons  per 
sq.  ft. 


Mean. 
Tons. 


75  to  150 
15  to  30 


12  to  18 
48  to  72 
74  to  120 


100  to  150 
15  to  35 


112.5 
22.5 


15 
60 
97 


125 
25 


i8or 


Crnsfiln^r  lieig-tit  of  Brick  and  Stone. 

If  we  assume  the  wt  of  ordinary  brickwork  at  112  lbs  per  cub  ft,  and  that  it  would 
crush  under  30  tons  per  sq  ft,  then  a  vert  uniform  column  of  it  600  ft  high,  would 
crush  at  its  base,  under  its  own  wt.  Caen  stone,  weighing  130  lbs  per  cub  ft,  would 
require  a  column  1376  ft  high  to  crush  it.  Average  sandstones  at  145  lbs  per  cub  ft, 
would  require  one  4158  ft  high  ;  and  average  granites,  at  165  lbs  per  cub  ft,  one 
of  8145  feet.  But  stones  begin  to  crack  and  splinter  at  about  half  their  ultimate) 
crushing  load;  and  in  practice  it  is  not  considered  expedient  to  trust  them  with  more 
than  %th  to  ^^th  part  of  it,  especially  in  important  works;  inasmuch  as  settlements, 
and  imperfect  workmanship,  often  cause  undue  strains  to  be  thrown  on  certain 
parts. 

The  Merchants'  shot-tower  at  Baltimore  is  246  ft  high ;  and  its  base  sustains  6}/4 
tons  per  sq  ft.  The  base  of  the  granite  pier  of  Saltash  bridge,  (by  Brunei,)  of  solia 
masonry  to  the  height  of  96  ft,  and  supporting  the  ends  of  two  iron  spans  of  455  ft 
each,  sustains  9]/^  tons  per  sq  ft.  The  base  of  u  brick  chimney  at  Glasgow,  Scotland, 
468  ft  high,  bears  9  tons  per  sq  ft ;  and  Professor  Rankine  considers  that  in  a  high 
gale  of  wind,  its  leeward  side  may  have  to  bear  15  tons.  The  highest  pier  of  Rocque» 
favour  stone  aqueduct,  Marseilles,  is  305  ft,  and  sustains  a  pressure  at  base  of  13^ 
tons  per  sq  ft. 

*  Trials  at  fSt.  liOnis   bridi^e,  by  order  of  Capt  James  B.  Eads,  C.  E., 

showed  that  some  maguesian  limestone  did  not  yield  under  less  than  1100  tons  per  sq  ft.  A  column 
8  ins  high,  2  ins  dium,   shortened  0.0025  inoh  under   pressure;    and   rernvered  -when  relieved. 

Experiments  made  ^vith  tbe  Oovt  testing*  macbine  at  l¥ater« 
town.  Mass,  1882-3,  gave  1400  tons  per  sq  ft  ultimate  crushg  load  for  white 
and  blue  marble  from  Lee,  Mass,  700  for  blue  marble  from  Montgomery  Co,  Pa,  960  for  limestone  from 
Consbohocken,  Pa,  500  for  limestone  from  Indiana,  840  for  red  sandstone  from  Hummelstowu,  Pa, 
260  to  1000  for  yellow  Ohio  sandstone  ;  Phila  bricks,  flatwise;  hard,  machine-made,  350  to  700  tons; 
hand-made,  700  to  1300;  pressed,  machine-made,  450  to  580;  Brickwork  columns,  13  ios  m<i  and  13  ins 
high  ;  in  lime,  100  tons  ;  in  cement,  150. 

t  Experiments  by  Col.  Wm.  Ludlow,  U.  S.  A.,  with  Govt  testing  machintft,  m  1381,  gave  Trom  3t 
to  64  tons  per  sq  ft  for  pure,  hard  ice ;  and  16  to  59  tons  for  inferior  grades.  The  ttpeoimsns  (•  aai 
12-inch  cubes)  compresaed  ^  to  1  inch  before  crusbinM* 


924 


STONE    BEAMS. 
STOIIfE   BEAMS. 


Table  of  safe  quiescent  extraneous  loads  for  beams  of  g^ood 
building'  g^ranite  one  inch  broad,  supported  at  both  ends,  and  loaded  at  the 
center;  assuming  the  safe  load  to  be  one-tenth  of  the  breaking  one;  and  the  latter 
to  be  100  lbs  for  a  beam  1  inch  square,  and  1  foot  clear  span.  The  half  weight  of 
the  beams  themselves  is  here  already  deducted  at  170 

lbs  per  cub  ft. 


i 

CL 

EAR 

SPANS  IN 

FEET 

1 

2 

3 

4 

5 

6 

7 

8 

10 

12 

15 

30 

Safe  center  loads  in  pounds. 

1 

10 

5 

2 

40 

20 

13 

10 

3 

90 

45 

29 

21 

17 

4 

160 

79 

52 

39 

31 

26 

21 

5 

250 

124 

82 

61 

48 

40 

34 

6 

360 

179 

119 

89 

70 

58 

48 

42 

32 

7 

490 

244 

162 

120 

96 

79 

67 

58 

45 

36 

27 

16 

8 

639 

319 

212 

158 

126 

104 

88 

76 

59 

47 

36 

22 

10 

999 

499 

331 

248 

197 

163 

139 

120 

94 

76 

58 

38 

12 

1439 

718 

478 

357 

284 

236 

201 

174 

137 

111 

85 

58 

14 

1959 

978 

650 

487 

388 

322 

274 

238 

188 

153 

118 

81 

16 

2559 

1278 

850 

636 

507 

421 

359 

312 

246 

201 

157 

109 

18 

3239 

1618 

1077 

806 

643 

534 

455 

396 

313 

257 

200 

141 

20 

3999 

1998 

1329 

995 

794 

660 

563 

490 

388 

319 

249 

176 

22 

4839 

2417 

1609 

1205 

961 

800 

682 

594 

470 

387 

303 

216 

24 

5758 

2877 

1916 

1434 

1145 

951 

813 

708 

562 

463 

362 

260 

27 

7288 

3642 

2425 

1815 

1450 

1205 

1030 

898 

713 

588 

462 

332 

30 

8998 

4496 

2995 

2243 

1791 

1489 

1273 

1110 

882 

728 

573 

415 

33 

10888 

5441 

3624 

2714 

2168 

1803 

1542 

1345 

1069 

883 

696 

505 

36 

12958 

6476 

4314 

3231 

2581 

2147 

1836 

1603 

1275 

1054 

832 

606 

If  uniformly  distributed  over  tlie  clear  span,  the  safe  extraneous 
loads  will  be  twice  as  great  as  those  in  the  table. 

For  good  slate  on  bed  the  safe  loads  may  be  taken  at  about  8  times ;  for 
good  sandstone  on  bed  at  about  one-half;  and  for  good  marble  or 
limestone  on  bed  at  about  the  same  as  those  in  the  table. 


MORTAR.  925 

MORTAE,  BRICKS,  &c. 

I.IME  MORTAR. 

Art.  1.  Mortar.  The  proportion  of  1  measure  of  quicklime,  either  in  ir- 
regular lumps,  or  ground,  and  5  measures  of  sand,  is  about  the  average  used  for 
common  mortar,  by  good  builders  in  our  principal  Atlantic  cities;  and  if  both 
materials  are  good,  and  well  mixed  (or  tempered)  with  clean  water,  the  mortar  is 
certainly  as  good  as  can  be  desired  for  such  ordinary  purposes  as  require  no  addi- 
tion of  hydraulic  cement.  The  bulk  of  the  mixed  mortar  will  usually  exceed  that 
of  the  dry  loose  sand  alone  about  ^  part. 

Quantity  reqaired.  20  cub  ft,  or  16  struck  bushels  of  sand,  and  4  cub  ft,  or 

3.2  struck  bushels  of  quicklime,  the  measures  slightly  shaken  in  both  cases,  will  make  abt  2214  oub  ft  of 
mortar ;  sufficient  to  lay  1000  bricks  of  the  ordinary  average  size  of  83^  by  4  by  2  ins,  with  the  coarss 
mortar  joints  usual  in  interior  house-walls,  rarying  say  from  %  to  14  inch.  With  such  joints,  lOOi 
8uch  bricks  make  2  cubic  yards  of  massive  work.  Nearly  one-third  of  the  mass  is  mortar.  For 
outside  or  showing  joints,  where  a  whiter  and  neater  looking  mortar  is  required,  house-builders  in- 
crease the  proportion  of  lime  to  1  in  4,  or  1  i0  3.  For  mortar  of  fine  screened  gravel,  for  cellar-walls 
of  stone  rubble,  or  coarse  brickwork,  1  measure  of  lime  to  6  or  8  of  gravel,  is  usual ;  and  the  mortar 
is  good.  In  average  rough  massive  rubble,  an  in  the  foregoing  brickwork,  about  one-third  the  mass  is 
mortar :  consequently  a  cubic  yard  will  require  about  as  much  as  500  such  bricks ;  or  10  cubic  feet,  (8 
struck  bushels)  of  sand ;  and  2  cub  ft,  or  1.6  bushels  of  quicklime.  Superior,  well-scabbled  rubble, 
carefully  laid,  will  contain  but  about  j  of  its  btilk  of  mortar ;  or  5^  cub  ft  sand,  and  1.1  cub  ft  lime, 
per  cub  yard. 

For  public  engineering  works,  especially  in  massive  ones,  or  where  exposed  to  dampness,  an  addi- 
tion should  be  made  in  either  of  the  foregoing  mortars,  of  a  quantity  of  good  liyd 
cement,  equal  to  about  3^  of  the  lime;  or  still  better,  }/^  of  the  lime  should  be 

omitted,  and  an  equal  measure  of  cement  be  substituted  for  it.  If  exposed  to  water  while 
quite  aewv  use  little  or  no  lime  outside. 

With  bricks  of  SJ^  by  4  by  2  ins,  the  following  are  the  quantities  of  mor- 
tar and  of  bricks  for  a  cubic  yard  of  massive  work. 

Thickness  Proportion  of  Mortar  No.  of  Bricks  No.  of  Bricks 

of  Joints.  in  the  whole  mass.  per  cub  yard.  per  cub  foot. 


i 


inch about  ^  638 23.63 

574 21.26 

522  19.33 


1 


t\: 


475  17.60 

433  16.04 


In  estimating  for  bricks  in  massive  work,  allow  2  or  3  per  ct  for  wa«te ; 

and  in  common  buildings,  5  per  ct,  or  more.  Much  of  the  waste  is  incurred  in  cutting  bricks  to  fit 
angles,  Ac.  In  Philadelphia  a  barrel  of  lump  lime  is  allowed  for  1000  bricks ;  or  for  2  perches  (25  cub 
ft  each)  of  rough  cellar-wall  rubble.  Somewhat  less  mortar  per  1000  is  contained  in  thin  walls,  than 
in  massive  engineering  structures  ;  because  the  former  have  proportionally  more  outside  face,  which 
does  not  require  to  be  covered  with  mortar;  but  thin  walls  involve  more  waste  while  building;  so  that 
both  require  about  the  same  quantity  of  materials  to  be  provided.  Careful  experiments  show  that 
mortar  becomes  harder,  and  more  adhesive  to  brick  or  stone,  if  the  proportion  of  lime  is  increased. 
Hence,  on  our  public  works  the  proportion  of  one  measure  of  quicklime  to  3  of  sand,  is  usually  spec- 
ified, but  probably  never  used. 

liime  is  usually  sold  in  lump,  by  the  barrel,    of  about  2S0  ibs  net, 

or  250  S)s  gross.    A  heaped  bushel  of  lump  lime  averages  about  75  fits.    Ground  qnlokllme, 

loose,  averages  about  70  Bbs  per  struck  bushel ;  and  3  bushels  loose  just  fill  a  common  flour  barrel ;  but 
from  3.5  to  3.75  bushels,  or  245  to  280  fts  can  readily  be  compacted  into  a  barrel. 

General  remarks  on  mortar  and  lime.     On  too  great  a  pro> 

portion  of  our  public  works,  the  common  lime  mortar  may  be  seen  to  be  rotten  and  useless,  where  it 
has  been  exposed  to  moisture  ;  which  will  be  carried  by  the  capillary  action  of  earth  to  several  feet 
above  the  natural  surface;  or  as  far  below  the  artificial  surface  of  embankments  deposited  behind 
abutments,  retaining-walls,  &c.  The  same  will  frequently  be  seen  in  the  soffits  of  arches  under  em- 
bankments. Common  lime  mortar,  thus  exposed  to  constant  moisture,  will  never  harden  properly. 
Even  when  very  old  and  hard,  it  absorbs  water  freely.     Cement  also  does  so,  but  hardens. 

Brickdnst,  or  burnt  clay,  improves  common  mortar ;  and  makes  it  hydraulic. 
In  localities  where  sand  cannot  be  obtained,  burnt  clay,  ground,  may  be  substituted;  and  will  gen- 
erally give  a  better  mortar. 

Protection  of  quicklime  from  moisture,  even  that  of  the  air,  is 
absolutely  essential,  otherwise  it  undergoes  the  process  of  air-slacking-,  or 


926 


MORTAR. 


apoDtaneous  slacking,  by  which  it  becomes  reduced  to  powder  as  when  slacked  by  water  as  usuaV 
but  without  heating,  and  with  but  little  swelling.  As  this  air  slacking  requires  from  a  few  months  t« 
a  year  or  more,  depending  on  quality  and  exposure,  it  gives  the  lime  time  to  absorb  sufficient  carbonia 

acid  from  the  air  to  injure  or  destroy  its  efficacy.  But  quicklime  ivlll  keep 
groofl  for  a  long*  time  if  first  ground,  and  then  well  packed  in  air-tight 

barrels.  The  grinding  also  breaks  down  refractory  particles  found  in  all  limes,  and  which  injure  the 
mortar  by  not  slacking  until  it  has  been  made  and  used.  For  the  same  reason  it  is  better  that  lime 
Bhoald  not  be  made  into  mortar  as  soon  as  it  is  slacked,  but  be  allowed  to  remain  slacked  for  a  day  or 
two  (or  CTtn  leTcral)  protected  from  rain,  sun,  and  dust. 

Itime  slacked  iu  great  bulk  may  char  or  even  set  fire  to  ^rood. 
liiuie  paste  and  mortar  urill  keep  for  years,  and  improve,  if  well 

buried  in  the  earth.  Also  for  months  if  merely  covered  m  heaps  under  shelter,  with  a  thick  layer  of 
eand.    The  paste  shrinks  and  cracks  in  drying ;  but  the  sand  in  mortar  prevents  this. 

As  approximate  avera^^es  varying  much  according  to  the  character  and 

degree  of  burning  of  the  limestone ;  and  to  the  fineness  or  coarseness  of  the  sand,  one  measure  of 
good  quicklime,  either  in  lump,  or  ground ;  if  wet  with  about  H  a  measure  of  water,  will  within  lesa 
than  an  hour,  slack  to  about  2  measures  of  dry  powder.  And  if  to  this  powder  there  be  added  about 
^  more  measures  of  water,  and  3  measures  of  dry  sand,  and  the  whole  thoroughly  mixed,  the  result 
will  be  about  3^  measures  of  mortar.  Or  the  same  slacked  dry  powder,  with  about  1  measure  of 
water,  and  5  measures  of  sand,  will  make  about  5%  measures  of  mortar.  In  both  cases  the  bulk  of 
the  mortar  will  be  about  ^  part  greater  than  that  of  thfe  dry  sand  alone.  If  5^  of  a  measure  of  water 
be  used  for  slacking,  the  result,  instead  of  a  dry  powder,  will  be  about  IH  measures  of  stifif  paste ;  or 
with  1  whole  measure  of  water  for  slacking,  the  result  will  be  about  l}4  measures  of  thin  paste,  of 
about  the  proper  consistence  for  mixing  with  the  sand.  Very  pure,  fat  limes,  slack  quickly,  and  make 
about  from  2  to  3  measures  of  powder;  while  poor,  meagre  ones,  require  more  time,  and  swell  less, 
Slow  slacking,  and  small  swelling,  in  case  the  lime  has  been  properly  burnt,  are  not  in  general  bad 
properties  ;  but  on  the  contrary,  usually  indicate  that  it  is  to  some  extent  hydraulic.  In  this  case  it 
makes  a  better  mortar ;  especially  for  works  exposed  to  moisture,  or  to  the  weather.  Very  pure  limes 
are  the  worst  of  all  for  such  exposures ;  or  are  bad  weather-limes  ;  and  in  important  works,  should 
never  be  used  without  cement. 

Sbell  lime  appears  to  be  about  the  same  as  that  from  the  purest  limestones; 

but  that  from  chalk  is  still  more  inferior,  and  will  not  bear  more  than  about  1]^  measures  of  sand; 
its  mortar  never  becomes  very  hard.  Madrepores  (commonly  called  coral)  appear  to  furnish  a  lime 
intermediate  between  those  of  chalk  and  limestone.     They  require  to  be  but  moderately  burnt. 

The  averag'e  w^eig'ht  of  common  hardened  mortar  is  about  105  to  115  lbs 

per  cub  ft. 
Orout  is  merely  common  mortar  made  so  thin  as  to  flow  almost  like  cream. 

It  is  intended  to  fill  interstices  left  in  the  mortar-joints  of  rough  masonry;  but  unless  it  contains  a 
large  amount  of  cement,  it  is  probably  entirely  worthless;  since  the  great  quantity  of  water  injures 
the  properties  of  lime ;  and  moreover,  its  ingredients  separate  from  each  other ;  the  sand  settling  be- 
low the  lime.  Besides  this,  it  will  never  harden  thoroughly  in  the  interior  of  thick  masses  of  ma- 
sonry ;  indeed,  the  same  may  probably  be  said  of  any  common  lime  mortar.  In  such  positions,  it  haa 
been  found  to  be  perfectly  soft,  after  the  lapse  of  many  years. 

Both  the  sand  and  the  water  for  lime  mortar,  should  be  free  from  clay  and 

salt.  The  clay  may  be  removed  by  thorougl>  washing ;  but  it  is  extremely  dif- 
ficult to  get  rid  of  the  salt  from  seashore  sand,  even  by  repeated  washings.  Enough  will  generally 
remain  to  keep  the  work  damp,  and  to  produce  efflorescences  of  nitre  on  the  surface ;  whether  with 
lime,  or  with  cement  mortar.     Slacking  by  salt  water  gives  less  paste  than  fresh. 

Mortar  should  not  be  mixed  upon  the  surface  of  clayey  ground ;  but  a  rough  board,  brick,  or  stone 
platform  should  be  interposed.  Pit  sand  sifted  from  decomposed  gneiss,  and  other  allied  rocks,  is  ex- 
cellent for  mortar;  its  sharp  angles  making  with  the  lime  a  more  coherent  mass  than  the  rounded 
grains  of  river  or  sea  sand.  Mortar  should  be  applied  wetter  in  hot  than  in  cold  weather ;  especially 
in  brickwork ;  otherwise  the  water  is  too  much  absorbed  by  the  masonry,  and  the  mortar  is  thereby 
injured. 

The  tenacity,  or  coliesive  streng'th,  that  is,  the  resistance  to  a  pull 

of  good  common  lime  mortar  of  the  usual  proportions  of  lime  and  sand,  and  6  months  old,  is  about 
from  15  to  30  fts  per  sq  inch ;  or  .%  to  1.9  tons  per  sq  ft.  With  less  sand,  or  with  greater  age,  it  will 
be  stronger. 

The  crushing:  streng-th  of  good  common  mortar  6  months  old  is  from  150 

lio  300  fts  per  sq  inch,  or  9.7  to  19.3  tons  per  sq  foot. 

The  sli<ling:  resistance,  or  that  which  common  mortar  opposes  to  any 

force  tending  to  make  one  course  of  masonry  slide  upon  another,  is  stated  by  Roudefet,  to  be  but  5  lbs 
per  sq  inch ;  or  about  one  third  of  a  ton  per  sq  ft,  in  mortar  6  months  old. 

TransTcrse  strength  of  good  common  mortar  6  months  old.    A  bar  1 

Inch  square  and  12  ins  clear  span,  breaks  with  a  center  load  of  4  to  8  Bbs. 

The  lime  in  mortar  decays  wood  rapidly,  especially  in  close, 

damp  situations.  Still  the  soaking  of  timber  for  a  week  or  two  in  a  solution  of  quicklime  in  water 
appears  to  act  as  a  preservative.  Iron,  so  completely  embedded  in  mortar  as  to  exclude  air  and 
moisture,  has  been  found  perfect  after  1400  years ;  but  if  the  mortar  admits  moisture  the  iron  decays. 
So,  probably,  with  other  metals. 

The  adhesion  to  common  bricks,  or  to  rougrh  rubble  at  any 

age  will  average  about  H  of  the  cohesive  strength  at  the  same  age;  or  say  12  to  24  ftsper  sq  inch,  oi 
.75  to  1.5  ton  per  sq  ft  at  6  months  old.  If  care  be  taken  to  exclude  dust  entirely,  by  dipping  each 
brick  into  water  before  laying  it,  or  by  sprinkling  the  stone  by  a  hose,  &c,  the  adhesion  will  be  in- 
treased.  On  the  other  hand,  much  dust  may  almost  prevent  any  adhesion  at  all.  The  precaution  of 
wetting  is  especially  necessary  in  very  hot  weather,  to  prevent  the  warm  bricks  or  stone  from  kill* 
Ing  the  mortar  by  the  rapid  absorption  and  evaporation  of  lis  water.  The  adhesion  to  rerf 
•mooth  hard  pressed  brl^Ub  or  to  smoothly  dressed  or  sawed  stone  is  considerably  less. 


BRICKS.  927 

BRICKS. 

Art.  2.  Bricks,  size,  weight,  «$:c.  A  brick  8.25  X  4  X  2  ins  contains 
66  cub  ins  ;  or  26.2  bricks  to  a  cub  ft ;  or  707  bricks  to  a  cub  yard. 

In  ordering  a  large  number,  a  minimum  limit  of  dimension  should  be  specified, 
in  order  to  prevent  fraud.  A  brick  y^  inch  less  each  way  than  the  above,  con- 
tains but  52.5  cub  ins  ;  thus  requiring  full  25  per  cent  more  bricks  to  do  the  same 
work,  and  25  per  ct  more  cost  for  laying,  which  is  generally  paid  by  the  1000. 

The  ivei^ht  of  a  good  common  brick,  8.25  X  4  X  2  ins,  will  average  about 
4.5  fl>s ;  or  118  lbs  per  cub  ft  =  3186  lbs  or  1.42  tons  per  cub  yard ;  or  2.01  tons  per 
1000.  A  good  pressed  brick  of  the  same  size  will  average  about  5  lbs,  =  131  iba 
per  cub  ft  =  3537  ft)s  or  1.58  tons  per  cub  yd ;  or  2.23  tons  per  1000.  Since  the 
weight  of  hardened  mortar  averages  but  little  less  than  that  of  good  common 
brick,  we  may  for  ordinary  calculations  assume  the  -weight  of  brichworh, 
with  common  bricks,  at  1.4  tons  per  cub  yard,  or  116  ft)S  per  cub  ft ;  and,  with 
pressed  brick,  at  1.56  tons  per  cub  yd,  or  129  ibs  per  cub  ft. 

In  water,  either  brick  will  in  a  few  minutes  absorb  from  >^  to  %  ft)  of 
water;  or  0.1  to  one-seventh  of  the  weight  of  a  pressed  brick,  or  3^  to  one-third 
of  its  bulk. 

]^uiiiber  of  bricks  83^  X  4  X  2,  required  per  sq  foot  of  wall,  allow- 
ing for  the  usual  waste  in  cutting  bricks  to  fit  corners,  jambs,  &c.: 

Wall   %Va  ins,  or  1     brick 14  bricks        Wall  21K  ins,  or  V/^  brick 35  bricks 

"     123|    "     or  13^      "     21       "  "    25^    "    or  3  "     42      " 

"    17       "    or  2         "     28      " 

liaylng,  per  day.  A  bricklayer,  with  a  laborer  to  keep  him  supplied  with 
materials,  will,  in  cornmon  house  walls,  lay  on  an  average  about  1500  bricks  per 
day  of  10  working  hours.  In  the  neater  outer  faces  of  back  buildings,  from  1000 
to  1200;  in  good  ordinary  street  fronts,  800  to  1000;  or  of  the  very  finest  lower 
story  faces  used  in  street  fronts,  from  150  to  300,  depending  on  the  number  of 
angles,  &c.  In  plain  massive  engineering  work,  he  should  average  about  2000 
per  day,  or  4  cub  yds ;  and  in  large  arches,  about  1500,  or  3  cub  yds. 

Since  bricks  shrink  about  ^3  part  of  each  dimension  in  drying  and  burning, 
the  moulds  should  be  about  Jj  part  larger  each  way  than  the  burnt  brick  is 
intended  to  be.     Good  well-burnt  bricks  will  ring  when  two  are  struck  together. 

At  the  brick-yards  about  Philadelphia,  a  brick-moulder's  work  is  2333  bricks 
per  day ;  or  14000  per  week.  He  is  assisted  by  two  boys,  one  of  whom  supplies 
the  prepared  clay,  moulding  sand,  and  water ;  while  the  other  carries  away  the 
bricks  as  they  are  moulded.  A  fourth  person  arranges  them  in  rows  for  drying. 
About  ^  of  a  cord,  or  96  cub  ft  of  wood,  is  allowed  per  1000  for  burning.  Where 
coal  is  used,  the  kilns  are  fired  up  with  anthracite,  and  the  finishing  is  done  with 
bituminous.     One  ton  of  coal,  in  all,  makes  4500  bricks. 

For  paving  sidewalks  the  bricks  are  laid  on  a  6-inch  layer  of  gravel, 
which  should  be  free  from  clay,  and  well  consolidated.  With  bricks  of  834  X  4 
X  2  ins,  with  joints  from  3^  to  34  inch  wide,  a  square  yard  requires,  flatwise, 
38  bricks;  edgewise,  73;  endwise,  149.  An  average  workman,  with  a  laborer  to 
supply  the  bricks  and  gravel,  will  in  10  hours  lay  about  2000  bricks  ;  or  53  sq  yds 
flat,  27  edgewise,  13  endwise.     When  done,  sand  is  brushed  into  the  joints. 

Art.  3.  The  crushing  strengtli  of  bricks  of  course  varies  greatly. 
A  rather  soft  one  will  crush  under  from  450  to  600  lbs  per  sq  inch  ;  or  about  .30 
to  40  tons  per  sq  ft ;  while  a  first-rate  machine-pressed  one  will  require  about  20O 
to  400  tons  per  sq  ft,  or  about  the  crushing  limit  of  the  best  sandstone;  two- 
thirds  that  of  the  best  marbles  or  limestones ;  or  3^  that  of  the  best  granites, 
or  roofing  slates.  But  masses  of  brickwork  crush  under  much  smaller  loads 
than  single  bricks.  In  some  English  experiments,  small  cubical  masses,  only 
9  inches  on  each  edge,  laid  in  cement,  crushed  under  27  to  40  tons  per  sq  ft. 
Others,  with  piers  9  ins  square,  and  2  ft  3  ins  high,  in  cement,  only  two  days 
after  being  built,  required  44  to  62  tons  per  sq  ft  to  crush  them.  Another, 
of  pressed  brick,  in  best  Portland  ceinent,  is  said  to  have  withstood  202  tons 
per  sq  ft;  and  with  common  lime  mortar  only  34  ^^  much. 

It  must,  however,  be  remembered,  that  cracking  and  splitting  usually  com- 
mence under  about  one-half  the  crushing  loads.  To  be  safe,  the  load  should  not 
exceed  3^  of  the  crushing  one  ;  and  so  with  stone.  Moreover,  these  experiments 
were  made  upon  low  masses;  and  the  strength  decreases  with  the  proportion 
of  the  height  to  the  thickness. 

The  pressure  at  the  base  of  a  brick  shot-tower  in  Baltimore,  246  feet  high,  is 
estimated  at  63^  tons  per  sq  ft;  and  in  a  brick  chimney  at  Glasgow,  Scotland, 
468  feet  high,  at  9  tons.  Professor  Pvunkine  calculates  that  in  heavy  gales  this  is 
increased  to  15  tons,  on  the  leeward  side. 


928 


BRICKS. 


With  our  present  imperfect  knowledge  on  this  subject,  it  cannot  be  considered  safe  to  expose  even 
first-class  pressed  brickwork,  in  cement,  to  more  than  12  or  15  tons  per  sq  ft ;  or  good  hand-moulded, 
to  more  than  two-thirds  as  much. 

Tensile  strengrtli  of  brick,  40  to  400  fts  per  sq  inch ;  or  2.6  to  26  tons  per  sq  ft 
The  £ng^lish  rod  of  brickwork  is  306  cub  feet,  or  n%c\ib  yards;  and 

requires  about  4500  bricks  of  the  English  standard  size ;  with  about  75  cub  ft  of  mortar.  The  English 
hundred  of  lime,  is  a  cub  yd. 

Frozen  mortar.  There  is  risk  in  using  common  mortar  in  cold  weather.  If  the  cold 
should  continue  long  enough  to  allow  the  frozen  mortar  to  set  well,  the  work  may  remain  safe ;  but  if 
a  warm  day  should  occur  between  the  freezing  and  the  setting  of  the  mortar,  the  sun  shining  on  one 
side  of  the  wall  may  melt  the  mortar  on  that  side,  while  that  on  the  other  side  may  remain  frozen 
hard.  In  that  case,  the  wall  will  be  apt  to  fall ;  or  if  it  does  not,  it  will  at  least  always  be  weak ;  for 
mortar  that  has  partially  set  while  frozen,  if  then  melted,  will  never  regain  its  strength.  By  the 
writer's  own  trials  hydraulic  cements  seemed  not  to  be  injured  by  freeiing. 

Experiments  for  rendering?  brick  masonry  imperTions  t9 

water.  Abstract  of  a  paper  read  before  the  American  Society  of  Civil  Engineers,  May  4, 1870, 
by  William  L.  Dearborn,  Civil  Engineer,  member  of  the  Society. 

The  face  walls  of  the  Back  Bays  of  the  Gate-houses  of  the  new  Croton  reservoir,  located  north 
of  Eighty-sixth  Street,  in  Central  Park,  were  built  of  the  best  quality  of  hard-burnt  brick ;  laid  ia 
mortar  composed  of  hydraulic  cement  of  New  York,  and  sand  mixed  in  the  proportion  of  one  measure 
of  cement  to  two  of  sand.  The  space  between  the  walls  is  4  ft ;  and  was  filled  with  concrete.  The  face 
walls  were  laid  up  with  great  care,  and  every  precaution  was  taken  to  have  the  joints  well  filled  and 
insure  good  work.  They  are  12  ins  thick,  and  40  ft  high  ;  and  the  Bays  when  full  generally  have  36  ft 
•f  water  in  them. 

When  the  reservoir  was  first  filled,  and  the  water  was  let  into  the  Gate-houses,  it  was  found  to  filter 
through  these  walls  to  a  considerable  amount.  As  soon  as  this  was  discovered,  the  water  was  drawn 
out  of  the  Bays,  with  the  intention  of  attempting  to  remedy  or  prevent  this  infiltration.  After  care- 
fully considering  several  modes  of  accomplishing  the  object  desired,  I  came  to  the  conclusion  to  try 
*'  Sylvester's  Process  for  Repelling  Moisture  from  External  Walls." 

The  process  consists  in  using  two  washes  or  solutions  for  covering  the  surface  of  brick  walls ;  one 
composed  of  Castile  soap  and  water ;  and  one  of  alum  and  water.  The  proportions  are  :  three-quar- 
ters of  a  pound  of  soap  to  one  gallon  of  water ;  and  half  a  pound  of  alum  to  four  gallons  of  water 
both  substances  to  be  perfectly  dissolved  in  the  water  before  being  used. 

The  walls  should  be  perfectly  clean  and  dry  ;  and  the  temperature  of  the  air  should  not  be  beloir 
60  degrees  Fahrenheit,  when  the  compositions  are  applied. 

The  first,  or  soap  wash,  should  be  laid  on  when  at  boiling  heat,  with  a  flat  brush,  taking  care  not 
to  form  a  froth  on  the  brickwork.  This  wash  should  remain  twenty-four  hours  ;  so  as  to  Decome  dry 
and  hard  before  the  second  or  alum  wash  is  applied ;  which  should  be  done  in  the  same  manner  aa 
the  first.  The  temperature  of  this  wash  when  applied  may  be  60°  or  70° ;  and  it  should  also  remain 
twenty-four  hours  before  a  second  coat  of  the  soap  wash  is  put  on ;  and  these  coats  are  to  be  repeated 
alternately  until  the  walls  are  made  impervious  to  water. 

The  alum  and  soap  thus  combined  form  an  insoluble  compound,  filling  the  pores  of  the  masonry, 
and  entirely  preventing  the  water  from  penetrating  the  walls. 

Before  applying  these  compositions  to  the  walls  of  the  Bays,  some  experiments  were  made  to  tert 
the  absorption  of  water  by  bricks  under  pressure  after  being  covered  with  these  washes,  in  order  to 
determine  how  many  coats  the  wall  would  require  to  render  them  impervious  to  water. 

To  do  this,  a  strong  wooden  box  was  made,  put  together  with  screws,  large  enough  to  hold  2  bricks; 
and  on  the  top  was  inserted  an  inch  pipe  forty  feet  long. 

In  this  box  were  placed  two  bricks  after  being  made  perfectly  dry,  and  then  covered  with  a  coat  of 
each  of  the  washes,  as  before  directed,  and  weighed.  They  were  then  subjected  to  the  pressure  of  a 
column  of  water  40  feet  high  ;  and,  after  remaining  a  sufficient  length  of  time,  they  were  taken  oat 
and  weighed  again,  to  ucertain  the  amount  of  water  they  had  absorbed. 

The  bricks  were  then  dried,  and  again  coated  with  the  washes  and  weighed,  and  subjected  to  press- 
ure as  before ;  and  this  operation  was  repeated  until  the  bricks  were  found  not  to  absorb  any  water. 
Vour  coatings  rendered  the  bricks  impenetrable  under  the  pressure  of  40  ft  head. 

The  mean  weight  of  the  bricks  (dry)  before  being  coated,  was  S%  Ba ;  the  mean  absorption  was 
one-half  pound  of  water.     An  hydrometer  was  used  in  testing  the  solutions. 

As  this  experiment  was  made  in  the  fall  and  winter,  (1863,)  after  the  temporary  roofs  were  put  on 
to  the  Gate-house,  artificial  heat  had  to  be  resorted  to,  to  dry  the  walls  and  keep  the  air  at  a  proper 
temperature.  The  cost  was  10.06  cts  per  sq  ft.  As  soon  as  the  last  coat  had  become  hard,  the  water 
was  let  into  the  Bays,  and  the  walls  were  found  to  be  perfectly  impervious  to  water,  and  they  still 
remain  so  in  1870,  after  about  6>^  years. 

Brick  arch  (footway  op  High  Bbidgb).  The  brick  arch  of  the  footway  of  High  Bridge  is  the 
arc  of  a  circle  29  ft  6  in  radius ;  and  is  12  in  thick ;  the  width  on  top  is  17  ft ;  and  the  length  covered 
was  1381  ft. 

The  first  two  courses  of  the  brick  of  the  arch  are  composed  of  the  best  hard-burnt  brick,  laid  edge« 
wise  in  mortar  composed  of  one  part,  by  measure,  of  hydraulic  cement  of  New  York,  and  two  parts 
of  sand.  The  top  of  these  bricks,  and  the  inside  of  the  granite  coping  against  which  the  two  top 
courses  of  brick  rest,  was,  when  they  were  perfectly  dry,  covered  with  a  coat  of  asphalt  one-half  an 
inch  thick,  laid  on  when  the  asphalt  was  heated  to  a  temperature  of  from  360°  to  518°  Fahrenheit. 

On  top  of  this  was  laid  a  course  of  brick  flatwise,  dipped  in  asphalt,  and  laid  when  the  asphalt  was 
bot;  and  the  joints  were  run  full  of  hot  asphalt. 

On  top  of  this  a  course  of  pressed  brick  was  laid  flatwise  in  hydraulic  cement  mortar,  forming  the 
paving  and  floor  of  the  bridge.  This  asphalt  was  the  Trinidad  variety:  and  was  mixed  with  10  per 
cent,  by  measure,  of  coal  tar ;  and  25  per  cent  of  sand.  A  few  experiments  for  testing  the  strength 
of  this  asphalt,  when  used  to  cement  bricks  together,  were  made,  and  two  of  them  are  given  below. 

Six  bricks,  pressed  together  flatwise,  with  asphalt  joints,  were,  after  lying  six  months,  broken. 
The  distance  between  the  supports  was  12  ins  ;  breaking  weight,  900  lbs ;  area  of  single  joint,  i6H  sf 
Ins.    The  asphalt  adheS'.d  so  strongly  to  the  brick  as  to  tear  away  the  surface  in  many  places. 


BRICKS.  929 

Two  bricks  pressed  together  end  to  end,  cemented  with  asphalt,  were,  after  lying  fi  months,  broken. 

The  distance  between  the  supports  was  10  ins;  area  of  joint,  8j^  sq  ins;  breaking  weight,  150  fl)«- 

The  area  of  the  bridge  covered  with  asphaltea  brick,  was  23065  sq  ft.  There  was  used  94200  fts  «t 
asphalt,  33  barrels  of  coal  tar,  10  cub  yds  of  sand,  93800  bricks. 

The  time  occupied  was  109  days  of  masons,  and  148  days  of  laborers.  Two  masons  and  two  labor- 
ers  will  melt  and  spread,  of  the  first  coat,  1650  sq  ft  per  day.  The  total  cost  of  this  coat  was  5.24 
cents  per  sq  ft,  exclusive  of  duty  on  asphalt.  There  were  three  grooves,  2  ins  wide  by  4  ins  deep, 
made  entirely  across  the  brick  arch,  and  immediately  under  the  first  coat  of  asphalt,  dividing  the 
arch  into  four  equal  parts.     These  grooves  were  filled  with  elastic  paint  cement. 

This  arrangement  was  intended  to  guard  against  the  evil  effects  of  the  contraction  of  the  arch  in 
winter;  as  it  was  expected  to  yield  slightly  at  these  points,  and  at  no  other  point;  and  then  the 
elastic  cement  would  prevent  any  leakage  there. 

The  entire  experiment  has  proved  a  very  successful  one,  and  the  arch  has  remained  perfectly  tight. 

In  proposing  the  above  plan  for  working  the  asphalt  with  the  brickwork,  the  object  was  to  avoid 
depending  on  a  large  sontinued  surface  of  asphalt,  as  is  usual  in  covering  arches,  which  very  fre- 
quently oracks  from  the  greater  contraction  of  the  asphalt  than  that  of  the  masonry  with  which  it  ic 
in  oontact ;  the  extent  of  the  asphalt  on  this  work  being  only  about  one-quarter  of  an  inch  to  eacti 
brick.     This  is  deemed  to  be  an  essential  element  in  the  success  of  the  impervious  covering." 

A  cheap  and  effective  process  for  preventing  the  percolation  of  water  through  the  arches  of  aque 
ducts,  and  even  of  bridges,  is  a  great  desideratum.  Many  expensive  trials  with  resinous  compounds 
have  proved  failures.  Hydraulic  cement  appears  to  merely  diminish  the  evil.  Much  of  the  troable 
is  probably  due  to  cracks  produced  by  changes  of  temperature. 

The  Kv^hite  efflorescence  so  common  on  walls,  especially  on  those  of  brick, 
is  due  to  the  presence  of  soluble  salts  in  the  bricks  and  mortar.  These  are  dissolved, 
and  carried  to  the  face  of  the  wall,  by  rain  and  other  moisture.  Sulphate  of  magne- 
sia (Epsom  Salt)  appears  to  be  the  most  frequent  cause  of  the  disfiguration.  In  many 
places  mortar  lime  is  made  from  dolomite,  or  magnesian  limestone,  which  often  con- 
tains 30  per  cent  or  more  of  magnesia ;  which  also  occurs  frequently  in  brick  clay, 
^al  generally  contains  sulphur,  most  frequently  in  combination  with  iron,  forming 
Ae  well-known  "  iron  pyrites  ".  The  combustion  of  the  coal,  as  in  burning  the  lime- 
stone or  clay,  in  manufactures,  in  cooking  etc,  converts  the  sulphur  into  sulphurous 
acid  gas,  which,  when  in  contact  with  magnesia  and  air,  as  in  the  lime  or  brick  kiln, 
or  in  the  finished  wall  or  chimney,  becomes  sulphuric  acid  and  unites  with  the  mag- 
nesia, forming  the  soluble  sulphat*.  We  are  not  aware  of  any  remedy  that  will  pre- 
vent its  appearance  under  such  circumstances ;  but  the  formation  of  the  sulphate  may 
be  prevented  by  the  use  of  limestone  and  brick-clay  free  from  magnesia. 


59 


930  ^  CEMENT. 

CEMENT. 

General    Principles. 
The   elements   chiefly   concerned  in  the  action  of  lime  and  cement 
mortars  are — 

Calcium,       Ca    "j 
Aluminum,  Al 

Carbon,  C     >  Oxygen,  O. 

Silicon,  Si 

Hydrogen,     H    J 
Oxygen  combines  with  each  of  the  others,  forming  oxides.     Thus: 
Calcium  oxide,  CaO,  is  lime; 
Aluminum  oxide,  AI2O3,*  is  alumina; 
Carbon  dioxide,  CO2,  is  carbonic  acid ; 
Silicon  oxide,  Si02,  is  silica,  or  silicic  acid  ;t 
Hydrogen  oxide,  H2O,  is  water. 

Limestone  is  a  calcium  carbonate,  or  combination  of  lime  and  carbonic 
acid,  CaO  +  CO2,  or  CaCOg. 

Clay  (including  argillaceous  minerals  in  general)  is  an  aluminum  silicate, 
•r  combination  of  alumina  and  silicic  acid,  AI2O3  -\-  SiOo. 

Ijime.  When  limestone  (without  clay)  is  "  burned,"  its  CO2  is  driven  off, 
and  the  remaining  ("quick")  lime  has  a  strong  affinity  for  water,  absorlx 
ing  it  with  such  avidity  as  to  develop  heat  sufficient  to  produce  steam, 
the  generation  of  which  disintegrates  and  swells  the  mass.  Combining  thus 
with  the  water,  the  lime  forms  calcium  hydrate,  CaO.HoO,  or  CaH202. 
This  process  is  called  slaking  or  slacking;  and  lime  which  has  satisfied  its 
affinity  for  water  is  called  slaked  (or  slack)  lime.  When  slaked  lime  is  used 
as  mortar,  it  gradually  absorbs  carbonic  acid  from  the  air,  forming  calcium 
carbonate,  the  water  being  liberated  and  evaporated.  Hardened  lime  mortar 
may  thus  be  regarded  as  an  artificial  limestone. 

Cement.  When  aluminum  silicate,  such  as  clay,  in  sufficient  quantities, 
is  burned  with  calciurn  carbonate,  such  as  limestone,  the  burned  product, 
called  cement,  is  deficient  in,  or  devoid  of,  the  slacking  property;  but,  on 
the  other  hand,  when  it  is  made  into  mortar,  the  combinations,  formed  be- 
tween the  elements  of  the  lime,  the  alumina,  the  silica  and  the  water,  during 
the  burning,  and  afterward  in  the  mortar,  are  such  that  they  readily  proceed 
under  water.  Chemists  differ  as  to  the  nature  of  these  combinations.  If 
free  lime  remains  in  the  mortar,  calcium  carbonate  is  formed  by  reabsorption 
of  carbonic  acid  from  the  air,  as  in  the  case  of  lime  mortar. 
^  Setting,  or  the  loss  of  plasticity,  usually  occurs  within  a  few  hours  (some- 
times within  a  few  minutes)  after  mixing  cement  with  water;  whereas 
hardening  (which  appears  to  result  from  a  different  set  of  cherAical  pro- 
cesses) often  proceeds  for  months  or  even  years. 

The  property  of  setting  and  hardening  under  water  is  called  hydraulic^ 
Ity;  and  cements  which  do  not  slack,  but  which  harden  under  water,  are 
called  hydraulic  cements;  or,  more  briefly,  cements. 

Hydraulic  lime  is  a  name  given  to  cements  (much  used  in  Europe) 
which,  while  to  some  extent  hydraulic,  do  not  contain  enough  of  the  hydrau- 
lic elements  to  prevent  slaking.  The  slaking,  however,  is  slower,  and  the 
swelling  less,  than  with  lime  proper. 

The  ratio  of  the  weight  of  aluminum  silicate  to  that  of  the  lime,  in  a  ce- 
ment, is  called  its  hydraulic  index.  Other  things  being  equal,  it  may 
be  used  as  an  indication  of  the  hydraulicity  of  the  cement. 

Natural  and  Portland  Cements.  Many  natural  limestones  contain 
clay  in  such  proportion  that  they  afford  cements  when  burned.  Cements 
so  made  are  called  natural ;  while  those  which  result  from  the  burning  of  arti- 
ficial mixtures  of  lime  carbonates  and  aluminum  silicates  are  called  artificial, 
and,  when  certain  refinements  (see  below)  are  observed,  "Portland"  cement. 

In  making  natural  cement,  the  material  is  burned  in  lumps;  but  for 
Portland  cement  the  material  is  finely  ground  before  burning,  and  the  :^ 

*  The  subscripts  indicate  the  combining  ratios  of  the  several  elements.  .| 
Thus,  in  alumina,  AI2O3  means  a  compound  of  2  atoms  of  aluminum  with  3  i\ 
of  oxygen. 

t  Quartz  is  silica;  and  most  of  the  sand  used  in  mortar  is  quartz  sand. 


CEMENT  MORTAR. 


931 


biirning  is  done  at  a  high  temperature,  producing  incipient  vitrif action.  In 
both  natural  and  Portland  cements,  the  burned  product  is  ground  to  an 
impalpable  powder.    ' 

In  natural  cement,  the  hydraulic  index  usually  varies  between  0.60  and 
1.50;  in  Portland  cements,  between  0.40  and  0.60. 

The  higher  cost  of  Portland  cement  is  due  to  the  more  careful  selection  of 
the  materials  and  to  the  more  elaborate  and  expensive  treatment  given  them, 
resulting  in  the  ultimate  attainment  of  much  greater  strength. 

The  name  Rosendale,  originally  and  properly  restricted  to  natural 
cements  made  in  Ulster  County,  New  York,  is  often  applied  indiscriminately 
to  American  natural  cements  in  general. 

Limestones  containing  magnesia  are  called  Dolomitic  Limestones  or 
Dolomites.  The  presence  of  more  than  3  per  cent,  of  magnesia,  in  the 
finished  product,  is  usually  considered  objectionable. 

Cement    Mortar. 

Cement  miortar  consists  of  cement  and  some  inert  granular  material, 
as  sand,  fine  gravel  or  ground  cinder,  mixed  with  water. 

Owing  to  the  cheapness  with  which  cements  are  now  manufactured,  and 
the  superiority  of  the  mortars  made  from  them,  the  latter  have  to  a  great 
extent  superseded  lime  mortars,  even  in  ordinary  building  operations. 

Amount  of  Mortar  Required  for  a  Cubic  Yard  of  Masonry.* 

Mortar. 
Description  of  Masonry.  Cu.  yd. 

Min.     Max.- 

Ashlar,  18*  courses  and  Y  joints 0.03     0.04 

12*        "  "       "        "         0.06     0.08 

Brickwork  (bricks  of  standard  size,  8i  X  4  X  2i  ins.) : 

Y  joints 0.10     0.15 

Y  to  Y  joints 0.25     0.35 

Y  to  Y  joints 0.35     0.40 

Rubble,  of  small,  rough  stones, 0.33     0.40 

"  "  large  stones,  rough  hammer-dressed,    0.20     0.30 

Squared-stone  masonry,   18*  courses  and  Y  joints, 0.12     0.15 

12*         "  "     "         "      0.20     0.25 


Cement  and  Sand  Required  for  1  Cubic  Yard  of  Mortar.* 


3^ 

ij 

Mortar  Proportioned 
by  Weight. 

Mortar  Proportioned  by  Volumes 

of  Packed  Cement  and 

Loose  Sand. 

Portland. 

Natural. 

Portland. 

Natural. 

Cement. 

Sand. 

Cement. 

Sand. 

Cement. 

Sand. 

Cement. 

Sand. 

l^ 

Bbl. 

Cu.Yd. 

Bbl. 

Cu.Yd. 

Bbl. 

Cu.Yd. 

Bbl. 

Cu.Yd. 

0 

7.40 

0.00 

7.91 

0.00 

7.40 

0.00 

7.91 

0.00 

1 

4.05 

0.57 

4.92 

0.51 

4.17 

0.57 

4.58 

0.58 

2 

2.80 

0.78 

3.43 

0.72 

2.91 

0.78 

3.04 

0.76 

3 

2.00 

0.85 

2.54 

0.80 

2.08 

0.85 

2.24 

0.81 

4 

1.60 

0.89 

2.04 

0.84 

1.66 

0.89 

1.70 

0.86 

5 

1.30 

0.91 

1.64 

0.86 

1.35 

0.91 

1.39 

0.88 

6 

1.10 

0.93 

1.40 

0.88 

1.14 

0.93 

1.28 

0.89 

*  Taken,  by  permission,  from  "A  Treatise  on  Masonry  Construction,"  by 
Prof.  Ira  O.  Baker.     New  York,  John  Wiley  «&  Sons.     9th  edition,  1899. 


932  CEMENT. 

The  eflPects  of  cold  upon  Portland  cements,  although  it  retards  th« 
setting,  do  not  appear  to  be  serious  otherwise.  Even  if  Portland  cement 
mortar  freezes  almost  as  quickly  as  the  masonry  is  laid  A^ith  it,  it  does  not  seem 
to  depreciate  materially.  We  have  found  this  to  be  the  case  also  with  lime 
mortar ;  even  when,  a  few  hours  after  freezing,  the  temperature  became  so  high 
as  to  soften  the  frozen  mortar  again.  But  although  the  mortar  of  either  lime 
or  cement  may  not  thereby  be  injured,  the  work,  especially  in  thin  brick  walls, 
may  be  ruined  and  overthrown.  Thus,  if,  soon  after  the  mortar,  through  the 
entire  thickness  of  such  a  wall,  be  frozen,  the  sun  shines  on  one  face  of  it,  so  as  • 
to  soften  the  mortar  of  that  face,  while  the  mortar  behind  it  remains  hard, 
it  is  plain  that  the  wall  will  be  liable  to  settle  at  the  heated  face,  and  at  least 
bend  outward  if  it  does  not  fall.  Coatings  of  cement,  applied  to  the  backs 
of  arches  on  the  approach  of  winter,  and  left  unprotected,  have  been  found 
entirely  broken  up  and  worthless  on  resuming  work  the  next  spring. 

Alternate  freezing  and  thawing  are  apt  to  disintegrate  both  natural 
and  Portland  cement  mortars. 

The  heating  of  sand  and  cement,  in  freezing  weather,  seems  to  be  a 
bad  practice,  especially  if  they  be  placed  in  cold  water.  But  for  use  out  of 
water  Mr.  Maclay  says  they  may  be  heated  to  50°  or  60°.  Cold  water  for 
mixing  is  probably  no  farther  injurious  than  that  it  retards  the  setting. 

Strengths. 

Factors  Affecting  Strength.  The  strength  of  samples,  under  test,  is 
much  affected  by  the  temperature  of  the  air  and  water,  as  also  by  the  degree 
of  force  with  which  the  cement  is  pressed  into  the  molds ;  by  the  extent  of 
setting  before  being  put  into  the  water,  and  of  drying  when  taken  out;  and 
still  more  ^y  the  pressure  under  which  it  sets,  which  increases  the  strength 
materially.  On  this  account,  cements  in  actual  masonry  may,  under  ordi- 
nary circumstances,  give  better  results  than  in  tests  of  samples.  The 
causes  named,  together  with  the  degree  of  thoroughness  of  the  mixing  or 
gaging,  the  proportion  of  water  used,  and  other  considerations,  may  easily 
affect  the  results  100  per  cent,  or  even  much  more.  Hence  the  discrepancies 
in  the  reports  of  different  experimenters.  Specimens  of  the  same  cement, 
tested  under  apparently  similar  conditions,  may  give  widely  different  results. 

The  Bureau  of  Surveys,  Philadelphia,  requires,  1901,  the  follow- 
ing tensile  strengths,  in  lbs.  per  sq.  inch;  1  day  in  air,  remainder  in  water: 

7  days.  28  days. 

Portland,  neat,     500  600 

"  3  parts  sand,  laboratory  test, .......    170  240 

"  3       "  "     mortar  from  mixing  box,   125  175 

Natural,  neat, 200  300 

"  2  parts  sand,  laboratory  test, 120  200 

"         2       "  "      mortar  from  mixing  box,     50  125 

See  also  diagrams,  p.  933,  and  Requirements,  p.  942. 

Portland  and  Natural  Cements.  Effect  of  Age.  The  diagram  * 
opposite  illustrates  approximately  the  strengths  of  average  Portland  and  of 
average  natural  cements,  neat  and  with  proper  doses  of  sand,  up  to  an  age 
of  two  years.  Tests  may  readily  vary  10  per  cent,  or  more  either  way  from 
the  average. 

Cements  of  the  same  class  differ  much  in  their  rapidity  of  hardening. 
At  the  end  of  a  month  one  may  gain  nearly  one-half  of  what  it  will  gain  in  a 
year,  and  another  not  more  than  one-sixth ;  yet  at  the  end  of  the  year  both 
may  have  about  the  same  strength.  Hence,  tests  for  1  week  or  1  month 
are  by  no  means  conclusive  as  to  the  final  comparative  merits  of  cements. 

Many  Tears  are  required  to  attain  the  greatest  hardness; 
but  after  about  a  year  the  increase  is  usually  very  small  and  slow,  especially 
with  neat  cement.  Moreover,  any  subsequent  increase  is  a  matter  of  little 
importance,  because  generally  by  that  time,  and  often  much  sooner,  the  work 
is  completed  and  exposed  to  its  maximum  stresses. 

There  seems  to  be  a  period,  occurring  from  a  few  weeks  to  several  months 
after  laying,  during  which  cement  and  its  mortars  for  a  short  time  not 
only  cease  from  hardening,  but  actually  lose  strength.  They  then  recover, 
and  the  hardening  goes  on  as  before.  This  feature  is  not  indicated  in  our 
diagram. 

*  See  Richard  L.  Humphrey,  in  "Cement,"  Chicago,  May,  1899. 


STRENGTH. 


933 


One  advantage  of  strong  cements  is  their  economy,  even  at  a 
higher  cost,  in  allowing  the  use  of  a  larger  proportion  of  the  cheaper  ingre- 
dients, sand,  gravel,  and  broken  stone.  Almost  any  common  cement,  if  of 
good  quality,  will,  with  1 .5  or  2  measures  of  sand,  give  a  mortar  strong  enough 
for  most  engineering  purposes;  but  a  good  Portland  will  give  one  equally 
strong  with  3  or  4  measures  of  sand ;  and  will,  therefore,  be  equally  cheap 
at  twice  the  price ;  besides  requiring  the  handling,  storing,  and  testing  of  only 
half  the  number  of  packages. 


%700 

Of 

^600 

8, 


^400 
%300 


yoo 


—- 

— - — j^^^r^ 

1 

y^ 

f 

j.verage^ 

raturalj^^Zt 

^' 

■-■ 

-" 

•"" 

Average  rortl 

fcdMo;^*^^^  l*"*-**  ^''''^ 

} 

-• 

Average  Natu 

ral'Mortar,     ^  pai-ts  Sand 

1 

i 

900  ^ 
800  i 
700  I 
GOO  J. 


•} 


14    2  3  4        G 
Weeks       Months 


1 
Tear 


2 
Years 


Any  addition  of  sand  weakens  cement,  especially  as  regards  ten- 
sion ;  as  it  does  also  lime  mortar.     But  economy  requires  its  use. 

Although,  with  sand,  the  strength  of  the  mortar  may  never  attain  to  that  of 
the  neat  paste,  yet  it  increases  with  age  in  a  greater  proportion ;  so  that  a  neat 
paste  which  at  the  end  of  a  year  would  be  but  twice  as  strong  as  in  7  days, 
may  with  sand  yield  a  mortar  which  at  the  end  of  a  year  will  be  3,  4  or  5 
times  as  strong  as  it  was  in  7  days.  At  the  end  of  a  year  good  Portlands  neat 
usually  have  from  1.5  to  2  times  their  strength  at  the  end  of  7  days;  and 
natural  cements  from  2.5  to  3.5  times;  but,  inasmuch  as  Portlands  average 
about  5  or  6  times  the  strength  of  the  others  in  7  days,  they  still  average 
about  2.5  to  3  times  as  strong  in  a  year  or  longer. 

Diagram*  exliibiting,  approximately,  tlie  effect  of  sand,  in 
different  proportions,  upon  " 

'  ^       '       '       i700] 1 1 1 . . .— — I ,'rnn.^ 


the  strengths  of  Portland 
and  natural  cements,  at 
different  ages  from  1  week 
to  1  year.  The  four  solid 
curves  represent  average 
Portland  cements,  and  the 
four  dotted  curves  repre- 
sent average  natural  ce- 
ments. For  each  kind  of 
cement,  the  curves  repre- 
sent ages  of  1  year,  6 
months,  1  month  and  1 
week,  respectively,  begin- 
ning at  the  top.  The 
curves  for  natural  cement 
are  carried  only  to  5  parts 
sand. 


700^ 


*  Compiled,  by  permis- 
sion, from  Prof.  Baker's  "Masonry  Construction, 


0      1        2       3       4       5       6       7       8 

Parts  of  Sand  to  1  Part  Cement 


934  CEMENT. 

Mr.  Wm.  W.  Maclay,  C.  E.,*  found  that,  in  the  testing  of  cements,  the 
temperature  of  the  air  and  water  had  far  more  influence  than  had  before  been 
suspected,  but  the  ultimate  effects  of  temperature,  within  certain  limits,  are 
fortunately  not  so  important  in  actual  practice  as  the  first  experiments 
might  lead  us  to  infer.  Work  must  go  on  notwithstanding  changes  of  tem- 
perature, but  we  must  take  care  that  our  mortar  shall  at  all  times  be  strong 
enough,  even  under  their  most  injurious  influences.  Cements  in  open  air  are 
certainly  more  or  less  injured  by  drying  instead  of  setting  when  the  tempera- 
ture exceeds  about  65°  to  70°.  But  if  mixed  only  in  small  quantities  at  a 
time,  and  quickly  laid  in  masonry  of  dampened  stone,  so  as  to  be  sheltered 
from  the  air,  the  injury  is  much  reduced.  The  sand  and  stone  should  both 
be  damp,  not  wet,  in  hot  weather,  and  a  little  more  water  may  be  used  in  the 
cement  paste ;  also,  if  possible,  not  only  the  mortar  while  being  mixed,  but 
the  masonry  also,  should  then  be  shaded. 

The  compressive  strengths  of  cements  and  cement  mortars,  in  cubes, 
appear  to  be  about  8  to  10  times  their  tensile  strengths.  The  crushing 
strength,  with  sand,  increases  with  age  much  more  rapidly  than  the  tensile 
strength,  and  the  more  so,  the  greater  the  proportion  of  sand.  Cements  are 
seldom  tested  in  compression. 

The  shearing  strength  of  neat  cements  averages  about  one-fourth  of 
the  tensile  strength. 

The  adhesion  of  cements  to  bricks  or  rough  rubble,  at  differ- 
ent ages,  and  whether  neat  or  with  sand,  may  probably  be  taken  at  an  aver- 
age of  about  three-fourths  of  the  cohesive  or  tensile  strength  of  the  cement  or 
mortar  at  the  same  age.  If  the  bricks  and  stone  are  moist  and  entirely  free 
from  dust  when  laid,  the  adhesion  is  increased;  whereas,  if  very  dry  and 
dusty,  especially  in  hot  weather,  it  may  be  reduced  almost  tO  nothing.  The 
adhesion  to  very  hard,  smooth  bricks,  or  to  finely  dressed  or  sawed  masonry, 
is  less  than  the  adhesion  to  rough  and  porous  surfaces. 

Abrasion.     See  "Mr.  Eliot  C.  Clarke,"  p.  937. 

Weight.  See  also  Weight,  p.  939,  and  (2)  Specific  Gravity,  p.  940. 
Weight  is  an  uncertain  indication.  A  coarse-ground  cement  weighs  heavier, 
but  gives  less  strength,  than  the  same  cement  more  finely  ground. 

Color.     See  "Variations  in  Shade,"  p.  936. 

Fineness.  See  "Mr.  Eliot  C.  Clarke,"  p.  936,  "Sieves,"  p.  938, 
"Fineness,"  pp.  938,  940,  and  "Requirements,"  p.  942. 

Cement,  when  freshly  ground,  is  not  so  good  as  when  a  few  weeks  old. 

Precautions. 

The  engineer  should  reserve  the  right  to  take  a  sample  from  each 
package,  and  to  reject  every  package  of  which  the  sample  drawn  out  does 
not  satisfy  the  stipulations.  On  works  using  large  quantities,  one  person 
should  be  specially  detailed  to  this  duty. 

Protection  from  moisture,  even  that  of  the  air,  is  very  essential  for 
the  preservation  of  cements,  as  well  as  of  quicklime.  With  this  precaution, 
the  cement,  although  it  may  require  more  time  to  set,  will  not  other- 
wise very  appreciably  deteriorate  in  many  months. 

Setting. 

Slow  setting  does  not  indicate  inferiority;  for  many  of  the  best  cements 
are  the  slowest  setting.  A  layer  of  very  quick-setting  cement  may  partially 
set,  especially  in  warm  weather,  before  the  masonry  is  properly  lowered  and 
adjusted  upon  it,  and  any  disturbance,  after  setting  has  commenced, 
is  prejudicial.  Such  cements  are  to  be  regarded  with  suspicion,  and  sub- 
mitted to  longer  tests  than  slow  ones.  Still,  quick-setting  cements  are  best 
in  certain  cases,  as  when  exposed  to  running  water,  etc.  They  may  be  ren- 
dered slower  by  adding  a  bulk  of  lime  paste  equal  to  5  or  15  per  cent,  of  the 
cement  paste,  without  weakening  them  seriously. 

As  a  general  rule,  cements  set  and  harden  better  in  water  than  in  air, 
especially  in  warm  weather.  If,  however,  the  temperature  for  the  first  few 
days  does  not  exceed  55°  to  65°  Fahrenheit,  there  seems  to  be  no  appreciable 
difference  in  this  respect;  but  in  warm  air,  setting  cement,  in  drying,  loses 
the  moisture  upon  which  the  operation  of  hardening  depends.  It  therefore 
sets  without  hardening.  In  hot  weather  every  precaution  should  be  used 
against  this. 
V 

*  "Transactions  American  Society  of  Civil  Engineers,"  Dec,  1877. 


I 


SAND. 


935 


Sand  Retards  Setting.  In  our  experiments  with  various  hydraulic 
cements,  of  the  consistence  of  mortar,  even  without  sand,  we  have  detected 
no  change  of  bulk  in  setting.  But  Mr.  Clarke  (see  p.  936)  found  an 
expansion  of  not  more  than  0,001  part  in  any  dimension. 


Sand. 

The  best  sand  isthat  with  grains  of  very  uneven  sizes,  and  sharp.  The 
more  uneven  the  sizes,- the  smaller  are  the  voids,  and  the  heavier  is  the  sand. 
It  is  generally  considered  that  the  sand  should  be  well  washed  if  it  contains 
clay  or  mud.  But  see  "Adding  Clay,"  p.  936.  Mr.  Clarke  says,  "the 
finer  the  sand,  the  less  is  the  strength." 

Proportion  of  Sand.  As  a  general  rule,  with  cements  of  good  quality, 
we  shall  have  mortars  fit  for  most  engineering  purposes  if  we  do  not  exceed 
from  1.5  to  2  measures  of  dry  sand  to  1  of  the  common  cements;  or  from  2  to 
3  of  sand  to  1  of  Portland. 

Voids  in  Sand.  Since  a  cubic  foot  of  pure  quartz  weighs  165  lbs.,  it 
follows  that,  if  we  weigh  a  cubic  foot  of  pure  dry  sand,  either  loose  or  rammed ; 
then,  as  165  is  to  the  weight  found,  so  is  1  to  the  solid  part  of  the  sand.  And,  if 
this  solid  part  be  subtracted  from  1,  the  remainder  will  be  the  voids,  as  below. 

Wt.,  lbs.  per  cub.  ft.,  dry,    80         85         90         95      100       105     110      115 
Proportion  of  solid,    . .   0.485  0.515  0.546  0.576  0.606  0.636  0.667  0.697 
Proportion  of  voids,    ..   0.515  0.485  0.454  0.424  0.394  0.364  0.333  0.303 

But  the  sand,  when  wet  in  mortar,  occupies  about  from  5  to  7  per  cent,  less 
space  than  when  dry;  the  shrinkage  averaging  say  6  per  cent. ;  thus  making 
the  voids  0.304  of  the  105  lb.  sand  when  wet;  and  0.364  of  the  95  lb.;  the 
mean  of  which  is  0.334.  But,  to  allow  for  imperfect  mixing,  etc.,  it  is  better 
to  assume  the  voids  at  0.4  of  the  dry  sand.  Moreover,  since  the  cements, 
as  before  stated,  shrink  more  or  less  when  mixed  with  water,  and  worked  up 
into  mortar,  it  would  be  as  well  to  assume  that,  in  order  to  make  sufficient 
paste  to  fill  the  voids  thoroughly,  we  should  use,  of  dry  common  cement 
slightly  shaken,  not  less  than  half  the  bulk  of  the  dry  sand ;  and  not  less  than 
45  per  cent,  if  Portland. 

To 'find  the  percentage  of  voids,  pour  into  a  graduated  cylindrical 
measuring-glass  100  measures  of  dry  sand.  Pour  this  out,  and  fill  the  glass 
up  ^to  60  measures  with  water.  Into  this  sprinkle  slowly  the  same  100 
measures  of  dry  sand.  These  will  now  be  found  to  fill  the  glass  only  to  say 
94  measures,  having  shrunk  say  6  per  cent. ;  while  the  water  will  reach  to  say 
121  measures;  of  which  121  —  94  =  27  measures  will  be  above  the  sand; 
leaving  60  —  27  =  33  measures  filling  the  voids  in  94  measures  of  wet  sand; 
showing  the  voids  in  the  wet  sand  to  be  ||  =0.351  of  the  wet  mass.  If  the 
sand  is  poured  into  the  water  hastily,  air  is  carried  in  with  it,  the 
voids  will  not  be  filled,  and  the  result  will  be  quite  different. 

Compressibility  of  Sand.  Careful  experiments  of  our  own,  with  or- 
dinary pure  sand  from  the  seashore,  both  dry  and  moist  (not  wet),  gave  the 
following  results.  The  dry  sands  were  compacted  by  thorough  shaking  and 
jarring;  the  moist  sands  by  ramming  in  thin  layers.  Sand  B  was  of  much 
finer  grain  than  A.     C  consisted  of  the  finest  sifted  grains  from  B. 


Perfectly  dry. 


Moist. 


Lbs.  per  cubic  ft. 


Sand. 
A 
B 
C 


Reduction  Lbs.  per  cubic  ft. 
of  bulk,  , 

Loose. 


69 


Rammed. 
107.5 
107.5 
103.5 


Reduction 
of  bulk, 
per  cent. 

20 

33.3 


None  of  these  sands,  when  dry  and  loose,  if  poured  gently  into  water  to  a 
depth  of  15  inches,  settled  more  than  about  one-fifteenth  part;  the  coarsest 
one.  A,  considerably  less. 

Water  Required.     See  "Mr.  Eliot  C.  Clarke,"  p.  936. 
Cold  wftter  for   mixing  is  probably  no  further  injurious  than  that  it 
retards  setting. 

Salt.     See  "Mr.  Eliot  C.  Clarke,"  p.  936. 


936  CEMENT. 

For  pointing,  the  best  Portland  is  none  too  good,  and  is  best  used  neat, 
but  it  is  often  used  with  from  1  to  2  parts  of  sand.  Mix  under  shelter,  and  in 
quantities  of  only  2  or  3  pints  at  a  time,  using  very  little  water;  so  that  the 
raortar,  when  ready  for  use,  shall  appear  rather  incoherent,  and  quite  defi- 
cient in  plasticity.  The  joints  being  previously  scraped  out  to  a  depth  of  at 
least  half  an  inch,  the  mortar  is  put  in  by  trowel ;  a  straight-edge  being  held 
just  below  the  joint,  if  straight,  as  an  auxiliary.  The  mortar  is  then  to  be 
well  calked  into  the  joint  by  a  calking-iron  and  hammer;  then  more  mortar 
is  put  in,  and  calked,  until  the  joint  is  full.  It  is  then  rubbed  and  polished 
under  as  great  pressure  as  the  mason  can  exert.  If  the  joints  are  very  fine, 
they  should  be  enlarged  by  a  stonecutter,  to  about  1-4  inch,  to  receive  the 
pointing.  The  wall  should  be  well  wet  before  the  pointing  is  put  in,  and  kept 
in  such  condition  as  neither  to  give  "water  to,  nor  take  it  from,  the  mortar. 
In  hot  weather,  the  pointing  should  be  kept  sheltered  for  some  days  from 
the  sun,  so  as  not  to  dry  too  quickly. 

Preservation  of  3Ietals. 

We  have  found,  by  ten  years'  trial,  that  if,  after  setting,  dampness  is  abso- 
lutely excluded,  cements  preserve  iron,  lead,  zinc,  copper  and  brass;  and  that 
plaster  of  Paris  preserves  all  except  iron,  which  it  rusts  somewhat  unless  the 
iron  is  galvanized.  Lime-mortar  probably  preserves  all  of  them,  if  kept 
free  from  damp. 

Efflorescence. 

Natural  cements,  when  used  as  mortar  for  brickwork,  often  disfigure  it, 
especially  near  sea-coasts,  and  in  damp  climates,  by  white  efiiorescence 
which  sometirnes  spreads  over  the  entire  exposed  face  of  the  work,  and  also 
injures  the  bricks.  This  also  occurs  in  stone  masonry,  but  to  a  much  less 
extent,  and  is  confined  to  the  mortar  joints.  It  injures  only  porous  stone. 
It  is  usually  a  hydrous  carbonate  of  soda  or  of  potash,  or  sulphate  of  lime 
(Epsom  salts),  often  with  other  salts.  As  a  preventive.  General  Gilmore  re- 
comniends  to  add,  to  every  300  lbs.  (1  barrel)  of  the  cement  ponder,  100  lbs. 
of  quicklime,  and  from  8  to  12  lbs.  of  any  cheap  animal  fat ;  the  fat  to  be  well 
incorporated  with  the  quicklime  before  slacking  it,  preparatory  to  adding 
it  to  the  cement.  This  addition  will  retard  the  setting,  and  somewhat  dimin- 
ish the  strength  of  the  cement.  It  is  said  that  linseed  oil,  at  the  rate  of  2 
gallons  to  300  lbs.  of  dry  cement,  either  with  or  without  lime,  will  in  all 
exposures  prevent  efiiorescence ;  but,  like  the  fat,  it  greatly  retards  setting, 
and  weakens  the  cement.     See  also  Bricks. 


Mr.  Eliot  C  Clarke  has  published  *  the  results  of  a  series  of  experi- 
ments made  for  the  Boston  Main  Drainage  Works.  From  his  paper  we  con- 
dense as  follows,  by  permission: 

Variations  in  shade,  in  a  given  kind  of  cement,  may  indicate  differences 
in  the  character  of  the  rock  or  degree  of  burning.  Thus,  with  Rosendale,  a 
light  color  generally  indicates  an  inferior  or  underburned  rock.  A  coarse- 
ground  cement,  light  in  color  and  weight,  would  be  viewed  with  suspicion. 

The  highest  strength  was  obtained  by  the  use  of  just  enough  water 
to  dampen  the  cement  thoroughly.  An  excess  of  water  retards  setting. 
Natural  cements  need  more  water  than  Portland ;  fine-ground  more  than 
coarse;  quick-setting  more  than  slow.  Neat  Rosendale,  a  year  old,  was 
strongest  with  35  per  cent,  water.  Neat  Portland,  same  age,  with  20  per 
cent. 

The  finer  the  sand,  the  less  the  strength. 

Salt,  either  in  the  water  used  for  mixing,  or  in  that  in  which  the  cement 
is  laid,  retards  setting  somewhat,  but  has  no  important  effect  upon  the 
strength. 

Adding  clay  gives  a  much  more  dense,  plastic,  water-tight  paste,  useful 
for  plaster  or  for  stopping  leaks.  Half  a  part  of  clay  did  not  seem  to  weaken 
mortar  materially,  except  in  the  case  of  sample  blocks  exposed  to  the  weather 
for  2^  years  after  a  week's  hardening  in  water. 

A  year's  saturation  in  fresh  or  salt  water,  and  in  contact  with  oak,  hard 
pine,  white  pine,  spruce  or  ash,  did  not  affect  the  mortars. 

With  sand,  fine-ground  cements  make  the  strongest  mortar;  but  when 
tested  neat,  coarse-groi|nd  cements  are  strongest.  This  is  especially  the 
ease  with  Portlands. 

*  "Trans.  Am.  Soc.  C.  E.,"  April,  1885. 


RECOMMENDATIONS.  937 

Good  results  were  obtained  from  mixing    different     cements.     A 

mortar  of  half  a  part  each  of  Rosendale  and  Portland,  and  two  parts  sand, 
was  stronger,  at  1  week,  1  month,  6  months  and  1  year,  than  the  average  of 
two  mortars,  one  of  1  part  Rosendale  and  one  of  1  part  Portland ;  each  with  2 
parts  sand.  Mixtures  of  Roman  (quick-setting)  and  Portland  (slow)  set 
about  as  quickly  as  Roman  alone,  and  were  much  stronger. 

Portland  resisted  abrasion  best  when  mixed  with  2  parts  sand;  Rosen- 
dale with  1  part.  A  little  more  or  less  sand  rapidly  reduced  the  resistance  in 
both  cases. 

Cements  expand  in  setting ;  but  not  more  than  1  part  in  1000  of  any 
given  dimension. 


Sand  cement  or  silica  cement  is  made  by  mixing  cement  with 
quartz  sand  (silica)  and  grinding  the  mixture.  It  is  claimed  that  the  cement^ 
in  the  mixture,  becomes  much  more  finely  ground,  and  that  a  mixture  of  1 
part  cement  and  3  parts  sand  can  therefore  carry,  in  mortar,  nearly  as  much 
sand  as  could  the  pure  cement  alone  before  this  treatment. 


Tlie  fineness  of  cement  and  sand  is  indicated  as  follows,  where  the 
large  nuinerals  represent  the  sieve  numbers;  the  small  numeral,  to  the  left 
of  each  sieve  number,  represents  the  percentage  retained  upon  that  sieve; 
and  the  final  small  numeral,  to  the  right  of  the  last  sieve  number,  represents 
the  percentage  passed  by  the  last  sieve.  The  sum  of  the  small  numerals 
=  100.  Thus,  '^  20  IS  30  ^^  40  '*''  means  that  5  per  cent,  was  retained  on  a 
No.  20  sieve,  15  per  cent,  on  No.  30,  and  35  per  cent,  on  a  No.  40,  while 
the  remaining  45  per  cent,  passed  the  No.  40  sieve. 


Properties  and  tests  of  cement.  Recommendations  of 
American  Society  of  Civil  Eng^ineers.  Digest  of  Final  Report  of 
the  Committee*  on  a  Uniform  System  for  Tests  of  Cement,  Trans.  Am.  Soc.  C.  E., 
Vol.  xiv,  November,  1885. 

The  first  tests  of  inexperienced,  though  intelligent  and  careful  persons,  are 
usually  very  contradictory  and  inaccurate,  and  no  amount  of  experience  can 
eliminate  the  variations  introduced  by  the  personal  equations  of  the  most  con- 
scientious observers.  Many  things,  apparently  of  minor  importance,  exert  so 
marked  an  influence  upon  the  results,  that  it  is  only  by  the  greatest  care  in 
every  particular,  aided  by  experience  and  intelligence,  that  trustworthy  tests 
can  be  made. 

Only  a  series  of  tests  for  a  considerable  period,  and  with  a  full  dose  of  sand, 
will  show  the  full  value  of  any  cement ;  and  it  would  be  safer  to  use  a  trust- 
worthy brand  without  applying  any  tests  whatever,  than  to  accept  a  new  article 
which  had  been  tested  only  as  neat  cement  and  for  but  one  day. 

It  is  recommended  that  tests  be  confined  to  methods  for  determining  (1) 
fineness,  (2)  liability  to  checking  or  cracking,  and  (3)  tensile  strength;  and,  for 
the  latter,  for  tests  of  7  days  and  upward,  that  a  mixture  of  1  part  of  cement  to 
1  part  of  sand  for  Natural  f  cements,  and  3  parts  of  sand  for  Portland  f  cements, 
be  used,  in  addition  to  trials  of  the  neat  cement.  The  quantities  used  in  the 
mixture  should  be  determined  by  weight. 

The  tests  should  be  applied  to  the  cements  as  offered  for  sale.  If  satisfactory 
results  are  obtained  with  a  full  dose  of  sand,  the  trials  need  go  no  further.  If 
not,  the  coarser  particles  should  first  be  excluded  by  using  a  No.  100  sieve, 
in  order  to  determine  approximately  the  grade  the  cement  would  take  if  ground 
fine,  for  fineness  is  always  attainable,  while  inherent  merit  may  not  be. 

The  amount  of  material  needed  for  making  five  briquettes  of  the  stand- 
ard size  recommended  is,  for  the  neat  cements,  about  1.66  pounds,  and  for  those 
with  sand,  in  the  proportion  of  3  parts  of  sand  to  1  of  cement,  about  1.25  pounds 
of  sand  and  6.66  ounces  of  cement. 

*Q.  A.  Gillmore,  Chairman,  D.  J.  Whittemore,  J.  Herbert  Shedd,  Eliot  C. 
Clarke,  Alfred  Noble,  F.  O.  Norton,  W.  W.  Maclay,  Leonard  F.  Beck  with, 
Thomas  C.  McCollom. 

t  Where  the  word  "  natural"  is  used  in  this  connection,  it  is  to  be  understood  as 
being  applied  to  the  lightly  burned  natural  American  or  foreign  cements,  in 
contradistinction  to  the  more  heavily  burned  Portland  cement,  either  natural 
or  artificial. 


938  CEMENT. 

Be  commendations  of  Am,  Soc.  Civil  Engrs.     Continued. 

Sampling;.  Usually,  where  cement  has  a  good  reputation,  and  is  used  in 
large  masses,  as  in  heavy  concrete  foundations,  or  in  the  backing  or  heart- 
ing of  thick  walls,  the  testing  of  every  fifth  barrel  seems  to  be  sufficient ;  but  in 
very  important  work,  where  the  strength  of  each  barrel  may  in  a  great  measure 
determine  the  strength  of  that  portion  of  the  work  where  it  is  used,  or  in  the 
thin  walls  of  sewers,  etc.,  etc.,  every  barrel  should  be  tested,  one  briquette  being 
made  from  each. 

In  selecting  cement  for  experimental  purposes,  take  the  samples  from  the 
interior  of  the  original  packages,  at  sufficient  depth  to  insure  a  fair  exponent  of 
the  quality.  Store  the  samples  in  tightly  closed  receptacles,  impervious  to  light 
or  dampness,  until  required  for  manipulation,  when  each  sample  of  cement 
should  be  so  thoroughly  mixed,  by  sifting  or  otherwise,  that  it  shall  be  uniform 
in  character  throughout  its  mass. 

Sieves.  For  ascertaining  the  fineness  of  cement  it  will  be  convenient  to  use 
three  sieves,  viz.:  (Sizes  of  wire  by  Stubs  gage.) 

No.  50—  50  meshes  per  linear  inch  (  2,500  meshes  per  square  inch),  No.  35  wire. 
No.  74—74  "  "  ♦'  "  (5,476  "  *'  *'  "  ),  No.  37  wire. 
No.  100-100      "        "        «        "    (10,000       "       "        "        "    ),Nor40wirel 

For  sand,  two  sieves  are  recommended,  viz.: 

No.  20—20  meshes  per  linear  inch  (400  meshes  per  square  inch),  No.  28  wire. 
No.  30—30        "        "        "        "     (900      "  "         *'         "    ),  No.  31  wire. 

Standard  sand.  Sands  looking  alike  and  sifted  through  the  same  sieves 
may  give  results  varying  within  rather  wide  limits.  The  use  of  crushed  quartz 
is  recommended,  the  degree  of  fineness  to  be  such  that  the  sand  will  all  pass  a 
No.  20  sieve  and  be  caught  on  a  No.  30  sieve. 

Mixing,  etc.  The  proportions  of  cement,  sand,  and  water  should  be  care- 
fully determined  by  weight,  the  sand  and  cement  mixed  dry,  and  all  the  water 
added  at  once.  The  mixing  must  be  rapid  and  thorough  ;  and  the  mortar,  which 
should  be  stiff  and  plastic,  should  be  firmlv  pressed  into  the  molds  with  the 
trowel,  without  ramming,  and  struck  off  level ;  the  molds  in  each  instance,  while 
being  charged  and  manipulated,  to  be  laid  directly  on  glass,  slate,  or  some  other 
non-absorbent  material.  The  molding  must  be  completed  before  incipient  set- 
ting begins.  As  soon  as  the  briquettes  are  hard  enough  to  bear  it,  they  should 
be  taken  from  the  molds  and  be  kept  covered  with  a  damp  cloth  until  they  are 
immersed.  For  the  sake  of  uniformity,  the  briquettes,  both  of  neat  cement  and 
those  containing  sand,  should  be  immersed  in  water  at  the  end  of  24  hours, 
except  in  the  case  of  1  day  tests. 

Fresh,  clean  water,  having  a  temperature  between  60°  and  70°  F.,  should  be 
used  for  the  water  of  mixture  and  immersion  of  samples. 

The  proportion  of  water  required  varies  with  the  fineness,  age,  or  other  condi- 
tions of  the  cement,  and  the  temperature  of  the  air,  but  is  approximately  as 
follows : 

For  neat  cement :  Portland,  about  25  per  cent.;  natural,  about  30  per  cent. 

For  1  cement,  1  sand  :  about  15  per  cent,  of  total  weight  of  sand  and  cement. 

For  1  cement,  3  sand  :  about  12  per  cent,  of  total  weight  of  sand  and  cement. 

The  object  is  to  produce  the  plasticity  of  rather  stiff  plasterer's  mortar. 

(1)  Fineness.  It  is  recommended  that  the  tests  be  made  with  cement  that 
has  passed  through  a  No.  100  sieve.  See  p  937. 

(2)  Cliecking'  or  cracking;.  Make  two  cakes  of  neat  cement,  2  or  3 
inches  in  diameter,  about  3^  inch  thick,  with  thin  edges.  Note  the  time  in 
minutes  that  these  cakes,  when  mixed  with  water  to  the  consistency  of  a  stiff 
plastic  mortar,  take  to  set  hard  enough  to  stand  the  wire  test  recommended  by 
Gen.  Gillmore,  viz.,  -—^  inch  diameter  wire  loaded  with  3^  pound  for  initial  set, 
and  2~r  inch  loaded  with  1  pound  for  final  set.  One  of  these  cakes,  when  hard  \ 
enough,  should  be  put  in  water  and  examined  from  day  to  day  to  see  whether  it 
becomes  contorted,  and  whether  cracks  show  themselves  at  the  edges.  Such  con- 
tortions or  cracks  indicate  that  the  cement  is  unfit  for  use  at  that  time.  In 
some  cases  the  tendency  to  crack,  if  caused  by  the  presence  of  too  muchj 
unslaked  lime,«will  disappear  with  age.    The  remaining  cake  should  be  kept  inj 


TESTING. 


939 


Recominendations  of  Am.  Soc.  Civil  Engrs.     Continued. 

the  air,  and  its  color  observed,  which,  for  a  good  cement,  should  be  uniform 
throughout ;  yellowish  blotches  indicating  a  poor  quality  ;  the  Portland  cements 
being  of  a  bluish-gray,  and  the  natural  cements  being  light  or  dark,  according 
to  the  character  of  the  rock  of  which  they  are  made.  The  color  of  the  cements 
in  air  indicates  the  quality  much  better  than  when  they  are  put  in  water. 

(3)  Tensile  streng-tli.  An  average  of  5  briquettes  may  be  made  for  each 
test,  only  those  breaking  at  the  smallest  section  to  be  taken.  The  briquettes 
should  always  be  broken  immediately  after  being  taken  out  of  the  water,  and 
the  temperature  of  the  briquettes  and  of  the  testing  room  should  be  constant 
between  60°  and  70°  F. 

The  stress  should  be  applied  at  a  uniform  rate  of  about  400  pounds  per  min- 
ute, starting  at  0.     With  a  weak  mixture  use  lialf  the  speed. 

The  molds  furnished  are  usually  of  iron  or  brass.  Wooden  molds,  if  well 
oiled  to  prevent  absorption  of  water,  answer  a  good  purpose  for  temporary  use, 
but  speedily  become  unfit  for  accurate  work.  Our  figures  show  the  form  of 
briquette  and  of  metal  mold  recommended. 


Section  through  a-h 


The  clips  should  be  hung  on  pivots,  so  as  to  avoid,  as  far  as  possible,  cross 
strain  upon  the  briquettes. 

Weight.  The  relation  of  the  weight  of  cement  to  its  tensile  strength  is  an 
uncertain  one.  In  practical  work,  if  used  alone,  it  is  of  little  value  as  a  test, 
while  in  connection  with  the  other  tests  recommended  it  is  unnecessary,  except 
when  the  relative  bulk  of  equal  weights  of  cement  is  desired. 

Setting'.  The  rapidity  of  setting  furnishes  no  indication  of  ultimate 
strength.     It  simply  shows  the  initial  hydraulic  activity. 

Q.uick'Setting'  cements  are  those  which  set  in  less  than  half  an  hour;  and 
slO'W-settin^  cements  are  those  requiring  half  an  hour  or  more  to  set.  The 
cement  must  be  adapted  to  the  work  required,  as  no  one  cement  is  equally 
good  for  all  purposes.  In  submarine  worft  a  quick-setting  cement  is  often 
imperatively  demanded,  and  no  other  will  answer,  while  for  work  above  the 
water-line  less  hydraulic  activity  will  usually  be  preferred.  Each  individual 
case  demands  special  treatment.  The  slow-setting  natural  cements  should  not 
become  warm  while  setting,  but  the  quick-setting  ones  may,  to  a  moderate 
extent,  within  the  degree  producing  cracks.  Cracks  in  Portland  cement  indi- 
cate too  much  carbonate  of  lime,  and  in  the  Vicat  cements  too  much  lime  in  the 
original  mixture. 


940  CEMENT. 

Properties  and  Tests  of  Cement.  Report  of  Board  U.  S.  A. 
£ng-ineer  Officers.  Properties  and  tests  of  Portland,  Natural  and  Puz- 
zolau*  cements.  Digest  of  a  Report  of  Majors  W.  L.  Marshall  and  Smith  S. 
Leach  and  Capt.  Spencer  Cosby,  Board  of  Engineer  Officers,  on  testing  Hydraulic 
Cements.    Professional  Papers,  No.  28,  Corps  of  Engineers,  U.  S.  A.,  1901. 

Unfortunately,  tests  for  acceptance  or  rejection  must  be  made  on  a  product 
which  has  not  reached  its  final  stage.  A  cement,  when  incorporated  in  masonry, 
undergoes  chemical  changes  for  months,  whereas  it  is  seldom  possible  to 
continue  tests  for  more  than  a  few  weeks  at  the  most. 

A  few  tests,  carefully  made,  are  more  valuable  than  many,  made  with  less  care. 

Cement  which  has  been  in  storage  for  a  long  time  should  be  carefully 
tested  before  use,  in  order  to  detect  deterioration. 

A  cement  should  be  rejected,  without  regard  to  the  proportion  of  failures 
among  samples  tested,  if  the  samples  show  dangerous  variation  in  quality  or 
lack  of  care  in  manufacture,  and  resulting  lack  of  uniformity  in  the  product. 

The  practice  of  offering  a  bonus  for  cement  showing  an  abnormal  strength 
is  objectionable,  as  it  leads  to  the  production  of  cements  with  defects  not 
easily  detected. 

For  Portland  or  Puzzolan  cement,  make  tests  for  (1)  fineness  of  grinding ;  (2) 
specific  gravity;  (3)  soundness,  or  constancy  of  volume  in  setting;  (4)  time  of 
setting,  and  (5)  tensile  strength.     For  Natural  cements  omit  tests  (2)  and  (3). 

(1)  Fineness.  Cementitious  quality  resides  principally,  if  not  wholly,  in 
the  very  finely  ground  particles.  Use  a  No.  100  sieve,  woven  from  brass  v^ire 
No.  40  Stubs  gage;  sift  until  cement  ceases  to  pass  through.  The  percentage 
that  has  passed  through  is  determined  by  weighing  the  residue  on  the  sieve. 
The  screen  should  be  frequently  examined  to  see  that  no  wires  have  been 
displaced.  See  p  937. 

(2)  {Specific  gravity.  The  specific  gravity  test  is  of  value  in  determining 
whether  a  Portland  cement  is  unadulterated.  The  higher  the  burning,  short  of 
vitrification,  the  better  the  cement  and  the  higher  the  specific  gravity^  If  under- 
burned,  the  specific  gravity  of  Portland  cement  may  fall  below  0  ;  if  overburned, 
it  may  reach  3.5.  Natural  cement  has  a  specific  gravity  of  about  2.5  to  2.8,  and 
Puzzolan  about  2.7  to  2.8. 

The  temperature  may  vary  between  60°  and  80°  F.  Any  approved  form  of 
volumenometer  or  specific  gravity  bottle  may  be  used,  graduated  to  cubic  centi- 
meters with  decimal  subdivisions.  Fill  the  instrument  to  zero  of  scale  with 
benzine.  Take  100  grams  of  sifted  cement  that  has  been  previously  dried  by 
exposure  on  a  metal  plate  for  20  minutes  to  a  dry  heat  of  212°  F.,  aiid  allow  it  to 
pass  slowly  into  the  benzine,  taking  care  that  the  powder  does  not  stick  to  the 
sides  of  the  graduated  tube  above  the  fluid,  and  that  the  funnel,  through  which 
it  is  introduced,  does  not  touch  the  fluid.  The  approximate  specific  gravity  will 
be  represented  by  100  divided  by  the  displacement  in  cubic  centimeters.  The 
operation  requires  care. 

(3)  Soundness,  and  (4)  setting  qualities.  The  temperature  should 
not  vary  more  than  10°  from  62°  F.  For  Portland  cement  use  20,  for  Natural  30, 
and  for  Puzzolan  18  per  cent,  of  water  by  weight.  Mix  thoroughly  for  5  minutes. 
On  glass  plates  make  two  cakes  about  3  inches  in  diameter,  3^  inch  thick  at  the 
middle  and  drawn  to  thin  edges,  and  cover  them  with  a  damp  cloth.  At  the  end 
of  the  minimum  time  specified  for  initial  set,  apply  needle  -r^  ,inch  diameter, 
weighted  to  %  pound.  If  an  indentation  is  made,  the  cement  passes  the  require- 
ment for  initial  setting.  Otherwise  the  setting  is  too  rapid.  At  the  end  of  the 
maximum  time  specified  for  final  set,  apply  the  needle  -J^  inch  diameter,  loaded 
to  one  pound.  If  no  indentation  is  made,  the  cement  passes  the  requirement  for 
final  set.     Otherwise  the  setting  is  too  slow. 

(jenerally  speaking,  both  periods  of  set  are  lengthened  by  increase  of  moisture, 
and  shortened  by  increase  of  temperature. 

*By  Portland  cement,  in  this  report,  is  meant  the  product  obtained  by 
calcining  intimate  mixtures,  either  natural  or  artificial,  of  argillaceous  and 
calcareous  substances,  up  to  incipient  fusion.  By  Natural  cement  is  meant 
one  made  by  calcining  natural  rock  at  a  heat  below  incipient  fusion,  and  grind- 
ing the  product  to  powder.  By  Puzzolan  is  meant  the  product  obtained  by 
grinding  slag  and  slaked  lime,  without  subsequent  calcination. 


TESTING.  941 

Becommendations  of  Board  of  U.  S.  A.  Engineer 
Officers.     Continued. 

In  gaging  Portland  cement  in  damp  weather,  the  samples  should  be  thoroughly 
dried  before  adding  water.  This  precaution  is'  not  deemed  necessary  with 
Natural  cement.  Sufficient  uniformity  of  temperature  will  result  if  the  testing 
room  be  comfortably  warmed  in  winter,  and  if  the  specimens  be  kept  out  of  the 
sun  in  a  cool  room  in  summer,  and  under  a  damp  cloth  until  set.  Temperatures 
may  vary  between  60°  and  80°  F.,  without  affecting  results  more  than  the 
probable  error  in  the  observation. 

Boiling-  test.  Place  the  two  cakes  under  a  damp  cloth  for  24  hours.  Place 
one  of  them,  still  attached  to  its  plate,  in  water  28  days  ;  immerse  the  other  in 
water  at  about  70°  F.,  and  let  it  be  in  a  rack  above  the  bottom  of  the  receptacle; 
heat  the  water  gradually  to  the  boiling  point,  maintain  the  heat  for  6  hours  and 
then  let  cool.  The  boiled  cake  should  not  warp  or  become  detached  from  the 
plate,  or  show  expansion  cracks.  If  the  cold-water  cake  shows  evidences  only 
of  swelling,  the  cement  may  be  used  in  ordinary  work  in  air  or  fresh  water  for 
lean  mixtures,  but  if  distortion  or  expansion  cracks  appear  in  it,  the  cement 
should  be  rejected. 

Accelerated  tests  are  not  generally  recommended,  but  where  a  test  must 
be  made  in  a  short  time,  the  boiling  test  is  considered  about  the  best.  It  not 
only  gives  short-time  indications,  but  at  once  directs  attention  to  the  presence 
'of  ingredients  which  might  lead  to  disintegration.  On  the  other  hand,  it  may 
lead  to  the  rejection  of  a  cement  which  would  behave  satisfactorily  in  actual 
work  and  which  would  stand  the  test  after  air-slaking.  Sulphate  of  lime,  while 
enabling  cements  to  pass  the  boiling  tests,  introduces  an  element  of  danger. 

(5)  Tensile  tests  are  preferred  to  flexural  or  compressive  tests.  Sand 
tests  are  the  more  important  and  should  always  be  made ;  and  neat  tests  should 
be  made  if  time  permits. 

A  cement  which  tests  moderately  high  at  7  days,  and  shows  a  substantial 
increase  in  strength  in  28  days,  is  more  likely  to  reach  the  maximum  strength 
slowly  and  retain  it  indefinitely  with  a  low  modulus  of  elasticity,  than  a  cement 
which  tests  abnormally  high  at  7  days  with  little  or  no  increase  at  28  days. 

Use  briquettes  of  the  form  recommended  by  the  American  Society  of  Civil 
Engineers,*  measuring  1  inch  square  in  cross-section  at  place  of  rupture,  and 
held  by  close-fitting  metal  clips,  without  rubber  or  other  yielding  contacts.  The 
tests  should  be  made  immediately  after  taking  the  briquettes  from  the  water. 

Neat  tensile  tests.  Use  unsifted  cements.  For  Portland  cement,  use 
20;  for  Natural,  30;  and  for  Puzzolan,  18  per  cent,  water  by  weight.  Place  the 
cement  on  a  smooth  non-absorbent  slab ;  in  the  middle  make  a  crater  sufficient  to 
hold  the  water ;  add  nearly  all  the  water  at  once,  the  remainder  as  needed  ;  mix 
thoroughly  by  turning  with  the  trowel,  and  vigorously  rub  or  work  the  cement 
for  5  minutes. 

Place  the  briquette  mold  on  a  glass  or  slate  slab.  Fill  the  mold  with  consecu- 
tive layers  of  cement,  each  to  be  3^  inch  thick  when  rammed.  Give  each  layer 
30  taps  with  a  soft  brass  or  copper  rammer  weighing  1  pound,  having  a  face  % 
inch  diameter  or  0.7  inch  square,  and  falling  about  ^  inch. 

After  filling  the  mold  and  ramming  the  last  layer,  strike  smooth  with  a  trowel, 
tap  mold  lightly  on  side,  to  free  cement  from  plate,  remove  the  plate,  and  leave 
for  24  hours,  covered  with  a  damp  cloth.  Then  remove  the  briquette  from  the 
mold  and  immerse  it  in  fresh  water,  which  should  be  renewed  either  continu- 
ously or  twice  in  each  week  during  the  specified  time. 

Tensile  tests  uritli  sand.  For  Portland  and  Puzzolan  cements,  use  1 
part  cement  to  3  parts  sand ;  for  Natural  or  Rosendale,  1  to  1.  Use  crushed 
quartz  sand,  passing  a.  No.  20  standard  sieve,  and  being  retained  on  a  No.  30 
standard  sieve. 

After  weighing  carefully,  mix  dry  the  cement  and  sand  until  the  mixture  is 
uniform,  add  the  water  as  in  neat  mixtures,  and  mix  for  5  minutes.  The  con- 
stituents should  be  well  rubbed  together. 

For  maximum  strength  in- tested  briquettes,  Portland  cements  require 
"water  =  11  to  123^  per  cent,  by  weight  of  constituent  cement;  Natural,  15  to 
17;  and  Puzzolan,  9  to  10. 

A  machine  which  applies  the  stress  automatically  and  at  a  uniform  rate 

♦See page 939.   . 


942  CEMENT. 

Becommendations  of  Board  of  U.  S.  A.  Engineer 
Oflacers.     Continued. 

of  increase  is  preferable  to  one  controlled  entirely  by  hand.  The  stress 
should  be  increased  at  the  rate  of  about  400  lbs.  per  minute.  A  rate  materially 
greater  or  less  than  this  will  give  different  results. 

The  highest  tensile  strength  from  each  set  of  briquettes  made  at  any  one  time 
is  to  be  considered  the  governing  test. 

Field  tests  are  recommended,  whether  or  not  the  more  elaborate  tests 
above  described  have  been  made.  In  connection  with  tests  of  weight  and  fine- 
ness, and  observations  of  texture  and  hardness  in  the  work,  field  tests  often 
suflfice  for  well-known  brands,  showing  whether  the  cement  is  genuine  and 
whether  it  is  reasonably  sound  and  active.  Pats  and  balls  of  neat  cement  from 
the  storehouse,  and  of  mortar  from  the  mixing  platform  or  machine,  should  be 
frequently  made.  Estimate  roughly  the  setting  and  hardening  qualities  by 
pressure  of  the  thumb-nail ;  hardness  of  set  and  strength  by  breaking  wath  the 
hand  and  by  dropping  upon  a  hard  surface.  The  boiling  test  may  also  be  used. 
Should  the  simple  tests  give  unsatisfactory  or  suspicious  results,  theu  a  full  series 
of  tests  should  be  carefully  made. 

A  cement  may  be  rejected  if  it  fails  to  meet  any  of  the  following  requirements. 

Requirements. 

Portland.    Natural.    Puzzolan. 

Slow.  Quick. 
Fineness.    Percentage  to  pass  through  a  No. 

100  sieve  as  in  (1) 87  to  92*  80  97 

Specific  gravitv.     Between 3.10  3.10      Not                  2.7 

and 3.25  3.25    given                 2.8 

Time  of  setting.     Initial,  not  less  than 45  m.  20  m.    20  m.  45  m. 

nor  more  than 30  m 

Final,  not  less  than 45  m 

nor  more  than    10  h.  2.5  h.     4  h.  10  h. 

Tensile  strength,  neat, 

^                  .                               f  7davst 450  400           90                  350 

lbs.  per  sq.  in.                           1 28  daVs f 540  480          200                  500 

Tensile  strength.      With  sand,  as  in    (5). 

^                   .                              f  7dayst 140  120           60                  140 

fts.  per  sq.  m.                          |28  daysf 220  180         150                  220 

*92  per  cent,  is  quite  commonly  attained  by  high-grade  American  Portlands, 
but  rarely  by  imported  brands.     For  the  latter,  use  87.  . 
t  Reject  any  cement  not  showing  an  increase  at  28  days  over  7  days. 


■ 


CONCRETE.  943 

CONCRETE. 

Cement  concrete  is  an  artificial  stone,  composed  of  cement  mortar 

mixed  with  an  "aggregate"  consisting  generally  of  broken  stone,  but  often  of 

gravel  and  sometimes  of  brick-bats,  oyster  shells,  etc.,  or  of  all  these  together. 

In  concrete,  as  in  mortar,  it  is  advisable,  on  the  score  of  strength,  that  all 

the  voids  be  filled  or  overfilled. 

Taking  the  voids,  in  broken  stone,  in  gravel  and  in  sand,  at  50  per  cent, 
of  the  mass,  or  equal  to  the  solid  portion,  we  have,  for  1  cubic  yard  of 
concrete: 

1  cu.  yd.  broken  stone    "|  (4  parts  stone 

0.5  cu.  yd.  sand  >     =     <2  parts  sand 

0.25  cu.  yd.  cement         j  1 1  part  cement 


1  cu.  yd.  broken  stone     1  f  8  parts  sandstone 

0.5  cu.  yd.  gravel  I      ^      J  4  parts  gravel 


0.25  cu.  yd.  sand,  [  ]  2  parts  sand 

0.125  cu.  yd.  cement       j  [l  part  cement. 

Caution.  When  a  solid  body  is  reduced  to  a  mass  consisting  of  broken 
pieces  separated  by  voids,  the  increase  in  bulk  is  due  solely  to  the  voids,  and 
is  equal  to  the  space  occupied  by  them.  Hence  the  ratio  between  the  in- 
crease of  bulk,  or  **  swelling,'*  and  the  original  bulk  is  that  of  the  voids 
to  the  original,  and  not  to  the  final  bulk.  Thus,  if  a  solid  cubic  yard  of  stone, 
after  being  broken  into  pieces,  occupies  twice  as  much  space* as  before,  then 
the  increase  in  bulk,  or  the  space  occupied  by  the  voids,  is  =  that  occupied 
by  solid  pieces  =  half  that  occupied  by  the  entire  broken  mass. 

It  is  doubtful  whether  hard  rock,  when  blasted  and  made  into  embank- 
ment, settles  to  less  than  1.67  cubic  yards  for  each  original  cubic  yard.  As 
the  result  of  certain  embankment  in  hard  sandstone,  Mr.  EUwood  Morris 
gives  1.417  yards  of  embankment  for  each  solid  yard;  but  the  rough  sides  of 
rock  excavations  make  it  difficult  to  measure  them  with  accuracy,  and  this 
may  have  affected  his  result. 

Stone  Crushers.  Principal  sizes.  From  catalogue  of  Parrel  Foundry 
and  Machine  Co.     For  prices,  see  Price-list. 

Weight, 
Lbs. 

8,000 

16,400 

29,000 

59,000 

Concrete  mixers  are  of  various  types.  Some  have  a  fixed  trough, 
with  a  revolving  worm ;  others  a  square  box  revolving  about  its  diagonal 
axis,  or  a  revolving  drum.  The  gravity  mixer  is  simply  a  steel  trough,  set  at 
an  angle  of  18°  to  22°  with  the  vertical,  and  armed  internally  with  fixed 
steel  pins  and  deflectors.  The  materials  for  the  concrete  are  fed  in  at  the  top 
of  the  trough,  and,  passing  downward  through  it,  are  mixed  by  their  contact 
with  the  pins  and  deflectors.  The  following  figures,  from  the  catalogue  of 
the  Drake  Standard  Machine  Works,  refer  to  the  type  first  named. 
Capacity,  Horse-power  Weight, 

Cu.  Yds.  per  Day.  Required.  Lbs. 

400  20  5,500 

200  25  4,000 

100  15  2,700 

75  10  1,800 

25  6  1,600 

Compressive  Strength. 

The  strength  of  concrete  is  affected  by  the  quality  of  the  broken  stone,  as 
well  as  by  that  of  the  cement,  the  degree  of  ramming,  etc. 
Ramming  adds  about  50  per  cent,  to  the  strength. 
Slow-setting  cements  are  best  for  concrete,  especially  when  to  be  rammed* 


Size  of  Stone, 
Ins. 

7X  10 

Capacity 
Tons. 

f   50 

1    15 

'  per  Day, 
Size,  Ins. 

Horse-power 
Required. 

8 

16  X  10 

(120 
1   60 

f} 

15 

24  X  13 

/300 
(150 

?*} 

30 

36  X  24 

(800 
1650 

t} 

65 

944 


CONCRETE. 


Natural 
14 
20 
36 


6  Mos.     1  Year. 


Experiments  on  13-lnch  concrete  cubes,  rammed  in  cast  iron 
molds,  by  A,  W.  Dow,  U.  S.  Inspector  of  Asphalt  and  Cement.*  The 
cubes  were  thoroughly  wet  twice  daily. 

The  results  for  one  year  are  means  of  five  cubes ;  the  rest  are  means  of  two 
cubes.     Deduct  from  3  to  8  per  cent,  for  friction  of  press. 

The  materials  were  as  follows: 

Cement.  Portland 

Per  cent,  retained  on  sieve  of  100  meshes  per  linear  inch,       8.5 
Time  for  initial  set,  minutes 190 

*'     ■*'    hard       "  "         305 

Tensile  strength  as  follows,  lbs.  per  square  inch : 

1  Day.     7  Days.     1  Mo.     3  Mos. 

Portland,  neat, 441  839 

"        3  parts  stan- 
dard broken  quartz, 

Natural,  neat 9' 

2  parts  stan- 
dard broken  quartz, 

Sand  used  in  concrete. 

No  residue  on  a  No.  3  sieve;  0.5  per  cent,  passed  No.  100.  Voids  44  per 
cent.,  with  4.4  per  cent,  water. 

Broken  Stone.  Gneiss.  Of  Nos.  6  and  12  (table  below)  3  per  cent, 
retained  on  2.S  inch  mesh;  all  on  li  inch.  Others,  0  retained  on  2.5  inch; 
nearly  all  on  0.1  inch.     For  voids,  see  table,  below. 

Gravel.  Clean  quartz,  passing  a  li-inch  mesh,  2  per  cent,  passing  a  No. 
10  mesh.     Voids,  29  per  cent. 

Water.  With  Portland  cement,  0.09  cu.  ft.  (  =  5.7  lbs.)  per  cu.  ft.  of 
rammed  concrete;  with  natural  cement,  0.12  cu.  ft.  (  =  7.5  lbs.). 


248 
180 


91 


429 


188 


398 


327 


428 


414 


474 


48S 


Crushing  Strength  of  13  in.  Concrete  Cubes,  in  lbs.  per  sq. 

Experiments  by  A.  W.  Dow,  as  above: 
Parts  by  voltime ;  cement,  1;  sand,  2;  aggregate,  6. 


in. 


Aggregate 

Voids  in  Aggregate. 

Crushing  Strength, 
lbs.  per  sq.  in.,  after 

No. 

6 

4 
1 

Per 
Cent, 
of  Vol. 

Per  Cent, 
of  Voids 

10 

45 

3 

6 

1 

1 

*« 

s 

Filled  with 
Mortar. 

Days. 

Days. 

Mos. 

Mos. 

Year. 

n 

O 

-g     8 

6 

45.3 

83.9 

908 

1790 

2260 

2510 

3060 

3 

3 

35.5 

107.0 

950 

1850 

2070 

2750 

^     9 

4 

2 

37.8 

100.6 

2840 

t  10 

6 

39.5 

96.2 

2700 

^  12 

6 

29.3 

129.1 

694 

1630 

2680 

1840 

2820 

6 

45.7 

83.9 

1630 

1530 

1850 

1 

6 

45.3 

83.9 

228 

539 

375 

795 

915 

-a  2 

3 

3 

35.5 

107.0 

108 

364 

593 

632 

841 

n     3 

4 

2 

37.8 

100.6 

915 

^     4 

6 

39.5 

96.2 

800 

^     5 

6 

29.3 

129.1 

87 

421 

361 

344 

763 

6 

6 

45.7 

83.9 

596 

829 

To  test  the  effect  of  the  quantity  of  water  used,  Mr.  Dow  made  four 
12-inch  cubes,  of  1  part  Portland  cement,  2  parts  sand,  6  parts  coarse  gravel ; 
two  so  wet  that  water  ran  from  the  concrete  when  tamped  in  the  mold;  the 
other  two  with  just  sufficient  water  to  make  a  plastic  paste,  as  in  the  te^ts  re- 

*  Report  of  the  Operations  of  the  Engineering  Department  of  the  District 
of  Columbia  for  the  year  ending  June  30.  1897. 


CONCRETE   BEAMS.  945 

corded  in  the  table.  At  the  end  of  a  year  the  two  wet  cubes  gave  2300  and 
2500  lbs.  per  square  inch  respectively;  average,  2400.  The  drier  cubes  gave 
2513  and  3037  respectively;  average,  2775. 


Tn  the  floors  of  graving  docks  at  Genoa,  Italy,  concrete  of  1  part  Portland 
cement,  2  parts  sand,  3  parts  small  gravel,  carries  safely  107  lbs.  per  square 
inch;  safety  factor  15.  In  the  concrete  bridge  over  the  Danube  at  Munder- 
kingen,  Germany,  the  maximum  pressure  is  said  to  vary  from  500  to  560  lbs. 
per  square  inch.  Test  pieces  broke  with  from  2000  to  4650  lbs.  per  sq.  inch, 
varying  with  their  age. 

Transverse   Strength. 
Concrete    Beams.     Tests  by  Boston  Transit   Commission,    1895-96. 
Beams  6  ins.  square;  spans  30  and  60  ins.;  age  from  29  to  37  days,  24  hours 
in  air,  remainder  in  water. 

S  =  modulus  of  rupture  =  unit  stress  in  extreme  fibers  at  instant  of  rup- 
ture, in  lbs.  per  sq.  inch, 
g 

W  ==  r-^  =  extraneous  center  breaking  load,  in  lbs.,  on  a  beam  1  inch 
square,  1  ft.  span. 

Mixture.  S. 

Portland     Sand.      Stone.  , ' < 

cement.  Max.          Min.         Av'ge. 

1               2  >  4i               40  tests               531              158             304 

1                2i  4                  38  tests                441              131              280 

Effect  of  size  of  stone.  Mixture,  1  —  2^  —  4;  24  hours  in  air,  29  days  in 
■wet  ground.     6  tests  for  each  size. 

Wt.,Lbs.  perCu.  Ft.  S. 


Size,  Ins. 

Stone. 

Concrete. 

Voids. 
Per  Cent. 

Max. 

Min. 

Av'ge. 

2   to  1.5 

95 

156 

48 

328 

263 

298 

1.5  to  1 

95 

156 

45 

349 

272 

311 

1    to  0.5 

99 

150 

40 

305 

219 

263 

0.5  to  0.25 

100 

150 

40 

312 

229 

261 

0.25  and  less 

94 

142 

37 

248 

214 

227 

mixed 

108 

152 

33i 

349 

230 

293 

When  masonry  is  backed  by  concrete,  the  two  are  liable  in  time 
to  crack  apart,  owing  to  unequal  settlement,  especially  if  the  ramming  has 
not  been  thorough. 

In  variable  climates,  cast-iron  cylinders,  filled  with  concrete,  are  , 

frequently  split  horizontally  by  unequal  expansion  and  contraction.     In  ' 
3uch  structures  it  is  safest  to  consider  the  cylinders  as  mere  molds  for  the 
concrete ;  and  to  depend  only  upon  the  concrete  for  sustaining  the  load. 

Molded  bloclis  of  Portland  concrete,  of  even  50  tons  weight,  can 
generally  be  handled  and  removed  to  their  places  in  from  1  to  2  weeks. 

Ramming  of  concrete,  when  properly  done,  consolidates  the  mass 
about  5  or  6  per  cent.,  rendering  it  less  porous,  and  very  materially  stronger. 
The  rammers,  like  those  used  in  street  paving,  are  of  wood,  about  4  ft.  long, 
6  to  8  ins.  diameter  at  foot,  with  a  lifting  handle,  and  shod  with  iron ;  weight 
about  35  lbs.  They  are  let  fall  6  or  8  ins.  The  men  using  them,  if  standing 
on  the  concrete,  should  wear  gum  boots  to  preserve  their  feet  from  corrosion 
by  the  cement. 

Ramming  cannot  be  done  under  water,  except  partially,  when  the  con- 
crete is  inclosed  in  bags.  A  rake  may,  however,  be  used  gently  for  leveling 
concrete  under  water. 

The  size  of  the  broken  stone  for  concrete  is  generally  specified  not 
to  exceed  about  2  ins.  on  any  edge ;  but  if  it  is  well  freed  from  dust  by  screen- 
ing or  washing,  all  sizes,  from  0.5  to  4  ins.  on  any  edge,  may  be  used,  care 
being  taken  that  the  other  ingredients  completely  fill  the  voids. 

60 


946  CONCRETE. 

Concrete  is  used  for  bringing  up  uneven  foundations  to  a  level 
before  starting  the  masonry.  By  this  means  the  number  of  horizontal 
joints  in  the  masonry  is  equalized,  and  unequal  settlement  is  thereby  pre- 
vented. 

Concrete  may  readily  be  deposited  under  water  in  the  usual 
way  of  lowering  it,  soon  after  it  is  mixed,  in  a  V-shaped  box  of  wood  or  plate 
iron,  with  a  lid  that  may  be  closed  while  the  box  descends.  The  lid,  however, 
is  often  omitted.  This  box  is  so  arranged  that,  on  reaching  bottom,  a  pin  may 
be  drawn  out  by  a  string  reaching  to  the  surface,  thus  permitting  one  of  the 
sloping  sides  to  swing  open  below,  and  allow  the  concrete  to  fall  out.  The 
box  is  then  raised  to  be  refilled.  In  large  works  the  box  may  contain  a  cubic 
yard  or  more,  and  should  be  suspended  from  a  traveling  crane,  by  which  it 
:  can  readily  be  brought  over  any  required  spot  in  the  work.  The  concrete 
•  may  if  necessary  be  gently  leveled  by  a  rake  soon  after  it  leaves  the  box.  Its 
consistency  and  strength  will  of  course  be  impaired  by  falling  through  the 
water  from  the  box ;  and  moreover  it  cannot  be  rammed  under  water  with- 
out still  greater  injury.  Concrete  has  been  safely  deposited  in  the  above- 
mentioned  manner  in  depths  of  50  ft. 

The  Tremie,  sometimes  used  for  depositing  concrete  under  water,  is  a 
box  of  wood  or  of  plate  iron,  round  or  square,  open  at  top  and  bottom, 
and  of  a  length  suited  to  the  depth  of  water.  It  may  be  about  18  ins.  diam. 
Its  top,  which  is  always  kept  above  water,  is  hopper-shaped,  for  receiving  the 
concrete  more  readily.  It  is  moved  laterally  and  vertically  by  a  travehng 
crane  or  other  device  suited  to  the  case.  Its  lower  end  rests  on  the  river 
bottom,  or  on  the  deposited  concrete.  In  commencing  operations,  its  lower 
end  resting  on  the  river  bottom,  it  is  first  entirely  filled  with  concrete,  which 
(to  prevent  its  being  washed  to  pieces  by  falling  through  the  water  in  the 
tremie)  is  lowered  in  a  cylindrical  tub,  with  a  bottom  somewhat  like  the  box 
before  described,  which  can  be  opened  when  it  arrives  at  its  proper  place. 
When  filled,  the  tremie  is  kept  so  by  fresh  concrete,  thrown  into  the  hopper 
to  supply  the  place  of  that  which  gradually  falls  out  below,  as  the  tremie  is 
lifted  a  little  to  allow  it  to  do  so.  The  weight  of  the  filled  tremie  compacts 
the  concrete  as  it  is  deposited.  A  tremie  had  better  widen  out  downward 
to  allow  the  concrete  to  fall  out  more  readily. 

The  area  upon  which  the  concrete  is  deposited  must  previously  be  sur- 
rounded by  some  kind  of  inclosure,  to  prevent  the  concrete  from  spreading 
beyond  its  proper  limits ;  and  to  serve  as  a  mold  to  give  it  its  intended  shape. 
This  inclosure  must  be  so  strong  that  its  sides  may  not  be  bulged  outward  by 
the  weight  of  the  concrete.  It  is  usually  a  close  crib  of  timber  or  plate  iron 
without  a  bottom ;  and  will  remain  after  the  work  is  done.  If  of  timber  it 
may  require  an  outer  row  of  cells,  to  be  filled  with  stone  or  gravel  for  sink- 
ing it  into  place.  Care  must  be  taken  to  prevent  the  escape  of  the  concrete 
through  open  spaces  under  the  sides  of  the  crib  or  inclosure.  To  this  end 
the  crib  may  be  scribed  to  suit  the  inequalities  of  the  bottom  when  the  latter 
cannot  readily  be  leveled  off.  Or  inside  sheet  piles  will  be  better  in  some 
cases ;  or  an  outer  or  inner  broad  flap  of  tarpaulin  may  be  fastened  all  around 
the  lower  edge  of  the  crib,  and  be  weighted  with  stone  or  gravel  to  keep  it  in 
place  on  the  bottom.  Broken  stone  or  gravel  or  even  earth  (the  last  two 
where  there  is  no  current),  heaped  up  outside  of  a  weak  crib,  will  prevent  the 
bulging  outward  of  its  sides  by  the  pressure  of  the  concrete.  After  the  con- 
crete has  been  carried  up  to  within  some  feet  of  low  water,  and  leveled  off,  the 
masonry  may  be  started  upon  it  by  means  of  a  caisson,  or  by  men  in  diving 
suits.  Or,  if  the  concrete  reaches  very  nearly  to  low  water,  a  first  deep 
course  of  stone  may  be  laid,  and  the  work  thus  brought  at  once  above  low 
water  without  any  such  aids. 

The  concrete  should  extend  out  from  2  to  5  feet  (according  to  the 
case)  beyond  the  base  of  the  masonry.  All  soft  mud  should  be  removed 
before  depositing  concrete. 

Bags  partly  filled  with  concrete,  and  merely  thrown  into  the  water, 
are  used  in  certain  cases.  If  the  texture  of  the  bags  is  slightly  open,  a  por- 
tion of  the  cement  will  ooze  out,  and  bind  the  whole  into  a  tolerably  compact 
mass.  Such  bags,  by  the  aid  of  divers,  may  be  employed  for  stopping  leaks, 
underpinning,  and  various  other  purposes,  that  may  suggest  themselves. 
Such  bags  may  be  rammed  to  some  extent. 

Tarpaulin  may  be  spread  over  deep  seams  in  rock  to  prevent' 
the  loss  of  concrete;  and,  in  some  cases,  to  prevent  it  from  being  washed 
away  by  springs. 


LAITANCE.  947 

"When  concrete  is  deposited  in  water,  especially  in  the  sea,  a  pulpy  gela- 
tinous fluid  exudes  from  the  cement,  and  rises  to  the  surface.  This  causes 
the  water  to  assume  a  milky  hue;  hence  the  term  laitance,  which  French 
engineers  apply  to  this  substance.  As  it  sets  very  imperfectly,  and,  with 
some  varieties  of  cements,  scarcely  at  all,  its  interposition  between  the  layers 
of  concrete,  even  in  moderate  quantities,  will  have  a  tendency  to  lessen,  more 
or  less  sensibly,  the  continuity  and  strength  of  the  mass.  It  is  usually  re- 
moved from  the  inclosed  space  by  pumps.  Its  proportion  is  greatly  dimin- 
ished by  reducing  the  area  of  concrete  exposed  to  the  water,  by  using  large 
boxes,  say  from  1  to  li  cu.  yds.  capacity,  for  immersing  the  concrete." 


948  MODERN   EXPLOSIVES. 


MODEEN  EXPLOSIVES. 


Art.  1.  Most  of  the  explosives,  which,  of  late  years,  have  been  taking 
the  place  of  gunpowder,  consist  of  a  powdered  substance,  partly  saturated 
«rith  nitro-glycerine,  a  fluid  produced  by  mixing  glycerine  with  nitric  and  sul- 
phuric acids. 

Art.  2.  Pure  nitro-srlycerine,  at  60°  Fah,  has  a  sp  grav  of  1.6.  It  is 
odorless,  nearly  or  quite  colorless,  and  has  a  sweetish,  burning  taste.  It  is  poison- 
ous, even  in  very  small  quantities.  Handling  it  is  apt  to  cause  headaches.  It  is 
insoluble  in  water.  At  about  306°  Fah  it  takes  fire,  and,  if  unconfined,  burns 
harmlessly,  unless  it  is  in  such  quantity  that  a  part  ol  it,  before  coming  in  con- 
tact with  air,  becomes  heated  to  the  exploding  point,  which  is  about  380°  Fah. 

N-G,  and  the  powders  containing  it,  are  always  exploded  by  means  of 
Sharp  percussion.  See  Arts  36,  &c.  After  N-G  is  made,  great  care  is  required  to 
vrash  it  completely  from  the  surplus  acids  remaining:  in  it  from  the 
process  of  manufacture.  Their  presence,  either  in  the  liquid  N-G,  or  in  the 
powders  containing  it,  renders  tne  N-G  liable  to  spontaneous  decomposition, 
wrhich,  by  raising  the  temperature,  increases  the  danger  of  explosion. 

Art.  3.  N-O  freezes  at  about  45°  Fah.  It  is  then  very  difificult  of 
explosion,  and  must  be  thawed  gradually,  as  by  leaving  it  for  a  suflBcient 
length  of  time  in  a  comfortably  warm  room,  or  by  placing  the  vessel  containing  it 
In  a  second  vessel  containing  hot  water,  not  over  100°  Fah  ;  but  never  by  exposing 
It  to  intense  heat,  as  in  placing  it  before  a  fire,  or  setting  it  on  a  stove  or  boiler. 
Extra  strong  caps  are  made  for  exploding  N-G  and  its  powders  when  frozen. 

Art.  4.  N-G,  owing  to  its  incompressibility,  is  liable  to  explosion 
throug-h  accidental  percussion.  This,  and  its- liability  to  leak- 
ag'e,  render  it  inconvenient  to  transport  and  handle.  Hence  it  is  rarely  used  in 
the  liquid  state  in  ordinary  quarrying  and  other  blasting.  In  the  oil  regions  of 
Penna,  it  is  largely  used  in  oil  wells,  in  order  to  increase  the  flow.  For  this 
purpose  it  is  confined  in  cylindrical  tin  casings,  from  1  to  5  inches  diara,  called 
torpedo-shells.  These  are  suspended  from,  and  lowered  into  the  well  by  means 
if,  a  cord  or  wire  wound  on  a  reel ;  and  are  destroyed  when  the  charge  is  ex- 
ploded. They  are  about  1  inch  less  in  diam  than  the  well,  and  contain  usually 
from  one  to  twenty  quarts  =  3  lbs,  5%oz  to  66  lbs,  63^ oz  of  N-G.  They  are 
pointed  at  their  lower  ends,  in  order  to  facilitate  their  passage  through  the  oil  or 
water  which  may  be  in  the  well.  When  a  greater  charge  than  about  663^  lbs  is 
required,  two  or  more  of  these  shells  are  placed  in  the  well,  one  on  top  of  another, 
the  conical  point  on  the  lower  end  of  each  one  fitting  into  the  top  of  the  one  next 
below.  In  this  case,  the  N-G  is  fired  by  means  of  a  cap  or  series  of  caps  placed 
In  the  top  of  the  charge  before  it  is  lowered.  When  the  charge  ie  in  place,  the 
caps  are  exploded  by  electricity  led  to  them  by  conducting  wires,  as  in  Art  37,  or 
(as  in  the  method  more  commonly  practised)  by  letting  a  weight  fall  on  them. 

When  a  well  has  been  repeatedly  torpedoed,  and  a  cavity  has  thus  been  formed 
in  it  so  large  that  the  space  surronnding  a  torpedo  would  interfere  too  greatly 
with  the  effect  of  the  explosion  of  the  N-G  on  the  walls  of  the  well,  the  latter  is 
placed  directly  in  the  well,  by  lowering  a  tin  cylinder,  filled  with  it,  and  pro- 
vided with  an  automatic  arrangement  which  allows  the  N-G  to  escape  when  at' 
the  bottom  of  the  well.  The  N-G  is  then  fired  by  a  torpedo  suspended  on  a  line, 
and  having  caps  placed  in  its  top.  These  caps  are  exploded  by  a  leaden  or  iron 
weight  sliding  down  the  line,  or  by  electricity.  When  the  rock  is  seamy,  the 
N-G  is  confined  in  short  cylindrical  tin  shells,  lowered  into  the  cavity,  and  fired  by 
a  torpedo.  It  is  also  used  for  increasing  the  flow  of  springs  of  water.  It  of  course 
cannot  be  used  in  hor  or  upivard  holes,  such  as  often  occur  in  tunneling,  &c. 

Art.  5.  N-G  explodes  so  suddenly  that  very  little  tamping>  is  re- 
quired. Moist  sand  or  earth,  or  even  water,  is  sufiicient.  This,  with  the  fact 
that  N-G  is  unaffected  by  immersion  in  water,  and  is  heavier  than  water,  render 
it  particularly  suitable  for  sub-aqueous  work,  or  for  holes  containing 
water,  provided  the  rock  has  no  seams  which  would  permit  the  N-G  to  escape. 
If  the  roek  is  seamy,  the  N-G  must  be  confined  in  a  water-tight  casing. 
Such  casings,  however,  necessarily  leave  some  spaces  between  the  rock  and  the 
explosive,  and  these  diminish  considerably  the  effect  of  the  latter. 

Art.  6.  The  great  explosive  force  of  X-O  is  due  partly  to  the  very 
large  volume  of  gas  into  which  a  small  quantity  of  it  is  converted  by  explosion,  and 


MODERN    EXPLOSIVES.  949 

partly  to  the  suddenness  with  which  this  conversion  takes  place,  the  gases  being 
liberated  almost  instantaneously,*  while  with  gunpowder  their  liberation  require? 
a  longer  time.  The  suddenness  of  the  explosion  increases  its  effect,  not  only  by 
fipplying  all  of  its  force  practically  at  one  instant,  but  also  by  greatly  healing  the 
gases  produced,  and  thus  still  further  increasing  their  volume. 

Art.  7.  The  liquid  condition  of  N-G  is  useful  in  causing  it  to  fill  the  drill- 
hole completely,  so  that  there  are  no  vacant  spaces  in  it  to  waste  the  force 
of  the  explosion.  On  the  other  hand,  the  liquid  form  is  a  disadvantage,  because, 
when  thus  used  without  a  containing  vessel  in  seamy  rock,  portions  of  the  N-G  leak 
away  and  remain  unexploded  and  unsuspected,  and  may  cause  accidental  explosioi 
at  a  future  time. 

Art.  8.  N-O  is  stored  in  tin  cans  or  earthenware  jars.  If 
properly  washed  from  acid  it  does  not  injure  tin.  For  transportation,  these  cans  o^ 
jars  are  packed  in  boxes  with  sawdust,  or  in  padded  boxes,  and  loaded  in  wagons 
The  R  R  companies  do  not  receive  it.  . 

Art.  9.  "When  N-G  and  its  compounds  are  completely  exploded,  the  g^ase(^ 
^iven  out  are  not  troublesome,  but  those  resulting  from  incomplete  explosioD 
guch  as  generally  takes  place,  or  from  combustion,  are  very  offensive. 

Art.  10.  For  convenience,  we  apply  the  name  ^^  dynamite  "  to  any  expla 
sive  Avhich  contains  nitro-glycerine  mixed  with  a  granular  absorbent;  *'tru^ 
dynamite"  to  those  in  which  the  absorbent  of  the  N-G  is  "  Kie8elguhr,"t  o^, 
some  other  inert  powder  which  takes  no  part  in  the  explosion :  and  ••'  false 
dynamite  "  to  those  in  which  the  absorbent  itself  contains  explosive  substance* 
other  than  N-G. 

Art.  11.  The  absorbent,  by  its  granular  and  compressible  condition, 
fusts  as  a  cushion  to  the  N-G,  and  protects  it  from  percussion,  and  from 
the  conseqfuent  danger  of  accidental  explosion. 

N-G  undergoes  no  change  in  composition  by  being  absorbed ;  and  it  then  freezes^ 
burns,  explodes,  &c,  under  the  same  conditions  as  to  pressure,  temperature,  Ac,  as 
when  in  the  liquid  form.  The  cushioning  effect  of  the  absorbent  merely  renders  it 
more  difficult  to  bring  about  auflBcient  percussive  pressure  to  cause  explosion.  The 
absorption  of  the  N-G  in  dyn  enables  the  latter  to  be  u«ed  in  hor  holes,  or  in  holes 
drilled  upward. 

Art.  12.  N-G  and  dyn  explode  much  more  readily  when  riji^idly 
confined,  as  by  a  metallic  vessel,  or  by  the  walls  of  a  hole  drilled  in  rock,  than 
when  confined  by  a  yielding  substance,  as  wood.  Therefore  the  fact  that  dyn,  not 
being  liquid,  can  be  packed  in  wooden  boxes,  renders  it  safer  than  N-G  which  has  to 
be  kept  in  Itone  or  metal  vessels. 

Art.  13.  True  dynamites  must  contain  at  least  about  50  per 
cent  of  N-G.  Otherwise  the  latter  will  be  too  completely  cushioned  by  the  absorbent, 
and  the  powder  will  be  too  difficult  to  explode.  False  dynamites,  on  the  contrary, 
may  contain  as  small  a  percentage  of  N-G  as  may  be  desired;  some  containing  aa 
little  as  15  per  cent.  The  added  explosive  substances  in  the  false  dynamites  generally 
contain  large  quantities  of  oxygen,  which  are  liberated  upon  explosion,  and  aid  in 
effecting  the  complete  combustion  of  any  noxious  gases  arising  from  the  N-G. 

Art.  14.  Dynamites  which  contain  lar^e  percentages  of 
N-Cr  explode  (like  the  liquid  N-G.  Art  6)  with  great  suddenness,  tending  to  shatter 
the  rock  in  their  vicinity  into  small  fragments.  They  are  most  useful  in  very  hard 
rock.  In  such  rock.  No  1  dynamite,  or  that  containing  75  per  cent  of  N-G,  is 
roughly  estimated  to  have  about  6  times  the  force  of  an  C'qual  w^t 
of  g-unpoi^der. 

For  soft  or  decomposed  rocks,  sand,  and  earth,  the  low^er  g'rades 
of  dynamite,  or  those  containing  a  smaller  percentage  of  N-G,  are  more  suitable. 
They  explode  with  less  suddenness,  and  their  tendency  is  rather  to  upheave  large 
masses  of  rock,  Ac,  than  to  splinter  small  masses  of  it.  They  thus  more  nearly  re- 
semble gunpowder  in  their  action. 

Judgment  must  be  exercised  as  to  the  g-rade  and  quantity  of  explosive 
to  be  used  in  any  given  case.  Where  it  is  not  objectionable  to  break  the  rock  into 
small  pieces,  or  where  it  is  desired  to  do  so  for  convenience  of  removal,  the  higher, 
shattering  grades  are  useful.  Where  it  is  desired  to  get  the  rock  out  in  large  masses, 
as  in  quarrying,  the  lower  grades  are  preferable. 

For  very  difficult  work  in  hard  rock,  and  for  submarine  blasting,  the  highest 
grades,  containing  70  to  75  per  cent  of  N-G,  are  used.  A  small  charge  of  these  does 
the  same  execution  as  a  larger  charge  of  lower  grade,  and  of  course  does  not  require 

*  Such  sudden  liberation  of  gas  is  called  "  detonation." 

t  KiecelKuhr  is  an  earthy,  silicious  limestone,  composed  of  the  fossil  remains  of  small  shelly 
receptacle  for  nitro-glycerine.  Kieself nhr  is  found  in  Hanover,  'German|r 


■ad  in  New  Jersey. 


950  MODERN   EXPLOSIVES. 

the  drilling  of  so  large  a  hole.    In  submarine  work  their  sharp  explosion  iM  nol 
deadened  by  the  water. 

For  general  railroad  ^ork,  ordinary  tunneling,  mining  of  ores,  <L'c,  the  aver* 
ag'e  grade,  containing  4:U  per  cent  of  N-G,  is  used;  for  quarrying*,  35  per 
cent;  lor  blaistiug  stumps,  trees,  piles,  Ac,  30  per  cent;  for  sand  and 
eartli,  15  per  cent. 

Art.  15.  Dynamite,  like  N-G,  can  be  readily  exploded  under 
water,  provided  it  is  so  immersed  as  not  to  be  scattered ;  but  lon^  exposure 
to  'water  is  injurious  to  it.  In  the  higher  grades,  the  water,  by  its  greater 
affinity  for  the  absorbent,  drives  out  the  N-G.  In  the  lower  grades  it  is  apt  to  wash 
away  the  salts  used  as  additional  explosives. 

Art.  16.  In  dyns  containing  a  large  percentage  of  N-G,  the  latter  is  liable 
to  exude  in  liquid  form,  or  to  "  leak,"  especially  in  warm  weather,  and  then  to 
explode  through  accidental  percussion.  The  same  danger  exists,  even  though  the 
percentage  of  N-G  be  small,  if  the  absorbent  has  but  small  absorbing  power,  and  is, 
consequently,  easily  saturated. 

Art.  17.  True  dyn  resembles  moist  brown  sugar.  Its  properties  are 
generally  those  of  the  N-G  contained  in  it.  Thus,  it  takes  fire  at  about  350°  F,  and 
burns  freely.  It  freezes  at  45°  F,and  is  then  difficult  to  explode.  It  is  not  exploded 
by  friction,  or  by  ordinary  percussion,  but  requires,  for  general  purposes,  a  strong 
cap,  or  exploder,  containing  fulminating  powder,  see  Arts  36,  38,  &c.  It  may,  how- 
ever, be  exploded  by  a  priming  of  gunpowder,  tightly  tamped,  and  fired  by  an  ordi- 
nary safety -fuse. 

Art.  18.  Tlie  charge  should  fill  the  cross  section  of  the 
hole  as  completely  as  possible.  If  water  is  not  standing  in  the  hole,  the  cartridge 
should  be  cut  open  before  insertion,  so  that  th  e  powder  may  escape  from  it  and  fill  the 
hole ;  or  the  powder  may  be  simply  emptied  from  the  cartridge  into  the  hole. 

Art.  19.  For  blasting  ice  in  place,  holes  are  cut  in  it,  and  a  number  of  dyn 
cartridges  (one  of  which  must  contain  an  exploding  cap)  are  tied  together  and  low- 
ered from  1  to  5  ft  into  the  water.  They  are  fired  as  soon  as  possible  after  immer- 
sion, to  avoid  the  danger  of  freezing^  Electrical  exploders  (Arts  37,  &c,)  are  best 
for  sub-aqueous  work. 

Art.  20.  Dyn  is  useful  for  breaking:  up  pieces  of  metal,  such  as  old 
cannon,  condemned  machinery,  "salamanders"  (masses  of  hardened  slag)  in  blast 
furnaces,  &c.  In  cannon,  the  dyn  is  of  course  exploded  in  the  bore.  In  other  pieces, 
small  holes  are  generally  drilled  to  receive  it;  but  plates,  even  of  considerable  thick- 
ness, may  be  broken  by  merely  exploding  dyn  upon  their  surface.  , 

Art.  21.  For  blasting-  trees  or  stumps,  one  or  more  cartridges  are 
fired  in  a  hole  bored  in  the  trunk  or  roots,  or  under  the  latter.  This  shatters  both 
trunk  and  roots.  A  tree  may  be  felled  neatly  by  boring  a  number  of 
small  radial  holes  into  it,  at  equal  short  dists  in  a  hor  line  around  its  circumf,  and, 
by  means  of  an  electric  battery  (Arts  37,  Ac),  exploding  simultaneously  a  small 
charge  of  dyn  in  each.  Or  a  single  long  cartridge  may  be  tied  around  the  trunk  of 
a  small  tree,  and  fired. 

Art.  22.  Piles  may  be  blasted  in  the  same  way  as  trees ;  or  a  hole  may 
be  bored  for  the  cartridge  in  the  axis  of  the  pile;  or  the  cartridge  may  be  simply 
tied  to  the  side  of  the  pile  at  any  desired  ht. 

Art.  23.  The  higher  grades  of  dyn,  like  N-G,  require  but  little  tamp« 
ing*.  Une  a  wooden  tamplng-bar,  never  a  mdallic  one,  for  any  explosive.  If 
a  charge  of  dyn  *•  hangs  fire,"  it  is  dangerous  to  attempt  to  remove  it.  Remove 
the  tamping,  all  but  a  few  ins  in  depth,  on  top  of  which  insert  another  cartridge, 
containing  an  exploder,  and  try  again.  See  electrical  exploders,  Arts  37,  Ac.  Dyn, 
like  N-G,  if  frozen,  must  be  thawed  gradually,  by  leaving  it  in  a  warm  room,  far 
from  the  fire;  or  by  placing  it  in  a  metallic  vessel,  which  is  then  placed  in  another 
vessel  containing  hot  water.  The  water  should  not  be  hotter  than  can  be  borne  by 
the  hand.  Otherwise  the  N-G  is  liable  to  separate  from  the  absorbent.  The  N-G  iu 
dyn  may  freeze  without  cementing  together  the  particles  of  the  absorbent;  ia 
which  case  the  powder  of  course  is  still  soft  to  the  touch.  An  overcharge  of 
N-G,  or  of  dyn,  is  liable  to  be  burned,  and  thus  wasted,  giving  off  offensive  gases. 

Art.  24.  Dyn  is  sold  in  cylindrical,  paper-covered  cart- 
ridges, from  %  to  2  ins  in  diam,  and  6  to  8  ins  long,  or  longer.  They  are  fur- 
nished to  order  of  any  required  size,  and  are  packed  in  boxes  containing  25  Bt)s  or  5Q 
ft)s  each.     The  layers  of  cartridges  are  separated  by  sawdust. 

Art.  25.  Some  of  the  R  R  companies  decline  to  carry  dyn  or  N-G  in  any 
shape.  Others  carry  dyn  under  certain  restrictions,  based  upon  State  Jaws;  pro- 
viding that  it  must  be  dry  (t  e,  that  no  N-G  shall  be  exuding  from  it) ;  that  boxe« 
and  cars  containing  it  shall  be  plainly  marked  with  some  cautionary  words,  as  "ex- 
plosive," "dangerous,"  &c;  that  the  cartridges  shall  be  so  packed  in  the  boxes,  and 
the  boxes  so  loaded  in  the  cars,  that  both  shall  lie  upon  their  sides,  and  the  boxes 


MODERN   EXPLOSIVES. 


951 


be  i  a  no  danger  of  falling  to  the  floor ;  that  caps,  &c,  shall  not  be  loaded  in  the 
same  car  with  dyn,  &c,  &c. 

Art.  26.  A  great  many  varieties  of  dyn  are  made.  They  differ 
(generally  but  sliglitly)  in  the  composition  of  the  absorbent,  and  in  the  method 
of  manufacture.  Each  maker  usually  makes  a  number  of  grades,  containing 
different  percentages  of  N-G,  &c,  and  gives  to  his  powders  some  fanciful  name. 

Art.  27.  The  following  table  of  explosives,  made  by  the  Repauno 
Chemical  Co,  Wilmington,  Del,  and  known  as  "  Atlai^"  powders,  gives  the 
percentage  of  N-G  in  each. 


Brand. 

Percentage 
of  N-G., 

Brand. 

Percentage 
of  N-G. 

A 

75 

D+ 

33 

B+ 

60 

B 

50 

E  + 

27 

c+ 

45 

E 

20 

c 

40 

The  absorbents  contain ;  in  •*  A"  brand,  18  per  cent  wood  pulp  and  7 
per  cent  carbonate  of  magnesia;  in  "  C"  brand  (the  average  grade),  46  per  cent 
nitrate  of  soda  (soda  saltpetre),  11  per  cent  wood  pulp,  and  3  per  cent  carbonate 
of  magnesia ;  in  "  E  "  brand,  62  per  cent  nitrate  of  soda,  16  per  cent  wood  pulp, 
&c,  and  2  per  cent  carbonate  of  magnesia. 

Art.  28.  "  Miner's  Friend  "  powder  contains  nitrate  of  soda,  wood 
pulp,  resin,  and  carbonate  of  magnesia.  It  freezes  at  42°,  and  is  then,  like  other 
dyn,  difficult  to  explode.  When  used  under  water,  the  cartridges  should  not  be 
broken,  because  the  powder  is  injured  by  direct  contact  with  water.  Their 
*'  Heela"  powder  is  a  lower  grade.  It  is  in  granulated  form,  like  ordinary 
blasting  powder,  but  is  said  to  be  much  stronger.  It  is  intended  as  a  substitute 
for  it. 

Art.  29.  '*  Oianf  powder  is  dyn  proper,  containing  75  per  cent  N-G, 
and  25  per  cent  Kieselguhr  obtained  near  their  works  in  New  Jersey.  The 
lowest  grade,  branded  "M,"  contains  20  per  cent  N-G.  The  name  "giant 
powder"  was  originally  applied  to  dynamite  in  general. 

Art.  30.  Other  brands  are  "  Hercules  "  powder  and  "  Judson  R  R  P 
powder,"  a  substitute  for  ordinary  blasting  powder.  It  is  put  up  in  water- 
proof paper  bags,  of  634. 123^>  <'i"d  25  fibs  each,  and  these  are  packed  in  wooden 
boxes  holding  50  lbs  each.  "  Judson  F  F  F  dynamite  "  is  a  higher  grade, 
in  cartridges  of  the  usual  shape,  packed  in  50-ib  boxes. 

Art.  31.  ''^  Rackaroek. "  cartridges  are  said  to  contain  no  N-G,  and  to 
be  entirely  inexplosive  until  immersed,  for  a  few  seconds,  in  an  iuexplosive 
liquid  furnished  by  the  same  Co.  They  are  then  allowed  to  stand  for  15  mins, 
after  which  they  may  be  used  at  any  time.  They  are  fired  in  the  same  way  as 
dyn,  and  can  be  used  under  water.  Th€  mfrs  claim  that  they  "  approximate 
N-G  in  strength,  and  are  stronger  than  dyn.'* 

Art.  32.  The  following-  explosives  are  made  and  used  in 
Europe,  but  have  not  yet  been  regularly  imported  into  the  TJ.  S. 

Compressed  g-un-cotton,  is  cotton  dipped  in  a  mixture  of  nitric  and 
sulphuric  acids,  then  reduced  to  a  fine  pulp,  and  made  into  discs  1  to  2  ins  thick, 
and  J^  to  2  ins  diam,  or  larger.  It  is  generally  used  wet,  for  the  sake  of  greater 
safety.  It  then  requires  extra  strong  caps  or  primers.  Roughly  speaking,  it  is 
about  as  strong  as  dyn  No  1,  but  is  less  shattering  in  its  effect.  Being  lighter 
than  dyn,  it  requires  larger  holes ;  and,  owing  to  its  rigidity,  is  less  easily  in- 
serted, and  does  not  fit  the  hole  so  completely.  When  dry,  it  is  very  inflam- 
mable, but,  if  not  confined,  it  burns  harmlessly.  It  contains  no  liquid,  to  freeze 
or  to  exude ;  and  is  safe  to  handle. 

Art.  33.  Tonite  consists  of  finely  divided  gun-cotton  mixed  with  nitrate 
of  baryta.  It  is  compressed  into  candle-shaped  cartridges  having,  at  one  end,  a 
recess  for  the  reception  of  an  exploder  containing  fulminate  of  mercury.  Th« 
cartridges  weigh  about  the  same  as  dyn.    They  are  generally  made  waterproof. 


952  MODERN   EXPLOSIVES. 

Art.  34.  Forclte,  lilthofractenr,  and  Dualin  are  foreign  makes  of 
nitro-glycerine  explosives.  In  Dual  in  the  absorbent  is  sawdust.  It  has  greater 
bulk  than  dyn  for  a  given  wt,  and  requires  larger  holes. 

Art.  35.  Explosive  gelatine  is  a  transparent,  pale  yellow,  elastic 
substance,  and  is  composed  of  90  per  cent  N-G  and  10  per  cent  gun-cotton.  It  is 
less  sensitive  than  dyn  to  percussion,  friction,  or  pressure,  and  is  not  affected  by 
water.  Its  specific  gravity  is  1.6.  It  burns  in  the  open  air.  For  complete 
detonation  a  special  primer  is  required.  The  addition  of  a  small  proportion  of 
camphor  renders  it  still,  less  sensitive,  and  increases  its  explosive  force.  The 
camphor  evaporates  to  some  extent. 

In  some  experiments  on  the.  power  of  different  explosives  to  increase  the  contents 
of  a  small  cavity  in  a  leaden  block,  explosive  gelatine  caused  an  increase  50  per  cent 
greater  than  that  caused  by  dyn  No  1.  In  hard  rock  the  diflf  would  probably  have 
been  greater.    The  increase  was  10  per  cent  less  than  that  caused  by  N-G. 

Art.  36.  Tlie  cap  or  exploder,  used  with  ordinary  safety  fuse  for  ex« 
ploding  N-G  and  dyn,  is  a  hollow  copper  cylinder,  about  y^  inch  diam,  and  an  inch 
or  two  in  length.  It  contains  from  15  to  20  per  cent,  or  more,  of  fulminate  of  mer 
cnry,  mixed  with  other  ingredients  into  a  cement,  which  fills  the  closed  end  of  the 
cap.  The  cap  is  called  *' single-force,"  "triple-force,"  &c,  according  to  the  quantity 
of  explosive  it  contains. 

The  end  of  the  fuse,  cut  off  square,  Is  inserted  Into  the  open  end  of  thii  cap,  far 
enough  to  touch  the  fulminating  mixture  in  it.  In  doing  this,  care  must  be  taken 
not  to  roughly  scratch  the  latter.  The  neck  of  the  cap  is  then  pinched,  near  its 
open  end,  so  as  to  hold  the  fuse  securely.  -The  cap,  with  the  fuse  thus  attached,  is 
then  inserted  into  the  charge  of  N-G  or  dyn,  care  being  taken  not  to  let  the  fuse 
come  into  contact  with  the  explosive,  "which  would  then  be  burned  and  wasted.  If 
%  dyn  cartridge  is  used,  the  fuse,  with  cap,  is  first  inserted  into  it.  The  neck  of  the 
cartridge  is  then  tied  around  the  fuse  with  a  string,  and  the  cartridge  is  then  ready 
to  be  placed  in  the  hole  and  fired. 

Art.  37.  The  Siemens  magrneto-electric  blasting  appa* 
ratus,  now  in  general  use,  consists  of  a  wooden  box  about  as  large  as  a  transit- 
box.  Outside  it  has  two  metallic  binding-posts  with  screws,  for  attaching  the  two 
wires  leading  to  the  exploder.  From  the  top  of  the  box  projects  a  handle  at  the 
end  of  a  vert  bar.  This  bar,  which  is  about  as  long  as  the  box  is  high,  Is  made  so 
as  to  slide  up  and  down  in  it,  and  is  toothed,  and  gears  with  a  small  pinion  inside 
the  box.  When  a  blast  is  to  be  fired,  the  bar  is  diawn  up,  by  means  of  the  handle, 
as  far  as  it  will  come.  It  is  then  pressed  quickly  down  to  the  bottom  of  the  box. 
In  its  descent  it  puts  into  operation,  by  means  of  the  pinion,  a  magneto-electric 
machine  inside  the  box.  This  generates  a  current  of  electricity,  which  increases  in 
force  with  the  downward  motion  of  the  bar,  but  which  is  confined  to  a  short  circuit 
of  wire  within  the  box^  until  the  foot  of  the  bar  strikes  a  spring  near  the  bottom  of 
the  box,  breaking  the  short  circuit  and  forcing  the  electricity  to  travel  through  the 
two  longer  "leading  wires,'*  which  lead  it  from  the  two  binding-posts  on  the  outsid* 
♦f  the  box  to  the  cap  or  exploder  placed  in  the  charge. 

Art.  38.  The  cap  used  with  this  machine  is  similar  t6  that  used  with  safety 
fuse  (Art  36),  except  that  its  mouth  is  closed  with  a  cork  of  sulphur  cement,  through 
Iv^hich  pass  the  two  wires  leading  from  the  electric  machine.  The  ends  of  these 
wires  project  into  the  fulminating  mixture  in  the  cap.  They  are  J/g  inch  apart,  but 
are  connected  by  a  platinum  wire,  which  is  bo  fine  as  to  be  heated  to  redness  by  the 
turrent  from  the  battery.   Its  heat  ignites  the  fulminate  and  thus  explodes  the  cap. 

Art.  39.  Where  a  number  of  holes  are  to  be  fired  simnl- 
taiieonsly  (thus  increasing  their  effect),  each  hole  has  a  platinum  cap  inserted 
into  its  charge,  and  one  of  the  short  wires  attached  to  each  cap  is  joined  to  one  of 
those  of  the  next  cap,  so  that  at  each  end  of  the  series  of  caps  there  is  one  free  end 
of  a  short  wire.  Each  of  these  two  ends  is  fastened  to  the  end  of  one  of  the  leading 
wires,  placing  the  whole  series  "  in  one  circuit."  Where  the  holes  are  too  far  apart 
for  the  caps  to  be  thus  joined  by  the  short  wires  attached  to  them,  the  ends  of  tb* 
latter  are  connected  by  cotton-covered  "  connecting;  wires.** 


MODERN   EXPLOSIVES. 


953 


Art.  40.  The  magneto-electrical  macliine  weigbs  about  16  lbs.  It  can 
fire  about  12  caps  at  once. 

Caps  for  ordinary  fuse  and  for  electrical  firing,  fuses,  wires,  electrical  machines, 
&c,  are  sold  by  most  of  the  makers  of,  and  dealers  in,  explosives,  rock-drilling 
machines,  &c. 

Art.  41.  Simultaneous  firing  of  a  number  of  holes  can  be  conveniently 
accomplished  only  by  electricity.  Electric  blasting  apparatus  is  specially  useful 
for  blasting  under  water,  where  ordinary  fuses  are  apt,  especially  at  great  depths, 
to  become  saturated  and  useless. 

If  an  electrical  machine  fails  to  fire  a  charge,  it  is  known  that  the  charge 
cannot  explode  until  the  attempt  is  repeated.  Therefore  no  time  need  be  lost, 
and  no  risks  run,  on  account  of  *'  hanging  fire." 

OFNPOWDEB. 

Tlie  explosive  force  of  powder  is  about  40000  fts,  or  18  tons,  per  squar« 
inch.  Its  iveig-lit  averages  about  the  same  as  that  of  water,  or  62V^  Bbs  per 
cubic  foot ;  hence,  1  ft)  =  about  28  cubic  inches.  In  ordinary  quarrying^  a  cubic 
yard  of  solid  rock  in  place,  (or  about  1.9  cubic  yards  piled  up  after' being  quar- 
ried,) requires  from  }yito%fb.  In  very  refractory  rock,  lying  badly  for  quarry- 
ing, a  solid  yard  may  require  from  1  to  2  ft)s.  In  some  of  the  most  successful 
great  blasts  for  stone  for  the  Holyhead  Breakwater,  Wales  (where  several 
thousands  of  lbs  of  powder  were  usually  exploded  by  electricity  at  a  single 
blast,)  from  2  to  4  cubic  yards  solid  were  loosened  per  S) ;  but  in  many  instances 
riot  more  than  1  to  1}4  yards.  Tunnels  and  shafts  require  2  to  6  ft)s  per  solid 
yard;  usually  3  to  5  Ids.  Soft,  partially  decomposed  rock  frequently  requires 
more  than  harder  ones.    Usually  sold  in  kegs  of  25  B)s.* 


Weigbt  of  powder 

in  one 

foot  deptli  of  bole. 

Dtameter  of  liole 

VtTelglit  of  powder 
avoirdupois 

lin 
Ofi)  5oz 

m  ins     1}4  ins 
Oft  8oz    Oft  lloz 

2  ins       2}^  ins       3  ins 
1ft  4oz        2ft        2ft  13oz 

Diameter  of  hole 

MTeigl&t  of  Powder 
avoirdupois 

33^  in 
3lbl4oz 

4  ins 
5ft  Ooz 

41^  ins 
6ft  60Z 

5  ins 
7ft  14oz 

51/^  ins 
9ft  80Z 

6  ins 
llft5o» 

954  TIMBER. 

PRESERVATION  OF  TIMBER. 

Art.  1.     (a)    The  decay  of  timber   is   caused   by  the  growth  and 

activities  of  fungi.  The  minute  spores  of  one  of  these  fungi,  germinating 
on  a  piece  of  wood,  send  out  fine  threads,  which  enter  the  wood  cells  and 
soon  give  otf  a  complex  compound  called  a  ferment  or  enzyme,  which  dis- 
solves certain  parts  of  the  wood  fibre.  The  dissolved  fibre  serves  as  food 
for  the  fungus.  The  threads  throw  out  branches  and  sub-branches,  and 
soon  the  timber  is  permeated  by  a  mass  of  such  threads,  the  growing  parts 
of  which  give  off  ferment.  The  action  of  the  ferment  changes  the  chemical 
and  physical  properties  of  the  wood,  rendering  it,  in  some  cases,  like  browu 
charcoal,  in  others  white,  soft  and  stringy,  and  the  wood  is  said  to  be  rotten 
or  decayed.  Eventually  some  of  the  threads  grow  out  from  the  surface 
of  the  timber,  and  form  toadstools  and  other  excrescences.  Under  these 
are  found  cavities  containing  thousands  of  spores,  which,  when  ripe,  are 
blown  off  into  the  air  and  settle  upon  other  timbers,  where  the  process  is 
repeated.  Moisture  and  heat  are  favorable  to  the  growth  of  the  fungi,  as 
are  also  the  starches,  sugars  and  oils  found  in  the  cells  of  the  sapwood  but 
wanting  in  the  heartwood.  If  protected  from  the  action  of  these  fungL 
wood  will  last  indefinitely.  Hence  the  accumulation  of  deadwood  should 
be  avoided.*  If  air  is  excluded,  as  when  timber  is  kept  constantly  and 
entirely  immersed  in  salt  or  fresh  water,  the  fungi  cannot  thrive.  Sap, 
confined  in  timber  with  air,  ferments,  producing  dry  rot;  as  where  beams 
are  enclosed  air-tight  in  brickwork,  etc.;  and  where  green  timber  is  painted 
or  varnished,  or  treated  with  creosote,  etc.  The  sap  then  not  only  prevents 
the  thorough  penetration  of  the  oil,  etc,  but  may  cause  the  greater  part  of 
the  wood  to  rot  although  its  firm  outer  shell  gives  it  a  deceptive  appearance 
of  strength,  (b)  Sap  should  therefore  be  first  removed  by  seasoning; 
t  e,  either  by  drying  the  wood  in  air  at  natural  or  higher  temperatures, 
or  by  first  steaming  the  wood  under  pres  so  as  to  vaporize  the  sap,  and 
then  removing  the  latter  by  means  of  a  vacuum.  Thorough  seasoning  of 
large  timbers  in  dry  air  at  ordinary  temperatures  may  require  years;  and 
too  rapid  kiln-drying  cracks  and  weakens  the  wood.  But  it  is  questionable 
whether  steaming  and  vacuum  remove  sap  as  thoroughly  as  do  the  slower 
dry  processes,  (c)  Alternate  exposure  to  water  and  air  is  very  destructive. 
It  causes  wet  rot. 

Art.  2.  Sea-worms.  The  limnoria  terebrans  works  from  near  high- 
water  mark  to  a  little  below  the  surface  of  mud  bottom;  the  teredo  navalis 
within  somewhat  less  limits.  The  teredo  is  said  to  be  rendered  less  active 
by  the  presence  of  sewage  in  water. 

Art.  3.  (a)  The  best  timber-preserving  processes  are  practically  useless 
unless  thoroughly  well  done.  If  the  gain  in  durability  will  not  war- 
rant the  expenditure  of  time  and  money  reqd  for  this,  it  is  more  economical 
to  use  the  wood  in  its  natural  state,  (b)  The  woods  best  adapted  to 
treatment  are  those  of  an  open  or  porous  texture.  They  absorb  the  oil 
etc  better  than  the  denser  woods;  and  their  cheapness  renders  the  use  of 
<;he  treatment  more  economical,  (c)  Most  of  the  processes  in  common  use 
seem  to  render  wood  less  combustible,  (d)  After  treatment  by  any  process, 
the  wood  should  be  well  dried  before  using. 

Art.  4.  (a)  Creosote  oil,  or  dead  oil,  is  the  best  known  preservative. 
Against  sea-worms  it  is  effective  for  15  to  25  years,  and  is  the  only  known 
protection,  (b)  As  temporary  expedients,  piles  are  sometimes  covered  with 
sheet  metal,  or  with  broad-headed  nails  driven  close  together.  These  rust  or 
wear  away  in  a  few  years.  Oak  piles,  cut  in  January,  and  driven  with  the  bark 
on,  have  resisted  the  teredo  for  4  or  5  years ;  and  cypress  piles,  well  charred, 
for  9  years,  (c)  For  ordinary  exposures  on  land,  8  to  10  lbs  of  creosote  oil 
per  cub  ft  are  reqd  =  say  670  to  830  lbs  per  1000  ft  board  measure  =  30 
to  40  lbs  per  cross  tie  of  4  cub  ft.  For  protection  against  sea-worms  10  to 
12  lbs  per  cub  ft  suffice  in  climates  like  those  of  Great  Britain  and  the 
Northern  U  S ;  but  in  warmer  waters  where  the  teredo  is  very  active,  from 
14  to  20  lbs  per  cub  ft  are  used.  Large  timbers  may  not  require  saturation 
throughout,  and  thus  may  take  less  per  cub  ft.  But  see  (i)  and  end  of  Art. 
1  (a),  (d)  Creosote  oil  weighs  about  8.8  lbs  per  U.  S.  gallon,  (e)  The 
sticks  should  be  reduced  to  their  intended  final  dimensions  and  framed  (if 
framing  is  reqd)  before  treatment ;  especially  if  for  exposure  to  teredo,  which 

*  See  paper  by  Dr.  Hermann  von  Schrenk,  read  before  the  American 
Railway  Engineering  and  Maintenance  of  Way  Association,  March,  1901. 


TIMBER.  955 

is  sure  to  attack  any  spots  which  (as  by  subsequent  cutting)  are  left  unpro- 
tected, (f)  Creosoted  ties  have  remained  sound  after  22  years'  exposure. 
The  creosote  protects  the  spiices  from  rusting,  (g)  Spruce  and  tamarack, 
owing  to  their  irregular  density,  are  unsuitable  for  creosoting.  (h)  Creosote 
renders  wood  stiffer  and  slightly  more  brittle.  In  hot  weather  it  exudes  to 
some  extent  and  discolors  the  wood.  Its  smell  excludes  it  from  dwellings. 
(i)  It  does  not  wash  out  from  the  wood,  but  often  fails  to  penetrate  the 
heart-wood.  Then,  if  any  sap  remains,  decay  begins  at  the  center.  See 
end  of  Art  1  (a) .  Burnettizing  the  cen  of  the  stick  (see  Art  7)  and  using  a 
coating  of  creosote  outside,  has  long  been  suggested  as  the  best  possible 
method.  It  is  the  principal  feature  of  the  Allardyce  process.  This  is 
cheaper  than  thorough  creosoting.  In  the  Rutgers  process,  which  has 
been  successfully  employed  for  ties  in  Germany  since  1874,  the  creosote  and 
a  solution  of  zinc  chloride  are  injected  simultaneously,  (j)  In  the  creo- 
resinate  process  *  the  preservative  fluid  consists  of  creosote  38  per.cent, 
forrnaldehyde  2  per  cent,  and  melted  resin  60  per  cent. 

Art.  5.  (a)  3Iineral  solutions  are  inferior  to  creosote,  even  on  land; 
and  useless  in  running  water  or  against  sea-worms ;  but  they  approximately 
double  the  life  of  inferior  timber  under  ordinary  land  exposures;  and  their 
cheapness  permits  their  use  where  that  of  creosote  is  too  expensive,  (b) 
They  render  wood  harder;  and  brittle  if  the  solution  is  too  strong.  They 
are  liable  to  be  washed  out  by  rain,  etc.  Hence  the  outer  wood  decays  first. 
See  Art  4  (i)  Art  8  (b)  (c)  (d).  (c)  A  committee  of  the  American  Soc  of  Cir 
Engrs,t  after  collating  a  large  number  of  experiments,  recommended  Bur- 
nettizing (Art  7)  for  damp  exposure,  as  that  of  cross  ties,  damp  floors, 
etc;  and  Kyanizing  (Art  6)  for  comparatively  dry  situations  with 
exposure  to  air  and  sun-light,  as  in  bridge  timbers,  for  which  it  is  better 
suited  than  Burnettizing  because  it  seems  to  weaken  wood  less.  In  such 
exposures  it  preserves  wood  sometimes  for  20  to  30  years. 

Art.  6.  (a)  Kyanizing  consists  in  steeping  the  wood  in  a  solution  of 
1  lb  of  bi-chloride  of  mercury  (corrosive  sublimate)  in  100  lbs  of  water,  (b) 
It  is  usual  to  allow  the  wood  to  soak  a  day  for  each  inch  of  the  thickness  or 
least  dimension  of  the  piece,  and  one  day  in  addition,  whatever  the  size. 
(c)  Gen'l  Cram  found  the  process  very  unhealthy,  "salivating  all  the  men": 
but  Mr.  J.  B.  Francis,  at  Lowell,  and  Mr.  H.  Bissell,  of  the  Eastern  R  R  of 
Mass,  had  little  or  no  trouble  in  this  respect.  The  sublimate,  however, 
which  is  very  poisonous,  is  apt  to  effloresce,  and  the  use  of  the  timber  is  thus 
rendered  dangerous,  (d)  The  process  is  valuable  for  timber  placed  in 
moderately  damp  situations,  but  the  salt  is  liable  to  be  washed  out  by  run- 
ning water.  Kyanized  spruce  fence  posts,  planted  4  ft  in  the  ground,  at 
Lowell,  Mass,  in  1850,  were  examined  in  1891,  and  most  of  them  were  found 
very  sound  both  above  and  below  the  surface  of  the  groimd. 

Art.  7.  (a)  Burnettizing  consists  in  immersing  the  wood  for  several 
hours  in  a  solution  of  2  lbs  chloride  of  zinc  in  100  lbs  of  water,  under  a 
pres  of  from  100  to  300  lbs  per  sq  inch. 

Art.  8.  Other  preventives,  (a)  Steeping  iji  a  solution  of  sulphate 
of  copper  (blue  vitriol)  has  been  extensively  used,  but  does  not  seem  to 
have  been  permanently  successful.  The  blue  vitriol  washes  out  readily. 
(b)  In  the  Barschall  or  Hasselmann  process,!  introduced  in  Germany 
in  1887,  in  the  U  S  in  1899,  the  wood  is  boiled,  at  a  temperature  from  212** 
to  284°  Fahr.  and  under  a  pressure  of  from  15  to  45  lbs  per  sq  inch,  in  a 
solution  of  iron,  copper  and  aluminum  sulphates  and  "Kainit,"  a  sulphate 
of  magnesia  and  potash,  mined  at  Stassfurt,  Germany.  The  solution  is  said 
to  carry  off  the  sap  (timber  being  more  readily  treated  by  this  process  when 
green  than  when  seasoned),  while  the  copper  destroys  the  fungi,  and  the 
iron  forms  an  insoluble  compound  with  the  cellulose  or  woody  fibre.  It  is 
claimed  that  the  process  greatly  hardens  the  wood,  especially  the  softer 
varieties,  rendering  them  suitable  for  ties,  without  impairing  their  strength, 
elasticity  or  pliability,  (c)  The  Wellhouse  process  injects  first  a  solu- 
tion of  chloride  of  zinc  with  glue,  and  then  one  of  tannin  (both  under 
f pressure),  in  order  to  diminish  the  subsequent  washing  out  of  the  chloride, 
n  a  later  modification,  the  zinc,  glue  and  tannin  solutions  are  injected 
separately.     Several  millions  of  ties  have  been  treated  in  this  way.    The 

*  See  *'A  Proposed  Method  for  the  Preservation  of  Timber,"  by  F.  A. 
Kummer,  Transactions,  Am  Soc  C  E,  Vol  XLIV,  December,  1900. 
t  See  Transactions,  Am  Soc  C  E,  July,  Au«  and  Sept.  1885. 
X  Railroad  Gazette,  February  9.  1900. 


956  TIMBER. 

process  is  not  recommended  for  sub-aqueous  use.  (d)  Processes  in  which 
the  wood  is  treated  by  painting  or  soaking  *  are:  Carbolineum  Avenariua 
(Tar-oil,  chlorine,  etc),  Ligni  Salvor  (Tar-oil,  etc),  Woodiline  and  Spirit- 
tine  (chemical  solutions)  and  a  distillate  of  pine  used  by  the  Penns5/ Ivania 
Railroad  Co.  for  car  work,  (e)  Fence-posts  etc  seem  to  be  preserved,  to 
some  extent,  by  having  only  their  lower  ends  dipped  in  tar  well  boiled  to 
remove  the  ammonia,  which  last  is  destructive  to  wood.  The  upper  end 
must  be  left  untarred  to  let  the  sap  evaporate,  (f )  Attempts  at  wood  pre- 
servation by  means  of  vapor  of  creosote  etc  have  proved  failures, 
(g)  While  wood  remains  thoroughly  saturated  with  petroleum  it  does 
not  decay.  But  unless  the  supply  is  kept  up  the  oil  evaporates  and  leaves 
the  wood  unprotected,  (h)  Cottonwood  ties  laid  upon  a  soil  contain- 
ing about  2  per  cent  carbonate  of  lime,  1  per  cent  salt  and  0.5  per  cent 
each  of  potash  and  oxide  of  iron,  on  the  Union  Pacific  R.  R.  in  1868,  were 
found  in  1882  "as  sound  and  a  good  deal  harder  than  when  first  laid,*' 
although  such  ties  in  other  soils  lasted  but  from  2  to  5  years,  (i)  The  use 
of  solutions  of  lime  and  of  salt;  and  charring  the  surface;  are  sometime* 
found  useful  in  damp  situations. 

*  See  Report  by  O.  Chanute  to  the  American  Railway  Engineering  and 
Maintenance  of  Way  A£sociation,  March,  1901. 


STRENGTH   OF   MATERIALS. 


957 


Art.  4.    Ultimate  averag-e  tensile  or  cobesive  stren^^th  of 
Timber, 

Being  tke  least  weights  in  pounds  which,  if  attached  to  the  lower  end  of  a  vert  rod 
one  inch  square,  firmly  upheld  at  its  upper  end,  would  break  it  by  tearing  it  apart. 
For  large  timbers  we  recommend  to  reduce  these  constants  ^  to  3^  part. 


The  atrengthi  in  all  these  tables  may 
readily  be  one- third  part  more  or  leis 
than  our  averages. 


Alder 

Ash,  English 

"  American  (author)  abt.. 
Birch 

"    Amer'n  black 

Bay-tree 

Beech,  English 

Bamboo 

Box 

Cedar,  Bermuda 

"       Guadaloupe  

Chestnut 

"  horse 

Cyprus , 

Elder 

Elm 

"    Canada 

Fir,  or  Spruce 

Hawthorn 

Hazel 

Holly  

Hornbeam 

Hickory,  Amer'n 

Lignum  Vitae,  Amer'n 

Lancewood 

Larch,  Scotch 

Locust 

Maple... 


Lbs  per 
sq.  inch. 


14000 
16000 
16500 
15000 

7000 
12000 
11600 

6000 
20000 

7600 

9500 
13000 
10000 

6000 
10000^ 

600(f 
13000 
10000 
10000 
18000 
16000 
20000 
11000 
11000 
23000 

7000 
18000 
10000 


Mahogany,  Honduras 

"     Spanish.. 

Mangrove,  white,  Bermuda... 

Mulberry 

Oak,  Amer'n  white 

"        "         basket 

"        "         red 

"    Dantzic,  seasoned  .. 

"    Riga 

"    English 

"    live,  Amer'n 

Pear 

Pine,   Amer'n,  white,   red,  \ 

and  Pitch,  Memel,  Riga*,  j" 

Plane 

Plum 

Poplar 

Quince.. 

Spruce,  or  Fir 

Sycamore 

Teak 

Walnut 

Yew 


Across  tlie  grain.  Oak. 

"         "        "  Poplar 

"         "        "  Larch,900to 
"  "        "  Fir,  &  Pines. 


Lbs  per 
sq.  inch. 


8000 
16000 
10000 
12000 


10000 


10000 

10000 

11000 
11000 
7000 
7000 
10000 
12000 
16000 
8000 
8000 

2300 

1800 

1700 

550 


These  are  averages.  The  strengths  vary  much  with  the  age  of  the  tree ;  th« 
locality  of  its  growth ;  whether  the  piece  is  from  the  center,  or  from  the  outer  por- 
tions of  the  tree ;  the  degree  of  seasoning ;  straightnessof  grain ;  knots,  «fec,  &c.  Also, 
inasmuch  as  the  constants  are  deduced  from  experiments  with  good  specimens  of 
small  size,  whereas  large  beams  are  almost  invariably  more  or  less  defective  from 
knots,  crookedness  of  fibre,  Ac,  it  is  advisable  in  practice  to  reduce  these  constant! 
as  recommended  above. 


*  Eflfect  of  Tappings  Trees  for  Turpentine.  Preliminary  experi- 
ments by  the  Forestry  Division  of  the  U.  S.  Department  of  Agriculture  upon 
long-leaf  pine  from  Alabama  indicate  that  (contrary  to  the  generally  received 
impression)  "  turpentine  timber,"  i.  e.  the  timber  of  trees  that  have  been 
•'  boxed  "  (robbed  of  their  turpentine),  while  it  has  slightly  less  tensile  and 
shearing  strength,  is  from  20  to  30  per  cent,  stronger  in  compression  (whether 
with  or  across  the  grain)  and  under  transverse  strain.  In  the  "  turpentine  tim- 
ber," however,  the  resin  collects  in  spots,  gumming  the  tools,  and  thus  rendering 
the  timber  harder  to  work  than  that  of  trees  which  have  not  been  deprived  of 
their  turpentine.  The  specimens  tested  were  taken  mostly  at  heights  of  from  7 
to  33  feet  above  ground.  (Circular  No.  8.  Issued  1892.)  Boxed  and  unboxed 
timber  are  frequently  called  "  bled  "  and  "  unbled  "  respectively. 


958 


STRENGTH  OP  MATERIALS. 


Art.  1.     CompressiTe  strengtlis  of  American  woods,  whm 

tlovoly  and  carefully  seasoned.  Approximate  averages  deduced  from  many  experi- 
ments made  with  the  U  S  Govt  testing  machine  at  Watertown,  Mass,  by  Mr.  S.  P. 
Sharpies,  for  the  census  of  1880.  Seasoned  i¥Oodis  resist  crushing  much  better 
than  green  ones ;  in  many  cases,  twice  as  well.  This  must  be  allowed  for  when 
building  bridges,  &c,  of  timber  recently  cut.  Different  specimen?  of  the  same  wood 
vary  greatly ;  frequently  as  6  to  8,  9,  or  more. 


The  strengths  In  all  these 
tables  may  readily  vary  as 
much  as  one-third  part  more 
or  leas  than  our  average. 


End- 
I  wise.* 

lbs  per 
sqin. 


Ash,  red  and  white 

Aspen .■ 

Beech 

Birch 

Buckeye 

Butternut 

BuUonwood  (sycamore).. 
Cedar,  red 

"     white  (arbor  vitae) 
Catalpa  (Indian  bean)... 

Cherry,  wild 

Chestnut 

Coffee  tree,  Kentucky. 

Cypress,  bald 

Elm,  Am'n  or  white.. 

"    red 

Hemlock 

Hickory 

Jjignum  vitse 

Linden,  American 

Locust,  black  and  yellow 

"       honey 

Mahogany 


6800 
4400 
7000 
8000 
4400 
5400 
6000 
6000 
4400 
5000 
8000 
5300 
5200 
6000 
6800 
7700 
5300 
8000 
10000 
5000 
9800 
7000 
9000 


Side- 
wise. f 
lbs  per 
sq  in. 


1800 

800 
1100 
1300 

600 

700 
1300 

700 

500 

700 
1700 

900 
1300 

500 
1300 
1300 

600 
2000 
1600 

500 
1900 
1600 
1700 


3000 
1400 
1900 
2600 
1400 
1600 
2600 
1000 
900 
1300 
2600 
1600 
2600 
1200 
2600 
2600 
1100 
4000 
130001 
900 
4400 
2600! 
53001 


The  strengths  in  all  these 
tables  may  readily  vary  as 
much  as  one-third  part  more 
or  less  than  our  average. 


End- 
wise.* 
lbs  per 
sq  in. 


Maple,      broad  -  leafed, 

Oregon 

*      sugar  and  black. 

**      white  and  red 

OafcjWhite,  post  (or  iron) 
swamp  white,  red 

and  black 

"    scrub  and  basket... 
^"    chestnut  and  live... 

"    pin 

Pl'ne,  white 

•*      Red  or  Norway 
"      pitch    and  Jersey 

scrub 
"     Georgia 

Poplar 

Sassafras 

Spruce,  black 
**       white. 
Sycamore  (buttonwood).. 

Walnut,  black 

**      white     (butter- 
nut) 


Slde- 

wiBe.% 

lbs  per 

aqin. 


5300 
8000 
6800 


7000 

6000 
7500 
6500 
5400 
6300 

5000 

8500 
5000 
5000 
5700 
4500 
6000 
8000 

5400 


Willow 4400 


1400 
1900 
1300 


2600 
4300 
2900 


1600 
1700 
1600 
1300 
600 
600 

1000 
1300 
600 
1300 
700 
600 
1300 
1300 

700 
700 


4000 
4200 
4500 
3000 
1200 
1400 

2000 

2600 
1100 
2100 
1300 
1200 
2600 
2600 

1600 
1400 


Hence  it  appears  that  seasoned  white  and  yellow  pines,  spruce,  and  ordinary  oaks, 
which  are  the  woods  most  employed  in  the  United  States  for  bridges,  roofs,  etc.,  crush 
endwise  with  from  5000  to  7000  Bbs  per  sq  inch,  in  shovt  blochs^  average,  6000. 

But  it  is  well  to  bear  in  mind  that  in  practice  perfectly  equable  pressure  is  rarely 
secured.  In  a  few  trials  on  sidewise  compression,  with  fairly  seasoned  white  pine 
blocks,  6  ins  high,  5  ins  long,  and  2  ins  wide,  we  found  that  under  an  equally  dis- 
tributed pressure  of  5000  fts  total  or  500  !bs  per  sq  inch,  they  compressed  about  from 
l>^  to  3^  inch;  which  is  equal  to  from  ^  to  V^  inch  per  foot  of  height;  or  from  ^ 
to  5^  of  the  height;  the  mean  being  about  %  inch  to  a  foot,  or  ^  of  the  height. 
Under  10000  ibs  total,  or  1000  lbs  per  sq  inch,  they  split  badly ;  and  in  some  cases 
large  pieces  flew  off. 

The  tensile  or  cohesive  strengths  of  pine  and  oak  average  about  10000  Bbs  per  sq 
inch,  or  3>^  as  much  as  average  cast-iron,  or  nearly  double  their  resistance  to  crush- 
ing. The  tensile  strength  does  not  change  with  the  length  of  the  piece;  so  that  in 
practice  we  may  take  its  safe  strain  at  from  1000  to  2000  ft)s  per  sq  inch,  depending 
npon  the  character  of  the  structure,  &c.,  without  regard  to  the  length,  except  when 
this  is  so  great  that  two  or  more  pieces  have  to  be  spliced  together  to  make  it;  thus 
weakening  the  piece  very  much. 

*  Specimens  4  centimetres  (1.57  inch)  square,  32  centimetres  (12.6  ins)  long. 
When  the  length  exceeds  10  times  the  least  side,  see  Wooden  Pillars. 

^  Specimens  4  centimetres  (1.57  inch)  square,  16  centimetres  (6.3  ins)  long;  laid 
upon  platform  of  testing  machine.  Pressure  applied  at  their  mid-length,  by  means 
of  an  iron  punch  4  centimetres  square,  or  just  covering  the  entire  width  of  the 
specimen,  and  one-fourth  of  its  length.  The  first  column  (headed  *'.01 ")  gives  the 
loads  producing  an  indentation  of  .01  inch.  The  second  column  (headed  ".1")  gives 
those  producing  an  indentation  of  .1  inch. 


STRENGTH   OP   MATERIAI^. 


959 


Table  of  safe  quiescent  loads  ft>r  horizontal  rec- 
tangrular  beams  of  white  pine  or  spruce,  one  inch  broad, 
supported  at  both  ends,  and  loaded  at  the  center;  tog^ethev 
with  theii  deflections  under  said  loads. 

.    The  safe  load  is  here  one-sixth  of  the  breaking  load. 

For  the  neat  loads,  deduct  3^  the  wt  of  the  beam  itself.  The  deflections, 
however,  are  the  actual  ones;  the  wts  of  the  beams  having  been  introduced  in  cal- 
culating them,  by  the  rule  in  Art  27. 

liOads  applied  suddenly  will  double  the  deflections  in  the  table ;  aa 
when,  for  instance,  if  a  load  is  held  by  hand,  just  touching  a  beam,  the  hold  should  be 
suddenly  loosed. 

Caution.  Inasmuch  as  this  table  was  based  upon  well  seasoned,  straight 
grained  pieces,  free  from  knots,  and  other  defects,  we  must  not  in  practice  tak« 
more  than  about  two-thirds  of  the  loads  in  the  table  for  a  safety  of  6  in  ordinary 
building  timber  of  fair  quality ;  and  with  these  reduced  loads  should  not  reduce 
the  deflections. 

Observe  also  that  our  table  is  for  safe  center  loads,  but  it  is  plain  that 
in  practice  we  cannot  always  apply  the  term  in  its  utmost  strictness  ;  otherwise 
the  load  would  have  to  be  sustained  by  a  mere  knife-edge,  at  the  very  center  of 
the  beam.  Now,  in  the  instance  Rem.  p.  960f  if  we  attempted  to  sustain  the  center 
load  of  6075  ft)s  upon  such  a  knife-edge,  it  would  at  once  cut  the  beam  in  two.  If 
we  even  applied  it  along  3  or  4  ins  of  the  length,  it  would  cut  into  it,  and  we  should 
not  have  a  safety  of  6  against  crushing  the  top  of  the  beam  until  as  in  the  case  of 
the  ends  we  distributed  the  load  along  full  46  ins  of  length,  or  about  32  ins  for  a 
•afety  of  4. 

The  safe  load  is  here  V^  of  the  breakg  one;  and  the  last  at  450  lbs  at  the 

center  of  a  beam  1  inch  square,  and  1  foot  clear  length  between  its  fiujiports.  For 
mere  temporary  purposes,  3^  part  may  be  added  to  the  loads  in  the  table,  thus  mak- 
ing them  equal  to  the  }/^  of  the  breakg  load.  But  in  important  structures,  subject 
to  vibration,  %  P*rt  should  be  deducted  from  the  tabular  loads,  thus  reducing 
them  to  %  of  the  breaking  load.  This  is  especially  necessary  if  the  timber  is  not 
well  seasoned. 

With  the  safe  loads  In  this  table  a  beam  may  bend  too 
much  for  many  practical  purposes.  When  this  is  the  case,  we  may,  by  reducing 
the  loads,  reduce  the  deflections  in  nearly  the  same  proportion. 

All  the  loads  in  the  Table  are  superabundantly  safe  ag-atnst  shearing:. 
Ag-ainst  crushing:  at  the  ends,  &c,  see  '^  Cautions  "  below  the  Table. 

Original 


Bept? 

Span  4  ft. 

Span  6  ft.|Span  8  ft. 

SpanlOftt 

Spanl2ft| 

Span  lift! 

Wt!T? 

of 
beam. 

■ 

1 

J 

1 

10  ft  of 
beam. 

load 

def. 

load 

def.|load 

def. 

load 

def. 

load 

def. 

load 

def. 

load 

def. 

Ins. 

S)8. 

ins. 

fts. 

ins.|ft)8. 

ins. 

fts. 

ins. 

lbs. 

ins. 

S)S. 

ins. 

fibs. 

ins. 

ft)S. 

1 

19 

.39 

13 

.92 1  10 

1.8 

8 

3.0 

6 

4.4 



2 

2 

75 

.22 

50 

-.45 

38 

.82 

30 

1.3 

25 

1.9 

'"21 

2.7 

""19 

3.7" 

4 

3 

170 

.13 

114 

.30 

85 

.53 

67 

.84 

57 

1.3 

48 

1.7 

42 

2.3 

6 

4 

300 

.10 

200 

.22 

150 

.39 

120 

.63 

100 

.92 

86 

1.3 

75 

1.7 

8 

5 

469 

.08 

312 

.18 

234 

.31 

187 

.50 

156 

.72 

134 

1.0 

117 

1.3 

10 

6 

675 

.06 

450 

.15 

337 

.26 

270 

.41 

225 

.60 

193 

.83 

168 

1.1 

12 

7 

919 

.06 

612 

.12 

460 

.22 

367 

.35 

306 

.51 

262 

.70 

230 

.93 

14 

8 

1200 

.05 

800 

.11 

600 

.19 

480 

.31 

400 

.45 

343 

.61 

300 

.81 

16 

9 

1520 

.04 

1014 

.10 

760 

.17 

607 

.27 

507 

.40 

434 

.54 

380 

.72 

18 

10 

1875 

.04 

1250 

.09 

937 

.16 

750 

.24 

625 

.35 

536 

.49 

468 

.64 

20 

11 

2270 

.04 

1514 

.08 

1135 

.14 

907 

.22 

757 

.32 

648 

.44 

567 

.58 

22 

12 

2700 

.03 

1800 

.07 

1350 

.13 

1080 

.20 

900 

.29 

772 

.40 

675 

.53 

24 

14 

3675 

.03 

2450 

.06 

18.37 

.11 

1470 

.17 

1225 

.25 

1050 

.34 

918 

.45 

28 

16 

4800 

.02 

3200 

.05 

2400 

.10 

1920 

.15 

1600 

.22 

1372 

.30 

1200 

.40 

32 

18 

6075 

.02 

4050 

.05 

3037 

.09 

2430 

.14 

2025 

.20 

1736 

.27 

1518 

.35 

36 

20 

7500 

.02 

5000 

.04 

3750 

.08 

3000 

.12 

2500 

.18 

2145 

.24 

1875 

.31 

40 

22 

9075 

.02 

6050 

.04 

4537 

.07 

3630 

.11 

3025 

.16 

2593 

.22 

2268 

.29 

44 

24 

10800 

.02 

7200 

.04 

5400 

.06 

4320 

.10 

3600 

.15 

3088 

.20 

27(X) 

.26 

48 

(Continued  on  next  page.) 


960 


STRENGTH   OF   MATERIALS. 


Table. 

continued. 

(Original.) 

Depth 

Span  18  ft. 

Span  20  ft. 

Span  25  ft. 

Span  30  ft. 

Span  35  ft. 

Span  40  ft. 

Wt.  of 

of 

■\(\fi-r\f 

beam. 

load 

def. 

load 

def. 

load 

def. 

load 

def. 

load 

def 



load 

def 

iUIlOI 

beam. 

Ins. 

ft)S. 

ins. 

lbs. 

ins. 

ft)8. 

ins. 

S)s. 

ins. 

fl)8. 

ins. 

fts. 

ins. 

fts. 

6 

150 

1.4 

135 

1.8 

108 

2.9 

90 

4.5 

77 

6.5 

67 

9.2 

12 

7 

204 

1.2 

184 

1.5 

147 

2.5 

122 

3.9 

105 

5.8 

92 

7.6 

14 

8 

267 

1.0 

240 

1.3 

192 

2.1 

160 

3.2 

137 

4.6 

120 

6.4 

16 

9 

338 

.92 

304 

1.2 

243 

1.9 

202 

2.8 

174 

4.0 

152 

5.5 

18 

10  . 

417 

.82 

375 

1.0 

300 

1.7 

250 

2.5 

214 

3.5 

188 

4.9 

20 

11 

505 

.74 

454 

.93 

363 

1.5 

302 

2.2 

259 

3.2 

227 

4.3 

22 

12 

600 

.68 

540 

.85 

432 

1.4 

360 

2.0 

308 

2.9 

270 

3.9 

24 

14 

817 

.58 

735 

.72 

588 

1.2 

490 

1.7 

420 

2.4 

367 

3.2 

28 

16 

1067 

.50 

960 

.63 

768 

1.0 

640 

1.5 

548 

2.1 

480 

2.8 

32 

18 

1360 

.45 

1215 

.56 

972 

.90 

810 

1.3 

694 

1.8 

607 

2.5 

36 

20 

1666 

.40 

1500 

.50 

1200 

.79 

1000 

1.2 

857 

1.6 

750 

2.2 

40 

22 

2017 

.37 

1815 

.45 

1452 

.72 

1210 

1.1 

1037 

1.5 

907 

2.0 

44 

24 

2400 

.33 

2160 

.41 

1728 

.65 

1440 

.96 

1234 

1.3 

1080 

1.8 

48 

26 

2817 

.31 

2526 

.38 

2018 

.60 

1684 

.88 

1449 

1.2 

1263 

1.6 

62 

28 

3267 

.28 

2940 

.35 

2352 

.65 

1960 

.81 

1680 

1.1 

1470 

1.5 

56 

30   3760 

.26 

3375 

.33 

2700 

.50 

2250 

.76 

1928 

1.1 

1687 

1.4 

60 

32   4267 

.25 

;]840 

.30 

3072 

.45  2560 

.71 

2194 

1.0 

1920 

1.3 

64 

34   4817 

.23 

4335 

.29 

3468 

.44  2890 

.67 

2477 

.92 

2167 

1.2 

68 

36   MOO 

.22 

4860 

.27 

3888 

.43  3240 

.63 

2777 

.86 

2430 

1.1 

72 

White  oak,  and  best  Soutliern  pitch  pine  will  bear  load*  }>^ 
greater. 

For  cast  iron,  mult  the  loads  in  the  table  by  4.5 ;  and  for  wroug^ht  by 

6.3.    For  these  new  loads,  mult  the  dels  by  .4  for  cast;  and  by  .3  for  wrought. 

If  Ihe  load  is  equally  distributed  over  the  apan,  it  niay  be  twice  as 
great  as  the  center  one,  and  the  defs  will  be  1}^  times  those  in  the  table.  If  the 
loads  in  the  table  be  equally  distributed  along  the  whole  beam,  the  defs  will 
be  but  five-eighths  as  great  as  those  in  the  table.  When  more 

accuracy  is  reqd,  half  the  wt  of  the  beam  itself  must  be  deducted  from  the  center 
load;  and  the  whole  of  it  from  an  equally  distributed  load.  The  wt  of  the  beam,  in 
the  last  column,  supposes  the  wood  to  be  but  moderately  seasoned,  and  therefore  to 
weigh  28.8  lbs  per  cub  ft. 

Uses  of  the  foregoing:  table.  Ex.  1.  What  must  be  the  breadth 
<^f  a  hor  rect  beam  of  wh  pine,  lo  ins  deep,  supported  at  both  ends,  and  of  20  ft  elear 
length  between  its  supports,  to  bear  safely  a  load  of  6  tons,  or  11200  fbs  at  its  center  ? 
Here,  opposite  the  depth  of  IS  ins  in  the  table,  and  in  the  column  of  20  feet  lengths. 


the  reqd  breadth ;  for  the  strength  is  in  the  same  proportion  as  the  breadth. 

Ex.  2.  What  will  be  the  safe  load  at  the  center  of  a  joist  of  white  pine,  18  ft  long, 
3  ins  broad,  and  12  ins  deep?  Here,  in  the  col  for  18  ft,  and  opposite  12  ins  in  depth, 
we  find  the  safe  load  for  a  breadth  of  1  inch  to  be  600  fi)s ;  consequently,  600  X  3  = 
1800  ft)s,  the  load  reqd. 

Rem.  Cautions  in  the  use  of  the  above  table.  For  instance,  in 
placing  very  heavy  loads  upon  short,  but  deep  and  strong  beams,  we  must  take  care 
that  the  beams  rest  for  a  sufficient  dist  on  their  supports  to  prevent  all  danger  from 
crushing  at  the  ends.  Thus,  if  we  place  a  load  of  6075  Bbs  at  the  center  of  a  beam 
of  4  feet  spaa,  18  ins  deep,  and  only  1  inch  thick,  each  end  of  the  beam  sustains  » 

vert  crushing  force  of  —^  =  3037  ft)s,  and  that  sidewise  of  the  grrain,  in 

which  position  average  white  pine,  spruce,  and  hemlock  crush  under  about  800 
fcs  per  sq  inch,  and  do  not  have  a  safety  of  6  until  the  pressure  is  reduced  to  about 
133  ft)S  per  sq  inch.  Therefore  our  beam,  in  order  to  have  a  safety  of  6  against 
crushing  at  its  -ends,  must  rest  on  each  support  3037  -j-  133  =  23  sq  ins  ;  or  for  a 
safety  of  4  nearly  16  sq  ins.  When  a  pressure  is  equally  distributed  side- 
wise  (that  is,  at  right  angles  to  the  general  direction  of  the  fibres)  over  the  entire 
pressed  surface  of  a  block  or  beam  (to  ensure  which,  the  opposite  surface  must  be 
supported  throughout  its  entire  length)  the  resulting  compression  might  readily 
escape  detection  unless  actually  measured.  But  when  a  considerable  pressure  is 
applied  to  only  a  portion  of  the  surface,  as  of  caps  and  sills  where  in  contact  with 
the  heads  and  feet  of  posts,  or  at  the  ends  of  loaded  joists  or  girders,  the  com- 
pression becomes  evident  to  the  eye,  because  the  pressed  parts  sink  below  the 
unpressed  ones,  in  consequence  of  the  bending  or  breaking  ol"  the  adjacent  fibres. 
What  in  the  first  case  (especially  if  slight)  would  be  called  compression,  would 


STRENGTH    OF    MATERIALS.  961 

in  the  second  be  called  crushing^  ;  even  when  neither  might  be  so  great  as 
to  be  unsafe. 

Owing  to  the  resistance  which  said  adjacent  fibres  oppose  to  being  bent  or 
broken,  it  is  plain  that  a  given  pressure  per  sq  incli,  or  per  sq  foot,  &c., 
will  cause  somewhat  less  compression  or  crushing  when  applied  to  only  a  part  of 
a  surface,  than  when  to  the  whole  of  it. 

The  writer  lias  seen  40  half  seasoned  hemlock  posts,  each  12  ins  square, 
footing  at  intervals  of  5  ft  from  center  to  center,  upon  similar  12  X  12  inch  hem- 
lock sills,  to  which  they  were  tenoned,  and  which  rested  throughout  their  entire 
length  on  stone  steps.  Each  post  was  gradually  loaded  with  32  tons,  or  equal  to 
say  500  lbs  per  sq  inch ;  and  their  feet  all  crushed  into  the  sills  from  ^  to  ^  inch. 
Their  heads  crushed  into  the  caps  to  the  same  extent.  In  practice  the  pres- 
sure at  the  heads  and  feet  of  posts  is  rarely,  if  ever,  perfectly  equable ;  and  the 
same  remark  applies  to  the  ends  of  loaded  joists,  girders,  &c.,  in  which  a  slight 
bending  will  throw  an  excess  of  pressure  upon  the  inner  edges  of  their  supports 


til 


STRENGTH   OF   MATERIALS. 


Oft  iNCIS 


»ftc<ici5i-ic<5«ooooe«e«i 


SSS^SSI2S^^|ggg|||||2g|| 


S8SSSS?:SS?3^tgS 


S'*«ft«oo«i 


h^SSnoSw 


aso»>-Hifti-HQi-nfto5e<ir-->*<«Or-ioao>-ii-(i-<»o 


3SSSSSS||S|||| 


-  i'^SSSSgSSgsl 


"  f-i  e<5  >o  o>  eo  o»  to 


«-i  ^1^ 


•     "2a 

:-iil3 

®  S    ••o  5; 

SHoog  g   . 
"    N-a,  a.to 

5 II  •  S 


^•*«^«^^^o^»^r-^«^*^o^j.^£j«2-j2t^22l3S88Sg 


STRENGTH   OF   WOODEN   PILLARS.  963 

WOODEN  PILLARS. 

The  strengths  of  pillars,  as  well  as  of  beams  of  timber,  depend  much  on  their  de- 
giree  of  seasoning-.  Hodgkinson  found  that  perfectly  seasoned  blocks,  2  diama 
long,  required,  in  many  cases,  twice  as  great  a  load  to  crush  them  as  when  only 
moderately  dry.     This  should  be  borne  in  mind  when  building  with  green  timber. 

In  important  practice,  timber  should  not  be  trusted  with  more  than  %to%ofvta 
calculated  crushing  load ;  and  for  temporary  purposes,  not  more  than  3^  to  34. 

Mr.  diaries  Slialer  Smith,  €.  £.,  of  St.  liOnis,  prepared  tlie 
following  formula  for  the  breaking  loads  of  sither  square  or  rectangular 
pillars  or  posts,  of  moderately  seasoned  white,  and  common  yellow  pine,  with  flat 
ends,  firmly  fixed,  and  equally  loaded,  based  upon  experiments  by  himself. 

It  is  Gordon's  formula  adapted  to  those  woods;  and  gives  results  considerably 
smaller  than  Hodgkinson's,  It  is  therefore  safer. 

Call  either  side  of  the  square,  or  the  least  side  of  the  rectangle,  the  breadth.   Then, 

Breakg  load  in  ft)s,  per  ^        ^^^ 

Rule.      sq  inch  of  area,  of  a     V  ==  /sq  of  length  in  ins  ^  ^    ^- .  \ 

pillar  of  W  or  Y  pine    J        1  +  (sq  of  breadth  in  ins  ^  '^^V 
Or  in  words,  square  the  length  in  ins ;  square  the  breadth  in  ins ;  div  the  first  square 
by  the  second  one ;  mult  the  quot  by  .004 ;  to  the  prod  add  1 ;  div  6000  by  the  sum. 

Ex.  Breakg  load  per  sq  inch,  of  a  white  pine  pillar  12  ins  square,  and  30  ft,  or  360 
ins  long.    Here  the  sq  of  length  in  ins  is  3602  =  129600.    The  square  of  the  breadth  is 

,^      ...         ,129600  5000 

122  =  144 ;  and  -^^  =  900 ;  and  900  X  .004  =  3.6;  and  3.6  +  1  =  4.6.  Finally,  -j^- 

=  1087  S)8,  the  reqd  breakg  load  per  sq  in.  As  the  area  of  the  pillar  is  144  sq  ins, 
the  entire  breakg  load  is  1087  X  l-^^  =  156628  lbs,  or  69.9  tons. 

Recent  experiments  on  wooden  pillars  20  ft  long,  and  13  ins  square,  by  Mr. 
Kirkaldy,  of  England,  confirm  the  far  greater  reliability  of  Mr.  Smith's  formula. 
Hence  we  present  the  following  new  set  of  original  tables  based  upon  it. 

For  solid  pillars  of  east  iron  and  of  pine,  whose  heights  range 
from  5  to  60  times  their  side  or  diam,  we  may  say,  near  enough  for  practice,  that  a 
cast  iron  one  is  about  16^^  times  as  strong  as  a  pine  one ;  but  no  such  approximato 
ratio  holds  good  between  wrought  iron  and  pine,  or  between  cast  and  wrought  iron. 

t  The  breaking  load  in  lbs  per  sq  inch  io  short  blocks,  by  Mr.  Smith. 


964 


STRENGTH    OF    WOODEN    PILLARS. 


Table  of  breakings  loads  in  tons  of  square  pillars  of  balf 
seasoned  wbite  or  eommon  yellow  pine  firmly  fix:ed  and 
equally  loaded.     By  C.  Shaler  Smith's  formuha.     (Original.) 


Side  of  square  pine  pillar,  in  inches. 


1      (    13^    I   l>^    I    1%    I     2      I    2M    I    2J^    I    2M    I      3      I   3>i    I   3J4    I 


I 


BREAKING  LOAD. 


Tons. 
2.54 
2.22 
1.93 
1,66 
1.44 
1-24 
1.07 
.93 
.82 


Tons, 
3.99 
3.59 
3.19 
2.81 
2.48 
2.19 
1.93 
1.71 
1,52 
1.21 


Tons. 
5.73 
5.26 
4.80 
4.35 
3.92 
3.53 
3.16 
2.85 
2.55 
2.07 
1.70 
1.42 


Tons. 
7.80 
7.25 
6.74 
6.19 
5.66 
5.1T 
4.70 
4.29 
3.89 
3.23 
2.70 
2.28 
1.94 
1.67 
1.44 


Tons. 

12.8 

12.2 

11.6 

10.9 

10.2 

9.6 

8.9 

8.3 


1.5 

1.4 

1.3 

1.2 

1.1 

1.0 
.84 
.70 
.63 
.57 
.50 


Tons, 

15.7 

15.1 

14.5 

13.7 

12.9 

12.3 

11.5 

10,8 

10.0 

8.8 

7.7 


Tons. 

Tons. 

Tons. 

18.9 

22.3 

26.1 

18.3 

21.7 

25.4 

17.6 

21.0 

24.7 

16.8 

20.2 

23.9 

15,9 

19.3 

23.0 

15.2 

18.5 

22.0 

14.3 

17.6 

21.1 

13.5 

16.7 

20.1 

12.7 

15.8 

19.2 

11.3 

14.2 

17.4 

9,9 

12.7 

15,7 

8,8 

11.4 

14,1 

7.7 

10.0 

12.6 

6.8 

8.9 

11,3 

6.1 

8,0 

10.2 

5.4 

7.2 

9.2 

4.9 

6.5 

8,3 

4.4 

5,9 

7.6 

3.9 

5,2 

6.8 

3.5 

4.8 

6.2 

3.2 

4.3 

5.6 

2.9 

3.9 

5.1 

2.7 

3.6 

4.7 

2.5 

3.4 

4.4 

2.3 

3.1 

4,1 

2.1 

2.9 

3.8 

2.0 

2.7 

3.4 

1.7 

2.3 

3,1 

1.4 

2.0 

2.7 

1.2 

1.7 

2.4 

1.1 

1.5 

2,1 

1.0 

1.4 

1,9 

.92 

1.3 

1,7 

.76 

1.0 

1.4 

Tons. 
34.5 
33,7 
33,0 
32.1 
31.2 
30.1 
29.1 
28.0 
27.0 


II 

Side  of  square  pine  pillar,  in  incbes. 

u 

ns 

^H  1 

4«    1 

^H  1 

5      1 

5J^    1 

^H  \  5H  \    6    \  eyi  \  en 

1  6H 

1    ^ 

^yi 

ma 

BREAKING  LOAD. 

Tons. 

Tons. 

Tons. 

Tons, 

Tons. 

Tons. 

Tous, 

Tons. 

Tons, 

Tons. 

Tons. 

Tons, 

Tons. 

7 

35.8 

40,6 

45.7 

51.1 

56.8 

62.8 

69.0 

75.5 

82,3 

89.4 

96.8 

104,5 

112.4 

2 

3 

31.4 

36,1 

41.1 

46.2 

51.8 

57,7 

63.8 

70.2 

76.9 

84.0 

91,3 

98.9 

106.7 

3 

4 

268 

31,2 

35.9 

40.8 

46,1 

51.7 

57,6 

63.8 

70.4 

77.3 

84,5 

92.1 

99,9 

4 

5 

22,6 

26.5 

30.8 

35.4 

40,5 

45,8 

51,5 

57,4 

63.7 

70.3 

77.2 

84,5 

92,1 

5 

« 

18.9 

22.5 

26.4 

30.5 

35,2 

40.1 

45,4 

51.0 

57.0 

63.3 

69.9 

76,8 

84.1 

6 

7 

15.8 

19,0 

22.6 

26,2 

30,5 

35.0 

39,9- 

45.0 

50.5 

56,5 

62.8 

69,4 

76.3 

7 

8 

13.3 

16,1 

19,2 

22.5 

26.3 

30,4 

34,9 

39.7 

45.9 

50,4 

56,2 

62.4 

69,0 

8 

9 

11.3 

13,7 

16,5 

19,5 

22.9 

26.6 

30,7 

35.0 

39,9 

44.8 

50.2 

56.0 

62.1 

9 

10 

9.7 

11,8 

14,2 

16,9 

19.9 

23.2 

26,9 

30,9 

35.2 

39,9 

44.9 

50.3 

56,0 

10 

11 

8,3 

10,2 

12,4 

14.8 

17,5 

20,4 

23,8 

27,4 

31.3 

35,6 

40.2 

45,1 

50.4 

11 

12 

7,2 

8.8 

10,7 

12,9 

15.4 

18.0 

21.1 

24.3 

27,9 

31.8 

36.0 

40,6 

45,5 

12 

13 

6.2 

7.7 

9,4 

11,4 

13,6 

16.0 

18.8 

21,7 

24.9 

28,5 

32.4 

36.6 

41,1 

l^ 

14 

5.5 

6.8 

8,3 

10,1 

12.1 

14,2 

16.7 

19.4 

22,4 

25.7 

29.2 

33.1 

37,3 

14 

15 

4,8 

6.0 

7,4 

9,0 

10.8 

12,7 

15,0 

17.5 

20.2 

26.4 

30.0 

33,9 

15 

16 

4.4 

5.4 

6,7 

8,1 

9*8 

11,5 

13,6 

15,8 

18.3 

21.0 

24.0 

27,3 

30.8 

16 

17 

4,0 

4.9 

6,1 

7,3 

8,8 

10.4 

12,3 

14.3 

16.6 

19.1 

21.9 

24,9 

28,3 

17 

18 

3.6 

4.4 

5,5 

6.6 

8,0 

9,4 

11,2 

13,0 

15.1 

17.4 

19.9 

22.^ 

25.8 

18 

19 

3,3 

4.0 

5.0 

6,0 

7,3 

8.6 

10-2 

11.9 

13.8 

16,0 

18.3 

20.9 

23,7 

19 

20 

3.0 

3.7 

4,6 

5.5 

6,6 

7.8 

9.3 

10.9 

12.6 

14,6 

16,8 

19.2 

21,8 

20 

22 

2.5 

3,0 

3,8 

4.6 

5,6 

6.6 

7,9 

9.2 

10.7 

12,4 

14,3 

16,3 

18.6 

22 

24 

2,1 

2.6 

3.2 

3.9 

4,7 

5,6 

6.7 

7,9 

9.1 

10.6 

12.2 

14.1 

16.0 

24 

26 

1,8 

2.2 

2.8 

3.4 

4,1 

4.9 

5.8 

6,8 

7.9 

9,2 

10.6 

12.2 

13,9 

26 

28 

1.5 

1.9 

2.4 

2.9 

3.5 

4.2 

5,1 

5.9 

6.9 

8,0 

9.3 

10.7 

12.2 

28 

30 

1.3 

1,7 

2,1 

2.6 

3.1 

3.7 

4.4 

5,2 

6.1 

7,1 

8,2 

9.4 

10.8 

30 

32 

1.2 

1,5 

1,9 

2.3 

2.7 

3.2 

3.9 

4,6 

5.4 

6.3 

7.3 

8.4 

9.6 

32 

34 

1.1 

1,3 

1,7 

2.0 

2.4 

2.9 

3.5 

4,1 

4.8 

5.6 

6.5 

7.5 

8.6 

34 

36 

1,0 

1,2 

1.5 

1.8 

2.2 

2.6 

3.1 

3.7 

4.3 

5.0 

5.8 

6.7 

7,7 

36 

38 

.9 

1.1 

1.3 

1.6 

2.0 

2.4 

2.8 

3.3 

3.9 

4,5 

5,3 

6,1 

7,0 

38 

40 

,8 

1.0 

1,2 

:.5 

1.8 

2.1 

2.6 

3.0 

3.5 

4,1 

4,8 

5.6 

6.3 

40 

Continued  on  next  page. 


STRENGTH  OP  WOODEN  PILLAES. 


965 


Table  of  breaking^  loads  in  tons  of  square  pine  pillars,  witli ' 
flat  ends  firmly  fixed,  and  equally  loaded.    (Cuntiuued.) 

Original. 


u 

Side  of  square  pine  pillar,  in  inches. 

II 

Sa 

^}4 

1    Wi 

1     8 

8% 

83^ 

1    85i   1    9     1    9)i    1    9«    1    9Ji    1     10     1  lOM  1  10« 

w.a 

BREAKING  LOAD. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

TDm. 

Tons. 

Tons. 

Tons. 

Tons. 

« 

120.6 

129.1 

137.9 

147.0 

156.3 

165.9 

175.8 

I86  0 

196.4 

207.2 

218.2 

229.5 

241.0 

2 

4 

107.9 

116-3 

125.0 

133.9 

143.0 

152.6 

162.4 

172.5 

182.8 

193.4 

204.2 

215.3 

226.6 

4 

6 

91.7 

99.7 

108.0 

116.5 

125.3 

134.3 

143.5 

153.0 

162.8 

173.5 

184.4 

195.6 

207.0 

6 

8 

75.9 

83.2 

90.8 

98.7 

106.8 

115.5 

124.4 

133.6 

143.0 

153.0 

163.2 

173.7 

184.4 

a 

10 

62.0 

68.5 

75.3 

82.4 

89.7 

97.7 

105.8 

114.3 

123.0 

132.3 

141.8 

151.6 

161.6 

10 

1? 

50.7 

56.5 

62.5 

68.7 

75.1 

82.2 

89.6 

97.2 

105.0 

113.5 

122.2 

131.2 

140.5 

12 

14 

41.8 

46.7 

51.9 

57.3 

62.9 

69.2 

75.7 

82.4 

89.4 

97.1 

105.0 

113.2 

121.6 

14 

16 

S4.7 

38.9 

43.4 

48.1 

53.0 

58.5 

64.3 

70.3 

76.5 

83.3 

90.4 

97.7 

105.3 

16 

I« 

29.1 

32.7 

36.6 

40.7 

45.0 

49.8 

54.9 

60.1 

65.6 

71.7 

78.0 

84.6 

91.4 

18 

'M) 

24.6 

27.7 

31.1 

34.7 

38.5 

42.7 

47.2 

51.8 

56.7 

62.0 

67.6 

73.5 

79.6 

20 

'23 

19.6 

22.1 

24.8 

27.7 

30.9 

34.4 

38.0 

41.9 

46.0 

50.5 

55.2 

60.2 

65.4 

23 

?fi 

15.8 

17.8 

20.1 

22.5 

25.2 

28.1 

31.1 

34.4 

37.9 

41.6 

45.6 

49.8 

54.3 

26 

n 

13.1 

14.8 

16.7 

18.7 

20.9 

23.4 

25.9 

28.6 

31.6 

34.9 

38.2 

41.9 

45.6 

29 

32 

10.9 

12.3 

13.9 

15.7 

17.6 

19.7 

21.8 

24.2 

26.7 

29.5 

32.3 

35.5 

38.7 

32 

35 

9.3 

10.6 

11.9 

13.4 

15.0 

16.8 

18.7 

20.7 

22.8 

25.2 

27.7 

30.5 

33.3 

35 

38 

8.0 

9.1 

10.2 

11.5 

12.9 

14.6 

16.3 

18.0 

19.7 

21.8 

23.9 

26.3 

28.8 

38 

41 

6.9 

7.9 

8.9 

10.0 

11.2 

12.6 

14.1 

15.6 

17.2 

19.1 

21.0 

23.1 

25.2 

41 

44 

6.0 

6.9 

7.8 

8.8 

9.8 

11.0 

12.3 

13.6 

15.0 

16.7 

18.4 

20.2 

22.1 

44 

60 

4.7 

5.4 

6.1 

6.9 

7.7 

8.8 

10.0 

11.3 

12.6 

13.7 

14.9 

16.2 

17.5 

50 

5« 

Side  of  square  pine  pillar,  in  incbes 

. 

%i 

ma 

lOH  1    11     1   llJi    1   11^   1   ll?i   1     12     1 

1  lOH 

11 

llM  1  UK  1  llH 

1    12 

fi.s 

BREAKI] 

SQ  LOAD. 

BREAKING  LO 

AD. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

4 

238.9 

251.0 

263.2 

275.9 

288.8 

302.1 

26.5 

28.6 

31.1 

33.8 

36.7 

39.9 

42 

6 

218.8 

230.6 

242.7 

255.2 

267.9 

281.0 

24.2 

26.4 

28.7 

31.2 

33.9 

36.8 

44 

8 

195.5 

207.0 

218.5 

230.4 

242.5 

255.1 

22.4 

24.3 

26.4 

28.7 

31.2 

33.9 

44i 

10 

172.2 

183.0 

194.1 

205.6 

217.3 

229.5 

20.7 

22.6 

24.5 

26.7 

28.9 

31.4 

m 

12 

150.3 

160.3 

170.7 

181.5 

192.5 

204.0 

19.2 

20.9 

22.7 

24.7 

26.8 

29.2 

no 

14 

130.5 

139.8 

149.4 

159.4 

169.6 

180.2 

17.8 

19.5 

21.1 

23.0 

25.0 

27.2 

52 

16 

113.2 

121.7 

130.4 

139.6 

149.0 

158.8 

16.6 

18.2 

19.7 

21.5 

23.3 

25.4 

54 

18 

98.7 

106.2 

114.1 

122.5 

131.0 

140.0 

15.5 

17.0 

18.4 

20.1 

21.8 

23.7 

56 

20 

86.2 

92.9 

100.0 

107.6 

115.4 

123.6 

14.5 

15.9 

17.2 

18.8 

20.4 

22.2 

58 

?2 

75.6 

81.7 

88.1 

95.0 

102.0 

109.5 

13.6 

14.9 

16.2 

17.7 

19.2 

20.9 

60 

24 

66.7 

72.2 

77.9 

84.1 

90.5 

97.3 

11.8 

12.9 

13.9 

15.2 

16.5 

18.0 

65 

26 

59.1 

64.0 

69.2 

74.9 

80.6 

86.8 

10.1 

11.1 

12.0 

13.2 

14.3 

15.6 

70 

28 

52.6 

57.1 

61.8 

66.9 

72,1 

77.7 

8.9 

9.8 

10.6 

11.6 

12.5 

13.7 

75 

30 

47.0 

51.1 

.55.3 

60.0 

64.7 

69.9 

7.8 

8.6 

9.4 

10.2 

11.1 

12.1 

HO 

32 

42.1 

46.0 

49.9 

54.0 

58.4 

63.0 

7.0 

7.7 

8.4 

9.1 

9.9 

10.7 

85 

34 

38.2 

41.5 

45.0 

48.8 

52.8 

57.1 

6.2 

6.8 

7.4 

8.1 

8.8 

9.6 

90 

36 

34.6 

37.7 

40.9 

44.3 

48.0 

51.8 

5.6 

6.1 

6.7 

7.3 

8.0 

8.7 

95 

.SH 

31.5 

.34.2 

37.2 

40.4 

43.9 

47.4 

5.1 

5.6 

6.1 

6.6 

7.2 

7.8 

100 

40 

38.8 

31.3 

34.0 

37.0 

40.1 

43.5 

3.5 

3.9 

4.2 

4.6 

5.0 

5.5 

120 

Continued  on  next  page. 
Remarks.    Mr  Kirkaldy  found  for  Ri^a  and  I>antzic  firOi 

20  ft  long,  and  13  ins  square,  (or  18^/^  sides  high,)  148  and  138  tons  total ;  or  .876 
and  .817  tons,  (1963  and  1829  lbs,)  per  sq  inch.  Mr  Smith's  rule  gives  for  common 
pine,  160  tons  total ;  or  .947  ton,  or  2121  lbs,  per  sq  inch.  Hodgkinson  would  give 
for  Riga  about  297  tons  total. 

Each  of  Mr  Kirkaldy's  20-ft  pillars  shortened  about  .6  of  an  inch  total;  or  .03 
inch  per  ft;  or  Vs. of  an  inch  in  4  ft  2  ins,  under  a  mean  of  1900  lbs  per  sq  inch. 

The  writer  has  Known  8  unbraced  pillars  of  hemlock,  tolerably  seasoned, 
12  ins  square,  and  42  ft  high,  to  be  gradually  loaded  each  with  32  tons,  or  71680 
lbs  total ;  (or  .2222  ton,  or  498  lbs  per  sq  inch)  without  appreciable  yielding.  As- 
suming their  strength  and  stiffness  to  be  about  as  for  Mr  Smith's  pine,  (as  in  all 
our  tables,)they  should  by  him  yield  at  39.9  tons  total.  With  these  same  data^ 
but  with  Hodgkinson's  formula,  they  should  yield  at  69.3  tons;  and  with 
Hodekinson's  own  data,  for  seasoned  red  deal,  at  91.6  tons. 


966 


STRENGTH    OF   WOODEN   PILLARS. 


.  Table  of  breaking^  loads  in  tons  of  square  pillars  of  half- 
seasoned  white  or  common  yellow  pine,  with  flat  ends 
0lrmly  fixed,  and  equally  loaded.  By  C.  Shaler  Smith's  formula. 
(Continued.) 

As  this  table  was  partly  made  by  interpolation,  the  last  figure  is  not  always  pre^ 
cisely  correct. 

Original. 


'11 

Side  of  square  pine  pillar  in  inches. 

W.S 

13  1  14  1  15  1  16  1  17  1  18  1  19 

20  1 

21  1  22  I  23  1  24 

K . 

BREAKING  LOAD. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

4 

358 

418 

482 

552 

625 

703 

786 

872 

964 

1060 

1161 

1265 

4 

6 

335 

394 

456 

526 

599 

676 

760 

847 

938 

1033 

1134 

1236 

6 

8 

308 

367 

429 

500 

572 

649 

732 

818 

910 

1005 

1106 

1208 

8 

10 

281 

339 

400 

466 

537 

612 

694 

780 

870 

964 

1064 

1166 

10 

12 

252 

307 

365 

432 

502 

576 

656 

740 

829 

922 

1022 

1124 

12 

14 

225 

277 

333 

397 

464 

536 

614 

696 

784 

876 

973 

1074 

14 

16 

201 

250 

303 

363 

428 

497 

573 

652 

739 

829 

926 

1024 

16 

18 

179 

224 

274 

331 

392 

458 

531 

608 

692 

780 

873 

972 

18 

20 

160 

201 

248. 

301 

359 

422 

492 

566 

647 

732 

822 

919 

20 

22  ■ 

143 

182 

224 

274 

329 

388 

455 

526 

604 

686 

773 

866 

22 

24 

127 

163 

203 

249 

301 

357 

421 

488 

563 

642 

726 

816 

24 

26 

115 

148 

184 

226 

275 

328 

389 

453 

523 

599 

680 

767 

26 

28 

103 

133 

167 

206 

252 

302 

359 

420 

490 

560 

638 

721 

28 

30 

93 

121 

152 

189 

231 

278 

332 

389 

453 

522 

597 

677 

30 

32 

84 

109 

138 

173 

212 

256 

307 

361 

421 

487 

558 

635 

32 

34 

76 

99 

126 

159 

196 

237 

284 

335 

392 

455 

523 

597 

34 

36 

69 

91 

116 

146 

180 

219 

264 

312 

366 

426 

490 

560 

36 

38 

63 

84 

107 

134 

166 

203 

245 

290 

341 

397 

458 

525 

38 

40 

58 

77 

99 

124 

154 

188 

227 

270 

318 

372 

429 

494 

40 

42 

54 

71 

91 

115 

143 

175 

212 

253 

298 

349 

403 

465 

42 

44 

50 

66 

84 

107 

133 

163 

198 

236 

280 

328 

380 

438 

44 

46 

46 

61 

78 

99 

123 

152 

185 

221 

263 

308 

358 

413 

46 

48 

43 

57 

73 

92 

115 

142 

173 

207 

247 

290 

337 

389 

48 

50 

40 

53 

68 

86 

107 

133 

162 

194 

231 

272 

317 

367 

50 

62 

37 

50 

64 

81 

101 

124 

152 

182 

217 

256 

300 

347 

52 

54 

35 

47 

60 

76 

95 

117 

144 

172 

205 

242 

283 

328 

54 

56 

33 

44 

56 

71 

89 

110 

135 

162 

193 

228 

267 

310 

56 

58 

31 

41 

52 

67 

84 

103 

127 

153 

182 

215 

253 

294 

bS 

60 

29 

38 

49 

63 

79 

98 

120 

144 

172 

204 

240 

280 

60 

65 

25 

33 

43 

55 

69 

86 

105 

126 

151 

179 

211 

246 

65 

70 

22 

29 

37 

48 

60 

74 

92 

111 

134 

159 

187 

218 

70 

75 

19 

25 

33 

42 

53 

66 

82 

98 

118 

141 

166 

195 

75 

80 

16 

22 

29 

37 

46 

58 

72 

87 

105 

125 

148 

174 

80 

85 

14 

19 

26 

33 

41 

52 

6.5 

78 

94 

112 

132 

156 

85 

90 

13 

17 

23 

30 

37 

46 

58 

70 

85 

102 

120 

141 

90 

95 

12 

16 

21 

27 

33 

42 

53 

64 

77 

93 

108 

127 

95 

100 

11 

14 

19 

24 

30 

38 

48 

58 

70 

84 

99 

117 

100 

110 

10 

12 

16 

20 

26 

33 

40 

48 

58 

70 

82 

97 

110 

120 

9 

11 

14 

17 

22 

28 

34 

41 

49 

60 

71 

83 

120 

130 

7 

9 

12 

14 

18 

23 

29 

36 

43 

52 

61 

72 

130 

140 

6 

8 

10 

12 

16 

20 

25 

31 

37 

44 

53 

62 

140 

150 

5 

7 

9 

11 

14 

18 

22 

27 

32 

38 

46 

54 

150 

160 

5 

6 

8 

10 

13 

16 

20 

24 

29 

34 

41 

48 

160 

170 

4 

5 

7 

9 

11 

14 

17 

21 

25 

30 

36 

43 

170 

180 

4 

5 

6 

8 

10 

12 

15 

19 

22 

27 

32 

88 

180 

190 

3 

4 

5 

7 

9 

11 

14 

17 

20 

24 

29 

34 

190 

200 

3 

4 

5 

6 

8 

10 

12 

15 

18 

22 

26 

31 

200 

STRENGTH    OF    WOODEN    PILLARS. 


967 


Breaking:  loads  of  lialf  seasoned  square  pine  pillars. 


tl 

BB  LOAD  PER  SQ  IN. 

'53  "Sc 

BK  LOAD  PKE  SQ  IN. 

II 

BB  LD  PER  8Q  IN. 

BR  LD  PEB 

SQ  IN. 

W.S 

W.H 

K  fl 

^  a 

Tons. 

Lbs. 

Tons. 

Lbs. 

Tons. 

Lbs. 

Tons. 

Lbs. 

1 

2.2232 

4980 

26 

.6027 

1350 

51 

.1960 

439- 

76 

.0924 

207 

2 

2.1969 

4921 

27 

.5697 

1276 

52 

.1888 

423 

77 

.0902 

202 

3 

2.1544 

4826 

28 

.5398 

1209 

53 

.1826 

409 

78 

.0879 

197 

4 

2.097S 

4699 

29 

.5116 

1146 

64 

.1763 

395 

/9 

.0862 

193 

5 

2.0290 

4545 

30 

.4853 

1087 

55 

.1705 

382 

80 

.0839 

188 

6 

1.9513 

4371 

31 

.4607 

1032 

56 

.1647 

369 

81 

.0821 

184 

7 

1.8665 

4181 

32 

.4379 

981 

67 

.1598 

358 

82 

.0799 

179 

8 

1.7772 

3981 

33 

.4165 

933 

58 

.1545 

346 

83 

.0781 

175 

9 

1.6857 

3776 

34 

.3969 

889 

69 

.1496 

335 

84 

.0763 

171 

10 

1..5942 

3571 

35 

.3781 

847 

60 

.1451 

325 

85 

.0746 

167 

11 

1.5040 

3369 

36 

.3599 

809 

61 

.1406 

315 

86 

.0728 

163 

12 

1.4165 

3173 

37 

.3447 

772 

62 

.1362 

305 

87 

.0714 

160 

13 

1.3317 

2983 

38 

.3295 

738 

63 

.1321 

296 

88 

.0696 

156 

14 

1.2513 

2803 

39 

.3152 

706 

64 

.1286 

288 

89 

.0683 

153 

16 

1.1745 

2631 

40 

.3018 

676 

66 

.1260 

280 

90 

.0670 

150 

16 

1.1027 

2470 

41 

.2889 

647 

66 

.1210 

271 

91 

.0656 

147 

17 

1.0353 

2319 

42 

.2772 

621 

67 

.1179 

264 

92 

.0638 

143 

18 

.9723 

2178 

43 

.2661 

596 

68 

.1147 

257 

93 

.0625 

140 

19 

.9134 

2046 

44 

.2554 

672 

69 

.1112 

249 

94 

.0616 

138 

20 

.8585 

1923 

45 

.2455 

650 

70 

.1085 

243 

96 

.0603 

135 

21 

.8076 

1809 

46 

.2357 

628 

71 

.1054 

236 

96 

.0589 

132 

22 

.7603 

1703 

47 

.2268 

608 

72 

.1027 

230 

97 

.0576 

129 

23 

.7165 

1605 

48 

.2183 

489 

73 

.1000 

224 

98 

.0567 

127 

24 

.6755 

1513 

49 

.2100 

472 

74 

.0973 

218 

99 

.0554 

124 

25 

.6380 

1429 

60 

.2031 

455 

75 

.0951 

213 

100 

.0545 

122 

968 


PLASTERING. 

PLASTEEIN6. 


The  plastering  of  the  inside  walls  ot  buildings,  whether  done  on  laths,  bricks,  or 
stone,  generally  consists  of  three  separate  coats  of  mortar.  The  first  of  these  is  called 
by  workmen  the  rough  or  scratch  coat;  and  consists  of  about  1  measure  of  quicklime, 
to  4  of  sand  ;  (which  latter  need  not  be  of  the  purest  kind ;)  and  3^  measure  of  bul- 
lock or  horse  hair;  the  last  of  which  is  for  making  the  mortar  more  cohesive,  and 
less  liable  to  split  off  in  spots.  This  coat  is  about  %  to  3^  inch  thick ;  is  put  on 
roughly ;  and  should  be  pressed  by  the  trowel  with  sufficient  force  to  enter  perfectly 
between  and  behind  the  laths;  which  for  facilitating  this  should  not  be  nailed 
nearer  together  than  }/^  an  inch.  In  rude  buildings,  or  in  cellars,  Ac,  this  is  often 
the  only  coat  used.  When  this  first  coat  has  been  left  for  one  or  more  days,  accord- 
ing to  the  dryness  of  the  air,  to  dry  slightly,  it  is  roughly  scored,  or  scratched,  (hence 
its  name,)  with  a  pointed  stick,  or  a  lath,  nearly  through  its  thickness,  by  lines  run- 
ning diagonally  across  each  other,  and  about  2  to  4  ins  apart.  This  gives  a  better 
hold  to  the  second  coat,  which  might  otherwise  peel  off.  If  the  first  coat  has  be- 
come too  dry,  it  is  well  also  to  dampen  it  slightly  as  the  second  one  is  put  on. 

The  second  coat  is  put  on  about  J^  to  %  inch  thick,  of  the  same  hair  mortar,  oi 
coarse  stuff.  Before  it  becomes  hard,  it  is  roughed  over  by  a  hickory  broom,  oi 
some  substitute,  to  make  the  third  coat  adhere  to  it  better. 

The  third  coat,  about  3^  inch  thick,  contains  no  hair;  and  forgiving  it  a  still 
whiter  and  neater  appearance,  more  lime  is  used,  say  1  of  lime,  to  2  of  sand ;  and 
the  purest  sand  is  used.  This  mortar  is  by  plasterers  called  stucco ;  a  nama 
also  applied  to  mortar  when  used  for  plastering  the  outsides  of  buildings.  Or  in- 
stead of  stucco,  the  third  coat  may  be,  and  usually  is,  of  hard  finish,  or  gauge  stuff; 
which  consists  of  1  measure  of  ground  plaster  of  Paris,  to  about  2  of  quicklime, 
without  sand.  Hard  finish  works  easier;  but  is  not  as  good  as  stucco,  for  walls  in- 
tended to  be  painted  in  oil.    The  plaster  of  Paris  is  for  hastening  the  hardening. 

Either  of  these  third  coats  is  smoothed  or  polished  to  a  greater  or  less  extent,  according  to  whether 
it  is  to  show,  or  to  be  papered,  painted,  &c.  The  polishing  tools  are  merely,  the  trowel ;  the  hand- 
loat,  (a  kind  of  wooden  trowel ;)  and  the  water-brush,  (a  short-handled  brush  for  wetting  the  surface 
part  at  a  time  with  water,  in  order  to  polish  more  freely.)  For  finer  polishing,  a  float  made  of  cork 
ia  used.  The  smooth  piece  of  board  about  10  to  12  ins  square,  with  a  handle  beneath,  on  which  the 
plasterer  holds  his  mortar  until  he  puts  it  on  to  the  wall  with  his  trowel,  is  called  a  hawk. 

The  more  thoroughly  each  coat  is  gone  over  with  the  water-brush  and  trowel,  (which  process  la 
called  hand-floating,)  the  firmer  and  stronger  will  it  be.  Frequently  only  two  coats  of  plastering  are 
put  on  in  inferior  rooms  ;  or  where  great  neatness  of  appearance  is  not  needed.  The  first  is  of  hair 
mortar,  or  coarse  stuff;  this  is  scraMhed  with  the  broom,  and  then  covered  by  the  finishing  coat  of 
finer  mortar,  cstucco.)  If  this  last  is  nearly  all  lime,  or  with  but  very  little  sand,  to  make  it  worli 
easier,  it  is  called  a  slipped  coat.  Without  any  sand  it  is  called  fine  stuff.  Neither  is  as  good  as 
stucco,  if  the  wall  is  to  be  papered.  When  this  is  the  case,  the  third  coat  also  may  have  a  little  hair, 
to  give  it  more  strength  ;  but  this  is  not  absolutely  necessary. 

A  very  good  eflfect  may  be  produced  in  station-houses,  churches,  &c,  by  only  two  coats  of  plaster  1b 
which  fine  clean  screened  gravel  is  used  instead  of  sand.  When  lined  into  regular  courses,  it  resem- 
bles a  bufiF-colored  sandstone,  very  agreeable  to  the  eye. 

In  purchasing  plastering  hair,  care  must  be  taken  that  it  has  not  been  taken  from  salted  hides-, 
inasmuch  as  the  salt  will  make  the  walls  damp.  For  the  same  cause  sea-shore  sand  should  not  be 
used.    It  is  almost  impossible  to  wash  it  entirely  free  from  salt. 

In  brick  walls  intended  to  be  plastered,  the  mortar  joints  should  be  left  very  rough,  to  let  the  plas- 
ter adhere.  If  it  is  put  on  smooth  walls,  without  first  raking  out  the  mytar  to  the  depth  of  nearly 
an  inch,  it  is  very  apt  to  fall  off ;  especially  from  outside  walls ;  as  can  be  seen  daily  in  any  of  our 
cities.  As  this  raking  out  of  brick  joints  is  tedious  and  expensive,  it  would  generally  be  better  to 
use  paint  rather  than  plaster.  The  walls  should  also  be  washed  clean  from  all  dust ;  and  should  b* 
slightly  dampened  as  the  plaster  is  put  on. 

To  imitate  granite  on  outer  walls :  after  the  second  or  smooth  coat  of  plaster  is  dry,  it  receives  a 
Doat  of  lime  wash,  slightly  tinted  by  a  little  umber,  or  ochre,  &c.  After  this  is  dry,  in  case  it  appears 
too  dark,  or  too  light,  another  may  i)e  applied  with  more  or  less  of  the  coloring  matter  in  it.  Finally, 
a  wash  of  lime  and  mineral-black  is  sprinkled  on  from  a  flat  brush,  to  imitate  the  black  specks  of 
granite.  By  this  simple  means,  a  skilful  workman  can  produce  excellent  imitations.  The  horizontal 
and  vertical  joints  of  the  imitation  masonry,  may  be  ruled  in  by  a  small  brush,  using  the  same  black 
wash,  and  a  long  straight-edge. 

The  rough  surfaces  of  all  walls  are  more  or  less  warped,  or  out  of  line ;  and  it  is  not  possible  for 
the  plasterer  to  rectify  this  perfectly  by  eye,  as  may  be  seen  in  almost  every  house.  Even  in  what 
are  called  first-class  ones,  a  quick  eye  can  generally  detect  unsightly  undulations  of  the  plastered 
surfaces. 

To  prevent  this,  the  process  of  screeding- is  resorted  io.    Screeds  are  a  kind  of 

gauge  or  guide,  formed  by  applying  to  the  first  rough  coat,  when  partly  dried,  horizontal  ^strips  of  the 
plastering  mortar,  about  8  ins  wide,  and  from  2  to  4  ft  apart  all  around  the  room-  These  are  made  t« 
project  from  the  first  coat,  out  to  the  intended  face  of  the  second  one  ;  and  while  soft  are  carefully 
made  perfectly  straight,  and  out  of  wind  with  each  other,  by  means  of  the  plumb-line,  straight-edge, 
&c.  When  they  become  dry,  the  second  coat  is  put  on,  filling  up  the  broad  horizontal  spaces  between 
them ;  and  is  readily  brought  to  a  perfectly  flat  surface,  corresponding  with  that  of  the  screeds,  by 
means  of  long  straight-edges  extending  over  two  or  more  of  the  latter. 

A  day's  work  at  plasteringTo 

A  plasterer,  aided  by  one  or  two  laborers  to  mix  his  mortar,  and  to  keep  his  hawk  supplied,  can 
average  from  100  to  200  scyiare  yards  a  daji  of  first  coat ;  about  %  as  much  of  second;  and  hoif  a« 


SLATING. 


969 


toneh  of  third,  which  requires  more  care.    The  amount  will  depend  upon  the  number  of  angles,  size 
•f  rooms,  whether  on  ceilings  or  on  wsms,  &c,  &c. 

Oen  Oillmore's  estimate  of  cost  of  plastering-    100  square  yarda 
,  with  2  or  with  3  coats.    Common  labor  $1  per  day. 


Materials. 

Three  Coats. 
Hard  finished  work. 

Two  Coats. 
Slipped  coat  finish. 

Quicklime 

4  casks. 

1  " 

2000 

4  bushels. 
7  loads. 
2^  bushels. 
13  lbs. 
4  days. 
3  days. 

$4.00 
.85 
.70 
4.00 
.80 
2.00 
.25 
.90 
7.00 
3.00 
2.00 

3K  casks. 

2000. 

3  bushels. 

6  loads. 

13  lbs. 
3H  days. 
2  days. 

$3.33 

Plaster  of  Paris 

Laths 

Hair. 

4.00 
.60 
1.80 

White  Sand 

Nails 

.90 
6  12 

2.00 

Cartage 

1.20 

$25.50 

$19.95 

This  amounts  to  25H  eta  per  sq  yd  for  3  coats ;  and  say  20  cts  for  2  coats. 

Plastering*  lattis  are  usually  of  split  white  or  yellow  pine,  in  lengths  of 
about  3  to  4  feet ;  and  hence  called  3  or  4  ft  laths.  They  are  about  114  ins  wide,  by  %  inch  thick. 
They  are  nailed  up  horizontally,  about  }4  inch  apart.  The  upright  studs  of  partitions  are  spaced  at 
such  distances  apart,  (generally  about  15  ins  from  center  to  center,)  that  the  ends  of  the  laths  may 
be  nailed  to  them.  Laths  are  sold  by  the  bundle  of  1000  each.  A  square  foot  of  surface  requires  134 
four  feet  laths  ,•  or  1000  such  laths  will  cover  666  sq  ft.  Sawed  laths  may  be  had  to  order,  of  any  re- 
quired length.  A  carpenter  can  nail  up  the  laths  for  from  40  to  60  sq  yds  of  plastering  in  a  day  oS 
10  hours  J  depending  on  the  number  of  angles  in  the  rooms,  &c. 


SLATING. 


Roofing  slates  are  usually  from  3^  to  i^  inch  thick;  about  y\  being  a  coramom 
average.  They  may  be  nailed  either  to  a  sheeting  of  rough  boards  (c,  g,  in  the  fig) 
from  %  to  1^  inch  thick,  (which  tshould  be,  but  rarely  are,  tongued  and  grooved,) 


\4rv-^ 

\              -J 

1 

i       1 

1 

1       1 

1 

1       1 

970 


SLATING. 


laid  horizontally  from  rafter  to  rafter  ;  or  sloping,  from  purlin  to  purlin  as  the 
case  may  be;  or  to  stout  laths  t  1 1  about  2  to  3  ins  wide,  and  from  1  to  1% 
thick,  nailed  to  the  rafters  at  distances  apart  to  suit  the  gauges  of  the  slates. 
Two  nails  are  used  to  each  slate ;  one  near  each  upper  corner.  They  may  be  either 
of  copper,  (which  is  the  most  durable,  but  most  expensive,)  of  zinc,  or  of  either 
galvanized  or  tinned  iron.  The  last  two  are  generally  used ;  or  in  inferior  work, 
merely  plain  irou  ones,  previously  boiled  in  liuseed  oil,  as  a  partial  preserva- 
tive from  rust.  Rust,  however,  sometimes  weakens  them  so  much  that  they 
break;  and  the  slates  are  blown  off  in  high  winds,  to  the  danger  of  passers  by. 
Since  good  slate  endures  for  a  long  series  of  years,  it  is  true  economy  to  use 
nails  that  are  equally  durable.  In  irou  roofs,  the  slates,  instead  of  being  nailed 
to  boards,  are  sometimes  tied  directly  to  the  iron  purlins,  by  wire.  A  square  of 
slating,  shingling,  &c,  is  100  sq  ft. 
In  laboratories,  chemical  factories,  &c,  subject  to  acici  fumes,  it  is  difficult  to 

provide  a  metal  fastening  that  will  not  be  eaten  away.  In  such  cases  it  is  best  to  depend  chiefly  upon 
alayerof  mortar  between  the  slates.  This  will  harden  before  the  metal  fastenings  give  way  ;  and 
will  hold  the  slates  in  place,  while  new  fastenings  are  being  inserted. 

The  least  pitcll  considered  advisable  for  a  roof,  to  prevent  rain  or  snow  from  being  driven 
through  the  interstices  between  the  slates,  is  about  263^°  ;  or  1  vert  to  2  hor ;  which  corresponds  to 
a  rise  of  J4  the  span  in  a  common  double  pitched  roof.  But  even  at  steeperpitches,  rain,  and  more 
particularly  snow,  will  be  forced  through  the  roof  by  violent  winds;  especially  if  laths  alone  be  used, 
or  even  boarding  alone.  To  avoid  this,  a  layer  of  mortar  about  %  inch  thick,  may  be  spread  over 
the  touching  surfaces  of  the  slates  if  on  laths.  If  on  boards,  the  same  process  may  be  adopted;  or 
the  more  common  one  of  first  covering  the  boards  with  a  layer  of  what  is  called  slating  felt ;  but 
which  in  reality  is  merely  thick  brown  paper,  soaked  in  tar.  TTiis  is  sold  in  long  continuous  rolls, 
28  ins  wide,  and  weighing  from  40  to  50 fi)s.  A  50  B)  roll  will  cover  about  300  sq  ft  of  roof.  With 
proper  precautions  against  the  admission  of  rain  and  snow,  a  pitch  as  flat  as  1  in  2^^,  or  even  1  in 
3,  may  be  adopted. 

The  thickness  of  slate  on  a  roof  is  double;  except  at  the  laps  is,  is,  &c,  where  it  is  triple.  The 
lapis  measured  from  the  nail  hole  (under  i)  of  the  lower  slate,  to  the  k)wer  edge  or  tail,  s,  of  the 
upper  one,  and  is  usually  about  3  ins.  In  order  that  the  showing  lower  edges  of  the  slates  shall, 
when  laid,  form  regular  straight  lines  along  the  roof,  the  nail  holes  are  made  at  equal  distances  from 
said  lower  edg^s  ;  so  that  any  irregularity  of  length  is  concealed  from  view  at  the  hidden  heads  of 
the  slates.  The  slater  estimates  the  length  of  his  slate  from  the  nail  hole  to  the  tail;  discarding  the 
narrow  strip  between  the  nail  hole  and  the  head.  If  from  this  reduced  length  the  lap  be  deducted, 
then  one-half  of  the  remainder  will  be  the  gauge,  weathering,  or  margin,  of  the  slating;  or,  in 
other  words,  the  showing  or  exposed  width  of  the  courses  of  slates.  The  gauge  in  ins  multiplied 
by  the  width  of  a  slate  in  ins,  gives  th«  area  in  sq  ins  of  finished  roof  covered  by  a  single  slate  ; 
and  if  1*4  (the  sq  ins  in  a  sq  foot)  be  divided  by  this  area,  the  quotient  will  be  the  number  of  slates 
required  per  sq  ft  of  roof.    The  upper  side  of  a  slate  is  called  its  back  ;  the  lower  one,  its  bed. 

Slating,  like  shingling,  must  evidently  be  commenced  at  the  eaves,  and  extended  upward.  Since 
the  beds  of  the  slates  are  not  exactly  parallel  to  the  boarding,  and  consequently  do  not  rest  flat  upon 
it,  those  at  the  lower  edge  w  would  easily  be  broken.  To  prevent  this,  a  tilting  strip  (a 
stout  wide  lath,  with  its  upper  side  planed  a  little  bevelling,  to  suit  the  slope  of  the  slates)  is  first 
nailed  around  near  the  eaves,  for  the  tails  of  the  lowest  course  of  slates  to  rest  on.  This  is  shown  oq 
a  larger  scale  at  T. 

Slate  of  the  best  quality  has  a  glistening  semi-metallic  appearance,  somewhat  like  that  of  a  sur- 
face of  paper  rubbed  with  black-lead  pencil.  That  of  a  dull  earthy  aspect,  is  softer,  more  absor- 
bent, and  consequently  more  liable  to  yield  to  atmospheric  influences,  rain,  frost,  &c.  Iron  pyrites 
frequently  occurs  in  slate;  and  since  it  always  decomposes.and  leaves  holes,  should  never  be  admitted 
on  a  roof.  Of  two  qualities  of  slate,  that  which  absorbs  the  least  weight  of  water,  when  pieces  of 
equal  size  are  soaked  for  an  hour  or  two,  is  generally  the  best ;  being  least  liable  to  split  by  frost, 
and  become  weather-worn.     This  test  is  easily  applied. 

In  England  the  different  Slzes  are  distinguished  by  absurd  names  of  no  meaning.  In  the 
United  States  they  are  called  6  byl2's;  16  by  24*8,  &c,  according  to  their  measures  in  inches.  They 
may  be  cut  to  order,  of  almost  any  prescribed  dimensions,  or  shape.  Those  in  common  use  vary  from 
about  7  by  14,  to  12  by  18.  The  first  forms  about  5  to  6  inch  courses ;  and  the  last  about  7  to  8  inch ; 
depending  upon  how  far  from  the  head  the  nail  holes  are  pierced.  The  farther  this  is,  the  firmer 
will  the  slating  be. 

Slate  roofs,  like  iron  ones,  heat  the  rooms  immediately  below  them  very  much.  This  is  somewhat 
diminished  when  the  slates  are  on  boards,  instead  of  laths ;  and  still  more  by  a  coat  of  plaster  be- 
neath.   They  are  also  liable  to  break  when  walked  on  ;  less  so  when  bedded  in  iportar. 

liVeiglit  of  slate  roofs.  Slate  weighs  about  175  fts.  per  cub  foot;  therefore, 

a  sq  ft,   %  inch  thick,  weighs  about  1.8  Sbs;   i^g,  2.7  Bbs  ;  and  34  thick,  3.6  fts.    But  owingto  the 
overlapping,  a  square  foot  of  roof  requires  about  23^  sq  ft  of  slate  of  ordinary  sizes;  and  if  the 
slate  is  laid  on  boards  an  inch  thick,  the  weight  per  sq  ft  of  roof  will  be  increased  about  234  B>s; 
or  with  13^  inch  boards,  2.8  lbs.    Laths  will  weigh  about  34  lb  per  sq  ft  of  roof. 
Hence, 

Approx  Weight 
of  one  sq  ft  of 
Slating,  in  S)s. 

Slate  %  inch  thick  on  laths 4.75 

"  "  on  1  inch  boards 6.75 

"  "  on  134  "       "      7.30 

"    3-16  "  on  laths 7.00 

"      "     "  on  1  inch  boards 9.00 

"      "     "  on  IM  "        "     9.55 

"      H    "  on  laths 9.25 

"       "     "  on  1  inch  boards 11.25 

"      "     "  on  134  "         "     11.80 

If  slating  felt  is  used,  add  J^ft;  or  if  the  slates  are  bedded  in  }i  inch  of  mortar,  add  3  lbs. 


SHINGLES.  971 

For  the  total  weight  bmne  by  the  roof  trusses,  that  of  the  purlins  also  must  be  added.  This  will 
not  vary  much  from  the  limits  of  1}^  to  3  tts  per  sq  ft  iu  roofs  of  moderate  span.  Add  for  wind  and 
snow,  say  20  B)s  per  sq  ft ;  and  finally  add  the  weight  of  the  truss  itself. 

For  Stoppiiig'  the  joints  between  slates  (or  shingles,  &c)  and  chimneys, 

dormer  windows,  &c,  a  mixture  of  stlfi"  white-lead  paint,  as  sold  by  the  keg,  with  sand  enough  to  pre- 
rent  it  from  running,  is  very  good ;  especially  if  protected  by  a  covering  of  strips  of  lead,  or  copper, 
tin,  &c,  nailed  to  the  mortar-joints  of  the  chimneys,  after  being  bent  so  as  to  enter  said  joints ;  which 
should  be  scraped  out  for  an  inch  in  depth,  and  afterward  refilled.  Mortar  protected  in  the  sam# 
way,  or  even  unprotected,  is  often  used  for  the  purpose;  but  is  not  equal  to  the  paint  and  sand.  Mor- 
tar a  few  days  old,  (to  allow  refractory  particles  of  lime  to  slack,)  mixed  with  blacksmith's  cind«rft 
and  molasses,  is  much  used  for  this  purpose  ;  and  becomes  very  hard,  and  effective. 


SHINGLES. 


Whitb  cedar  shingles  are  the  best  in  use ;  and  when  of  good  quality  will  last  40  or 
50  years  in  our  Northern  States.  They  are  usually  27  ins  long ;  by  from  6  to  7  ins 
wide ;  about  ^^  inch  thick  at  upper  end ;  and  about  %  at  lower  end  or  butt ;  and  are 
laid  in  courses  about  S}/q  ins  wide ;  so  that  not  quite  ^of  a,  shingle  is  exposed  to  the 
weather. 

They  are  usually  laid  in  three  thicknesses ;  except  for  an  inch  or  two  at  the  upper  ends,  where  there 
are  four.  They  are  nailed  to  sawed  shingling-laths  of  oak  or  yellow  pine;  about  16  ft  long;  2J^  in? 
wide,  and  1  inch  thick  ;  placed  in  horizontal  rows  about  8J^  ins  apart.  These  are  nailed  to  the  raft- 
ers, or  purlins ;  which,  for  laths  of  the  foregoing  size,  should  not  be  more  than  2  ft  apart  from  center 
to  center.  Two  nails  are  used  to  each  shingle,  near  its  upper  end.  They  should  not  be  of  less  size 
than  400  to  a  B>.  Wrought  nails  being  the  strongest,  are  the  best;  cut  ones  are  apt  to  break 

by  the  warping  of  the  shingles.  Two  poUnds  of  such  nails  will  suffice  for  100  sq  ft  of  roof,  including 
waste.  An  average  shingle  7>i  ins  wide,  in  8}4  inch  courses,  exposes  63%  sq  ins  ;  making  2}i  shingles 
to  a  sq  ft  of  roof;  but  to  allow  for  waste,  and  narrow  shingles,  it  is  better  in  practice  to  allow  about  3 
shingles  to  a  sq  ft. 

Shingling,  like  slating,  must  plainly  be  begun  at  the  eaves ;  and  extended  upward.  For  closing  the 
joints  between  the  shingles,  and  chimneys,  dormer  windows,  &c,  see  at  end  of  Slating. 

Cypress  and  white  pine  are  also  much  used  for  s'hingles,  being  much  cheaper,  but  scarcely  half  as 
durable.  All  shingles  wear  quite  thin  in  time  by  rain  and  exposure.  In  warm  damp  climates  they 
all  decay  within  6  to  12  years. 


PAINTING. 


The  principal  material  used  in  house-painting,  is  either  white  lead,  or  oxide  of 
zinc,  ground  in  raw  (unboiled)  linseed  oil,  by  a  mill,  to  the  consistency  of  a  thick 
paste.  In  this  condition,  it  is  sold  by  ftie  manufacturers  in  kegs  of  25,  50,  and  100 
ft)8.  To  prepare  it  for  actual  use,  merely  requires  the  addition  of  more  linseed  oil, 
■say  3  or  4  pints  to  10  Bbs  of  the  keg  paint,  for  thinning  it  suflSciently  to  flow  readily 
under  the  brush. 

Good  painting  requires  4  or  5  coats ;  but  usually  only  4  are  used  in  principal  rooms;  and  S  in  inferIo^ 
ones.  Each  coat  must  be  allowed  to  dry  perfectly  before  the  next  one  is  put  on.  One  E»  of  the  keg 
paint  will,  after  being  thinned,  cover  about  2  sq  yds  of  first  coat;  3  yds  of  second;  and  4  yds  of  each 
subsequent  coat ;  or  1  sq  yd  of  3  coats  will  require  in  all,  1.08  lbs  ;  of  4  coats,  l^  fts  ;  of  5  coats,  1.58 
S)s.  The  reason  why  the  first  coats  require  so  much  more  than  the  subsequent  ones,  is  that  the  bare 
surface  of  the  wood  absorbs  it  more. 

When,  as  is  usual,  raw  or  unboiled  oil  is  used  for  thinning,  dryers  must  be  added  to  it;  otherwise 
the  paint  might  require  several  weeks  to  harden  ;  whereas,  with  dryers,  from  1  to  3  days,  according 
to  the  weather,  suffice  for  each  coat  to  become  hard  enough  to  receive'  the  next  one.  The  dryers  most 
•ommonly  used,  are  powdered  litharge,  in  the  proportion  of  one  heaped  teaspoonful;  or  Japan  var- 
nish, 1  table-spoonful,  to  10  lbs  of  the  keg  paint.  Either  sugar  of  lead,  or  sulphate  of  zinc,  may  also 
be  used  instead  of  litharge ;  and  in  the  same  proportion.  Although  both  litharge  and  Japan  varnish 
are  dark-colored,  yet  the  quantity  is  so  small  as  not  to  appreciably  affect  the  whiteness  of  the  paint. 
If  the  varnish  is  used  in  excess,  as  is  often  done  in  the  hurry  to  have  work  finished,  it  produces 
cracks  all  over  the  surface.  No  dryer  is  necessary  if  painters'  boiled  oil  be  used  for  thinning.  Mere 
boiling  will  not  cause  oil  to  harden  more  rapidly;  but  that  intended  for  painters,  has  litharge  added 
to  it  previously  to  boiling ;  in  the  proportion  of  1  j^  fts  to  each  10  gallons  of  raw  oil.  In  some  works 
written  for  the  use  of  house  painters,  it  is  asserted  that  boiling  renders  the  oil  too  thick  for  any  but 
coarse  outdoor  work.  But  this  is  entirely  a  mistake ;  for  if  the  boiling  be  properly  done,  the  oil 
will  be  quite  thin  enough  for  the  best  inside  work ;  and  will  moreover  be  clearer  than  while  raw ;  aad 


972 


PAINTING. 


will  impart  to  the  painted  surface  a  more  shining  appearance.  The  heat  ihould  be  barely  sufiBcIeat 
to  produce  boiling  ;  or  about  600°  Fah.  The  boiling  should  continue  about  Ih  hours;  the  oil  being 
thoroughly  stirred  at  short  intervals,  to  prevent  the  litharge  from  settling  at  the  bottom.  The  fire 
may  then  be  allowed  to  subside;  when  the  operation  will  be  completed.  A  sediment  will  then  form 
at  the  bottom ;  which  must  be  left  behind  when  the  oil  is  poured  otf.  Although  no  dryer  is  necessary 
with  this  oil,  still  a  little  litharge  may  be  added  when  great  expedition  demands  it.  Painters  rarely 
use  this  oil,  on  account  of  its  trifling  increase  of  cost. 

Another  substance  much  used  with  the  thinning  oil,  (except  for  the  first  coat,j  is  spirits  of  turpen- 
tine ;  called  "  turp  "  by  the  workmen.  The  quantity  of  oil  may  be  diminished,  to  the  extent  of  the 
added  turp.  This  being  more  fluid  than  oil,  causes  the  paint  to  work  more  pleasantly  under  the  brush. 
It  moreover  diminishes  the  tendency  of  the  paint  to  become  yellow  ;  especially  in  rooms  kept  closed 
for  some  time.  It  is  also  much  cheaper  than  oil.  It  should  not  be  used,  or  but  sparingly,  for  exposed 
outdoor  work  ;  inasmuch  as  its  tendency  is  to  impair  the  tirmuess  of  the  paint ;  and  although  its 
effects  are  scarcely  appreciable  indoors,  they  are  quilfe  apparent  when  the  work  has  to  resist  the 
weather.  As  the  fashions  change  in  house-painting,  the  surface  is  at  times  required  to  present  a 
shining  or  glossy  finish :  at  other  times  a  dead  one  is  in  vogue.  The  glossy  one  is  that  which  the 
paint  will  naturally  have,  provided  that  no  more  turp  than  oil  be  used  in  the  thinning.  The  dead 
inish  is  obtained  by  using  no  oil,  but  turp  alone,  for  the  last  coat;  which  in  that  case  is  called  a 

Flatting  coat.  Although  turp  is  not  properly  a  dryer,  still,  as  it  evaporates  quickly,  it  facilitates  the 
ardening  of  the  paint. 

In  outdoor  work  it  is  usually  advisable  to  use  more  dryer  than  inside,  so  that  the  paint  may  soooM' 
become  hard  enoUgh  not  to  be  injured  by  dust  or  rain.    Otherwise  less  would  be  better. 

When,  instead  of  a  white  finish,  one  of  some  other  color  is  required,  the  coloring  ingredient  it 
mixed  with  the  white  paint  to  be  used  in  the  last  coat  only  ;  although  two  coloring  coats  are  some- 
times found  to  be  necessary  before  a  satisfactory  effect  is  produced.  The  coloring  ingredients  may  be 
Indigo,  lampblack,  terra  sienna,  umber,  ochre,  chrome  yellow,  Venetian  red,  red  lead,  &c,  &c;  which 
are  ground  in  oil,  ready  for  sale,  by  the  manufacturers  of  the  white-lead  and  zinc  paints.  They  art 
simply  well  stirred  into  the  white  paint. 

All  surfaces  to  be  painted,  should  first  be  thoroughly  dry,  and  free  from  dust.  If  on  wood,  all 
plane-marks,  and  other  slight  irregularities,  should  first  be  smoothed  off  by  sand-paper,  when  the 
neatest  finish  is  required.  Also,  all  heads  of  nails  must  be  punched  to  about  ^  inch  below  the  sur- 
face. To  prevent  knots  from  showing  through  the  finished  work,  (as  those  in  white  or  yellow  pine 
would  do,  on  account  of  the  contained  turpentine,)  they  must  first  be  killed,  as  it  is  termed.  A  usual 
and  effective  way  of  doing  this,  is  by  covering  them  with  two  coats  of  shellac  varnish;  which,  when 
dry,  should  be  smoothed  by  sand-paper.  Another  mode,  not  quite  so  certain,  is  by  one  or  two  coats 
of  white  lead  mixed  with  thin  glue- Water,  or  size,  as  it  is  called. 

After  these  preparations,  the  first,  or  priming  coat,  is  put  on  ;  in  which  there  should  be  no  turp; 
because  it  would  sink  at  once  into  the  bare  wood,  leaving  the  white  lead  behind  it,  in  a  nearly  dry 
friable  condition.  After  this  the  nail  holes,  cracks,  Ac,  must  be  filled  with  common  glaziers'  putty, 
made  of  whiting  (fine  clean  washed  chalk)  and  raw  linseed  oil;  boiled  oil  will  not  answer;  the  putty 
#would  be  friable.  The  putty  would  be  apt  to  fall  out,  if  put  in  before  priming;  because  the  wood 
would  absorb  the  oil,  and  the  putty  would  then  shrink.  After  the  first  coat  is  perfectly  dry,  the 
second  one  is  put  on  ;  and  for  it  about  1  measure  of  turp  may  be  mixed  with  3  measures  of  the  thin- 
ning oil.  In'the  third,  and  any  subsequent  coats,  equal  measures  of  turp  and  oil,  may  be  used  for 
thinning,  if  the  work  is  required  to  dry  with  a  gloss ;  but  if  it  is  to  finish  dead,  the  last  coat  must 
be  a  flatting  one;  or  one  in  which  the  thinning  oil  is  entirely  omitted,  and  turp  alone  substituted 
for  it. 

Painters  generally  clean  their  brushes  by  merely  pressing  out  most  of  the  paint  with  a  knife ;  and 
then  keep  them  in  water  until  further  use.  If  to  be  put  away  for  some  time,  they  may  be  thoroughly 
cleaned  by  turp ;  or  by  soap  and  water.  To  prevent  a  hard  skin  from  forming  on  the  top  of  their 
paint  when  not  used  for  some  days,  they  pour  on  a  little  oil. 

The  best  paints  for  preserving;  iron  exposed  to  the  weather, 

appear  to  be  pulverized  oxides  of  iron,  such  as  yellow  and  red  iron  ochres  ;  or  brown  hematite  iron 
ores  finely  ground;  and  simply  mixed  with  linseed  oil,  and  a  dryer.  White  leail  applied  directly  to 
the  iron,  requires  incessant  renewal ;  and  indeed  probably  exerts  a  corrosive  effect.  It  may,  how- 
ever, be  applied  over  the  more  durable  colors,  when  appearance  requires  it.  Red  lead  is  said  to  be 
very  durable,  when  pure.  An  instance  is  recorded  of  pump-rods,  in  a  well  200  ft  deep,  near  London, 
which,  having  first  been  thus  painted,  were  in  use  for  45  years  ;  and  at  the  expiration  of  that  time, 
their  weight  was  found  to  be  precisely  the  same  as  when  new;  thus  showing  that  rust  had  not 
affected  them. 

When  the  size  of  the  exposed  iron  admits  of  it,  its  freedom  from  rust  may  be  very  much  promoted 
by  first  heating  it  thoroughly;  and  then  dipping  it  into,  or  washing  it  well  with,  hot  linseed  oil; 
which  will  then  penetrate  into  the  interior  of  the  iron.  For  tinned  iron  exposed  to  the  weather,  on 
roofs,  rain  pipes,  <fec,  Spanish  brown  is  a  very  durable  color.  The  tin  Is  frequently  found  perfectly 
(bright  and  protected,  when  this  color  has  been  used,  after  an  exposure  of  40  or  50  years.  Whit« 
paint  washes  off  in  a  few  years  by  rain. 

Plastered  walls  should  if  possible  be  allowed  to  dry  for  at  least  a  year,  before  being  painted  in  oi{ 
otherwise  the  paint  will  be  liable  to  blister.  They  may,  if  preferred,  be  frescoed  (water-coloit, 
mixed  with  size)  to  the  desired  tint  during  the  interval. 

The  painting  of  unseasoned  wood  hastens  its  decay.  If  the  surface  to  be  painted  is  greasy,  the 
grease  must  first  be  removed  by  water  in  which  is  dissolved  some  lime. 

Washes  for  ontside  i^orlt.  Downing,  in  his  work  on  country  houses, 
recommends  the  following:  For  wood-work ;  in  a  tight  bushel,  slack  half  a  bushel  of  fresh  lime,  by 
pouring  over  it  boiling  water  suflRcient  to  cover  it  4  or  5  ins  deep  ;  stirring  it  until  slacked.  Add  2 
lbs  of  sulphate  of  zinc  (white  vitriol)  dissolved  in  water.  Add  water  enough  to  bring  all  to  the  con- 
sistence of  thick  whitewash.  Apply  with  a  whitewash  brush.  This  wash  is  white;  but  it  may  b« 
colored  by  adding  powdered  ochre,  Indian  rod,  umber,  &c.    If  lampblack  is  added  to  water-colors,  it 


GLASS,  AND   GLAZING. 


973 


should  first  be  thoroughly  dissolved  in  alcohol.  The  sulphate  of  zinc  causes  the  wash  to  become  hard 
in  a  few  weeks. 

For  brick,  masonry,  or  rong-H-cast.    Slack  3^  a  bushel  of  lime  as 

before ;  then  All  the  barrel  %  full  of  water,  and  add  a  bushel  of  hydraulic  cement.  Add  3  D>s  of  suU 
phate  of  zinc,  previously  dissolved  in  water.  The  whole  should  be  of  the  thickness  of  paint;  and 
may  be  put  on  with  a  whitewash  brush.  The  wash  is  improved  by  stirring  in  a  peck  of  white  sand, 
just  before  using  it.     It  may  be  colored,  if  desired,  like  the  preceding. 

He  also  gives  the  following  cheap  oil-paint  for  outside  work  on  wood,  brick,  stone,  <fec  ;  and  says  it 
becomes  far  harder  and  more  durable  than  common  paint :  One  measure  of  ground  fresh  quicklime ; 
add  the  same  quantity  of  fine  white  sand,  or  fine  coal  ashes ;  and  twice  as  much  fresh  wood  ashes ; 
all  the  foregoing  to  be  passed  through  a  fine  sieve.  Mix  well  together  dry.  Mix  with  as  much  raw 
linseed  oil  as  will  make  the  mixture  as  thin  as  paint.  Apply  with  a  painter's  brush.  It  may  be  col- 
ored like  the  foregoing,  taking  care  to  mix  the  colors  well  with  oil  before  adding  them.  It  is  best  t« 
put  on  two  coats  ;  the  first  thin,  and  the  second  thick. 

Also,  another,  said  to  stand  15  to  20  years  :  50  lbs  best  white  lead ;  10  quarts  raw  linseed  oil ;  J^  ft 
dryer;  50  BC>s  finely  sifted  sharp  clean  sand  ;  2  fi)s  raw  umber.  Add  very  little,  say  J^  pint  of  tur- 
pentine.    Apply  with  a  large  brush. 

Cement  for  stoppings  joints,  such  as  around  chimneys,  &c,  &c.     White 

lead  ground  in  oil,  as  sold  by  the  keg;  mixed  with  enough  pure  sand  to  make  a  stifif  paste  that  will 
not  run.  It  grows  hard  by  exposure,  and  resists  heat,  cold,  and  water.  Pieces  of  stone  may  b« 
strongly  cemented  together  by  it,  allowing  a  few  months  for  proper  hardening. 

"WliiteiFasli  for  inside  work,  according  to  Mr.  Downing,  "is  made  more 
fixed  and  permanent,  by  adding  2  quarts  of  thin  8ize  to  a  pailful  of  the  wash,  just  before  using. 
The  best  size  for  this  purpose  is  made  of  shreds  of  glove  leather ;  but  any  clean  size  of  good  quality 
will  answer,"  as  thin  glue- water.  We  will  add,  that  the  common  practice  of  mixing  salt  with  white- 
wash, should  not  be  permitted.  Paper  pasted  on  a  wall  which  has  previously  been  covered  with  salt 
whitewash,  is  very  apt  to  become  wet,  and  loose,  and  to  fall  ofiF  during  damp  weather.  The  white- 
wash should  be  scraped  off,  and  the  wall  or  partition  covered  with  a  coat  or  two  of  thin  size,  to  pro- 
tect  the  paper  from  the  effect  of  the  salt  that  may  still  adhere  to  the  plaster. 


GLASS,  AND  GLAZING. 

Window  glass  is  sold  by  the  box.  Whatever  may  be  the  size  of  the  panes,  a  box 
contains  as  nearly  50  sq  ft  of  glass  as  the  dimensions  of  the  panes  will  admit  of. 

Fanes  of  any  size  may  be  made  to  order  by  the  manufacturers.  The  sizes  given  in  the  following 
table,  as  well  as  many  others,  are  generally  to  be  had  ready  made.  Ordinary  window  glass  of  all  the 
sizes  in  the  table,  is  about  one-sixteenth  of  an  inch  thick  ;  and  this  is  the  thickness  supposed  to  be 
intended  when  a  greater  one  is  not  specified.  Double-thick  glass  is  nearly  3^  inch  ;  and  its  price  is 
50  per  ct  more  than  the  single  thick.     It  is  of  course  much  stronger  than  the  single. 

The  panes  are  confined  to  the  sash  by  glaziers'  putty,  made  of  whiting  (powdered  chalk)  and  raw 
linseed  oil  ;  and  by  small  triangular  pieces  of  thin  tin,  about  }4  i°cli  on  a  side,  which  uphold  the 
glass  while  the  putty  is  being  put  on  ;  and  are  allowed  to  remain  afterward,  as  a  protection  while  the 
putty  continues  soft. 


TABI.E  OF  UriJMBERS  OF  PANES  IN  A  BOX 

Size  in 

Panes 

Size  in 

Panes 

Size  in 

Panes 

Size  in 

Panes 

Size  in 

Panes 

ins. 

to 
a  box. 

ins. 

to 
a  box. 

ins. 

to 
a  box. 

ins. 

to 
a  box. 

ins. 

to 
a  box. 

«X   8 

150 

12  X  36 

17 

16X42 

11 

24X24 

12 

30X66 

7X    9 

115 

13X14 

40 

48 

9 

26 

12 

70 

•  xio 

90 

16 

35 

54 

8 

30 

10 

32X34 

12 

75 

18 

31 

60 

8 

36 

9 

36 

»X12 

67 

20 

28 

18X20 

20 

42 

7 

42 

14 

57 

24 

23 

22 

18 

48 

6 

48 

16 

50 

32 

17 

24 

17 

54 

6 

60 

18 

45 

14  X  16. 

32 

30 

14 

60 

5 

66 

10X12 

60 

18 

29 

36 

11 

66 

5 

34X36 

14 

52 

20 

26 

42 

10 

26X28 

10 

44 

16 

45 

24 

22 

50 

8 

32 

9 

48 

18 

40 

30 

17 

60 

7 

36 

8 

54 

20 

.36 

36 

14 

20X22 

17 

42 

7 

60 

S4 

30 

42 

12 

24 

15 

48 

6 

66 

3 

30 

24 

46 

11 

30 

12 

54 

5 

36X40 

5 

11X12 

55 

15X16 

30 

38 

10 

60 

5 

44 

5 

14 

47 

18 

27 

42 

9 

28X30 

9 

48 

16 

41 

20 

24 

48 

8 

36 

7 

54 

18 

37 

24 

20 

54 

7 

42 

6 

60 

20 

33 

30 

16 

64 

6 

56 

5 

70 

34 

27 

36 

13 

22X24 

14 

66 

4 

38X44 

12  XH 

43 

40 

12 

30 

11 

30X34 

7 

52 

16 

38 

16X18 

25 

36 

9 

36 

7 

40X46 

18 

34 

20 

23 

42 

8 

42 

6 

54 

20 

30 

24 

19 

48 

7 

48 

5 

72 

24 

25 

30 

15 

56 

6 

54 

4 

44X50 

28 

22 

36 

13 

60 

5 

60 

4 

56 

30 

20 

974  GLASS. 

The  best  qualities  of  American  glass  made  in  the  yicinity  of  Philadelphia, 
Boston,  Pittsburg,  &c,  are  for  most  purely  useful  purposes,  as  good  as  those  from 
foreign  countries;  but  when  the  highest  degree  of  beauty  is  required,  as  in  the 
lower  front  windows  of  first-class  dwellings,  fancy  stores,  &g,  polished  plate- 
glass  of  England,  France,  or  Germany,  must  be  used,  although  the  price  for 
moderate  sized  panes  is  from  5  to  8  times  as  great  as  that  of  the  best  quality 
single-thick  American.  Its  perfectly  smooth  surface,  free  from  distorted  reflec- 
tions, also  makes  it  the  best  for  covering  pictures;  still,  if  carefully  selected 
American  panes  be  used  for  this  purpose,  few  except  critics  in  glass  will  detecl 
the  difference. 

A  tbick  s-lass  is  made  expressly  for  flooring,  up  to  1  inch  thick, 
and  up  to  50  inches  by  9  feet  dimensions.  Also,  for  skylights,  from  ^  to  3^  inch 
thick.  Tliis  can  be  furnished  to  order  of  any  size  up  to  40  inches  by  8  or  10  feet. 
The  smaller  sizes  can  also  be  had  ground.  Grinding  prevents  the  entrance  of 
the  full  glare  of  the  sun ;  and,  moreover,  diffuses  the  light  over  a  much  greater 
width  of  space  below. 

8treng:tli  of  g^lass.  Tensile  2500  to  9000  fbs  per  square  inch.  Boston  rods 
by  author,  3500  to  5200.  Crushing  strength,  6000  to  10000  ft)s  per  square  inch. 
Transversely,  (by  the  writer's  trials,)  flooring  glass,  1  inch  square,  and  1  foot 
between  the  end  supports,  breaks  under  a  center  load  of  about  170  lbs;  con- 
sequently, it  is  considerably  stronger  than  granite,  except  as  regards  crushing ; 
in  which  the  two  are  about  equal. 

Remark.  Window  and  other  glass  which  contains  an  excess  of  potash  or  oi 
soda  is  very  liable  to  become  dull  in  time,  owing  to  the  decomposition  of  those 
ingredients  by  atmospheric  influences. 


ROPE. 


975 


ROPE. 

Tlie  streng'tli  of  rope  varies  greatly.  Pieces  from  the  same  coil  may  vary 
25  per  cent.  The  table  below  supposes  an  average  quality  Manila.  Good 
Italian  hemp  is  considerably  stronger.  The  tarring'  of  ropes  is  said  to 
lessen  their  strength  ;  and,  when  exposed  to  weather,  their  durability  also.  We 
believe  that  its  use  in  standing  rigging  is  partly  to  diminish  contraction  and 
expansion  by  alternate  wet  and  drying  weather.  A  few  months  of  exposed 
work  weakens  ropes  20  to  50  per  cent. 

Table  of  Manilla  rope. 


Diam. 
Ins. 

Circ. 
Ins. 

wtper 
foot, 
lbs. 

Break 
Tons. 

ng  load, 
lbs. 

Diam. 
Ins. 

Giro. 
Ins. 

Wtper 
foot, 
lbs. 

Breaking  load. 
Tons.  1     lbs. 

.239 

^4 

.019 

.25 

560 

1.91 

■  6 

1.19 

11.4 

25536 

.318 

1 

.033 

.35 

784 

2.07 

^V. 

1.39 

13.0 

29120 

.477 

i>^ 

.074 

.70 

1568 

2.23 

7 

1.62 

14.6 

32704 

.636 

2 

.132 

1.21 

2733 

2.39 

7^^ 

1.86 

16.2 

36288 

.795 

1V^ 

.206 

1.91 

1278 

2.55 

8 

2.11 

17.8 

39872 

.955 

3 

.297 

2.73 

6115 

2.86 

9 

2.67 

21.0 

47040 

1.11 

zv. 

.404 

3.81 

8534 

3.18 

10 

3.30 

24.2 

54208 

1.27 

4    ' 

.528 

5.16 

11558 

3.50 

11 

3.99 

27.4 

61376 

1.43 

W. 

.668 

6.60 

14784 

3.82 

12 

4.75 

30.6 

68544 

1.59 

5 

.825 

8.20 

18368 

4.14 

13 

5.58 

33.8 

75712 

1.75 

^y^ 

.998 

9.80 

21952 

4.45 

14 

6.47 

37.0 

8288G 

liWorking:  loads.  For  manila  ropes  from  1  to  1%  ins  diam,  running  at 
different  speeds  over  sheaves  of  the  diams  stated,  Mr.  C.  W.  Hunt  (Trans  Am 
Soc  Mechl  Engrs,  Vol.  XXIII,  1901)  gives  a  table  embodying  approximately  the 
following  results  of  experience.  Working  load  =  C  X  ultimate  strength  of  new 
rope.     D  =  minimum  diam  of  sheave,  in  ins. 


Speed 

ft  per  min 

as  for  work  on 

C 

V  rope 
D 

1^"  rope 
D 

Slow 

Medium 

Eapid 

50  to  100 
150  to  300 
400  to  800 

derrick,  crane,  quarry 
wharf,  cargo 

0.140 
0.056 
0.028 

8 
12 
40 

14 
18 
70 

Such  ropes  wear  out  rapidly.  A  rope  1>^  ins  diam  wears  out  in  lifting  from 
7,000  to  10,000  tons  of  coal.  On  the  other  hand,  13^  inch  transmission  ropes, 
running  5000  ft  per  min  and  carrying  1000  H.  P.  over  sheaves  5  ft  and  17  ft  in 
diam,  last  for  years. 

Mr.  Hunt's  figures  for  ultimate  strength,  based  upon  tests  of  full-sized  speci- 
mens of  manila  rope  made  by  three  independent  rope-walks  and  purchased  in 
open  market,  are  practically  Identical  with  those  given  in  our  table  above,  as 
are  also  those  of  Prof.  B.  Kirsch,  of  the  Imperial  Royal  Technological  Industrial 
Museum,  Vienna,  quoted  by  Mr.  Hunt. 


976 


WEIGHTS   AND   STRENGTHS   OF   WIRE    ROPES. 


WEIOHTS  AXI>  STRENOTHS  OF  WIRE  ROPES. 

Wire  Rope  manufactured  by  Jobn  A.  Roeblin^'s  Sons  Co.,  Tren- 
ton, N.  J.  The  prices  and  weights  given  are  for  ropes  with  hemp  centers. 
When  made  with  wire  centers,  the  prices  per  foot  are  10  per  cent,  higher,  and 
the  weights  10  per  cent,  greater. 


Trade 

No. 


Diam. 


Approx. 
circum. 
in  ins. 


Wt. 
per  ft. 


Approx.  break- 
ing strength  *  in 
tons  of  2000  fibs. 


Minimum  diam. 
of  drum  in  feet. 


Iron.     C.  steel. 


Iron.     C.  steel. 


Price  in  cents 
•per  foot.t 


Iron.     C.  steel. 


Standard  Hoisting  Rope,  with  6  strands  of  19  wires  each. 


1 
2 
3 
4 
5 

6 
7 
8 
9 
10 

ic 

10a 
10  6 


8.00 

78 

6.30 

62 

4.85 

48 

4.15 

42 

3.55 

36 

3.00 

31 

2.45 

25 

2.00 

21 

1.58 

•  17 

1.20 

13 

0.89 

9.7 

0.62 

6.8 

0.50 

5.5 

0.39 

44 

0.30 

3.4 

0.22 

2.5 

156 
124 

96 

84 

72 

62 

50 

42 

34 

26 

19.4 

13.6 

11.0 
8.8 
6.8 
5.0 


13 

12 
10 

f^ 
6 


8>^ 
8 

6^ 
5 

r^ 

33^ 


1 


117 
92 
80 
63 
57 
48 
40 
33 
26 
20 
16 
12 
10 
8 


142 
111 

93 

74 

66 

56 

46 

38 

30 

23 

18 

14 

12 

11 

10 
9^ 


Transmission  or  Hanlag^e  Rope,  with  6  strands  of  7  wires  each. 


11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 


ly 

A% 

3.55 

34 

68 

13 

sy^ 

51 

134 

^}i 

3.00 

29 

58 

12 

8 

43 

4 

2.45 

24 

48 

10% 

734 

36 

iVn 

3K 

2.00 

20 

40 

9^ 

6H 
5% 

29 

3 

1.58 

16 

32 

^74 

23 

% 

2% 
2i| 

1.20 

12 

24 

5 

173^ 

0.89 

9.3 

18.6 

4>^ 

14 

It 

2>| 

0.75 

7.9 

15.8 

6 

4 

12 

2  ^ 

0.62 

6.6 

13.2 

P 

33^ 

10 

134 

0.50 
0.39 

5.3 

4.2 

10.6 

8.4 

3 
2K 

8 

0.30 

3.3 

6.6 

3^ 

y 

0.22 

2.4 

4.8 

2% 

2 

T^TT 

1 

0.15 

1.7 

3.4 

23I 
2% 

ly. 

i 

Vs 

0.125 

1.4 

2.8 

i>l 

Notes  on  the  Use  of  Wire  Rope,  by  the  Roebling's  Company. 

The  ropes  with  19  wires  per  strand  are  the  more  pliable,  and  therefore  best 
adapted  for  hoisting*  and  running  rope.  The  others  are  stiffer  and  better 
adapted  for  guys,  &c.  Ropes  of  iron  or  steel,  up  to  3  inches  diameter,  made  to 
order.  Hemp  center  rope  is  more  pliable  than  wire  center.  Wire  rope  must 
not  be  coiled  or  uncoiled  like  hemp  rope.  When  on  a  reel,  the  reel 
should  be  mounted  on  a  spindle  or  flat  turn-table  in  order  to  pay  oflf  the  rope. 
When  forwarded  in  a  small  coil  without  a  reel,  roll  the  coil  on  the  ground  like  a 
wheel,  and  thus  run  off  the  rope.  Avoid  untwisting  and  short  bends.  To 
preserve  wire  rope,  apply  raw  linseed  oil  (which  may  be  mixed  with  an 

*For  the  safe  ivorking  load,  take  one-fifth  to  one-seventh  of  the 
breaking  load,  according  to  speed.        1 

t  Discounts,  1901:  bright  rope,  30  per  cent,  and  73^  percent.;  galvanized,  25 
per  cent,  and  1%  per  cent. 


WEIGHTS   AND   STRENGTHS   OF   WIRE   ROPES.         977 


equal  quantity  of  Spanish  brown  or  lamp-black)  with  a  piece  of  sheepskin, 
keeping  the  wool  against  the  rope.    If  for  use  in  -water  or  under- 

g^round,  add  1  bushel  of  fresh-slacked  lime  and  some  sawdust  to  1  barrel  of 
tar.     Boil  the  mixture  well  and  saturate  th6  rope  with  it  while  hot. 

Oalvanized  wire  rope  for  rigging  is  cheaper  and  more  durable  than  hemp 
rope ;  and  does  not  stretch  permanently  under  great  strains.  Its  bulk  is  one- 
sixth  and  its  weight  one-half  that  of  hemp  rope.  'Roebling's  wire  rope  has  been 
made  the  si-andard  by  the  United  States  Navy  Department.  Shackles,  sockets, 
swivel-hooks,  and  fastenings,  &c.,  furnished  and  put  on  and  splices  made. 
Pulley-wheels  furnished.  Also  galvanized  steel  cables  for  suspension  bridges. 
Crucible  cast-steel  wire  ropes  are  much  more  durable  than  iron  ones.  They 
should  be  kept  well  lubricated. 

Patent  Flattened  Ntrand  IVire  Rope. 
Manufactured  by  A.  I^esclien  «fe  Sons  Rope  Co.,  St.  I^ouis,  Mo. 


Hoisting  Ropes. 


Haulage  and  Transmission  Ropes. 


Breaking  *  strength 

Minimum  diam.  of 

List  price  f  per 

s 

I., 

in  tons  of  2000  lbs. 

drum  in  feet. 

foot,  in  cents. 

1 

•g 

Sfi 

■I-+ 

a 

y 

i 

i 

S' 

o 

1 

i 

1% 

u 

1 

1 

s 

11 

< 

3 

HI 

2  o 

1 

02 

2 

3  J 

1 

Es 

4» 

a 

s 

" 

■* 

" 

Hoisting*  Rope. 


2^ 

8.50 

260 

176 

75 

«  w 

9 

10 

257 

182 

152 

2^4 

2 

6.50 

211 

140 

66 

5  3 

8 

9 

202 

144 

120 

2 

1^ 
IK 

5.00 

168 

109 

54 

^A\: 

7.25 

7.5 

173 

121 

104 

1% 
1>I 

3.71 

124 

81 

.  40 

t'^  S 

5.75 

6.5 

123 

86 

74 

13^ 

2.50 

84 

56 

28 

^^'c. 

5 

5 

82 

59.5 

52 

IH 

11^ 

2.15 

67 

47 

21 

«  m  2 

4.5 

4.5 

68 

50 

43 

iy« 

1 

1.70 

56 

38 

17 

4 

4 

56.5 

39.5 

34 

1 

4 

1.25 

40 

29 

13 

^5-^ 

3.5 

3.5 

45 

30 

26 

Vq 

0.96 

32 

21 

9 

fl.g3 

3 

3 

35 

24 

21 

^2 

0.67 

22 

15 

6 

'^z. 

O  S 

2 

2.5 

26 

18.25 

15.5 

K^ 

y^ 

0.44 

13 

9 

4 

1.5 

1.75 

19.5 

14.5 

10.5 

K 

Haulagfe  and  Transmission  Rope. 


2.40 
2.00 
1.64 
1.20 
0.93 
0.68 
0.40 

80 
64 
53 
38 
30 
21 
13 

54 
45 
36 
27 
20 
14 
9 

81 

54 

45 

67 

45 

36.5 

53.5 

35 

29 

42 

27.5 

22 

34 

20.5 

17.5 

24 

14 

12.5 

15.5 

10 

8.25 

*  Working  load  =  0.2  X  breaking  strength, 
t  Discount,  1901,  30  per  cent,  and  1%  per  cent. 

I  "Hercules."    *' Made  from  a  specially  drawn  and  patent  tempered  steel, 
which  is  solely  made  for  this  brand  of  rope." 

62 


978  PAPEatt. 


PAPEE. 


24  sheets  1  quire.        20  quires  1  ream, 
Sizes  of  drawing^  papers. 

Ins.    Tub. 

Antiquarian 31  X  52 

Double  Elephant 26  X  40 

Atlas 26  X  34 

Imperial 21  X  30 


Ins.    Ini. 

Super  Royal 19  X  27 

Royal 19  X  24 

Medium 17  X  22 

Demy 15  X  20 

Cap 13  X  n 


The  English  drawing-papers  are  stronger  and  superior  to  the  American.  Those 
by  Whatman  have  a  high  reputation  ;  they  are,  however,  of  different  qualities.  When 
paper  is  pasted  on  muslin,  the  difference  in  quality  is  not  so  important.  Of  paper 
iu  rolls,  the  German  makes  are  the  best.   There  is  but  little  of  other  makes  imported. 

Both  white  and  tinted  papers,  for  the  use  of  engineers,  are  made 
in  continuous  rolls,  without  seams.  Widths  36,  42,  54,  58,  and  62  ins;  usual 
lengths  40  yds ;  but  can  be  had  to  order  to  400  yds  or  more.  These  may  also  be 
had  mounted  on  muslin,  iu  rolls  10  to  40  yds  long. 

Cartridg-e  or  pattern  paper  is  furnished  in  long  rolls,  of  same  lengths  as 
white  paper,  mounted  or  not ;  widths  up  to  54  ins.     Color,  a  light  bufi'. 

Tracing  paper.  Most  of  that  sold,  whether  domestic  or  foreign,  tears  so 
readily  as  to  be  of  comparatively  little  service.  Parchment  paper,  37  and  38  ins 
wide,  rolls  of  20  and  33  yds,  is  better,  but  does  not  take  ink  perfectly. 

Tracing*  cloth,  usually  called  tracing  muslin,  and  sometimes  vellum  cloth,  is 
altogether  preferable  to  tracing  paper,  on  account  of  its  great  strength.  Widths 
18,  30,  36,  and  42  ins  ;  lengths  to  24  yds. 

Profile  paper  is  made  in  widths  of  9  ins  and  20  ins,  and  in  single  sheets 
or  in  long,  continuous  rolls. 

Cross  section  paper,  mounted  or  unmounted,  tracing  paper  and  cloth, 
are  furnished  in  sheets  and  in  rolls,  ruled  in  quarters,  fifths,  eighths,  tenths, 
twelfths,  and  sixteenths  of  an  inch,  or  in  millimeters. 

Colors.  Since  the  introduction  of  blue  printing,  tinted  drawings  are  seldom 
made,  except  for  architectural  effect;  but  colors  may  be  used  to  advantage  on 
black-line  pHnts  from  tracings,  p  432  d.  A  good  draughtsman  needs  but  few 
colors;  say  India  ink,  Prus^an  blue,  lake,  or  carmine,  light  red,  burnt  umber, 
burnt  sienna,  raw  sienna,  gamboge,  Roman  ochre,  sap  green.  Winsor  &  Newton's 
colors  are  among  the  best  in  use.  Purchase  none  but  the  very  best  India  ink. 
Cakes  of  colors  should  always  be  wiped  dry  on  paper,  after  being  rubbed  in 
water;  and  but  little  water  should  be  used  while  rubbing;  more  being  added 
afterward. 

Lead  pencils.  Genuine  A.  W.  Faber's  Nos.  2,  3,  and  4,  are  very  good.  The 
hardness  increases  with  the  number.  Nos.  3  and  4  are  good  for  field-book  use :  which 
'to  prefer,  will  depend  on  the  character  of  the  paper;  No.  3  for  smooth,  and  No. 4  for 
the  coarser  or  more  granular  papers.  His  lettered  pencils  are  of  a  higher  grade  and 
better  suited  for  draughting.  "  H  "  stands  for  "  hard,".  "  B ''  for  "  soft."  The  degree 
of  hardness  or  of  softness  is  indicated  by  the  number  of  H's  or  of  B's.  *'F"  (inter- 
mediate) corresponds  with  No.  3.  Dixon's  American  pencils  are  good.  The  office 
draughtsman  should  have  a  flat  file,  or  a  piece  of  fine  emery  paper  glued  to  a  striy 
of  wood,  upon  which  to  rub  his  lead  to  a  fine  point  readily,  after  using  the  knife. 


BLUE-PRINTS.  979 


BLUE-PRINTS,  ETC 


* 


Art.  1.  (a)  In  order  to  obtain  the  best  results,  all  unnecessary  ex- 
posure, either  of  the  sensitized  paper  or  of  the  solutions,  to  sunlight  or  to 
other  white  light  should  be  avoided. 

(b)  Cleanliness  is  of  the  first  importance.  The  vessels  in  which  the  solu- 
tions are  made  and  mixed  must  be  scrupulously  clean,  and,  if  washed  with  soap, 
•must  be  carefully  rinsed  with  clean  water.  They  should  be  left  full  of  water 
when  not  in  use.  The  presence  of  free  alkali  of  any  kind  is  fatal  to  good  re- 
sults, and  immediately  destroys  the  blue  color  of  a  finished  print.  See  Art.  19  (b). 
The  solutions  must  not  be  allowed  to  come  in  contact  with  iron. 

Art.  2.  (a)  The  iSolution  used  in  sensitizing  the  paper  for  blue-prints 
is  usually  that  of  /erricyauide  of  potassium  {red  prussiate  of  potash)  f  and  am- 
moniocitrate  of  iron  (citrate  of  iron  and  ammonia)  in  water. 

(b)  The  two  suits  are  usually  dissolved  separately  and  the  two  solu- 
tions then  mixed.  The  potassium  salt  should  be  broken  up  fine.  The  iron  salt 
is  usually  quite  pure  and  dissolves  very  rapidly.  It  may  be  kept  indefinitely  in 
a  solid  state  if  perfectly  dry,  but  it  readily  absorbs  moisture,  and  then  becomes 
sticky  and  unfit  for  use;  and  the  solution  is  apt  to  become  mouldy  after  a  few 
days,  either  alone  or  when  mixed  with  the  potassium  solution.  Hence,  it  should 
be  prepared  (in  a  dark  room)  in  small  quantities  as  required. 

Art.  3.  (a)  The  following  is  an  average  of  several  recipes  that  give  ex- 
cellent results: 

Solution  A.  1  ounce  of  red  prussiate  of  potash  to  6  ounces  of  water,  or  2^ 
ounces  of  the  salt  to  a  pint  of  water. 

Dissolve  thoroughly  and  filter.  The  solutions  may  be  sufficiently  filtered 
through  raw  cotton,  and  much  more  rapidly  than  through  paper. 

Solution  B.  1^2  ounces  of  ammoniocitrate  of  iron  to  6  ounces  of  water,  or  4 
ounces  of  the  salt  to  1  pint  of  water. 

Dissolve  thoroughly.     Filter,  unless  the  solution  is  perfectly  clear. 

(b)  Keep  the  two  solutions  in  separate  glass-stoppered  bottles  in  a  dark  place 
until  they  are  to  be  used.  Then  mix  them  in  equal  parts,  ahd  filter  the  mix- 
ture. Take  care  that  no  undissolved  particles  of  the  red  prussiate  get  into  the 
double  solution.    It  must  be  rejected  when  its  brown  color  changes  to  bluish  green. 

(c)  The  combined  solution  will  cost  amateurs  Irom  1  to  2  cents  per  ounce  to 
make.    About  4  ounces  will  suffice  for  coating  100  square  feet  of  paper. 

(d)  If  a  lew  drops  of  strong  ammonia  solution  be  added  to  the  citrate  solu- 
tion, B,  until  the  odor  is  quite  perceptible,  the  addition  of  a  saturated  solution 
of  oxalic  acid  in  water  to  the  double  solution  will  hasten  the  print- 
ing in  cloudy  weather.  10  per  cent,  of  the  oxalic-acid  solution  will  in- 
crease the  rapidity  of  printing  about  23^  times ;  20  per  cent.,  5  times ;  30  per 
cent.,  10  times;  but  with  more  than  20  per  cent.,  it  is  difllcult  to  get  clear  white 
lines.  In  sunlight  the  difference  is  much  less  marked.  (Enqineerinq  News,  Dec. 
15,  1892.)  i/  . 

Art.  4.  (a)  Where  fine  work  is  not  essential,  any  well-sized  paper,  suffi- 
ciently tough  to  bear  the  washing,  will  answer.  For  important  work  use  paper 
of  fine  uniform  texture  and  smooth  hard  surface,  free  from  injurious  chemical 
substances.  If  the  solution  penetrates  below  the  surface,  a  portion  of  the  chem- 
icals may  remain  in  the  paper  in  spite  of  the  washing,  and  damage  the  result. 
Many  papers  are  made  especially  for  this  purpose.  The  Saxe  (German)  and 
Rives  (French)  papers  are  considered  among  the  best.  Johannot  and  Steinbach 
papers  give  good  prints,  but  are  not  very  strong.  Weston's  and  Scotch  linen 
papers  are  stronger,  and  the  latter  gives  excellent  prints.  Before  sensitizing 
a  large  quantity  of  paper  of  a  new  kind,  try  a  small  sheet  of  it.  liinen  for  sen- 
sitizing is  also  sold  by  dealers  in  photographic  material  and  engineers'  supplies. 

Art.  5.  (a)  The  solution  is  applied  (in  the  dark  room  of  course)  to 
one  side  only  of  the  paper.  This  is  sometimes  done  by  " floating"  the 
paper  upon  the  solution,  taking  care  that  none  gets  upon  the  hack  of  the  sheet. 

*  See  "Modern  Heliograpbic  Processes,"  by  Ernst  Leitze;  D.  Van  Nostrand 
Co  ,  New  York,  $3.00;  a  work  to  which  we  are  indebted  for  many  valuable  sug- 
gestions. 

"Modern  Reproductive  Graphic  Processes,"  by  Lieut.  J.  S.  Pettit,  D.  Van 
Nostrand  Co.,  Science  Series,  No,  76,  50  cents,  deals  chiefly  with  artistic  photog- 
raphy, lithography,  etc. 

See  also  paper  by  Benj.  H,  Thwaite,  Proc's,  Inst'n  Civ.  Eng'rs,  Vol.  Ixxxvi,  p. 
312,  reprinted  in  Engineering  News,  Nov.  27,  1886, 

t  Not  the  /errocyanide  or  yellov  prussiate. 


yoO  BLUE-PRINTS. 

The  paper  is  held  by  two  diagonally  opposite  corners,  and  the  diagonal  joining 
the  other  two  corners  is  then  allowed  to  touch  the  surface  of  the  liquid.  Then 
the  two  corners  held  in  the  hand  are  dropped,  first  one  and  then  the  other.  The 
paper  should  then  be  lifted,  one  half  at  a  time,  to  see  whether  any  air  bubbles 
have  been  formed  under  it.  If  so,  they  may  be  removed  by  drawing  over  the 
solution  that  half  of  the  sheet  under  which  they  occur,  while  the  other  half  is 
held  up  from  the  liquid.  One  or  two  minutes  suffice  for  floating,  and  the  paper 
is  then  drawn  out  over  an  edge  of  the  bath,  draining  otf  the  surplus  liquid. 
This  process  requires  a  tray  larger  than  the  sheet,  and  the  inner  surface  of  the 
tray  must  be  not  only  water-proof,  but  also  proof  against  chemical  action  from 
the  solution.    Considerable  care  is  required  in  the  manipulation. 

Art.  6.  (a)  The  solution  is  usually  applied  by  means  of  a  soft  wide  brnsli 
(such,  for  instance,  as  those  used  for  wetting  the  leaves  in  letter-copying  books) 
or  a  large  soft  js^poiig-e  entirely  free  from  sand  or  other  grit. 

Art.  7.  (a)  In  applying  the  solution,  the  paper  may  be  laid  upon  a  board  cov- 
ered with  soft  smooth  oil-cloth,  which,  after  each  sheet  is  sensitized,  should 
be  wiped  off,  to  avoid  smearing  the  back  of  the  next  sheet. 

(b)  The  operation  must  be  quickly  performed,  so  that  no  portion  of  a 
sheet  may  become  dry  before  its  entire  surface  has  been  coated.  For  very  large 
sheets  it  may  be  necessary,  for  this  reason,  to  employ  two  persons.  First  cover 
the  sheet  by  strokes  of  the  wet  sponge  or  brush,  moved  in  the  direction  of  the 
length  of  the  paper,  and  then,  immediately,  by  light  strokes  at  right  angles  to 
these  and  with  the  sponge  or  brush  squeezed  out,  so  that  the  solution  may  be 
uniformly  and  thinly  distributed  over  the  entire  "surface.  Wash  out  the  sponge 
immediately  in  the  dark  room. 

Art.  8.  (a)  The  paper  is  then  hung  up  to  dry  in  the  dark  room,  by  means 
of  clips,  of  any  convenient  form  and  free  from  iron.  Small  sheets  may  be  hung 
by  one  corner ;  larger  sheets  by  two  adjacent  corners,  or  by  three  or  more  places 
(according  to  size)  along  one  edge,  taking  care  to  buckle  this  edge  slightly,  so 
that  the  paper  may  not  be  stretched  in  drying.  If  the  sheets  are  hung  over  a 
rod  or  rail  the  solution  will  dry  unevenly  at  the  bend.  In  order  that  the  whites 
in  the  print  may  be  clear,  the  air  should  be  warm,  so  that  the  paper  may  dry 
quickly  and  thesolution  be  thus  prevented  from  penetrating  it  deeply. 

Art.  9.  (a)  Make  sure  that  the  paper  is  perfectly  dry  before  it  is  used 
or  put  away,  and  see  that  it  is  kept  both  dry  and  dark  until  it  is  wanted  for  use. 
If  carefully  prepared  and  preserved  it  will  retain  sensitiveness  for  a  long  time, 
but  the  best  results  are  obtained  with  fresh  paper,  and  it  is  best  not  to  keep  it 
more  than  a  month  or  two. 

Art.  10.  (a)  The  tracing  paper  or  tracing  cloth  should  be  of  a  bluish 
cast  (a  yellow  paper  delays  printing),  thin  (see  Art.'  15,  ),  and  as  nearly 

transparent  as  possible.  It  should  be  preserved,  both  before  and  after  drawing, 
from  long  exposure  to  light,  which  tends  to  render  it  opaque. 

(b)  Both  before  and  atter  drawing,  it  should  be  kept  either  flat  or  rolled,  and 
not  folded,  because  folds  render  it  difficult  to  bring  the  drawing  into  perfect  con- 
tact with  the  sensitive  paper  in  printing. 

Art.  11.  (a)  The  draivin^-  or  tracing  should  be  made  with  the  best 
India  ink,  rubbed  very  black.  The  addition  of  a  little  gamboge  or  chrome  yel- 
low increases  the  opacity.  Lines  drawn  in  chrome  yellow  and  in  gamboge  print 
well ;  but  Prussian  blue  or  carmine  should  be  rendered  more  opaque  by  the  addi- 
tion of  a  little  Chinese  white  or  flake  white.  Hold  the  tracing  up  to  a  strong 
light,  in  order  to  detect  any  weak  places  in  the  lines. 

Art.  12.  (a)  Printing  consists  in  exposing  the  sensitive  paper  to  the 
action  of  light,  the  drawing  being  placed  between  the  light  and  the  sensitive 
surface.  The  arc  electric  lig-ht  prints  more  slowly  than  direct  sunlight,  but 
has  the  advantage  of  constancy  in  all  weathers  and  at  all  hours,  and  of  fixedness 
of  position.    See  Art.  16  (a). 

(b)  Place  the  frame  with  its  face  perpendicular  to  the  rays  of  light,  as  nearly 
as  may  be,  and  see  that  no  shadows,  as  of  trees,  buildings,  etc.,  are  allowed  to  fall 
upon  a  portion  of  the  drawing. 

(c)  All  handling  of  the  paper,  such  as  cutting  it  to  size  or  placing  it  in  the 
frame,  should  be  done  in  a  weak  light. 

Art.  13.  (a)  To  secure  close  contact  between  the  tracing  and  the  sensitive 
paper  (see  Art.  15,  )  they  are  usually  placed  in  a  printing-frame.   The 

essential  parts  of  an  ordinary  frame  are :  the  frame  proper,  a  plate  of  clear  glass 
for  the  passage  of  the  light,  and  a  padded  back,  which,  by  means  of  clamps  and 
springs,  presses  the  two  sheets  closely  together  and  against  the  glass. 

(b)  The  tra<3ing  is  laid  in  the  frame,  with  its  drawn  side  next  to  the  glass 
(but  see  Art.  16    b),  and  then  the  sensitive  paper,  with  the  sensitive 

side  next  to  the  tracing.    Finally,  the  padded  back  is  placed  in  the  frame. 


BLUE-PRINTS.  981 

(c)  The  back  is  often  made  in  two  halves,  hinged  together  and  each  provided 
■with  a  spring,  so  that  one  half  may  be  raised  to  permit  examination  of  the 
progress  of  the  exposure,  while  the  other  half,  remaining  clamped,  holds  the 
tracing  and  the  sensitive  paper  in  position. 

(d)  By  using  a  frame  left  open  at  both. ends  long  strips  of  sensitive  paper  may 
be  used,  a  part  at  a  time,  the  rest  being  rolled  up  at  the  ends  of  the  frame  ai'd 
wrapped  for  protection  from  light. 

(e)  In  any  frame  it  is  important  that  the  glass  be  sufficiently  thick  to  with- 
stand the  pressure  required  in  order  to  secure  close  contact  between  the  two 
papers  (see  Art.  15,  below),  of  excellent  quality,  and  free  from  defects  which 
would  obstruct  or  unequally  refract  the  light.  The  glass  should  be  carefully 
cleaned  before  printing. 

(f )  Improved  forms  of  printing-frames  have  rubber  air-cushions  in  place  of 
flannel  pads.  In  others  the  necessary  pressure  is  secured  by  means  of  a  vacuum 
produced  between  the  tracing  and  the  glass  by  means  of  a  pump. 

{^)  Printing-frames  are  supplied  by  dealers.  The  prices,  including  glass,  vary 
from  about  S2  for  frames  10  x  12  inches,  to  $80  or  $45  for  frames  36  x  60  inches. 
Frames  running  on  rollers,  with  fittings  for  exposing  them  outside  of  windows, 
are  also  furnished,  at  prices  varying  with  the  dimensions  and  the  requirements. 

(li)  For  large  blue-prints,  Prof.  E.  C.  Cleaves,  of  Cornell  University,  uses, 
instead  of  a  frame,  a  wooden  cylinder  covered  with  felt  and  revolving  on  its 
axis.  Upon  this  cylinder  the  tracing  and  sensitive  paper  are  stretched  by  means 
of  a  suitable  clamping  device,  and  the  cylinder  is  then  revolved  in  the  sunlight. 
This  method  dispenses  with  the  use  of  glass.  It  of  course  requiies  a  longer  ex- 
posure than  the  ordinary  method.     {Trans.  Am.  Soc.  Meek.  Eng..,  vol.  viii,  p.  722.) 

(i)  For  still  larger  prints.  Prof.  R.  H.  Thurston  stretches  the  two  papers  upon 
a  thin  board,  which  is  then  sprung  into  a  curve  and  held  in  that  shape,  keeping 
the  papers  in  tension  upon  the  convex  side.  This' method  also  dispenses  with 
the  use  of  glass,  and,  the  curvature  of  the  board  and  the  papers  being  but 
slight,  the  whole  of  the  paper  is  exposed  to  the  light  at  one  and  the  same  time. 
{Trans.  Am.  Soc.  Meek.  Eng.,  vol.  ix,  p.  696.) 

Art.  14.  (a)  The  time  required  for  exposure  varies  with  the 
color,  directness,  and  intensity  of  the  light,  with  the  thickness  and  opacity  of 
the  tracing  paper,  with  the  blackness  of  the  drawing,  with  the  materials  and 
the  care  used  in  sensitizing  the  paper,  and  with  the  freshness  of  the  latter,  from 
two  or  three  minutes  to  hours  or  even  days.  Roughly,  we  may  say  that  in  full 
sunlight,  in  Philadelphia,  about  three  minutes  ordinarily  suffice  from  noon  to  2 
p.  M.,  and  ten  minutes  at  10  a.  m.  or  4  p.  m.;  in  the  shade,  thirty  to  forty-five 
minutes  at  noon ;  but  no  fixed  rules  can  be  given.  Experience  must  decide 
in  each  case.  A  preliminary  experiment  may  be  made  with  a  small  frame.  If 
the  back  of  the  frame  is  in  two  or  more  pieces,  the  process  may  be^nspected  from 
time  to  time. 

(b)  If  perfectly  opaque  ink  be  properly  used,  the  blue  background  may  be 
printed  very  dark  without  spoiling  the  lines,  but  over-exposure  in  printing  ren- 
ders the  background  first  blackish  and  then  of  a  dingy  shade.  See  Art.  17  (c)  and 
(d).  Drawings  in  pale  ink  must  be  printed  very  lightly,  in  order  that  the 

lines  may  remain  white,  and  it  is  best  to  use  with  them  a  weak  light,  or  to  pro- 
tect them  by  tissue  paper  or  ground  glass.    See  Art.  18  (a). 

Art.  15.  (a)  To  obtain  perfectly  sharp  impressions,  the  side  of  the  tracing 
upon  which  the  drawing  is  made  should  be  in  iminediate  contact  with 
the  sensitized  surface  of  the  blue-print  paper,  especially  if,  as  with  sunlight,  the 
direction  of  the  light  is  variable;  for,  if  any  appreciable  distance  intervenes 
between  the  two,  as  in  printing  through  cardboard  (see  Art.  16,  below),  the 
shadows  cast  by  the  lines  of  the  tracing  will  move  over  the  sensitized  surface  as 
the  direction  of  the  light  changes,  and  thus  give  a  blurred  impression.  In  most 
cases,  however,  it  is  practically  out  of  the  question  to  place  the  two  surfaces  in 
this  way,  because  that  position  gives  a  reversed  impression  as  regards  right  and 
left.*  Hence  a  thin  tracing  paper  or  linen  is  recommended  in  Art.  10  (a).  For 
the  same  reason  it  is  imperative  that  the  two  papers  be  firmly  and  evenly 
pressed  against  the  glass. 

Art.  16.  (a)  By  using  a  light  which  is  constant  in  position,  relatively  to 
the  surface  of  the  tracing,  such  as  an  arc  electric  light,  it  is  possible,  by  prolong- 
ing the  exposure  for  hours  or  even  days,  to  obtain  blue-prints  from  drau"- 
in^s  made  upon  stout  dl*ai¥in^  paper  or  even  upon  bristol  board. 

(b)  With  sunlight  the  same  object  may  be  accomplished,  either  by  placing  the 
original  with  its  back  to  the  glass,  and  the  sensitive  paper  (which  should  be  very 

*  A  print,  thus  reversed  in  position,  may  of  course  be  easily  read  by  means 
of  a  mirror.    This  is  commonly  done  with  Patent  Office  drawings. 


982  BLUE  PRINTS. 

thin)  with  its  back  to  the  sunlight,  or  by  placing  the  printing  frame  in  the  bottom 
of  a  deep  and  narrow  box,  so  that  the  light  can  shine  directly  upon  the  frame 
only  when  approximately  parallel  with  the  long  sides  of  the  box.  To  print 
rapidly,  the  sunlight  must  be  kept  full  upon  the  frame  by  frequently  moving 
the  box. 

Art.  17.  (a)  The  print,  when  sufficiently  exposed,  is  taken  from  the  frame, 
and  both  its  face  and  back  are  wa^lied  thoroug'hly  in  clean  water  until 
the  characteristic  blue  color  is  perfectly  developed. 

(b)  The  washing  should  be  done  in  a  tray  with  a  flat  bottom  larger  than  the 
largest  print  to  be  washed,  and  care  should  be  taken  not  to  injure  the  surface 
of  the  prints  by  hard  rubbing  or  by  sharp  bending,  or  otherwise.  It  is  better 
to  have  a  circulation  of  water  in  the'  tray,  not  only  to  keep  the  water  clean,  but 
also  to  bring  about  the  necessary  agitation  of  the  prints  without  handling  them. 

(c)  The  washing  may  be  hastened, and  dark  or  "over-exposed"  prints  m^y  be 
lightened  somewhat,  by  having  the  water  warm,  say  at  90°  or  100°  Fahrenheit. 

(d)  Over-exposed  prints  may  also  be  lightened  by  immersing  them  in  water 
rendered  slightly  alkaline  by  ammonia.    In  this  bath  they  at  once  assume  a 

Eurple  tint,  which  soon  becomes  weaker.  At  the  proper  moment,  which  must 
e  learned  by  experience,  the  alkaline  action  must  be  stopped  by  drawing  the 
print  rapidly  through  a  solution  of  1  part  of  hydrochloric  ("  muriatic  ")  acid 
(H.  CI.)  in  100  parts  of  water. 

(e)  Continue  washing  until  the  water  has  for  some  time  come  off  perfectly 
clear.    Then  hang  the  prints  up  smoothly  to  dry. 

Art.  18.  (a)  After  washing,  the  application  of  a  solution  of  from  1  to  5 
per  cent,  of  hydrochloric  acid,  or  of  oxalic  acid,  in  water,  intensifies  the  blue 
color,  and  is  therefore  useful  in  bringing  out  pale  or  "under-exposed"  prints; 
but  the  prints  must  then  be  afterward  washed  again  in  pure  water.  Hydro- 
chloric acid  applied  before  washing,  or  to  imperfectly  washed  prints,  will  make 
the  lines  show  blue. 

Art.  19.  (a)  To  erase  a  (white)  line  on  a  blue-print,  go  over  the  line  with 
the  sensitizing  solution  applied  with  a  clean  brush  or  quill  pen.  This  should  be 
done  in  a  weak  light.    Then  expose  the  entire  print  and  re-wash. 

(b)  White  lines  are  added  to  blue  prints,  usually  in  Chinese  white; 
but  the  blue  color  may  be  removed,  showing  the  white  paper  beneath,  by  apply- 
ing a  saturated  solution  of  concentrated  lye  (caustic  soda  or  potash)  of  of  car- 
bonate of  soda*  or  carbonate  of  potash,  with  a  fine  clean  pen  nearly  dry.  If 
laid  on  too  freely,  it  spreads  rapidly.  Even  if  the  pen  is  perfectly  clean,  the  sur- 
face thus  produced  has  a  yellowish  cast  as  compared  with  the  white  of  the 
paper.  The  carbonate  solutions  act  more  slowly  than  the  lye,  but  not  less 
surely,  and  they  are  not  injurious  to  the  skin,  whereas  the  lye  burns  badly. 
The  ordinary  lime-water  sold  by  druggists  makes  little  or  no  impression  upon 
the  blue  color,  tf  red,  instead  of  ivhite,  lines  are  desired,  mix  with  the 
soda  or  potash  solution  ordinary  carmine  writing-ink,  in  such  quantity  (to  be 
ascertained  by  trial)  as  will  give  the  desired  color. 

Art.  20.  "(a)  Blue  prints  which  are  to  be  subjected  to  much  handling  should 
be  mounted  upon  cloth,  or  the  prints  may  be  made,  in  the  first  place,  upon 
sensitized  tracing  linen. 

Art.  21.  (a)  Processes  ^ivin;::  a  white  ground,  with  either  blue 
or  black  lines,  are  usually  so  complicated  as  to  be  beyond  the  reach  of  most 
engineers.  Their  results,  also,  are  generally  uncertain,  eveh  when  applied  by 
experts ;  the  background  often  lacking  in  whiteness. 

(b)  Tandy  lie  paper  (Eugene  Dietzgen  Co.,  Chicago)  and  Maduro 
paper  give  excellent  dark  brown  lines  on  good,  smooth,  hard  paper.  The 
"  Nigrosine  "  and  other  so-called  black-line  prints,  furnished  by  dealers,  usually 
give  perishable  purple  lines  on  a  gray  and  somewhat  glossy  ground,  and  on  brittK-, 
unserviceable  paper, 

(c)  Francis  LicClere,  21  North  13th  Street,  Philadelphia,  furnishes  excel- 
lent blacK-line  prints  to  order  at  10  cents  per  square  foot.  The  lines  are 
perfectly  black  and  permanent,  and  the  prints  are  made  on  good  drawing 
paper,  the  color  and  durability  of  which  are  not  affected  by  the  process.  He 
also  furnishes  fine  blue-line  prints,  on  similar  paper,  at  5  cents  per  square  foot. 

♦  Either  carbonate  ("washing-soda")  or  bicarbonate  ("baking-soda")  will 
answer. 


PRICE    LIST   AND    BUSINESS   DIRECTORY.  983 


PEIOE  LIST  AND  BUSINESS  DIREOTORT. 


For  a  work  of  this  kind,  any  attempt  to  present  a  list  of  exact  or  even  of 
closely  approximate  prices  would  be  useless.  We  aim  merely  to  give  indica- 
tions of  the  average  costs  or  of  the  ranges  of  cost.  For  actual  quotations, 
apply  to  those  named  in  the  business  directory,  following  the  list  of  prices. 
See  the  numbers  given  in  the  line  or  lines  immediately  following  each  title 
in  the  price  list,  and  referring  to  said  names. 

In  selecting  names  for  the  business  directory  the  aim  has  been  merely  to 
furnish  a  useful  (though  by  no  means  exhaustive)  list  of  representative 
names.     No  other  consideration  has  been  entertained. 

Abbreviated  Outline  of  Classification. 

For  principle  of  classification,  see  Bibliography,  p.  1008. 

1.0  Materials  and  Elementary  Shapes. 

1.1  Chemicals,  etc.    1.13,  Preservatives;  Paints,  Impregnating,  etc.     1.14, 

Explosives. 

1.2  Wood,  Lumber,  Poles,  Posts,  and  Piles. 

1.3  Stone,    Concrete,    Asphalt,    etc.     1.34,    Cement.     1.35,    Brick,    Tile, 

Glass,  etc. 

1.4  Iron  and  Steel.     1.45,  Nails,  Rivets,  Screws,  Bolts,  etc.,  Chains.     1.46, 

Tubes.     1.47,  Wire,  etc. 

1.5  Other  Metals  and  Alloys. 

1.6  Paper.     1.7,  Ropes,  etc.     1.8,  Packings,  Gaskets,  Belting,  Lagging, 

etc. 

2.0  Constructions. 

2.1  Earthwork.     2.12,  Dredging.     2.13,  Foundations. 

2.2  Masonry.     2.21,  Brick.     2.22,  Stone.     2.23,  Concrete. 

2.3  Metal  Structures.     2.31,    Bridges.     2.32,   Turntables.     2.33,    Tanks, 

Stacks,  etc.   2.34,  Boilers.    2.35,  Fireproofing,  Concrete  Metal  Con- 
struction. 

2.4  Paving.     2.5,  Sewers.     2.6,  Chimneys.     2.7,  Wharves,  Docks,  Har- 

bor Improvement. 

3.0  Machinery. 

3.1  Electrical  Machinery. 

3.2  Tools.     3.22,  Machine  Tools. 

3.3  Engines,  Locomotives,  Cars.     3.35,  Water  Engines  and  Motors,  Tur- 

bines.    3.36,  Cars.     3.37,  W^agons. 

8.4  Blowing  and   Pumping   Machinery.     3.44,    Wind   Mills.     3.45,    Hy- 

draulic Rams.     3.46,  Pumps. 

3.5  Hoisting  and  Conveying  Machinery.     3.51,  Power  Transmission. 

3.6  Excavators,  Dredges,  Machinerv  for  Road  and  General  Construction. 

3.65,  Diving  Apparatus.     3.66,  Pile  Drivers.     3.67,  Wells  and  Weil 
Driving  Machinery.     3.68,  Road  Making  Machinery. 

3.7  Heating,  Ventilating,  and  Refrigerating. 

4.0  Engineering,  Surveying,  and  Scientific  Instruments  and  Supplies.  4.1, 
Testing  Machines.  4.2,  Surveying  Instruments.  4.3,  Computing 
Instruments.  4.4,  Drawing  Insts  and  Materials.  4.5,  Heliography. 
4.8,  Testing  Laboratories. 

9.0  Miscellanieous  Supplies  (Arranged  according  to  class  of  work). 

9.1  Railroad  Supplies, 

9.2  Hydraulic  Supplies.     9.22,  Filters.     9.24,  Water  Meters.     9.25,  Pipe 

and  Hose.     9-26,  Hydrants  and  Valvee. 


984  PRICE   LIST. 

PRICE  LIST. 

1.0  Materials  and  Elementary  Shapes. 

1.1  Chemicals,  etc. 
1.13  Preservatives. 

1.131  Coatings,  Paints. 

35,  76,  198,  274,  289,  327,  361,  433,  502,  564,  586,  635. 
Paints,  in  oil,  $1  to  $1.50  per  gal. 
In  cts  per  lb : 

Lead:    White,  foreign,  8  to  10;  American,  7.    Red,  foreign,  8;  Ameri- 
can, 6. 
Zinc :    American,  5 ;  Paris,  9  to  10 ;  Antwerp,  7  to  8. 
Lampblack,  12  to  14. 

Blue,  Chinese,  40;   Prussian,   35;   ultramarine,  15;   brown,  Vandyke, 
10  to  13.     Green,  chrome,   10  to  12.     Sienna,  burnt  and  raw,  10 
to  13.     Umber,  burnt  and  raw,  10  to  12. 
Metal  coatings,  $1.50  to  $2.50  per  gal. 
Preservatives,  fillers,  oils,  etc.,  25  to  50  cts  per  gal. 
Graphite  pipe-joint  compound,  13  to  20  cts  per  lb. 
Linseed  oil,  60  to  70  cts  per  gal.     Turpentine,  40  cts  per  gal. 
Plain  varnish,  30  cts  per  gal. 

Carbolineum  avenarius,  80  cts  per  gal.     Woodiline  or  spirittine,  25 
cts  per  gal.     Creosote  oil,  1  to  li  cts  per  lb. 

1.133  Creosoting,  Impregnating,  etc. 

48,  63,  133,  149.5,  229,  319,  373.5,  442,  453,  578,  612.5,  634,  663. 

Creo-resinate  and  creosote  process,  13  to  19  cts  per  cu  ft. 

Creosoting,  20  to  60  cts  per  cu  ft  of  material  treated,  depending  chiefly  on 

degree  of  saturation,  and  exclusive  of  cost  of  timber;  =  $16  to  $50  per 

loop  ft  B  M. 
Kyanizing  (mercury  bichloride  process),  8  to  9^  cts  per  cu  ft. 
Barschall  or  Hasselmann  process,  8  cts  per  cu  ft. 
Wellhouse  (zinc-tannin  process),  12  to  19  cts  per  tie. 
Burnettizing  (zinc  chloride  process),  8  to  18  cts  per  tie. 
The  treatment  of  ties  is  usually  cheaper  per  cu  ft  than  that  of  larger  lumber. 

1.14  Explosives. 

214,311,346,452,492,517. 

Gunpowder,  16  cts  per  lb. 

Smokeless  powder,  60  cts  per  lb. 

Rackarock,  18  to  25  cts  per  lb. 

Dynamite,  13  to  21  cts  per  lb  for  different  grades,  varying  between  20% 

and  75%  nitroglycerine. 
Percussion  caps,  30  to  60  cts  per  M. 
Blasting  machinery,  see  3.231. 
Drills,  see  3.23. 

1.3  Wood,  Lumber,  Timber. 

16,5,  268,  271,  330,  599,  636. 

Lumber,  in  dollars  per  1000  ft  board  measure  (B  M) : 

Yellow  pine,  short  leaf,  12  to  13;  flooring,  20  to  35;  long  leaf,  19  to  20; 

flooring,  22  to  25. 
Walnut,  110  to  130.     Poplar,  25  to  40.     Ash,  60  to  65. 
Oak,  culls,  20;  common,  28;  plain  sawed,  40;  boards,  60  to  70;  lO-inch 

and  wider,  100  to  125;  plank,  40. 
Hemlock  joists  and  boards,  15  to  20. 
Spruce,  30  to  40. 

Shingles,  cypress,  per  1000,  8  to  11. 
Studding,  joists,  rafters,  etc.,  hemlock,  15  to  18. 
Clearing  and  grubbing,  see  2.11. 
Wood  pipe,  see  9.254.     Ties,  9.14.     Piles,  1.23. 

1.33  Poles,  Posts,  Piles. 

408,  494,  599. 

Piles,  15  to  25  cts  per  linear  ft  of  pile. 

Piling,  round  or  sheet,  30  to  50  cts  per  linear  ft  of  pil©. 


MATERIALS.  985 

1^  Stone,  Concrete,  Asphalt,  etc. 

Earthwork,  dredging,  foundations,  see  2.1. 

1.33  Stone. 

92,  368,  388,  393,  620. 

Sand  and  gravel  (within  100  miles  of  seashore),  $1  to  $2  per  cu  yd. 

Broken  stone,  75  cts  to  S2  per  cu  yd. 

Rip-rap,  $1  to  $3  per  cu  yd. 

Trap  rock,  70  cts  per  ton  of  2000  lbs. 

Ordinary  building  stones,  $1  to  S5  per  cu  yd. 

Granite,  $15  to  $45  per  cu  yd 

Slate  roofing,  12  to  25  cts  per  sq  ft. 

1.33  Asphalt. 

17,  51,  61,  272,  437,  444,  449,  559,  649. 
Paving,  see  2.4. 

1.34  Cement. 

22,  23,  30,  54,  86,  97,  101,  106,  149,  170,  177,  179,  227,  252,  258,  310,  326, 
327,  336,  352,  358,  360,  366,  418,  445,  456,  569,  588,  616,  626,  642,  657. 

Portland  (artificial)  cements,  per  bbl  of  about  400  lbs  gross:  German, 
$2.25  to  $3.00;  American,  $1.10  to  $1.60. 

Rosendale  (natural)  cements,  per  bbl  of  about  300  lbs  net :  From  Rosen- 
dale  Township  and  vicinity,  Ulster  Co.,  N.  Y.,  95  cts  to  $1.10;  other 
Rosendales,  75  to  85  cts. 

About  $1  to  $2  worth  of  cement  mortar  required  per  cu  yd  of  masonry  in 
buildings,  $1.50  to  $3  per  1000  bricks. 

Lime,  60  to  90  cts  per  bbl  of  about  250  lbs. 

About  60  cts  worth  required  per  cu  yd  of  masonry  in  buildings.  $1  to  $1.50 
per  1000  bricks. 

Plaster,  $1.50  to  $2  per  bbl  of  varying  weight. 

Concrete  construction,  see  2.35. 

1.35  Brick,  Tile,  Glass,  etc. 

Sewer  pipe,  see  9.255. 

1.351  Brick. 

217,  232,  291,  299,  383,  448,  468,  505,  515. 

Paving,  see  2.4. 

Building  bricks,  per  1000:  Salmon,  $5  to  $7;  hard,  $7  to  $9;  stretchers, 

$9  to   $14;   pressed,  $17  to  $20;  colored,  $20  to  $30;  iron  spots,  $30; 

Pompeiian,  $35, 
Fire-brick,  $20  to  $24  per  M. 
Vitrified  paving  brick,  $15  to  $25  per  M. 
Sewer  pipe,  see  9.255.     Paving,  2.4. 

1.353  Tiling. 

Floors  and  walls,  35  and  40  cts  per  sq  ft  and  upward. 

Tile,  380,  0.13  to  0.8  ct  per  cu  in  of  material. 

Roofing  tile,  $7  to  $30  or  more  per  square.     1  square  -=  100  sq  ft. 

1.353  Glass. 

557. 

American  window.  $  per  box  of  about  50  sq  ft.  Discount,  80%  to  85%. 
"  United  inches '-' 25  50  80  100 

Single  AA    32  38  49 

Double  AA    43  56  68  88 

Single  A 27  32  45 

Double  A 38  50  62  80 

Single  B 26  30  39 

Double  B 36  46  56  75 

Or,  say,  single  thick,  ^^  to  i  ct  per  united  in :  double  thick,  i  to  i  ct  per 
united  in:  where  the  number  of  united  ins  equals  the  sum  of  the  two 
dimensions;    Thus,  a  sheet  of  glass  24  X  36  contains  60  united  ins. 


986  PRICE   LIST. 

1.4  Iron  and  Steel. 

98,  123,  128,  131,  134,  176,  189,  195,  233,  242,  283,  301,  310,  329,  526,  535, 

560,  612,  640,  662,  670. 
Scrap  iron  and  steel,  $12  to  $19  per  ton  of  2240  lbs. 

1.41  Cast  Iron  and  Steel. 

71,  145,  151.  185,  189,  192,  218,  249,  280,  281,  329,  372,  377,  463.2,  612, 

624,  652. 
Cast-iron  pipe,  see  9.251. 
Pig  iron,  per  ton  of  2240  lbs:  Foundry,  $13  to  $15;    Bessemer,  $16;  gray 

forge,  $14;  Lake  Superior  charcoal,  $17. 

1,43  Forged  Iron  and  Steel. 

16,  29,  69,  79,  128,  132,  137,  147,  153,  176,  219,  255,  269,  273,  312,  329, 
341,  343,  463.2,  463.6,  550,  553,  580,  597,  622,  662,  665,  668,  677. 

1.43  KoUed  and  Structural  Iron  and  Steel. 

16,  28;  29,  34,  41,  45,  73,  84,  137,  176,  ^48,  263,  324,  329,  349,  377,  434. 

435,  466,  547,  560,  571,  606. 
Iron  and  steel,  cts  per  lb : 

Refined  iron  bars  and  steel  bars,  ordinary  sizes. 

Angles,  ordinary  sizes,  T  shapes. 

Beams  and  channels,  structural  shapes.  )-    If  to  2^. 

Tank  plates,  structural  plates. 

Bessemer  machinery  steel. 
Steel  rails,  $28  per  ton  of  2240  lbs.     Old,  $15. 

1.431  Sheet  and  Plate  Iron  and  Steel. 

38,  41,  44,  73,  84,  189,  263,  324,  329,  382,  395,  435,  463.2,  471.5,  673,  675, 
681.  ^ 

Galvanized  iron  sheets : 
Discount,  60%  to  80%. 

Gage,  14  to  17  22  25  28  29  30 

Cts  per  lb,  12  to  13  14  16  17  19  21 

Extra,  for  additional  widths,  36  to  48  in,  1  to  4  cts  per  lb. 
Black  iron,  gage  16,  3  cts  per  lb;  gage  28,  about  4  cts. 

1.44  Bar  Steel. 

45,  84,  98,  189,  283,  301,  324,  329,  331,  342,  429,  435,  466,  518,  560,  612, 
662. 

1.45  Fastenings. 
28,  309. 

1.451  Nails  and  Spikes. 

40,  44,  45,  170.6,  202,  283,  329,  356,  434,  518,  621,  640. 
Nails,  etc.,  cts  per  lb:  Cut,  2  to  2i;  wire,  2^. 
Spikes,  railway.  If  to  2  cts  per  lb.  ;   i 

1A52  Rivets.  ^/-fffT 

28,  40,  81,  123,  154,  170.6,  262,  305,  310,  329,  356,  491,  529,  621. 

1.453  Screws. 

40,  81,  518,  629. 

1.454  Bolts  and  Nuts. 

28,  34,  40,  152,  237,  ■305,_  309,  329,  356,  431,  518,  519,  529. 

Bolts  and  nuts  for  machines,  price  per  100,  square  or  button  heads.  Length 
under  head,  2  ins.     Discount,  70%  to  75%.     See  list,  p.  884. 

Diameter,    ins i  ^  f  1 

Price  per  hundred $1.78  $3.86  $7.70  $16.00 

Extra  per  in  over  2  ins..   0.16  0.52  1.00  1.80 

1.455  Turnbuckles. 

157,  409,  518. 

Open,  price  each.     Discount,  67%. 

Rod,    ins ^12 

With  ends $0.80  $1.60  $5.35 

Without  ends 0.60  1.10  .3.10 

With  upset  ends,  30%  extra. 


ELEMENTARY  SHAPES.      MATERIALS.  987 

1.456  Washers. 

163,  305,  356,  529. 

1.457  Chains. 

28,  96,  100,  150,  262,  329,  398,  462,  585. 

American  coil  chaiu : 

Inch ^%  i  f  Ho  li 

Cts  per  lb 8  6  4  3.5 

1.46  Tubes. 

10,  321,  407.  424,  434,  556,  562.     See  also  9.25,  etc. 

1.47  Wire,  Wire  Rope,  and  Fencing. 

44,  332,  342,  355,  369,  384,  407,  530.5,  621,  623,  646. 
Hoisting  and  conveying  machinery,  see  3.5,  etc. 

1.471  Wire. 

44,  342,  407,  621,  623. 

Cts  per  lb : 

Iron.  Tinned.     Cast  Steel. 

Nos.  0  to  9 2.5  3.7  8.0 

No.   18    4.0  4.6  13.0 

1.473  Wire  Fencing. 

332,  355. 

For  the  simpler  patterns,  30  cts  to  $1  per  rod  (16^  ft),  according  to  style 
and  finish. 

1.473  Wire  Rope. 

36,  44,  369,  384,  646.  See  pages  976,  977.  Discoimt,  30%  and  7i%; 
galvanized,  25%  and  7i%. 

1.5  Other  3Ietals  and  Alloys. 

15.  46,  77,  407,  459,  471.5,  483,  609. 

Roll  and  sheet  brass,  random  lengths.     Discount,  20%  to  30%. 

Width  in  ins 2  to  18     18  to  24     24  to  32     32  to  40 

Cts  per  lb 22  to  30     29  to  39     36  to  49     50  to  75 

Extra  quality,  4  to  7  cts  per  lb  extra.     Bronze  metal,  7  cts  per  lb  extra. 

Soft  sheet  copper,  20  to  23  cts  per  lb  net  for  16  ozs  to  the  sq  ft  in  thickness, 
base  sizes. 

Lead,  pig,  4.3  to  4.4  cts  per  lb;  sheet,  6  cts.  Sheet  zinc,  6  to  7  cts  per  lb. 
Spelter,  3.75  to  4  cts  per  lb.  Antimony,  9  to  11  cts  per  lb.  Nickel,  56 
to  60  cts  per  lb.     Mercury,  about  $1.50  per  lb  in  large  lots. 

Tin  plates,  per  box  of  112  lbs: 

American  charcoal  plates,  $6  to  $8. 

American  coke  plates,  Bessemer,  $5  to  $7. 

American  terne  plates,  per  box  of  224  lbs,  $10  to  $12. 

1.6  Paper. 

Tar,  2  cts  per  lb,  50  to  70  cts  per  roll  of  108  sq  ft. 
Tarred  felt,  2^  cts  per  lb.     Straw  paper,  If  cts  per  lb. 
Rosin  sized  sheathing,  30  to  60  cts  per  roll  of  500  sq  ft. 


1.7  Ropes,  etc. 

36,  246. 

Rope,  cts  per  lb;  Manila,  plain  or  tarred,  10  to  12;  Sisal,  7  to  9;  hay,  7  to 
8;  cotton,  9  to  14;  jute,  6  to  7.  * 

Twine,  7  to  10.     Oakum,  best,  6  to  7. 

1.8  Packing,  Gasket,  Belting,  Lagging,  etc. 

25,  90,  256,  325,  334,  459,  632. 

Belting,  rubber,  cts  per  in  of  width,  per  ft  of  length:  2-ply,  7.5;  5-pIy, 
13.5;  8-ply,  21. 
Leather : 

Width,  ins  ...       1  6  12  24  72 

Per  ft   $0.12      $0.92  $1.88  $4.20  $15.00 

Pipe  jointing  supplies,  see  3.26. 


988  PRICE   LIST. 

2.0  Constructions. 

Water-works  supplies,  see  9.2.     Railroad  supplies,  9.1. 

3.1  Earthwork,  Dredging,  Foundations. 

2.11  Excavation  and  Embankment.  - 

See  also  pp.  800,  etc. 

Clearing  and  grubbing,  $50  to  $150  per  acre. 

Earth  excavation,  20  cts  to  $1  per  cu  yd.     In  trenches,  $1  to  $5,  according 

to  depth. 
Pipe  trenching,  see  also  pp.  658,  659. 

Embankment,  50  cts  to  $1.50  per  cu  yd.     Rolled,  $1  to  $2. 
Cut  and  fill,  10  to  25  cts  per  cu  yd. 
Sodding,  20  to  40  cts  per  sq  yd. 

Rock  excavation,  $1  to  $6  per  cu  yd,  depending  on  hardness  of  rock,  etc. 
Rock  filling,  $1  to  $4  per  cu  yd. 

2.12  Dredging. 

10  cts  to  $1  per  cu  yd,  according  to  length  of  haul,  etc. 
See  also  pp.  631,  632. 

2.13  Foundations. 
402,  419,  539,  600. 

Piles,  see  1.23. 

2.2  Masonry. 

2.21  Brick  Masonry. 

$10  to  $20  per  cu  yd.     $5  to  $10  per  1000  +  cost  of  bricks. 
Bricks,  see  1.351. 

2.22  Stone  Masonry. 

Dollars  per  cu  yd :  Rubble,  dry,  2  to  5 ;  in  cement,  3  to  6. 
Ashlar,  7  to  30.     Granite  coping,  30  to  45. 
Plain  cellar  work,  3  to  4. 
Stone,  see  1.32. 

2.23  Concrete  and  Cement  Masonry. 

In  place,  $4  to  $10  per  cu  yd. 

Cement,  see  1.34.     Floors  and  sidewalks,  2.4. 

2.24  Plastering. 

Three-coat  work,  25  to  50  cts  per  sq  yd. 

2.3  Metal  Structures. 

146,  215,  600,  603. 

2.31  Bridges. 

28,  70,  75,  91,  143,  146,  160,  161,  167,  190,  215,  216,  218,  328,  339,  344,  378, 
391,  394,  402,  419,  426,  439,  465,  466,  471,  472,  474,  482,  526,  539,  542, 
544,  598,  600,  603,  638,  639,  642.5. 

Riveters,  see  3.27.     Foundations,  2.13. 

2.32  Turntables. 

28,  91,  167,  190,  344,  394,  419,  466,  474,  482,  617,  639,  672. 

2.33  Tanks,  Stacks,  etc. 

64,  126,  146,  228,  250,  266,  302,  335,  463.4,  525,  542,  554,  615,  617,  633, 

648. 
Pipes,  see  9.25.     Wind  mills,  3.44. 
Stand-pipe,  22  X  60  ft,  including  foundations,  $4000  to  $7000. 

2.34  Boilers. 

21,  56,  180,  257,  266,  288,  294,  302,  335,  464,  467,  525,  545,  579,  617,  638. 

Boilers.     Upright  tubular.     Price  with  base  and  fixtures: 

HP 4  12  30  50 

Dollars 150  .225  375  550 

Riveters,  see  3.27.     Engines,  3.3. 


CONSTRUCTIONS.      MACHINERY.  989 

2*35  Fire-proofing,  Concrete-metal  Construction. 

50,  142,  223,  406,  410,    411,  530,  583. 

Concrete-metal  construction :  Concrete  and  wire,  flooring,  15  to  25  cts  per 
sq  ft,  exclusive  of  floor  beams.  For  covering  columns  and  girders,  10 
cts  per  sq  ft;  if  concreted,  15  cts.  Walls,  70  cts  to  $1.40  per  sq  yd. 
Wall  furring,  40  to  50  cts  per  sq  yd.    Wire  lathing,  12  to  20  cts  per  sq  yd. 

2.36  Wliarves,  Docl^s,  Harbor  Improvement. 

212,  330,  402,  533. 

Excavators,  etc.,  see  3.6.     Cement,  1.34. 

Dredging,  2.12. 

2.4  Paving. 

411,583. 

Per  sq  yd.     Macadam,  65  cts  to  $1 ;  brick,  $1  to  $2;  Belgian  block,  $2  to 

$3;  asphalt,  $2  to  $3. 
Cellar  floors,  $1.25  to  $2;  sidewalks,  $3  to  $5.50.     Cement,  see  1.34. 

2.5  Sewers. 

From  6  to  18  ins  diameter,  dollars  per  ft  run : 
Depth, 

ft    ...         5  10  13  15  17      .20 

0.30  to  0.90     0.65  to  1.00     0.90  to  1.20     1.30  to  1.40   .  1.50    2.00 

Laying  sewer  pipe,  cts  per  ft,  exclusive  of  excavations:  15  in,  30  to  60; 

4  in,  20  to  30. 
Brickwork  in  sewers,  $8  to  $12  per  cu  yd. 

2.6  Cliimneys. 

181. 

2.7  Roofing. 

Slate,  7  cts  and  upward  per  sq  ft.     Slag,  4  cts.     Tin,  6  to  8  cts.     Shingle, 

10  cts. 
Skylights,  complete  and  erected,  50  cts  per  sq  ft  and  upward. 

3.0  Macliinery. 

Testing  machines,  see  4.1.  Surveying  instruments,  4.0.  Miscella- 
neous, 9.0. 

3.1  Electrical  Machinery. 

253,  267,  454,  591,  660. 

Power  transmission,  mechanical,  see  3.51.  Blasting  apparatus.  See 
3.231. 

3.3  Tools,  Machine  Tools. 

Road-making  machinery,  see  3.68.     Pile  drivers,  3.66 

3.21  Small  and  Wood  Tools. 

53,  240,  498,  589. 
Shovels,  see  3.62. 

3.22  Machine  Tools. 

103.5,  375,  451,  498,  552.5,  674. 

3.23  Drills. 

32,  121,  148,  159,  194,  306,  315,  346,  400,  478,  481,  510,  531,  596,  605. 

Air  compressors,  see  3.43.     Explosives,  1.14. 

Rock  drills,  percussion : 

Diam  cylinder,  ins If  3i                    4^ 

Length  stroke,  ins 3f  6f                    71 

Depth  hole,  ft H  10  to  15         20  to  30 

Bottom  diam  hole,  ins 1  If                    2i 

Boiler  H  P  required 3  10                    15 

Price,  complete, $150  $300                $450  • 

Rock  drills,  diamond  (rotary).     Discount,  10%. 

Depth  hole,  ft 4000     1500  1000          600       400 

Diam  hole,  ins 2i%       2^  2j\            1^5         1^ 

Diam  core,  ins 2           l|  1^-               1            1 

Boiler  H  P  required 25          15  12  to  15        10       hand 

Card  price,  dollars 4000     2500  1900          1400      425 

Pumo.  extra,  dollars 3400  2800         1900 


990  PRICE    LIST. 

3.231  Blasting  Machinery, 

346,  492. 

Blasting  machine : 

To  fire  20  to    30  holes .♦ $25 

To  fire  75  to  100  holes : 75 

Connecting  wire,  40  cts  per  lb. 
Leading  wire,  li  cts  per  ft. 

3.;^  Presses. 

172,  242.5,  645. 

3.35  Punclies,  Shears. 
74,  375. 

3.36  Pipe-cutting,  Tapping,  and  Jointing  Machinery. 

3.361  Cutters. 

404,  541. 

3.363  Tappers. 

224,  466.5,  563. 

For  dry  pipe,  $20  to  $30  each.     For  pipes  under  pressure,  $100  to  $200 
each. 

3.363  Jointers. 

643. 

2-in,  $2  each;  12-in,  $9;  36-in,  $16;  72-in,  $44. 

3.37  Riveters. 

12,  16,  292,  478,  674. 

3.3  Engines,  Locomotives,  Cars. 

Pumps,  see  3.4.     Road  rollers,  3.681.  Boilers,  2.34. 

Portable  engines; 

H  P                    Cylinder.  Boiler  Diam.             Price. 

10                       7X8  ins  29  ins                      $600 

25                      9X10     "  35     "                        850 

50                    13  X  15     "  44     "                     -1300 

3.31  Stationary  and  Hoisting  Engines. 

33,  125,  136,  171.4,  200,  204,  231,  234,  241,  251,  257,  259,  333,  354,  370, 
397,  427,  512,  536,  545,  572,  602. 

Hoisting  engines : 

Single  cylinder,  friction  drum,  but  no  foot  brake,  4  to  6  H  P,  $250  to  $400. 
Double  cylinder: 

12  H  P     20  H  P     50  H  P 
Friction  drum  and  foot  brake,  or  re- 
versible         $600        $1100       $1300 

Same,  with  boiler 1000  2000         3000    • 

2-winch  engine 600  100  1300 

6-winch  engine 900  1400 

Single  cylinder  stationary  engines : 

HP 12  30  50  100  200         300 

From    $300         $450         $650      $1100       $2000     $3100 

To 450  600  800         1400         2000       3400 

With  base,  about  10%  extra. 

Portable  engines  on  wheels,  complete,  6  to  15  H  P,  $700  to  $1500. 

3.33  Locomotives. 

102,  103,  124,  171.6,  199,  371,  387,  486,  493,  520,  524,  543. 

3.33  Gas  and  Gasoline  Engines. 

47.  57,  172.2,  184,  247,  415,  440,  463.4,  581,  584,  593,  595,  601,  661,  674. 

3.34  ■  Oil  Engines. 

500. 

3.35  Water  Engines  and  Motors,  Turbines. 

18,  57,  365,  469,  514,  522,  565,  590,  610,  633,  674. 
Water-works  supplies,  see  9.2.     Pumps,  3.4. 

3.36  Cars. 

*29,  53.5.  102,  108,  278,  351,  405,  486,  499,  509,  571,  672. 


MACHINERY.      ENGINES.  991 

3.37  Wagons. 

160,  644. 
3.4  Blowing  and  Pumping  Machinery. 

3.41  Blowers,  Forges. 

572,  604. 

3.4:3  Mechanical  Draft. 

604. 

3.43  Air  Compressors. 

47,  156,  170.2,  315,  346,  353,  400,  443,  457,510,  552,  567,  594,  605. 

3.44  Wind  Mills. 

250,  633. 

3.45  Hydraulic  Bams. 

207,  265,  497. 

Drive  pipe : 

Diam,  ins 1  2  4  6 

Dollars,  net 5  to  7         10  to  12         35  to  40         60  to  70 

Certain  manufacturers  make  much  heavier  and  more  elaborate  machines 
at  several  times  these  prices. 

3.46  Pumps. 

20,  58,  59,  62,  82,  83,  129,  182,  183,  186,  187,  188,  200,  208,  221,  222,  231, 
261,  279,  284,  296,  297.5,  303,  319.5,  320.5,  345,  348,  353,  359,  396,  423, 
503,  504,  507,  508,  522,  532,  538,  551,  555,  567,  568,  572,  594,  674,  676. 

Prices  in  dollars  each : 


Capacity   in    gals 

per  min 5 

20 

100 

500 

2500 

10,000 

Single  cylinder: 

Boiler  feed  .  .  50 

200 

400 

Tank  and  low 

lift    

150 

350 

600 

Duplex  : 

High  pressure 

250 

750 

Low  pressure 

150 

500 

Centrifugal   . 

50  to  75 

80  to  175 

200  to  450 

800  to  1300 

3.5  Hoisting  and  Conveying  Machinery. 

234,  257,  344,  370,  373,  522. 

Electrical,  see  3.1.  Excavating  machinery,  3.6.  Hoisting  engines, 
3.31.     Cars,  3.36.     Wire  rope,  1.473. 

3.51  Power  Transmission. 

173,  205,  373. 

3.53  Cranes,  Derricks. 

33,  67,  104,  136,  140,  148,  171.4,  204,  293,  314,  333,  344,  674. 

3.53  Pulley  Blocks  and  Trucks. 

89,  333. 

3.54  Elevators,  Hoists. 

65,  88,  148,  171.4,  322,  370,  373,  390,  441. 
•     Hoisting  crabs  on  winches,  i  to  2^  tons  capacity,  S35  to  $100. 
Differential  hoists,  i  to  3  tons  capacity,  $10  to  $15  to  $40  each. 

3.55  Jacks. 

148,211,455,645. 

3.56  Tramways,  Conveyors. 

65,  88,  104,  127,  171,  251,  307,  322,  333,  344,  351,  354,  369,  370,  373.  405. 

441,  527. 
Track,  see  9.1.     Cars,  3.36.     Wire  rope,  1.473. 

3.6  Excavators,  Dredges,  Road  and  General  Construction 

Machinery,  etc. 

Hoisting  and  conveying,  see  3.5.  Wagons,  3.37.  Hoisting  engines, 
3.31.  Drills,  3.23.  Explosives,  1.14.  Excavation  and  embankment, 
2.11. 


L  xvxv^xj     ±j±aj. 


3.61  Excavators. 

67,  107,  333,  389,  405,  613,  618,  678. 

3.63  Trenching  Machinery. 

67,139,333,389,421,49^. 

3.63  Scrapers,  Plows,  etc. 

127,  160,  333,  558,  607,  658. 

Wheeled  scrapers.     Discount,  20% .     $40  to  $60  each. 

Drag  scrapers.     Discount,  50%  to  60%. 

Ordinary,  $10  to  $15  each.     Fresno  or  back,  $35  to  $40  each. 

Plows.     Discount  20%. 

Horses    2  4  6  8  10  12 

Each,  dollars,     15  to  30       30  35  40  45  55 

Hardpan  plow,  $70. 

3.64  Dredging  Machinery. 

52,  67,  94,  107,  222,  241,  293,  389,  447,  463,  539,  566. 
Pumps,  see  3.4. 

3.65  Diving  and  Diving  Apparatus.  / 

236,  425,  546. 

Air  pumps.     Discount  10%.     Each,  $125  to  $400,  according  to  depth. 

Helmets,  $100.  Rubber  suits,  40.  Weights,  underwear,  line,  tubing,  re- 
pair materials,  etc.,  per  outfit,  $110  to  $145.  i 

Complete  outfit :  Deep  sea,  $700  to  $750 ;  moderate  depths,  $550  to  $6001 
shallow  water,  $350  to  $400.  ] 

3.66  Pile-driving  Machinery. 

136,  314,  333,  641. 

3.67  Wells  and  Well-drivinfe  Machinery. 

47,  171.8,  282,  480,  484,  666. 
Drills,  see  3.23. 

3.68  Road-making  Machinery,  etc. 

13,  39,  160,  545. 

Rock  drills,  etc.,  see  3.23. 

3.681  Road  Rollers. 

110,  230,  290,  295,  320,  333,  337,  545. 

3.683  Concrete  Mixers. 

164,  171.2,  209,  230,  245,  260,  320,  333,  511,  545. 
Each,  $225,  $500  and  upward  to  about  $12i0. 

3.683  Rock  and  Ore  Crushers. 

55,  105,  155,  160,  194,  238,  264,  275,  333,  346,  390,  513,  658,  664,  680. 


Receiving 

Capacity, 

HP 

Price, 

Capacity, 

Tons  per 

Required. 

Dollars 

Inches. 

Hour. 

UX3 

Hand 

30 

7X10 

4  to  6 

8  to  10 

500 

10X20 

12  to  15 

18  to  20 

1000 

13X60 

40  to  50 

40  to  50 

4000 

24  X  72 

160  to  200 

150  to  180 

13000 

3.7  Heating,  Ventilating,  and  Refrigerating. 

26,  318,  604. 

4.0  Engineering,  Surveying,  and  Scientific  Instruments  and 

Supplies. 

14,  68,  72,  99,  120,  196,  201,  239,  243,  285,  297,  304,  340,  347,  350,  364,  367, 

386,  473,  501,  506,  523,  528,  569.5,  582,  625,  667,  679. 
Prices,  see  below. 

4.1  Testing  3Iachines. 

235,  460,  475,  521. 

4.2  Surveying  Instruments. 

Dealers,  see  4.0.     Prices,  see  below. 


INSTRUMENTS.     SUPPLIES.  993 

4.31  Transits,  Plane  Tables,  Compasses,  etc.     . 

Transits,  plain,  $150  to  $200.     Engineers',  complete,  $200  to  $300.     Min- 
ing, $200  to  $300  and  $700.     Mountain,  $150  to  $300. 

Theodolites  and  portable  alt-azimuth  astrorfcmical  instruments,  $500  to 
$1000. 

Solar  attachments,  $50  to  $70.     Sextants,  $60  to  $150.     Pocket  sextants, 
$50. 

Plane  tables,  complete,  $150  to  $300.     Compasses,  pocket,  $10  to  $25. 
4.33  Levels. 

Engineers'  levels,  $100  to  $200.     Hand  levels,  $5  to  $18;  usually  about  $9. 
4.33  Rods,  Tapes,  etc. 

381,  450. 

Leveling  rods,  $13  to  $16  each.     Range  poles,  $2  to  $5. 

Tapes,  from  5  to  10  cts  per  ft,  depending  upon  graduation  and  length. 

Chains,  8  to  12  cts  per  ft. 
4.39  Miscellaneous. 

Current  meters,  $65  to  $100.     Direction  meters,  $200  to  $250.     Velocity 
register  and  timepiece,  $50  to  $60. 

Hook  gages,  $15  to  $60. 

4.3  Computing  Instruments. 
317. 

Planimeters,  $25  to  $125. 

Slide  rules,  plain  Mannheim,  $1  to  $5;  other  forms,  $1  to  $50. 

Computing  machines,  $100  to  $300. 

4.4  Drawing  Instruments  and  Materials. 

24,  201,  243,  340,  386,  506,  569.5,  667. 

Drawing  instruments,  $10  to  $30  per  set,  very  elaborate  sets  as  high  as  $60 

and  even  $100.     Drawing  pens,  $1  to  $3  each.     Compasses  (drawing), 

$3  to  $9.     Dividers,  $2  to  $4.50. 
Triangular  boxwood  scales,  ordinary,  12  ins  long,  $1  to  $3. 
Metal  straight  edges,  36  ins  long,  $4  to  $5;  shorter,  as  low  as  $1.50. 
T-squares,  36  ins,  with  celluloid  edges,  about  $2.25;  wood,  $1  to  $2.40; 

steel,  $7  to  $10,  usually  with  adjustable  angle. 
Triangles,  celluloid,  6  ins,  50  cts;  12  ins,  $1.10.     French  curves,  celluloid, 

50  cts  to  $1.50. 
Protractors,  German  silver,  4  ins,  50  cts  to  $1.50;  6  ins,  about  $3;  with 

arms,  6  ins,  $9;  8  ins,  $20. 
Drawing  inks,  25  cts  per  bottle. 

4.5  Heliography. 
363,  438,  569.5,  573. 

Black-line  prints,  Le  Clare's,  10  cts  per  sq  ft.     Nigrosine,  8  cts. 

Blue  prints,  3  cts  per  sq  ft. 

Blue-print  paper,  30  ins  wide,  10  to  20  cts  per  yd. 

Blue-print  frames,  hand,  from  $2  for  10  X  12  ins  to  $40  for  36  X  60  ins. 

4.6  Tents,  etc. 
428,  637. 

Wall  tents.     Discount,  55%. 

Size,  Height, 

Ft.  Ft. 

9X9  7i 

12  X  12  8 

18X24  11 

24X28  13  105  205)  ^„„. 

30X70  15  265  515  T^P®^* 

U  S  folding  canvas  cot,  $1.50  to  $2. 

4.7  Drawing  Tables. 

19,  226,  357. 

4.8  Testing  Laboratories. 
308,  399,  488. 

9.0  Miscellaneous  Supplies. 

9.1  Railroad  Supplies. 

108,  158,  202,  300,  351,  405,  479,  508.5,  655,  664.5,  672. 
Bridges,    see    2.31.     Earthwork,    2.1.     Locomotives,    3.32.     Cars,    3.36. 
Tramways,  3.56. 

63 


8-oz. 

15-oz. 

Duck. 

Duck. 

$11 

$25 

15 

35 

44 

95 

105 

205) 

265 

515/ 

994  PRICE   LIST. 

9.11  Rails. 

45,  329,  434,  472,  495. 
Stegl  rails,  $28  per  ton.     01^,  $15. 
9.13  Joints. 

87,  128,  170.8,  329,  461,  653. 
1.5  to  2.6  cts  per  lb. 

9.13  Switches,  Frogs,  and  Signals. 

60,  158,  225,  286,  472,  509,  587,  611,  628,  655,  664.5,  672. 

9.14  Ties. 
66,  494,  599,  636. 

White  oak,  50  to  70  cts  each ;  chestnut,  35  to  45 ;  yellow  pine,  45  to  65. 
Tie  plates,  5  to  15  cts  each. 

9.15  Spikes. 

202,  283,  329,  356,  518,  640.     If  to  2  cts  per  lb. 
9.3  Hydraulic  Supplies. 

Gaskets,  see  1.8.     Stand  pipes,  tanks,  2.33.     Pipe-cutting,  tapping,  and 
jointing  machines,  3.26. 
9.31  Water-softening  Plants. 

313,  654. 
9.33  Filters. 

193.  288,  376,  385,  446,  534,  542. 

9.33  Water-wheel  Governors. 
374. 

Water-wheels,  see  3.35. 

9.34  Water  Meters. 

109,  122,  244,  298,  432,  436,  487,  590,  614,  629,  676. 

All  types  (except  piston ;  see  below) : 

Nom'l  diam,  pipe,  ins f  or  ^      1  3  6  12 

Dollars  each,  from 9         18  95         300         1000 

"      to 12         26         125         500         1500 

Reciprocating  piston  meters,  about  50%  more. 

Venturi  meters,  see  pp.  532,  etc.     Current  meters,  4.29. 

9.35  Pipe. 

37,  95,  169,  206,  362,  490,  514. 
Pipe-cutting  machinery,  etc.,  see  3.26. 
Pipe  laying,  see  pp.  658,  659. 
Lead-  and  tin-lined  iron  pipe : 

Lead-lined,  5  Cts  per  Lb.  Tin-lined,  10  Cts  per  Lb. 

Discount,  25%  10% 

Size  Price  Price 

IN  Ins.  per  Ft.  per  Ft. 

1  $0.30  $0.55 

2  0.65  1.35 
4  1.72  3.00 
6                                      3.28                                        4.25 

Block  tin  pipe,  35  cts  per  lb. 

Lead  pipe,  5  cts  per  lb. 

Boiler  tubes.     Discount,  40%  to  50%.     Prices,  see  p.  882. 

Seamless  brass  tubes,  base  price,  20  to  25  cts  per  lb.     Extras,  see  p.  919. 

Spiral  riveted  pipe,  in  dollars  per  linear  ft;     Discount,  40%. 

Diam,  Ins.  Black.  Asphalted.       Galvanized. 

Thickness,  0.022  in. 
3  0.20  0.23  0.30 

6  0.33  0.39  0.50 

12  0.68  0.80  1.05 

Thickness,  0.049  in. 
3  0.34  0.37  0.46 

6  0.57  0.63  0.85 

12  1.15  1.27  1.65 

Thickness,  0.109  in. 
6  1.25  1.31  1.90 

12  2.50  2.62  3.25 

24  4.70  4.94  6.25 


HYDRAULIC   SUPPLIES.  995 

9.251  Cast-iron  Pipe. 

29,  162,  203,  254,  270,  392,  416,  630,  640,  647,  674. 

Cast-iron  water  pipe,  $24  per  ton.     Pipe  aftd  laying,  $30  to  $35  per  ton. 
See  also  pp.  658,  659,  875,  876. 
9.253  Steel  Pipe. 

11,  42,  138,  270,  403,  469,  542. 
Steel  pipe,  4  cts  per  lb  at  works, 

9.253  Wrought-iron  Pipe. 

Wrought-iron  pipe,  per  ft.  Discount,  l^-in  and  smaller,  50%;  2-in  and 
larger,  60%.     Prices,  see  p.  882. 

9.254  Wood  Pipe. 
Prices  in  cts  per  ft : 

Inner  diam,  ins.  .1  3  6  10  16  30  84 

Plain,  square  ...    5  15         40 

Plain,  round 5  20         45 

Strengthened  for 

40  lbs  persq  in .  1 2  25         50 

160   "     "    ••    "  .18  30         70 
Wood  stave  pipe, 

40  lbs persq in.  50  90         200         350  1200 

160  "     *'    "    ••  .  70  140         270 

9.255  Sewer  Pipe,  etc. 

31,  80,  130,  175,  197,  232,  291,  416,  420,  448,  608,  619,  631. 
Sewer  pipe.     Discount,  70  to  80%.     Prices,  see  p.  575. 

9.256  Hose. 
90. 

Water  hose,  price  per  in  of  internal  diameter,  per  ft  of  length  : 

2-PLY  4-PLy  6-PLY 

S0.333  $0.50  $0.75 

Other  plies  at  nearly  proportional  rates. 
Air,  hot  water,  and  steam  hose,  price  per  in  of  internal  diam,  per  ft  of 
length : 

4-PLY  6-PLY  8-PLY 

$0.83  $1.24  $1.66 

Other  plies  at  nearly  proportional  rates. 

9.26  Hydrants  and  Valves. 

93,  141,  144,  162,  165,  166,  172.8,  210,  220,  254,  323,  338,  379,  401,  412, 
458,  489,  490,  514,  516,  563,  615,  627,  669. 

Gate  valves,  iron  body,  bronze  mounted,  bell  or  spigot  ends.  Discount, 
40%. 

Diam,   ins 4  6  12  18  24  36  48 

Dollars  each 20         30         90         200         350         900         2000 

Single  gate,  iron  body,  low  pressure  steam  and  water.     Discount,  40%. 

Diam,   ins 4  6  12  18  24  48 

Screw  ends $15         $25         $85 

Flange  ends 20  30  80         $160^       $280       $1600 

Double  gate,  iron  body,  brass  mountings,  steam  or  water.     Discount,  55%. 

Diam,   ins 2  4  6  12 

Screw  or  flange $10  $20  $30  $100 

Double  gate,  iron  body,  composition  or  bronze  mounted,  for  heavy  pres- 
sures (about  500  lbs  per  sq  in) : 

Diam,  ins 2  4  6  12 

From    $10       $20  $35  $90  Discount,  75% 

To 15         30  60  150  Discount,  60% 

Double  gate  valves,  all  bronze.     Discount,  50%. 

Screw  ends $1.40     $2.35  $6.25      $34.00  $76.00 

Flange  ends $3.40       4.15"        11.00        43.00  88.00 

Fire  hydrants.     Discount,  30%.     5i-in  stand  pipe,  $38;  6i-in,  $47. 

Compression  fire  hydrants.  Discount,  20%.  Length  iFrom  pavement  to 
bottom  of  connections,  5  ft.  1  to  2i-in  nozzles,  $28 ;  2  to  2i-in,  $33.  Add 
or  deduct  $1  for  each  6  ins  difference  in  height  from  5  ft.  2i-in  nozzles, 
$2.     Frost  case,  $5. 

9.27  Anti-bursting  Devices. 
48.5. 


996  BUSINESS   DIRECTORY. 

BUSINESS    DIRECTORY. 

10  Abb^,  Max  F.—  Mfg.  Co.,  220  Broadway,  New  York. 

11  Abendroth  &  Root  Mf^.  Co.,  99  John  St.,  New  York. 

12  Acme  Machinery  Co.,  Bolt  and  Rivet  Headers,  Cleveland,  Ohio. 

13  Acme  Road  Machinery  Co.,  Frankfort,  N.  Y. 

14  Ainsworth,  Wm. —  &  Sons,  Pocket  Transit,  Denver,  Col. 

15  Ajax  Metal  Co.,  Bearings,  Ingots,  Philadelphia. 

16  Albree,  Chester  B.—  Iron  Works,  14  to  30  Market  St.,  Allegheny,  Pa. 
16.5  Albro,  E.  D. —  Co.,  Hardwood  Lumber,  Cincinnati,  Ohio. 

17  Alcatraz  Co.,  Alcatraz  Asphalt,  Crocker  Bldg.,  San  Francisco,  Cal. 

18  Alcott's  High  Duty  Turbine  Water-wheel,  Mount  Holly,  N.  J. 

19  Alexander,  J.  S. —  Mfg.  Co.,  Grand  Rapids,  Mich. 
19.5  Allis-Chalmers  Co.,  Milwaukee.  Wis. 

20  Allis,  Edward  P.—  Co.,  Milwaukee,  Wis.     See  19.5. 

21  Almy  Water-tube  Boiler  Co.,  178. to  184  Aliens  Ave.,  Providence,  R.  I. 

22  Alpha  Portland  Cement  Co.,  Easton,  Pa. 

23  Alsen's  German  Portland  Cement.  143  Liberty  St.,  New  York. 

24  Alteneder,  T.—  &  Sons,  Philadelphia. 

25  American  Belting  and  Packing  Co.,  35  Warren  St.,  New  York. 

26  American  Blower  Co.,  Detroit,  Mich. 

27  American  Brake  Shoe  Co.,  Chicago. 

28  American  Bridge  Co.,  100  Broadway,  New  York. 

29  American  Car  and  Foundry  Co.,  St.  Louis,  Mo. 

SO  American  Cement  Co.,  22  S.  Fifteenth  St.,  Philadelphia. 

31  American  Clay  Mfg.  Co.,  Akron,  Ohio. 

32  American  Diamond  Rock  Drill  Co.,  120  Liberty  St.,  New  York. 

33  American  Hoist  and  Derrick  Co.,  St.  Paul,  Minn. 

34  American  Iron  and  Steel  Mfg.  Co.,  Lebanon,  Pa. 
34.5  American  Locomotive  Co.,  25  Broad  St.,  New  York. 

35  American  Lucol  Co.,  44  Broadway,  New  York. 

36  American  Mfg.  Co.,  65  Wall  St.,  New  York. 

37  American  Pipe  Mfg.  Co.,  112  N.  Broad  St.,  Philadelphia. 

38  American  Pressed  Steel  Co.,  333  Walnut  St.,  Philadelphia. 

39  American  Road  Machine  Co.,  Kennett  Square,  Pa. 

40  American  Screw  Co.,  Providence,  R.  I. 

41  American  Sheet  Steel  Co.,  Battery  Park  Bldg.,  New  York. 

42  American  Spiral  Pipe  Works,  64-66  Wabash  Ave.,  Chicago. 

43  American  Steam  Gage  Co.,  Jamaica  Plain,  Boston. 

44  American  Steel  and  Wire  Co.,  Chicago  and  New  York. 

45  American  Steel  Hoop  Co.,  Battery  Park  Bldg.,  New  York. 

46  American  Tin  Plate  Co.,  Battery  Park  Bldg.,  New  York. 

47  American  Well  Works,  Aurora,  111. 

48  American  Wood  Preserving  Co.,  806  Chestnut  St.,  Philadelphia. 

48.5  Anti-bursting  Pipe  Co.,  Pneumatic  Dome  to  Prevent  Bursting  from 
Freezing,  Standard  Bldg.,  Pittsburg,  Pa. 

49  Ashton  Valve  Co.,  Pop  Valves,  etc.,  Boston. 

50  Associated  Expanded  Metal  Cos.,  New  York,  Phila.,  Chicago,  etc. 

51  Assyrian  Asphalt  Co.,  Chicago. 

52  Atlantic  Gulf  and  Pacific  Co.,  Park  Row  Bldg.,  New  York. 

53  Atlantic  Works,   Incorporated,  Wood-working  Machinery,  Twenty- 

third  and  Arch  Sts.,  Philadelphia. 
53.5  Atlas  Car  and  Mfg.  Co.,  Cleveland,  Ohio. 

64  Atlas  Portland  Cement  Co.,  143  Liberty  St.,  New  York. 

65  Austin,  F.  C—  Mfg.  Co.,  Harvey,  111. 

56  Babcock  &  Wilcox  Co.,  29  Cortlandt  St.,  New  York. 

67  Backus  Water  Motor  Co.,  Newark,  N.  J. 

68  Bacon  Air  Lift  Co.,  100  Broadway,  New  York. 

59  Baldwinsville  Centrifugal  Pump  Works,  Syracuse,  N.  Y. 

60  Baltimore  Enamel  and  Novelty  Co.,  Signals,  Baltimore,  Md. 

61  Barber  Asphalt  Paving  Co.,  11  Broadway,  New  York. 

62  Barr  Pumping  Engine  Co.,  Germantown  Junction,  Philadelphia. 

63  Barschall  Impregnating  Co.,  31  Nassau  St.,  New  York. 

64  Bartlett,  Hayward  &  Co.,  Baltimore,  Md. 

65  Bartlett,  C.  O.—  &  Co.,  Cleveland.  Ohio. 

66  Baxter,  G.  S.—  &  Co.,  18  Wall  St.,  New  York. 

67  Beatty,  M. —  &  Sons,  Welland,  Ontario. 

68  Beckmann,  L. — ,  Toledo,  Ohio. 

69  Belden  Machine  Co.,  New  Haven,  Conn. 


BUSINESS   DIRECTORY.  997 

70  Bellefontaine  Bridge  and  Iron  Co.,  Bellefontaine,  Ohio. 

71  Belle  City  Malleable  Iron  Co.,  Racine,  Wis. 

72  Berger,  C.  L. —  &  Sons,  Boston. 

73  Berger  Mfg.  Co.,  Sheet  Flooring,  Canton,  Ohio. 

74  Berlin-Erfurt  Machine  Works,  66  and  68  Broad  St.,  New  York. 

75  Berlin  Iron  Bridge  Co.,  East  Berlin,  Conn. 

76  Berry  Bros.,  Limited,  Varnishes,  Detroit,  Mich. 

77  Bethlehem  Steel  Co.,  South  Bethlehem,  Pa. 

79  Billings  &  Spencer  Co.,  Hartford,  Conn. 

80  Blackmer  &  Post  Pipe  Co.,  St.  Louis,  Mo. 

81  Blake  &  Johnson,  Waterbury,  Conn. 

82  Blake,  George  F.—  Co.,  91  Liberty  St.,  New  York. 

83  Bliss,  E.  W.—  Co.,  Brooklyn,  N.  Y. 

84  Boker,  Hermann  —  &  Co.,  103  Duane  St.,  New  York. 

86  Bonneville  Portland  Cement  Co.,  1307  Real  Estate  Trust  Bldg.,  Philau 

87  Bonzano  Rail  Joint  Co.,  22  S.  Fifteenth  St.,  Philadelphia. 

88  .  Borden  &  Selleck  Co.,  Chicago. 

89  Boston  and  Lockport  Block  Co.,  146  Commercial  St.,  Boston. 

90  Boston  Belting  Co.,  256  Devonshire  St.,  Boston. 

91  Boston  Bridge  Works,  70  Kilby  St.,  Boston. 

92  Bound  Brook  Stone  Crushing  Co.,  Trap  Rock,  Bound  Brook,  N.  J. 

93  Bourbon  Copper  and  Brass  Works,  618  and  620  E.  Front  St.,  Cin..  O. 

94  Bowers,  A.  B.— ,  Hydraulic  Dredging,  First  Nat.  Bank  Bldg.,  San 

Francisco,  Cal. 

95  Bowes  &  Co.,  23  Lake  St.,  Chicago. 

96  Bradlee  &  Co.,  Philadelphia. 

97  Bradley  Pulverizer  Co.,  Boston. 

98  Braeburn  Steel  Co.,  Braeburn,  Pa. 

99  Brandis,  F.  E.—  Sons  &  Co.,  812  Gates  Ave.,  Brooklyn,  N.  Y. 

100  Bridgeport  Chain  Co.,  Bridgeport,  Conn. 

101  Brier  Hill  Iron  and  Coal  Co.,  Youngstown,  Ohio. 

102  Brill,  J.  G.—  Co.,  Philadelphia. 

103  Brooks  Locomotive  Works,  Dunkirk,  N.  Y.     See  34.5. 
103.5  Brown  &  Sharpe  Co.,  Providence,  R.  I. 

104  Brown  Hoisting  Machinery  Co.,  Incorporated,  Cleveland,  Ohio. 

105  Buchanan,  C.  G.— ,  141  Liberty  St.,  New  York. 

106  Buckeye  Portland  Cement  Co.  Bellefontaine,  Ohio. 

107  Bucyrus  Co.,  South  Milwaukee,  Wis. 

108  Buda  Foundry  and  Machine  Co.,  Hand  Cars,  Crossing  Gates,  Harvey, 

109  Buffalo  Meter  Co.,  363  Washington  St.,  New  York. 

110  Buffalo  Pitts  Co.,  Buffalo,  N.  Y. 

120  Buff  &  Buff,  Transits  and  Levels,  Boston. 

121  Bullock,  M.  C—  Mfg.  Co..  Diamond  Drills,  Chicago. 

122  Builders'  Iron  Foundry,  Venturi  Meter,  Providence,  R.  I. 

123  Burden  Iron  Co.,  Troy,  N.  Y. 

124  Burnham,  Williams  &  Co.,  Baldwin  Locomotive  Works,  Philadelphia. 

125  Byers,  John  F. —  Machine  Co.,  Ravenna,  Ohio. 

126  Caldwell,  W.  E.—  Co.,  Louisville,  Ky. 

127  California  Wire  Works,  9  Fremont  St.,  San  Francisco,  Cal. 

128  Cambria  Steel  Co.,  Philadelphia. 

129  Cameron  Steam  Pump  Works,  New  York. 

130  Camp,  H.  B. —  Co.,  Akron,  Ohio. 

131  Canton  Steel  Co.,  Tool  Steel,  etc..  Canton,  Ohio. 

132  Cape  Ann  Tool  Co.,  Pigeon  Cove,  Mass. 

133  Carbolineum  Wood  Preserving  Co.,  13  Park  Row,  New  York. 

134  Carbon  Steel  Co.,  Pittsburg,  Pa.  . 

135  Carborundum  Co.,  Niagara  Falls,  N.  Y. 

136  Carlin's,  Thomas —  Sons  Co.,  Allegheny,  Pa. 

137  Carnegie  Steel  Co.,  Carnegie  Bldg.,  Pittsburg,  Pa. 

138  Carroll-Porter  Boiler  and  Tank  Co.,  Pittsburg,  Pa. 

139  Carson  Trench  Machine  Co..  Boston. 

140  Case  Mfg.  Co.,  Columbus,  Ohio. 

141  Cayuta  Wheel  and  Foundry  Co.,  Sayre,  Pa. 

142  Central  Fire-proofing  Co.,  Fire-proofing  System,  874  Broadway,  Nei? 

York. 

143  Champion  Bridge  Co.,  Wilmington,  Ohio. 

144  Chapman  Valve  Mfg.  Co.,  Indian  Orchard,  Mass. 


998  BUSINESS   DIRECTORY. 

145  Chester  Steel  Castings  Co.,  407  Library  St.,  Philadelphia. 

146  Chicago  Bridge  and  Iron  Co.,  Chicago. 

147  Chicago  Drop  Forging  and  Foundry  Co.,  Kensington,  111. 

148  Chicago  Pneumatic  Tool  Co.,  Painting  Machines,  Casting  Cleaners* 

Chicago. 

149  Chicago  Portland  Cement  Co.,  513  Stock  Exchange  Bldg.,  Chicago. 
149.5  Chicago  Tie  Preserving  Co.,  Chicago. 

150  Chillcott-Evans  Chain  Co.,  Allegheny,  Pa. 

151  Chrome  Steel  Works,  Crucible  Steel  Castings,  Brooklyn,  N.  Y. 

152  Church,  Isaac — ,  Expansion  Bolts,  Toledo,  Ohio. 

153  Clapp,  E.  D.—  Mfg.  Co.,  Auburn,  N.  Y. 

154  Clark  &  Cowles,  Plainville,  Conn. 

155  Clark,  J.  C— ,  Atlanta,  Ga. 

156  Clayton  Air  Compressor  Works,  26  Cortland t  St.,  New  York. 

157  Cleveland  City  Forge  and  Iron  Co.,  Cleveland,  Ohio. 

158  Cleveland  Frog  and  Crossing  Co.,  Cleveland,  Ohio. 

159  Cleveland  Twist  Drill  Co.,  Lake  and  Kirkland  Sts.,  Cleveland,  Oh^. 

160  Climax  Road  Machine  Co.,  Marathon,  N.  Y. 

161  Clinton  Bridge  and  Iron  Works,  Clinton,  Iowa. 

162  Clow,  James  B. —  &  Sons,  Pipe  and  Specials,  Lake  and  Franklin  Sts., 

Chicago. 

163  Cobb  &  Drew,  Washers,  etc.,  Plymouth,  Mass.,  and  Rock  Falls,  111. 

164  Cockburn  Barrow  and  Machine  Co.,  240  Eleventh  St.,  Jersey  City, 

N.  J. 

165  Coffin  Valve  Co.,  Boston. 

166  Coldwell-Wilcox  Co.,  Newburgh,  N.  Y. 

167  Columbia  Bridge  Co.,  Pittsburg,  Pa. 

168  Columbian  Powder  Co.,  Hamilton  Bldg.,  Pittsburg,  Pa. 

169  Colwell  Lead  Co.,  63  Center  St.,  New  York. 

170  Commercial  Wood  and  Cement  Co.,  156  Fifth  Ave.,  New  York. 
170.2  Compressed  Air  Co.,  621  Broadway,  New  York. 

170.4  Continental  Compressed  Air  Power  Co.,  Bourse  Bldg.,  Philadelphia. 
170.6  Continental  Iron  Co.,  Niles,  Ohio. 

170.8  Continuous  Rail  Joint  Co.  of  America,  908  Lawyers'  Bldg.,  Newark, 
N.  J. 

171  Consolidated  Telpherage  Co.,  20-22  Broad  St.,  New  York. 
171.2  Contractors'  Plant  Co.,  172  Federal  St.,  Boston. 

171.4  Contractors'  Plant  Mfg.  Co.,  Buffalo,  N.  Y. 

171.6  Cooke  Locomotive  and  Machine  Co.,  Paterson,  N.  J.     See  34.5. 
171.8  Cook  Well  Co.,  System  of  Wells,  St.  Louis,  Mo. 

172  Cornell,  J.  B.  &  J.  M. — ,  Hydraulic  Presses,  etc.,  Twenty-sixth  St.  and 

Eleventh  Ave..  New  York. 
172.2  Cornell  Machine  Co.,  176  W.  Superior  St.,  Chicago. 

172.5  Corning  Brake  Shoe  Co.,  Corning,  N.  Y. 
172.8  Crane  Co.,  Chicago. 

173  Cresson,  George  V. —  Co.,  Eighteenth  St.  and  Allegheny  Ave.,  Phila. 

174  Crosby  Steam  Gage  and  Valve  Co.,  Boston. 

175  Crown  Fire  Clay  Co.,  Akron,  Ohio. 

176  Crucible  Steel  Co.  of  America,  Empire  Bldg.,  Pittsburg,  Pa. 

177  Cumberland  Hydraulie  Cement  and  Mfg.  Co.,  Cumberland,  Md. 

178  Cummer,  F.  D.—  &  Son  Co.,  Driers,  Cleveland,  Ohio. 

179  Cummings  Cement  Co.,  Ellicott  Square  Bldg.,  Buffalo,  N.  Y. 

180  Cunningham  Iron  Co.,  Summer,  B,  and  Fargo  Sts.,  Boston. 

181  Custodis,  Alphons  —  Chimney  Construction  Co.,  Bennett  Bldg.,  Ne\f 

York. 

182  D'Auria  Pumping  Engine  Co.,  Philadelphia. 

183  Davidson,  M.  T. — ,  133  Liberty  St.,  New  York. 

184  Dayton  Globe  Iron  Works  Co.,  Dayton,  Ohio. 

185  Dayton  Malleable  Iron  Co..  Dayton,  Ohio. 

186  Dean  Bros.  Steam  Pump  Works,  Indianapolis,  Ind. 

187  Deane  Steam  Pump  Co.,  Holyoke,  Mass. 

188  Deming  Co.,  The — ,  Pumps,  Salem,  Ohio. 

189  Denman  &  Davis,  Steel  Castings.  85  John  St.,  New  York. 

190  Detroit  Bridge  and  Iron  Works,  Detroit,  Mich. 

191  Detroit  Lubricator  Co.,  Detroit,  Mich. 

192  Detroit  Steel  and  Spring  Co..  Steel  Castings,  etc.,  Detroit,  Mich. 

193  Deutsch,  W^m.  M. — ,  New  York. 

194  Diamond  Drill  and  Machine  Co.,  Birdsboro,  Pa. 


BUSINESS    DIRECTORY.  999 

195  Diamond  State  Steel  Co.,  Wilmington,  Del. 

196  Dibble,  F.  J.—,  Weir  Gages,  Peabody,  Mass. 

197  Dickey,  W.  S.—  Clay  Mfg.  Co.,  Kansas  City,  Mo. 

198  Dixon,  Joseph^  Crucible  Co.,  Graphite  Pipe  Joint  Compound,  etc., 

^  Jersey  City,  N.  J. 

199  Dickson   Locomotive  Works,  Narrow  Gage   Locomotives,  Scranton, 

Pa.     See  34.5 

200  Dickson  Mfg.  Co.,  Scranton,  Pa.     See  19.5. 

201  Dietzgen,  Eugene—  Co.,  149,  151  Fifth  Ave.,  New  York. 

202  Dilworth,  Porter  &  Co.,  Limited,  Spikes,  Rail  Braces,  etc.,  Pittsburg, 

203  Dimmick  Pipe  Co.,  Birmingham,  Ala. 

204  Debbie  Foundry  and  Machine  Co.,  138  Dey  St.,  New  York. 

205  Dodge  Mfg.  Co.,  Mishawaka,  Ind. 

206  Donaldson  Iron  Co.,  Emaus,  Lehigh  Co.,  Pa. 

207  Douglas,  W.  &  B.~,  Middletown,  Conn. 

208  Downie  Pump  Co.,  Deep-well  and  other  Pumps,  Downieville,  Pa. 

209  Drake  Standard  Machine  Works,  298  W.  Jackson  Boulevard,  Chicago. 

210  Drummond,  M.  J.—  &  Co.,  192  Broadway,  New  York. 

211  Duff  Mfg.  Co.,  Allegheny,  Pa. 

212  Dutton  Pneumatic  Lock  and  Engineering  Co.,  Yonkers,  N.  Y. 

214  Dupont,  E.  I.—  &  Co.,  Wilmington,  Del. 

215  Eastern  Bridge  and  Structural  Co.,  Worcester,  Mass. 

216  Eastern  Construction  Co.,  Brooklyn,  N.  Y. 

217  Eastern  Paving  Brick  Co.,  Catskills,  N.  Y. 

218  Easton  Foundry  and  Machine  Co.,  Easton,  Pa. 

219  Eccles,  Richard—,  Auburn,  N.  Y. 

220  Eddy  Valve  Co..  Waterford,  N.  Y. 

221  Edson  Mfg.  Co.,  132  Commercial  St.,  Boston. 

222  Edwards,  Joseph —  &  Co.,  Centrifugal  Pumps  and  Hydraulic  Dredging 

Machinery,  414  Water  St.,  New  York. 

223  Electric  Fire-proofing  Co.,  P'ire-proofing  System,  119  W.  Twenty-third 

St.,  New  York. 

224  Eley,  Philip  N.— ,  Bayonne,  N.  J. 

225  Elliot  Frog  and  Switch  Co.,  East  St.  Louis,  111. 

226  Emerson,  F.  W.—  Mfg.  Co.,  Rochester,  N.  Y, 

227  Empire  Portland  Cement  Co.,  Warners,  N.  Y. 

228  Enterprise  Boiler  Co.,  Youngstown,  Ohio. 

229  Eppinger  &  Russell  Co.,  Creosote  and  Creosoting,  66  Broad  St.,  N.  Y. 

230  Erie  Machine  Shops,  Asphalt  Rollers  and  Mixers,  etc.,  Erie,  Pa. 

231  Erie  Pump  and  Engine  Co.,  Erie,  Pa. 

232  Evens  &  Howard  Fire  Brick  Co.,  St.  Louis,  Mo. 

233  Ewald  Iron  Co.,  Staybolt  Iron,  St.  Louis,  Mo. 

234  Exeter  Machine  Works,  Pittston,  Pa. 

235  Fairbanks,  The  —  Co.,  Philadelphia. 

236  Falcon,  Jos.  G. — ,  Submarine  Work,  Flexible  Joint,  Evanston,  111. 

237  Falls  Hollow  Staybolt  Co.,  Cuyahoga  Falls,  Ohio. 

238  Farrel  Foundry  and  Machine  Co.,  Havemeyer  Bldg.,  New  York. 

239  Fauth  &  Co.,  108  Second  St.,  S.  W.,  Washington,  D.  C. 

240  Fay,  J.  J.—  &  Egan  Co.,  Wood  Tools,  202  W.  Front  St.,  Cin.,  Ohio. 

241  Featherstone  Foundry  and  Machine  Co.,    1215  Monadnock  Block, 

Chicago. 

342  Federal  Steel  Co.,  Empire  Bldg.,  New  York.     See  Illinois  and  Lorain 

Steel  Cos. 

242.5  Ferracute  Machine  Co.,  Bridgeton,  N.  J. 

243  Ferrari,  Guido — ,  S.  E.  Cor.  Seventh  and  Chestnut  Streets,  Phila. 

244  Ferris,  Walter — ,  Ferris-Pitot  Water  Meter,  Drexel  Building,  Phila. 

245  Fisher  &  Saxton,  123  G  St..  N.  E.,  Washington,  D.  C. 

246  Fitler,  Edwin  H.—  &  Co..  Rope,  Philadelphia. 

247  Foos  Gas  Engine  Co.,  Station  F,  Springfield,  Ohio. 

248  Fort  Pitt  Bridge  Works,  Pittsburg,  Pa. 

249  Flagg,  Stanley  G.—  &  Co.,  Small  Steel  Castings,  Philadelphia. 

250  Flint  &  Walling  Mfg.  Co.,  Kendallville,  Ind. 

251  Flory,  S.—  Mfg.  Co.,  Cableways,  Bangor,  Pa. 

252  Fort  Scott  Cement  Association,  Kansas  City,  Mo. 

253  Fort  Wayne  Electric  Works.  Fort  Wayne,  Ind. 

254  Fox,  John —   &  Co.,  Special  Castings,  Water,  Gas,  and  Flange  Pipes, 

New  York. 


1000  BUSINESS   DIRECTORY. 

255  Frankford  Steel  and  Forging  Co.,  Shafting,  Ellwood  City,  Pa. 

256  Franklin  Mfg.  Co.,  Franklin,  Pa. 

257  Fraser  &  Chalmers,  Chicago.     See  19.5. 

258  French,  Sam'l    H. —    &  Co.,  Cements,  Plaster,  etc.,  York  Ave.  and 

Callowhill  St.,  Philadelphia. 

259  Fulton  Iron  and  Engine  Works,  Detroit,  Mich. 

260  Garden  City  Sand  Co.,  188  Madison  St.,  Chicago. 

261  Gardner  Governor  Co.,  Quincy,  111. 

262  Garland  Chain  Co.,  Rankin  Station,  Pa. 

263  Garry  Iron  and  Steel  Roofing  Co.,  202  Mervin  St.,  Cleveland,  Ohio. 

264  Gates  Iron  Works,  650  Elston  Ave.,  Chicago.    See  19.5. 

265  Gawthrop,  Allen —  Jr.,  Wilmington,  Del. 

266  Gem  City  Boiler  Co.,  Dayton,  Ohio. 

267  General  Electric  Co.,  Schenectady,  N.  Y. 

268  Georgia  Car  and  Mfg.  Co.,  Logging,  etc.,  Savannah,  Ga. 

269  General  Forging  Co.,  Boonton,  N.  J, 

270  Gillespie,  T.  A.—  &  Co.,  Westinghouse  Bldg.,  Pittsburg,  Pa. 

271  Gillingham,  Garrison  &  Co.,  Philadelphia. 

272  Gilson  Asphalt um  Co.,  Buena  Vista  Asphalt,  Wainwright  Bldg.,  St. 

Louis,  Mo. 

273  Glasgow  Iron  Co.,  Pressed  Shapes,  Pottstown,  Pa. 

274  Goheen  Mfg.  Co.,  Canton,  Ohio. 

275  Good  Roads  Machinery  Co.,  Kennett  Square,  Pa. 

278  Goodwin  Car  Co.,  Dumping  Cars  Leased,  New  York  and  Chicago. 

279  Goulds  Mfg.  Co.,  Seneca  Falls,  N.  Y. 

280  Grand  Rapids  Malleable  Co.,  Grand  Rapids,  Mich. 

281  Graphite  Metal  Co.,  Graphitic  Steel  Castings,  15  Cortlandt  St.,  l^evf 

York. 

282  Gregory,  Elisha — ,  Artesian  Wells,  60  Liberty  St.,  New  York. 

283  Green  Ridge  Iron  and  Spike  Works,  Spikes,  Scranton,  Pa. 

284  Guild  &  Garrison,  Brooklyn,  N.  Y. 

285  Gurley,  W.  &  L.  E.— ,  Troy,  N.  Y. 

286  Hall  Signal  Co.,  44  Broad  St.,  New  York. 

288  Hammond  Iron  Works,  Warren,  Pa. 

289  Harrison  Bros.    &  Co.,  Incorporated,  Thirty-fifth  and  Gray's  Ferry 

Road,  Philadelphia. 

290  Harrisburg  Foundry  and  Machine  Works,  Harrisburg,  Pa. 

291  Harvey,  Moland  &  Co.,  Conshohocken,  Pa. 

292  Havana  Bridge  Works,  Montour  Falls,  N.  Y. 

293  Hayward  Co.,  97  Cedar  St.,  New  York. 

294  Hazleton  Boiler  Co.,  120  Liberty  St.,  New  York. 

295  Heisler,  Chas.  L.— ,  Erie,  Pa. 

296  Heisler  Pumping  Engine  Co.,  Erie,  Pa. 

297  Heller  &  Brightly,  Spring  Garden  St.  and  Ridge  Ave.,  Philadelphia. 
297.5  Henion   &  Hubbell,  Deep-well  Pumps,  61  N.  Jefferson  St.,  Chicago. 

298  Hersey  Mfg.  Co.,  South  Boston. 

299  Higley,  Keplinger  &  Co.,  Canton,  Ohio. 

300  Hildenbrand,  W. — ,  Abt  System  of  Rack  Railways,  1  Broadway,  New 

York. 

301  Hobson,  Houghton  &  Co.,  Drill  Rods,  98  John  St.,  New  York. 

302  Hodge,  E.—  &  Co.,  East  Boston. 

303  Holly  Mfg.  Co.,  Lockport,  N.  Y. 

304  Holmes,  J.  W.— ,  Batavia,  N.  Y. 

305  Hoopes  &  Townsend,  1330  Buttonwood  St.,'Philadelphia, 

306  Howells  Mining  Drill  Co.,  Plymouth,  Pa. 

307  Hunt,  C.  W.—  Co.,  West  New  Brighton,  N.  Y. 

308  Hunt,  The  Robert  W.—  Co.,  1121  The  Rookery,  Chicago. 

309  Hurd  &  Co.,  Wall  Ties,  570  West  Broadway,  New  York. 

310  Illinois  Steel  Co.  Cement  Department,  Steel  Portland  Cement,  Th« 

Rookery,  Chicago. 

311  Independent  Powder  Co.,  Terre  Haute,  Ind. 

312  Indianapolis  Drop  Forging  Co.,  Indianapolis,  Ind. 

313  Industrial  Water  Co.,  15  Water  St.,  New  York. 

314  Industrial  Works,  Bay  City,  Mich. 

315  Ingersoll-Sergeant  Drill  Co.;  Park  Place,  New  York. 

317  International  Arithmachine  Co.,  141  La  Salle  St.,  Chicago. 

318  International  Cooling  Co.,  32  Pine  St.,  New  York. 


BUSINESS    DIRECTOEY.  1001 

319  International  Creosoting  and  Construction  Co.,  Galveston,  Texas. 
319.5  International  Steam  Pump  Co.      (Includes  Nos.  82,  187,  303,  348, 

353,  568,  676.)      114-118  Liberty  St.,  New  Vork. 

320  Iroquois  Iron  Works,  Asphalt  Machinery,  etc.,  Buffalo,  N.  Y. 

320  5  Jackson,  Byron —  Machine  Co.,  Centrif .  Pumps.  San  Francisco,  Cal. 

321  Janney,  Steinmetz  &  Co.,  Drexel  Bldg.,  Philadelphia. 

322  Jeffrey  Mfg.  Co.,  Columbus,  Ohio. 

323  Jenkins  Bros.,  New  York,  Philadelphia,  Chicago,  Boston. 

324  Jessop,  Wm.—  &  Sons,  91  John  St.,  New  York. 

325  Johns,  H.  W.—  Mfg.  Co.,  Packings,  100  William  St.,  New  York. 

326  Johnson  &  Wilson,  150  Nassau  St.,  New  York. 

327  Johnson  Cement  Coating  Co.,  Coating  for  Walls,  156  Fifth  Ave.,  Ne^f 

York. 

328  Joliet  Bridge  and  Iron  Co.,  Joliet,  111. 

329  Jones  &  Laughlins,  Limited,  Third  Ave.  and  Ross  St.,  Pittsburg,  Pa. 

330  Jones,  B.  F.— ,  147  E.  One  Himdred  and  Twenty-fifth  St.,  New  York. 

331  Jones,  B.  M.—  &  Co.,  81  Milk  St.,  Boston. 

332  Jones  National  Fence  Co.,  Columbus,  Ohio. 

333  Kaltenbach  &  Gri«ss,  1618  Williamson  Bldg.,  Cleveland,  Ohio. 

334  Keasby  &  Mattison  Co.,  Lagging,  Ambler,  Pa. 

335  Keeler,  E.—  Co.,  Williamsport,  Pa. 

336  Kelley  Island  Lime  and  Transport  Co.,  Lagerdorfer  Portland  Cement, 

Cleveland,  Ohio. 

337  Kelly,  O.  S.—  Co.,  Springfield,  Ohio. 

338  Kennedy  Valve  Mfg.  Co.,  50  Beekman  St.,  New  York. 

339  Kenwood  Bridge  Co.,  Chicago. 

340  Keuffel  &  Esser  Co.,  127  Fulton  St.,  New  York. 

341  Kevstone  Drop  Forge  Works;  office,  Harrison  Building,  Philadelphia. 

342  Kidd  Bros.  &  Burgher  Steel  Wire  Co.,  McKee's  Rocks,  Pa. 

343  Kilborn  &  Bishop  Co.,  New  Haven,  Conn. 

344  King  Bridge  Co.,  Cleveland,  Ohio. 

345  Kingsford  Foundry  and  Machine  Works,  Oswego,  N.  Y. 

346  Kirk,  Arthur—  &  Son,  Pittsburg,  Pa. 

347  Knight,  F.  C—  &  Co.,  400  Locust  St.,  Philadelphia. 

348  Knowles  Steam  Pump  Works,  91  Liberty  St.,  New  York. 

349  Koken  Iron  Works,  Koken  Bldg.,  715  Locust  St.,  St.  Louis,  Mo. 

350  Kolesch  &  Co.,  138  Fulton  St.,  New  York. 

351  Koppel,  Arthur — ,  Narrow  Gage  Railway  Materials,  68  Broad  St., 

New  York. 

352  Krause,  Wm.—  &  Sons  Cement  Co.,  Portland  Cement,  1623  N.  Fifth 

St.,  Philadelphia. 

353  Laidlaw-Dunn-Gordon  Co.,  Cincinnati,  Ohio. 

354  Lambert  Hoisting  Engine  C9.,  Newark,  N.  J. 

355  Lamb  Fence  Co.,  Adrian,  Mich. 

356  Lanz,  M.—  &  Sons,  Pittsburg,  Pa. 

357  Laughlin-Hough  Co.,  30  Broad  St.,  New  York. 

358  Lawrence  Cement  Co.,  1  Broadway,  New  York. 

359  Lawrence  Machine  Co.,  Lawrence,  Mass. 

360  Lawrenceville  Cement  Co.,  26  Cortlandt  St.,  New  York. 

361  Lawrence,  W.  W.—  &  Co.,  Pittsburg,  Pa. 

362  Lead-lined  Iron  Pipe  Co.,  Lead  and  Tin-lined  Pipe,  Wakefield,  Mass. 

363  Le  Clere,  F. — ,  Black-line  Prints  from  Tracings,  21  N.  Thirteenth  St., 

Philadelphia. 

364  Ledder,  G.  G.— ,  302  Washington  St.,  Boston. 

365  Leffel,  James —  &  Co.,  Automatic  Engines,  Springfield,  Ohio. 

366  Lehigh  Portland  Cement  Co.,  Allentown,  Pa. 

367  Leitz,  A. —  Co.,  422  Sacramento  St.,  San  Francisco,  Cal. 

368  Leopold,  J. —    &  Co.,  Granite  and  Trap  Rock,  18  Broadway,  New 

York.  * 

369  Leschen,  A.—  &  Sons  Co.,  920-922  N.  Main  St.,  St.  Louis,  Mo. 

370  Lidgerwood  Mfg.  Co.,  96  Liberty  St.,  New  York. 

371  Lima  Locomotive  and  Machine  Co.,  Lima,  Ohio. 

372  Lima  Steel  Casting  Co.,  Lima,  Ohio. 

373  Link  Belt  Engineering  Co.,  Nicetown,  Philadelphia. 
373.5  Locks  and  Canals  Co.,  Timber  Preservation,  Lowell,  Mass. 

374  Lombard  Water-wheel  Governor  Co.,  61  Hampshire  St.,  Boston. 


1002  BUSINESS   DIRECTORY. 

375  Long  &  Allstatter  Co.,  Hamilton,  Ohio. 

376  Loomis-Manning  Filter  Co.,  420  Chestnut  St.,  Philadelphia. 

377  Lorain  Steel  Co.,  Steel  Castings,  Lorain,  Ohio. 

378  Louisville  Bridge  and  Iron  Co.,  Incorporated,  Louisville,  Ky. 

379  Ludlow  Yalve  Mfg.  Co.,  Troy,  N.  Y. 

380  Ludowici  Roofing  Tile  Co.,  508  Chamber  of  Commerce  Bldg.,  Chicago* 

381  Lufkin  Rule  Co.,  Measuring  Tapes,  Saginaw,  Mich. 

382  Lukens  Iron  and  Steel  Co.,  Coatesville,  Pa. 

383  Mack  Mfg.  Co.,  New  Cumberland,  W.  Va. 

384  Macomber  &  Whyte  Rope  Co.,  21  South  Canal  St.,  Chicago. 

385  Maignen  Filtration  Co.,  1310  Arch  St.,  Philadelphia. 

386  Manasse,  L. —  Co.,  88  Madison  St.,  Chicago. 

387  Manchester  Locomotive  Works,  Manchester,  N.  H.      See  34.5. 

388  Manhattan  Trap  Rock  Co.,  11  Broadway,  New  York. 

389  Marion  Steam  Shovel  Co.,  Marion,  Ohio. 

390  Martin,  Wm.  R. — ,  Iron  Works,  Screens,  Lancaster,  Pa. 

391  Massillon  Bridge  Co.,  Massillon,  Ohio. 

392  Massillon  Iron  and  Steel  Co.,  Massillon,  Ohio. 

393  McClenahan  &  Bros.,  Granite,  Port  Deposit,  Md. 

394  McClintic-Marshall  Construction  Co.,  Pottstown  and  Pittsburg,  Pa. 

395  McCullough  Iron  Co.,  Sheet  Iron  and  Steel,  Wilmington,  Del. 

396  McGowen,  John  H. —  Co.,  Cincinnati,  Ohio. 

397  Mcintosh,  Seymour  &  Co.,  Auburn,  N.  Y. 

398  McKay,  Jas.—  &  Co..  Pittsburg,  Pa. 

399  McKenna,  Chas.  F.— ,  221  Pearl  St.,  New  York. 

400  McKiernan  Drill  Co.,  120  Liberty  St.,  New  York. 

401  McLean,  John—,  298  Monroe  St.,  New  York. 

402  McMullen,  Arthur—  &  Co.,  13  Park  Row,  New  York. 

403  McNeil,  James —  &  Bro.,  Twenty-ninth  and  Railroad  Sts.,  Pittsburg. 

404  Mc Williams  &  McConnell,  Shamokin,  Pa. 

405  Mead,  John  A. —  Mfg.  Co.,  7  Broadway,  New  York. 

406  Melan  Arch  Construction  Co.,  13  Park  Row,  New  York. 

407  Merchant  &  Co.,  Metals,  Seamless  Tubes,  etc.,  Philadelphia. 

408  Meredith,  J.  P. —  Cedar  Co.,  Memphis,  Tenn. 

409  Merrill  Bros.,  467  Kent  Ave.,  Brooklyn,  N.  Y. 

410  Merritt  &  Co.,  Expanded  Metal  Construction,  Philadelphia. 

411  Metalloid  Sidewalk  Co.,  Concrete  and  Expanded  Metal  Cement  Walks, 

606  Century  Bldg.,  St.  Louis,  Mo. 

412  Michigan  Brass  and  Iron  Works,  Detroit,  Mich. 

413  Michigan  Lubricator  Co.,  Detroit,  Mich. 

414  Michigan  Pipe  Co.,  Stand  Pipe,  Bay  City,  Mich. 

415  Mietz,  A.—,  128-138  Mott  St.,  New  York. 

41 6  Millar,  Charles—  &  Son  Co.,  Utiea  Pipe  Foundry  Co.,  Utica,  N.  Y. 

418  Milwaukee  Cement  Co.,  Milwaukee,  Wis. 

419  Missouri  Valley  Bridge  and  Iron  Works,  Roofs,  Leavenworth,  Kan. 

420  Monmouth  Mining  and  Mfg.  Co.,  Monmouth,  111. 

421  Moore  Mfg.  Co.,  Syracuse,  N.  Y. 

422  Moran  Flexible  Steam  .Joint  Co.,  Incorporated,  Louisville,  Ky. 

423  Morris  Machine  Works,  Centrifugal  Pumps,  Baldwinsville,  N.  Y. 

424  Morris,  Tasker  &  Co..  Incorporated,  Philadelphia. 

425  Morse,  A.  J.—  &  Son,  Diving  Outfits,  140  Congress  St.,  Boston. 

426  Mount  Vernon  Bridge  Works,  Mt.  Vernon,  Ohio. 

427  Mundy,  J.  S.— ,  Newark,  N.  J. 

428  Murray  &  Co.,  Tents,  329  S.  Canal  St.,  Chicago. 

429  Nash,  Geo.—  &  Co.,  15  Piatt  St.,  New  York. 

430  Nathan  Mfg.  Co.,  92  Liberty  St.,  New  York. 

431  National  Elastic  Nut  Co.,  Milwaukee,  Wis. 

432  National  Meter  Co.,  New  York. 

433  National  Paint  Works,  Williamsport,  Pa. 

434  National  Tube  Co.,  Pittsburg,  Pa.,  New  York,  Philadelphia,  Boston, 

Chicago. 

435  National  Steel  Co.,  Fattery  Park  Bldg.,  New  York. 
435.5  National  Wood  Pipe  Co.,  Los  Angeles,  Cal. 

436  Neptune  Meter  Co.,  Jackson  Ave.  and  Crane  St.,  New  York. 

437  Neuchatel  Asphalt  Co.,  Limited,  Val  de  Travers  Rock  Asphalt,  265 

Broadway,  New  York. 

438  New  Blue  on  White  Process  Co.,  Cincinnati,  Ohio. 
439,  New  England  Structural  Co.,  Boston. 


BUSINESS   DIRECTORY.  1003 

440  New  Era  Iron  Works,  Gas  Engines,  Dayton,  Ohio. 

441  New  Jersey  Foundry  and  Machine  Co.,  Overhead  Trackage  Systems, 

26  Cortlandt  St.,  New  York. 

442  New  Orleans  Wood  Preserving  Works,  Creosote  and  Creosoting,  New 

Orleans,  La. 

443  New  York  Air  Compressor  Co.,  95  Liberty  St.,  New  York. 

444  New  York  and  Bermudez  Co.,  Bowling  Green  Bldg.,  New  York. 

445  New  York  and  Rosendale  Cement  Co.,  200  Broadway,  New  York. 

446  New  York  Continental  Jewell  Filtration  Co.,  15  Broad  St.,  New  York. 

447  New  York  Dredging  Co.,  Park  Row  Bldg.,  New  York. 

448  New  York  State  Drain  Tile  and  Pipe  Works,  Albany,  N.  Y. 

449  New  York  Mastic  Works,  Seyssel  Rock  Asphalt,  11  Broadway,  New 

York. 

450  Nichols  Engineering  and  Contracting  Co.,  Steel  Measuring  Tape,  etc., 

1538  Monadnock  Block,  Chicago. 

451  Niles  Tool  Works  Co.,  Hamilton,  Ohio. 

452  Nitro-powder  Co.,  Kingston,  N.  Y. 

453  Norfolk  Creosoting  Co.,  Norfolk,  Va. 

454  Northern  Electrical  Mfg.  Co.,  Madison,  Wis. 

455  Norton,  A.  O.— ,  167  Oliver  St.,  Boston. 

456  Norton,  F.  O. —  Cement  Co.,  92  Broadway,  New  York. 

457  Norwalk  Iron  Works  Co.,  South  Norwalk,  Conn. 

458  Norwood  Engineering  Co.,  Florence,  Mass. 

459  Ogden,  J.  Edward —  Co.,  Yarn,  Pig  Lead,  147  Cedar  St.,  New  York. 

460  Olsen,  Tinius —  &  Co.,  500  N.  Twelfth  St.,  Philadelphia. 

461  100  Per  Cent.  Splice  Co.,  803  Land  Title  Bldg.,  Philadelphia. 

462  Oneida  Community,  Limited,  Niagara  Falls,  N.  Y. 

463  Osgood  Dredge  Co.,  Albany,  N.  Y. 
463.2  Otis  Steel  Co.,  Cleveland,  Ohio. 

463.4  Otto  Gas  Engine  Works,  Philadelphia  and  Chicago. 

463.6  Page-Storms  Drop  Forge  Co.,  Chicopee  Falls,  Mass., 

464  Parker  Engine  Co.  of  California,  1041  Drexel  Bldg.,  Philadelphia. 

465  Parker,  Thatcher  A. — ,  Terre  Haute,  Ind. 

466  Passaic  Rolling  Mill  Co.,  Paterson,  N.  J. 

466.5  Payne,  Walter  S.—  &  Co.,  Fostoria,  O. 

467  Peck,  C.  C. — ,  Rochester,  N.  Y. 

468  Peerless  Brick  Co.,  Philadelphia. 

469  Pelton  Water-wheel  Co.,  143  Liberty  St.,  New  York. 

470  Penberthy  Injector  Co.,  Detroit,  Mich. 

471  Penn  Bridge  Co.,  Beaver  Falls,  Pa.  ^ 

471.5  Pennsylvania  Galvanizing  Co.,  Limited,  Philadelphia. 

472  Pennsylvania  Steel  Co.,  Rails,  Switches,  etc.,  Steelton,  Pa. 

473  Pfister,  Herm. — ,.428  Plum  St.,  Cincinnati,  Ohio. 

474  Philadelphia  Bridge  Works,  Pottstown,  Pa. 

475  Philadelphia  Machine  Tool  Co.,  443-449  N.  Darien  St.,  Philadelphia. 

478  Philadelphia  Pneumatic  Tool  Co.,  Stephen  Girard  Bldg.,  Philadelphia. 

479  Philadelphia  Railway  Track  Equipment  Co.,  Philadelphia. 

480  Phillips  &  Worthington,  11  Broadway,  New  York. 

481  Phillips  Rock  Drill  Co.,  40  N.  Seventh  St.,  Philadelphia. 

482  Phoenix  Bridge  Co.,  410  Walnut  St.,  Philadelphia. 

483  Phosphor-Bronze   Smelting    Co.,    Limited,    2208   Washington    Ave., 

Philadelphia. 

484  Pierce  Well  Engineering  and  Supply  Co.,  136  Liberty  St.,  New  York. 

485  Pittsburg  Gage  and  Supply  Co.,  Steam  Specialties,  Pittsburg  and 

Philadelphia. 

486  Pittsburg  Locomotive  and  Car  Works,  Pittsburg,  Pa.     See  34.5. 

487  Pittsburg  Meter  Co.,  East  Pittsburg,  Pa. 

488  Pittsburg  Testing  Laboratory,  Limited,  325  Water  St.,  Pittsburg,  Pa. 

489  Pittsburg  Valve,  Foundry,  and  Construction  Co.,  Empire  Bldg.,  Pitts- 

burg, Pa. 

490  Pleuger  &  Henger  Mfg.  Co.,  St.  Louis,  Mo. 

491  Plymouth  Mills,  Washers,  Plymouth,  Mass. 

492  Pollard,  J.  G.— ,  Firing  Tools,  141  Raymond  St.,  Brooklyn,  N.  Y. 

493  Porter,  H.  K. —  Co.,  Light  and  Compressed  Air  Locomotives,  531 

Wood  St.,  Pittsburg,  Pa. 

494  Porter  Morse  Co.,  Saginaw,  Mich. 

495  Potomac  Steel  Co.,  Light  Steel  Rails,  Times  Bldg.,  Pittsburg,  Pa. 

496  Potter  Mfg.  Co.,  Sewer  Machines,  Indianapolis,  Ind. 


1004  BUSINESS    DIRECTORY. 

497  Power  Specialty  Co.,  126  Liberty  St.,  New  York. 

498  Pratt  &  Whitney  Co.,  Hartford,  Conn. 

499  Pressed  Steel  Car  Co.,  Pittsburg,  Pa. 

500  Priestman  &  Co.,  Incorporated,  Bourse  Building,  Philadelphia. 

501  Prince,  L.  M.— ,  108  W.  Fourth  St.,  Cincinnati,  Ohio. 

502  Prince  Mfg.  Co.,  71  Maiden  Lane,  New  York. 

503  Prindle  Engineering  Co.,  Centrifugal  Pumps,   120  Liberty  St.,  Ne^ 

York. 

504  Pulsometer  Steam  Pump  Co.,  135  Greenwich  St.,  New  York. 

505  Purington  Paving  Brick  Co.,  Galesburg,  111. 

505.5  Q.  &  C.  Co.,  see  508.5,  Railroad  Supply  Co. 

506  Queen  &  Co.,  1010  Chestnut  St.,  Philadelphia. 

507  Quimby,  Wm.  E.— ,  Screw  Pumps,  86  Liberty  St.,  New  York. 

508  Quimby  Engineering  Co.,  Ridge  Ave.,  Philadelphia. 

508.5  Railroad  Supply  Co.,  Bedford  Bldg.,  Chicago. 

509  Ramapo  Iron  Works,  Hillburn,  Rockland  Co.,  N.  Y. 

510  Rand  Drill  Co.,  128  Broadway,  New  York. 

511  Ransome  &  Smith  Co.,  Brooklyn,  N.  Y. 

512  Rawson  &  Morrison  Mfg.  Co.,  Cambridgeport  (Boston)",  Mass. 

513  Raymond  Bros.  Impact  Pulverizer  Co.,  1402  Monadnock  Block,  Chi- 

cago. 

514  Reading  Foundry  Co.,  Limited,  Reading,  Pa. 

515  Reese-Hammond  Fire  Brick  Co.,  912  Park  Bldg.,  Pittsburg,  Pa. 

516  Renssellaer  Mfg.  Co.,  1108-9  Monadnock  Bldg.,  Chicago. 

517  Repauiio  Chemical  Co.,  Wilmington,  Del. 

518  Republic  Iron  and  Steel  Co.,  Chicago. 

519  Rhode  Island  Tool  Co.,  Providence.  R.  I. 

520  Richmond  Locomotive  Works,  Richmond,  Va. 

521  Riehl^  Bros.,  1424  N.  Ninth  St.,  Philadelphia. 

522  Risdon  Iron  Works,  San  Francisco,  Cal. 

523  Ritchie,  E.  S. —  &  Sons,  Current  Meters,  Brookline,  Mass. 

524  Richmond  Locomotive  and  Machine  Works,  Richmond,  Va.   See  34.5. 
625  Riter-Conley  Mfg.  Co.,  55  Water  St.,  Pittsburg,  Pa. 

526  Roberts,  A  &  P.—  Co.,  Pencoyd,  Pa.,  261  S.  Fourth  St.,  Philadelphia. 

527  Robins  Conveying  Belt  Co.,  Park  Row  Bldg.,  New  York. 

528  Robinson,  Prof.  S.  W.— ,  Pitot  Meter,  Ohio  State  College,  Columbus, 

Ohio. 

529  Rockford  Bolt  Works,  Rockford,  111. 

530  Roebling  Construction  Co.,  121  Liberty  St.,  New  York. 
530.5  Roebling's  Sons  Co.,  John  A. — .  Trenton,  N.  J. 

531  Rogers,  John  M.— .  Gloucester,  N.  J. 

532  Roots,  P.  H.  &  F.  M.—  Co.,  Connersville,  Ind. 

533  Ross,  Sanford  P. — ,  Incorporated,  277  Washington  St.,  Jersey  City, 

N.J. 

534  Ross  Valve  Co.,  Feed  Water  Filter,  Troy,  N.  Y. 

535  Rowland,  Wm.   &  Harvey — ,  Norway  Iron,  Frankford,  Philadelphia. 

536  Ruger,  J.  W.—  Mfg.  Co.,  300  Chicago  St.,  Buffalo,  N.  Y. 

538  Rumsey  &  Co.,  Limited,  Centrifugal,  Hand,  and  Power  Pumps,  Seneca 

Falls,  N.  Y. 

539  San  Francisco  Bridge  Co.,  220  Market  St.,  San  Francisco,  Cal. 

540  Sargent  Co.,  Chicago. 

541  Saunders,  D, —  Sons,  Pipe  Threading  Machinery,  Yonkers,  N.  Y. 

542  Scaife,  Wm.  B.—  &  Sons,  Pittsburg,  Pa. 

543  Schenectady  Locomotive  Works,  Schenectady,  N.  Y.     See  34.5. 

544  Scherzer  Rolling  Lift  Bridge  Co.,  Monadnock  Block,  Chicago. 

545  Scholl,  Julian—  &  Co.,  126  Liberty  St.,  New  York. 

546  Schrader's  Son,  A. — ,  Diving  Apparatus,  32  Rose  St.,  New  York. 

547  Schreiber,  L. —  &  Sons  Co.,  Iron  Work,  Cincinnati,  Ohio. 

549  Scott,  Charles —  Spring  Co.,  Philadelphia. 

550  Scranton  Forging  Co.,  Scranton,  Pa. 

651  Scranton  Steam  Pump  Co.,  Scranton,  Pa. 

552  Sedgwick-Fisher  Co.,  Chicago, 

552.5  Sellers.  Wm.—  &  Co.,  Incorporated,  1600  Hamilton  St.,  Phila. 

553  Seward,  M. —  &  Son,  New  Haven,  Conn. 

554  Sharon  Boiler  Works,  Sharon,  Pa. 

555  Shaw,  Thomas-r-  (Quimby  Engineering  Co.,  Successor),  Compound 

Propeller  Pump,  915  Ridge  Ave.,  Philadelphia. 


BUSINESS   DIRECTORY.  1005 

556  Shelby  Steel  Tube  Co.,  Cleveland,  Ohio. 

557  Shoemaker.  Benj.  H. — ,  Window  Glass,  etc.,  205  N.  4th  St.,  Phila. 

558  Shuart  Grader  Co.,  Oberlin,  Ohio. 

559  Sicilian  Asphalt  Paving  Co.,  Trinidad  Rock  Asphalt,  Times  Bldg., 

New  York. 

560  Singer,  Nimick  &  Co.,  Incorporated,  Pittsburg,  Pa. 

561  Sipe,  James  B. —  &  Co.,  Allegheny,  Pa. 

562  Smidth,  F.  L. —  &  Co.,  66  Maiden  Lane,  New  York. 

563  Smith,  A.  P.—  Mfg.  Co.,  Newark,  N.  J. 

564  Smith,  Edward —  &  Co.,  45  Broadway,  New -York. 

565  Smith,  S.  Morgan—  Co.,  York,  Pa. 

566  Smith,  Theo.—  &  Sons  Co.,  Jersey  City,  N.  J. 
566.5  Smith-Vaile  Co.    See  594. 

567  Snider-Hughes  Co.,  Cleveland,  Ohio. 

568  Snow  Steam  Pump  Works,  Buffalo,  N.  Y. 

569  Snyder,  Hiram —  &  Co.,  229  Broadway,  New  York. 
569.5  Soitmann,  E.  G.,  125  E.  42d  St.,  New  York. 

570  Souther,  John —  &  Co.,  12  Post  Office  Square,  Boston. 

571  Southern  Car  and  Foundry  Co.,  Gadsden,  Ala. 

572  Southwark  Foundry  and  Machine  Co.,  Philadelphia. 

573  Spaulding  Print  Paper  Co.,  Blue  Print  Cloth,  44  Federal  St.,  Boston. 

578  Spirittine  Chemical  Co.,  Wilmington,  N.  C. 

579  Springfield  Boiler  and  Mfg.  Co.,  Springfield,  111. 

580  Springfield  Drop  Forging  Co.,  Springfield,  Mass. 

581  Springfield  Gas  Engine  Co.,  Springfield,  Ohio. 

582  Stackpole  &  Bro.,  39  Fulton  St.,  New  York. 

583  Stamsen  &  Blome,  Unity  Bldg.,  Chicago. 

584  Standard  Automatic  Gas  Engine  Co.,  Oil  City,  Pa. 

585  Standard  Chain  Co.,  Pittsburg,  Pa. 

586  Standard  Paint  Co.,  77  John'St.,  New  York. 

587  Standard  Railroad  Signal  Co.,  Troy,  N.  Y. 

688  Standard  Silica  Cement  Co.,  66  Maiden  Lane,  New  York. 

589  Standard  Tool  Co.,  Cleveland,  Ohio. 

690  Standard  Water  Meter  Co.,  22-26  Reade  St.,  New  York. 

691  Stanley  Electric  Mfg.  Co.,  Pittsfield,  Mass. 

692  Starr  Brass  Mfg.  Co.,  Boston. 

593  Stickney,  Charles  A. —  Co.,  St.  Paul,  Minn. 

594  Stilwell-Bierce  &  Smith-Vaile  Co.,  Dayton,  Ohio. 

595  Stover  Engine  Works,  Freeport,  111. 

596  Stow  Flexible  Shaft  Co.,  Twenty-sixth  and  Callowhill  Sts.,  Phila. 

697  Strieby  &  Foote  Co.,  Newark,  N.  J. 

698  Strobel,  C.  L. — ,  1744  Monadnock  Block,  Chicago. 

699  Strock,  S.  C. — ,  11  Broadway,  New  York. 

600  Structural  Engineering  Co.,  41  Cortlandt  St.,  New  York. 

601  Struthers,  Wells  &  Co.,  Warren,  Pa. 

602  Stuebner,  G.  L.— ,  Long  Island  City,  N.  Y. 

603  Stupp  Bros.  Bridge  and  Iron  Co.,  St.  Louis,  Mo. 

604  Sturtevant,  B.  F.—  Co.,  Boston. 

605  Sullivan  Machinery  Co.,  135  Adams  St.,  Chicago. 

606  Susquehanna  Iron  and  Steel  Co.,  Columbia,  Pa. 

607  Syracuse  Chilled  Plow  Co.,  Syracuse,  N.  Y. 

608  Syracuse  Stoneware  Co.,  Syracuse,  N.  Y. 

609  Tatham  &  Bros.,  Fifth  and  Locust  Sts.,  Philadelphia. 

610  Taylor,  John  W.—  Co.,  Mt.  Holly,  N.  J. 

611  Taylor  Signal  Co.,  Buffalo,  N.  Y. 

612  Tennessee  Coal,  Iron,  and    Railroad    Co.,  Basic  Open-hearth    Steel, 

Birmingham,  Ala. 
612.5  Texas  Tie  and  Lumber  Co.,  Timber  Preservation,  Somerville,  Tex. 

613  Thew  Automatic  Shovel  Co.,  Lorain,  Ohio. 

614  Thomson  Meter  Co.,  79  Washington  St.,  Brooklyn,  N.  Y. 

615  Thompson,  J. —    &  Co.,   Union  Hydraulic  Works,  Van  Horn  and 

Sophia  Sts.,  Philadelphia. 

616  Thorn  Cement  Co.,  112  Church  St.,  Buffalo,  N.  Y. 

617  Tippett   &  Wood,  Philipsburg,  N.  J. 

618  Toledo  Foundry  and  Machine  Co.,  Toledo,  Ohio. 

619  Tomkins  Brothers,  257  Broadway,  New  York. 

620  Tomkins,  Calvin—,  Broken  Stone,  120  Liberty  St.,  New  York. 

621  Townsend,  C.  C.  &  E.  P.—,  Wire,  New  Brighton,  Pa. 

622  Transue  &  Williams  Co.,  Alliance,  Ohio. 


1006  BUSINESS   DIRECTORY. 

623  Trenton  Iron  Co.,  Trenton,  N.  J. 

624  Troy  Malleable  Iron  Co.,  Troy,  N.  Y. 

625  Ulmer  &  Hoff,  224  Champlain  St.,  Cleveland,  Ohio. 

626  Union  Akron  Cement  Co.,  141  Erie  St.,  Buffalo,  N.  Y. 

627  Union  Hydraulic  Works  (J.  Thompson  &  Co.),  Philadelphia. 

628  Union  Switch  and  Signal  Co.,  Swissvale,  Pa. 

629  Union  Water  Meter  Co.,  Worcester,  Mass. 

630  United  States  Cast-iron  Pipe  and  Foundry  Co.,  217  La  Salle  St. 

Chicago. 

631  United  States  Clay  Mfg.  Co.,  Empire  Bldg.,  Pittsburg,  Pa. 

632  United  States  Mineral  Wool  Co.,  143  Liberty  St.,  New  York. 

633  United  States  Wind  Engine  and  Pump  Co.,  Batavia,  111. 

634  United  States  Wood  Preserving  Co.,  Creo-resinate  and  Creoaoto  Pro- 

cess, 29  Broadway,  New  York. 

635  Valentine  &  Co.,  Varnishes,  New  York. 

636  Vanderbilt    &  Hopkins,  Lumber,  Timber  Preservation,  120  Liberty 

St.,  New  York. 

637  Vanderherchen,  M.  F. —  &  Co.,  Vine  and  Water  Sts.,  Philadelphia. 

638  Variety  Iron  Works  Co.,  Cleveland,  Ohio. 

639  Virginia  Bridge  and  Iron  Co.,  Roanoke,  Va. 

640  Virginia  Iron,  Coal,  and  Coke  Co.,  Spikes,  Pig  Iron,  Bristol,  Tenn. 

641  Vulcan  Iron  Works,  Chicago. 

642  Vulcanite  Portland  Cement  Co.,  Vulcanite  Bldg.,  1710  Market  St., 

Philadelphia. 

642.5  Wabash  Bridge  and  Iron  Works,  Wabash,  Ind. 

643  Watkins,  Thomas — ,  Johnstown,  Pa. 

644  Watson  Wagon  Co.,  Canastota,  N.  Y.    _ 

645  Watson-Stillman  Co.,  453  Rookery,  Chicago. 

646  Waterbury  Rope  Co.,  69  South  St.,  New  York. 

647  Warren  Foundry  and  Machine  Co.,  Phillipsburg,  N.  J.,  160  Broadway, 

New  York. 

648  Warren  City  Boiler  Works,  Warren,  Ohio. 

649  Wamer-Quinlan  Asphalt  Co.,  Trinidad  Asphalt,  4  Warner  Bldg.,  Syra- 

cuse, N.  Y. 

652  Weaver-Hirsh  Co.,  Gray  Iron  Castings,  Allentown,  Pa. 

653  Weber  Railway  Joint  Mfg.  Co.,  Empire  Bldg.,  71  Broadway,  New 

York. 

654  Wefugo  Co.,  Cincinnati,  Ohio. 

655  Weir  Frog  Co.,  Cincinnati,  Ohio. 

656  Wells  Light  Mfg.  Co.,  Wells  Light,  44  Washington  St.,  New  York. 

657  Western  Cement  Co.,  Louisville,  Ky. 

658  Western  Wheeled  Scraper  Co.,  Aurora,  111. 

659  Westinghouse  Air  Brake  Co.,  Pittsburg,  Pa. 

660  Westinghouse  Electric  and  Manufacturing  Co.,  Pittsburg,  Pa. 

661  Westinghouse  Machine  Co.,  Pittsburg,  Pa. 

662  Westmoreland  Steel  and  Mfg.  Co.,  Tool  Steels,  Pittsburg,  Pa. 

663  West  Pascagoula  Creosote  Works,  West  Pascagoula,  Miss. 

664  West  Pulverizing  Machine  Co.,  220  Broadway,  New  York. 
664.5  Wharton  Railroad  Switch  Co.,  Philadelphia. 

665  Wilcox,  D. —  Mfg.  Co.,  Mechanicsburg,  Pa.  , 

666  Williams  Bros.,  Machinists,  Ithaca,  N.  Y. 

667  William,  Brown  &  Earle,  918  Chestnut  St.,  Philadelphia. 

668  Williams,  J.  H.—  &  Co.,  Brooklyn,  N".  Y. 

669  Williamsport  Valve  and  Hydrant  Co.,  Williamsport,  Pa. 

670  Wilmot    &  Hobbs  Mfg.  Co.,  Steel,  Strip-steel,  Crucible-steel,  Bridge- 

port, Conn. 

672  Wonham   &  Magor,  Switches,  29  Broadway,  New  York. 

673  Wood,  Alan—  Co.,  Sheet  Iron  and  Steel,  519  Arch  St.,  Philadelphia. 

674  Wood,  R.  D.—  &  Co.,  400  Chestnut  St.,  Philadelphia. 

675  Worth  Bros.  Co.,  Coatesville,  Pa. 

676  Worthington,  Henry  R.— ,  120  Liberty  St.,  New  York. 

677  Wyman  &  Gordon,  Worcester,  Mass. 

678  Wyoming  Shovel  Works,  Wyoming,  Pa. 

679  Young  &  Sons,  43  North  Seventh  St.,  Philadelphia. 

680  Young-Brennan  Crusher  Co.,  Ill  Hancock  St.,  Brooklyn,  N.  Y. 

681  Youngstown  Iron  and  Steel  Roofing  Co.,  Trough  Floor,  Youngstown; 

Ohio. 


1008 


BIBLIOGRAPHY. 


BIBLIOGKAPHY. 

The  following  list  of  books  makes  no  pretensions  to  completeness.  It 
aims  simply  to  be  usefully  suggestive  to  the  general  civil  engineer.  A  few 
works,  believed  to  be  specially  useful,  are  designated  by  single  and  double 
stars.  The  list  is  arranged  according  to  the  Decimal  Classification  of  Mel- 
ville Dewey.  In  this  classification  (see  outline  below)  all  subjects  are  ar- 
ranged under  ten  general  heads,  and  to  each  of  these  heads  is  assigned  a  num- 
ber in  the  hundreds  place,  as  Natural  Science,  500;  Useful  Arts,  600;  etc. 
Then  each  general  head  is  divided  into  ten  sub-heads,  to  each  of  which  a  num- 
ber in  the  tens  place  is  assigned.  Thus,  Natural  Science  (500)  is  subdivided 
into  Mathematics,  510;  Astronomy,  520;  Physics,  530;  etc.  Again,  each 
of  these  is  subdivided,  and  decimally  numbered,  and  this  successive  subdivi- 
sion and  decimal  enumeration  may  be  continued  indefinitely.  To  find  a  sub- 
ject, it  is  best  to  inquire,  first,  under  which  of  the  ten  general  heads  it 
belongs,  then  under  which  sub-head,  and  so  on.  Thus,  Plane  Geometry  is 
seen  to  belong  (1)  under  Natural  Science,  500,  (2)  under  Mathematics,  510, 
and  (3)  under  Geometry,  513.  However,  matter  on  a  special  subject  is 
often  contained  in  books  on  a  more  general  subject  which  embraces  the  spe- 
cial one.  Thus,  matter  pertaining  to  Geometry  (513)  is  found  in  many 
works  which  would  be  classified  under  Mathematics  (510).  Conversely,  in 
looking  for  books  on  Mathematics,  the  sub-heads,  Algebra,  Geometry,  etc., 
should  be  consulted  as  well.  In  general,  it  is  advisable  to  look  under  all 
heads  that  may  contain  the  information  wanted.  Thus,  information  on 
Locomotives  may  be  found  under  Mechanical  Engineering,  621,  or  under 
Railroads,  625.  To  avoid  duplication  in  such  cases,  however,  one  such  head 
has  usually  been  selected,  and  reference  to  it  made  under  the  other. 


Outline  of  Classification. 


500 

Natural  Science 

621.8 

TransmissionM'ch'm 

510 

Mathematics 

622 

Mining 

610.8 

Tables  Math.  Insts. 

624 

Bridges  and  Roofs 

512 

Algebra 

624.0s 

Specifications 

513 

Geometry 

624.a 

Trestles.  Viaducts 

514 

Trigonometry 

624.2 

Girders.    Beams 

515 

Descriptive    Geometry 

624.3 

Trusses 

516 

Analytical     Geometry 

624.6 

Arches 

517 

Calculus 

624.7 

Compound  Bridges 

519 

Least  Squares 

624.8 

Draw  Bridges 

520 

Astronomy 

625 

Roads  and  Railroads 

526 

Geodesy 

625.1 

Route.    Track 

526.9 

Surveying 

625.1a 

R.R.  Surveying 

530 

Physics 

625.1ao 

R.R.  Curves 

531 

Mechanics 

625.  lae 

R.R.  E'rthwork 

531.2 

Statics 

625.2 

Trains.     Cars 

531.3 

Dynamics.  Kinetics 

625.8 

Roads  &  Pavements 

531.4 

Work.     Friction. 

626-7 

Hydraulic  Engineering 

531.6 

Energy 

627.8 

Dams 

531.7 

Power 

628 

Sanitary  Engineering 

532 

Hydraulics 

628.1 

Water  Works 

533 

Pneumatics 

628.12-] 

L3               Reservoirs 

635-6 

Light  and  Heat 

628.15 

Pipes 

637-8 

Elec'y  and  Magnetism 

628.16 

Purification 

540 

Chemistry 

628.17 

Meters. 

650 

Geology 

660 

Chem.  Tech'y.Explosivet 

551.5 

Meteorology 

670 

Manufactures.  Iron  &  St'l 

600 

Useful  Arts 

690 

Building 

620 

Engineering 

691 

Materials 

620.C 

Civil    Engineering 

691.7 

Iron  and  Steel 

620.i 

Indexes 

693 

Masonry 

620.1 

Strength  of  Materials 

694 

Carpentry 

621 

Mechanical    Engin'g 

697 

Heating  and  Vent'g 

621.1 

Steam     Engineering 

700 

Fine  Arts 

621.13 

Locomotives 

720 

Architecture 

621.18 

Steam  Generation 

721 

Arch'l    Construction 

621.2 

Water  Eng'&  Motors 

721.1 

Foundations 

621.3 

Electrical  Engin'g 

725 

Public  Building* 

621.4 

MiscellaneousMotors 

740 

Drawing 

621.6 

Pumps  &  Blowers 

770 

Photography 

BIBLIOGRAPHY.  1009 

Abbreviations. 

A         D.  Appleton  &  Co.,  1,  3  &  5  Bond  St.,  New  York,  N.  Y. 
B         Henry  Carey  Baird  &  Co.,  810  Walnut  St.,  Philadelphia,  Pa. 
C  G     Charles  Griffin  &  Co.,  Limited,  Exeter  Street,  Strand,  London. 
C  H     Chapman  &  Hall,  11  Henrietta  St.,  Covent  Garden.  London,  W.  C. 
C  L      Crosby  Lockwood  &  Son,  7  Stationers'  Hall  Court,  London,  E.  C. 
E  N     Engineering  News  Publishing  Co.,  220  Broadway,  New  York,  N.  Y« 
L  J.  B.  Lippincott  Co.,  Philadelphia,  Pa. 

L  G     Longmans,  Green  &  Co.,  91  &  9.3  Fifth  Ave.,  New  York,  N.  Y. 
M         The  Macmillan  Company,  66  Fifth  Ave.,  New  York,  N.  Y. 
R  G     Railroad  Gazette,  83  Fulton  St.,  New  York,  N.Y. 
S  C      Spon  and  Chamberlain,  12  Cortlandt  St.,  New  York,  N.  Y. 
S  L      Sampson,  Low,  Marston  &  Co.,  London. 
V  N     D.  Van  Nostrand  Co.,  23  Murray  St.,  New  York,  N.  Y. 
W        John  Wiley  &  Sons,  43  &  45  E.  19th  St.,  New  York,  N.  Y. 
500       Natural  Science. 
510  Mathematics. 

**Bledsoe,  A.  T. — .     The  Philosophy  of  Mathematics.     L. 
Hutton,  Charles — .     Mathematics.      1818. 

Merriman,  Mansfield —  and  Robert  C.  Woodward.     Higher  Mathematics. 
1vol.     8vo.     Cloth.     $5.00.     W. 
510.8  Tables  and  Mathematical  Instruments. 

Babbage,  Chas. — .     Logarithms  of  Nos.  from  1  to  108,000.    $3.00.     S  C. 
Barlow's  Tables  of  Squares,  Cubes,  Square  Roots,  Cube  Roots,  Recipro- 
cals, to  10.000.     $2.50.      S  C. 
Buchanan,  E.  E.— .     Tables  of  Squares.     $1.00.     S  C. 
*Chambers's  Mathematical  Tables.     Logarithms  of  Numbers  from   1  to 
108,000,  Trig.,  Nautical  and  other  Tables.     8vo.    Cloth.    $1.75.    V  N. 
Compton,  Alfred  G. — .     Manual  of  Logarithmic  Computations.     3d  ed. 

12mo.     Cloth.     $1.50.     W. 
Hall,  John  L.— .     Tables  of  Squares.     $2.00.     E  N. 
*Hering,  Carl — .     Tables  of  Equivalents  of  Units  of  Measurement.     W.  J. 
Johnston,  New  York. 
Holman,  Silas  W. — .     Computation  Rules  and  Logarithms.     8vo.     Cloth. 
$1.00.     M. 
*Hutton,  Charles — .     Mathematical  Tables.     London.     1822. 
Johnson,  J.  B. — .     Three-Place  Logarithmic  Tables,  Numbers  and  Trigo- 
nometric Functions.     15c.  each,  $5.00  per  100.     W. 
Jones,  G.  W. — .     Logarithmic  Tables.     8vo.     $1.00.     M. 
Ludlow,  Lt.,  H.  H. —  and  Edgar  W.  Bass.     Logarithmic,  Trigonometric, 

and  other  Mathematical  Tables.     8vo.     Cloth.     $2.00.     W. 
Osborn,  Frank  C. — .     Tables  of  Moments  of  Inertia  and  Squares  of  Radii 

of  Gyration.     175  pp.     $3.00.     V  N  and  E  N. 
Pickworth.  Charles    N. — .      The   Slide    Rule.      12mo.      Flexible   cloth. 

$0.80.     V  N. 
Rankine,  W.  J.  M.— .     Rules  and  Tables.     C  G. 

Skidmore  and  Vidal.     Table  of  Tangents.     4to.     Cloth.     $2.00.     V  N. 
Unwin,    W.    C. — .     Short   Logarithmic   and   other  Tables.     Small    4to. 

Cloth.     $1.40.     S  C.  . 

Tables.     See  also  under  subject  in  question. 
Surveying  Instruments.     See  526.91. 
512  Algebra. 

Chrystal,  G.— .    Algebra.    Part  I.    8vo.    $3.75.   Part  II.   8vo.    $4.00.    M. 
Liibsen,  H.  B. — .     Mathematics  Self  Taught.     Arithmetic  and  Algebra. 
Translated  and  published  by  H.  H.  Suplee,  120  Liberty  St..  New  York. 
Smith,  Charles—.     Algebra.     $1.90.     M. 

Todhunter,  I. — .     A  Treatise  on  the  Theory  of  Equations.    $1.80.     M. 
Todhunter,  I. — .     Algebra.     16mo.     $0.75.     M. 
Wentworth,  G.  A. — .     Algebra.     G. 
613  Geometry. 

♦Chauvenet,  William — .     Geometry.     L. 
Halstead,  Geo.  Bruce — .    Elements  of  Geometry.    8vo.    Cloth.    $1.75.  W. 
Wentworth,  G.  A. — .     Geometry.     G. 
Analytical  Geometry.     See  516. 
Descriptive  Geometry.     See  515. 

*     **  Believed  ta  be  specially  useful. 
64 


1010  BIBLIOGRAPHY. 

514  Trigonometry. 

Todhunter,  I. — .     Trigonometry.     Plane — .     M. 
Todhunter,  I. — .     Trigonometry.     Spherical — .     M. 

515  Descriptive  Geometry. 

See  744.     Drawing. 

516  Analytical  Geometry. 

Johnson,   W.  W. — .     Curve  Tracing  in  Cartesian  Co-ordinates.     12mo. 

Cloth.     $1.00.     W. 
Todhunter,  I. — .     Treatise  on  Plane  Co-ordinate  Geometry.     $1.80.     M. 
Todhunter,  I. — .     Examples  of  Anal.  Geom.  of  Three  Dimensions.    $1.  M. 

517  Calculus. 

*Barker,  Arthur  H. — .     Graphical  Calculus.     197  pp.     8vo.    $1.50.     L  G. 
Greenhill,  A.  G. — .     Differential  and  Integral  Calculus.     $2.60.     M. 
Johnson,  W.  W. — .     An  Elementary  Treatise  on  the  In-tegral  Calculus. 

Small  Svo.     243  pp.     Cloth.     $1.50.     W. 
Rice,  J.  M. —  and  Johnson,  W.  W. — .     An  Elementary  Treatise  on  the 

Differential   Calculus.     Small   Svo.     485   pp.     Cloth.     $3.00.     Same, 

abridged,  208  pp.     $1.50.     W. 
Todhunter,  I. — .     Calculus,  Differential — Integral.     $2.60  each.     M. 
Wansbrough,   Wm.   D. — .     The  A,   B.  C.  of  the  Differential  Calculus. 

12mo.     Cloth.     $1.50.     V  N. 
519  Probabilities.     Least  Squares. 

Johnson,  W.  W. — .     The  Theory  of  Errors  and  the  Method  of    Least 

Squares.     12mo.     182  pp.     Cloth.     $1.50.     W. 
Merriman,  Mansfield — .     The  Method  of  Least  Squares.     7th  ed.     Svo. 

Cloth.     $2.00.     W. 
530  Astronomy. 

Doolittle,  C.  L. — .     A  Treatise  on  Practical  Astronomy.     Svo.     652  pp. 

Cloth.     $4.00.     W. 
Loomis,  Elias — .     Practical  Astronomy.     Harper  &  Bros.,  New  York. 
Young,  C.  A. — .     Astronomy.     G. 

536  Geodesy. 

Comstock,  George  C. — .    A  Text-Book  of  Field  Astronomy  for  Engineers. 

Svo.     213  pp.     Cloth.     $2.50.     W. 
Gore,  J.  H. — .     Elements  of  Geodesy.     Svo.     Cloth.     $2.50.     W. 
Hayford,  John  F.— .     Geodetic  Astronomy.     Svo.     Cloth.     $3.00.     W. 
Merriman,  Mansfield — ,     The  Elements  of  Precise  Surveying  and  Geodesy. 

Svo.     Cloth.     $2.50.     W. 
Least  Squares.     See  519. 
526.9  Surveying. 

Burt,  W.  A. — .     Key  to  the  Solar  Compass,  and  Surveyor's  Companion. 

5th  ed.     Pocket-book  form.     $2.50.     V  N. 
Clevenger,  S.  R. — .     A  Treatise  on  the  Method  of  Government  Surveying 

as  prescribed  bv  the  U.  S.  Congress  and  Commissioner  of  the  General 

Land  Office.      16mo.     Morocco.     $2.50.     V  N. 
Gillespie,  W.  M. —  LL.D.     Treatise  on  Land-Surveying.     A. 
Gribble,  Theodore  Graham — .    Preliminary  Survey  and  Estimates.    Svo. 

Cloth.     415  pp.     LG. 
*Johnson,  J.  B. — .     The  Theory  and  Practice  of  Surveying.  .  Revised  and 

enlarged.     900  pp.     15th  ed.     Small  Svo.     Cloth.     $4.00.     W. 
Merriman,  Mansfield —  and  John  P.  Brooks.     Hand-Book  for  Surveyors. 

2d  ed.     16mo.     Morocco.     $2.00.     W. 
Nugent,   Paul  C. — .     Plane  Surveying.      Svo.      593  pp.      320  figures. 

Cloth.     $3.50.     W. 
Plympton,  Geo.  W. — .     The  Aneroid  Barometer:  Its  Construction  and 

Use.     ISmo.     $0.50.     V  N. 
♦Williamson,  R.  S. — .     On  the  Use  of  the  Barometer  on  Surveys  and  Rec- 
♦  onnoissances.     4to.     Cloth.     $15.00.     V  N. 
Winslow,  Arthur — .     Stadia  Surveying.     ISmo.     $0.50.     V  N. 
Railroad  Surveying.     See  also  625. Ta. 

536.91  Instruments. 

Baker,  Ira  O. — .     Engineers'  Surveying  Instruments.     2d  ed.     400  pp. 
12mo.     Cloth.     $3.00.     W. 
♦United  States  Coast  and  Geodetic  Survey.     The  Plane  Table :    Its  Uses 
in  Topographical  Surveying.     Svo.     Cloth.     $2.00.     V  N. 

*     **  Believed  to  be  specially  useful. 


BIBLIOGRAPHY.  101 1 

Webb,  W.  L. — .  Problems  in  the  Use  and  Adjustment  of  Engineering 
Instruments.     Revised  and  enlarged.     16mo.     Morocco.     $1.25.     W. 

526.94  Leveling. 

Baker,  Ira  O. — .  Leveling:  Barometric,  Trigonometric,  and  Spirit. 
$0.50.     V  N. 

536.98  Topography. 

*Haupt,  Lewis  M. — .  The  Topographer:  His  Instruments  and  Methods. 
J.  M.  Stoddard,  Phila.,  1883. 

Reed,  Lt.,  Henry  A. — .  Topographical  Drawing  and  Sketching.  In- 
cluding Photography  Applied  to  Surveying,  lllus.  4th  ed.  4to. 
Cloth.     $5.00.     W. 

Specht,  George  J. — ,  A.  S.  Hardy,  John  B.  McMaster,  H.  F.  Walling.  Topo- 
graphical Surveying.      ISrrK).     $0.50.     V  N. 

Wilson,  Herbert  M. — .  Topographic  Surveying,  including  Geographic, 
Exploratory  and  Military  Mapping,  with  Hints  on  Camping,  Emer- 
gency Surgery  and  Photography.     8vo.     Cloth.     $3.50.     W. 

530  Physics. 

♦Barker,  George  F. — .     Physics,  Advanced  Course. 

*.Deschanel,  A.  Privat — .     Elementary  Treatise  on  Natural  Philosophy. 

Translated  by  J.  D.  Everott.     A. 
*Ganot.     Physics.     Translated  by  E.  Atkinson,  Ph.D.     Wm.  Wood  &  Co., 
New  York. 
Tait.  P.  G.— .     Properties  of  Matter.     3d  ed.     $2.25.     M. 
Meteorology.     See  551.5. 

531  Mechanics. 

Ball,  Robert  Stawell — .     Experimental  Mechanics.     M. 

Church,  I.  P. — .     Mechanics  of  Engineering.     8vo.     Cloth.     $6.00.     W. 

Church,  I,  P. — .  Notes  and  Examples  in  Mechanics.  135  pp.  Svo. 
Cloth.     $2.00.     W. 

DuBois,  A.  Jay — .  Mechanics.  Vol.  I,  Kinematics,  $3.50;  Vol.  II,  Sta- 
tics, $4.00;  Vol.  Ill,  Kinetics,  $3.50.     Svo.     Cloth.     W. 

DuBois,  A.  Jay — .  The  Mechanics  of  Engineering.  Small  4to.  Cloth. 
Vol.  I,  669  pp.     $7.50.     Vol.  II,  632  pp.     $10.00.     W. 

Goodeve,  T.  M. — .     Principles  of  Mechanics.     L  G. 

Greene,  Chas.  E. — .     Structural  Mechanics.     271  pp.     $3.00.     E  N. 
*Lanza,  Prof.  G. — .     Applied  Mechanics  and  Resistance  of  Materials.     7th 
ed.     8vo.     Cloth.     $7.50.     W. 

Mach,  Dr.  Ernst — .     The  Science  of  Mechanics.     Translated  by  Thos.  J. 
McCormack.     12mo.     534  pp.     $2.50.     Open  Court  Publishing  Co. 
*Maxwell,  J.  Clerk— .     Matter  and  Motion.     18mo.     $0.50.     V  N. 
*Merriman,  Mansfield — .     Text-Book  on  Mechanics  of  Materials.     8th  ed. 
Svo.     Cloth.     $4.00.'    W. 

Michie,  Peter  S. — .  Elements  of  Analytical  Mechanics.  4th  ed.  Svo. 
Cloth.     $4.00.     W. 

Morin,  A. — .  Fundamental  Ideas  of  Mechanics  and  Experimental  Data. 
Revised  and  translated  by  Joseph  Bennett.     1860.     A. 

Moseley,  Henry — .  The  Mechanical  Principles  of  Engineering  and  Archi- 
tecture.    With  additions  by  D.  H.  Mahan.     W. 

Nystrom,  John  W. — .  A  New  Treatise  on  Elements  of  Mechanics.  352 
pp.     Svo.     Cloth.     $3.00.     B. 

Perry,  John — .  Practical  Mechanics.  Edited  by  W.  R.  Ayrton.  3^ 
shillings.     V  N. 

Rankine,  W.  J.  M. — .     Mechanical  Text-Book.     9  shillings.     C  G. 

Rankine,  W.  J.  M. — .     Applied  Mechanics.     Svo.     Cloth.     $5.00.     C  G. 

Weisbach,  Julius — .  Mechanics  of  Engineering  and  Machinery.  Trans- 
lated by  J.  F.  Klein.     2d  ed.     Svo.     Cloth.     $5.00.     W.    ' 

Weisbach,  Julius — .  Mechanics  of  Engineering.  Translated  by  Eckley 
B.  Coxe.     1vol.     Large  Svo.      1112  pp.     902  illus.     $10.00.     V  N. 

Weisbach,  Julius — .  A  Manual  of  Theoretical  Mechanics.  Translated  by 
Eckley  B.  Coxe.     1100  pp.     Svo.     Cloth.     $10.00.     V  N. 

Wood,  De  Volson — .  The  Elements  of  Analytical  Mechanics.  6th  ed. 
500  pp.     Svo.     Cloth.     $3.00.     W. 

Ziwet,  Alexander — .  An  Elementary  Treatise  on  Theoretical  Mechanics 
— Kinetics — Statics — Dynamics.     $2.25  each.     In  one  vol.,  $5.00.     M. 

*     **  Believed  to  be  specially  useful. 


1012  BIBLIOGRAPHY. 

Hydromechanics.     See  532. 

Machinery  and  Applied  Mechanics.     See  also  621.8. 

Strength  of  Materials.     See  620. 1 . 

531.2  Statics. 

Johnson,   L.  J. — .     Statics  by  Algebraic  and  Graphic  Methods.     8vo. 
141  pp.     Cloth.     $2.00.     W. 
*Lock,  J.  B. — .     Elementary  Statics.     M. 
Todhunter,  I. — .     A  Treatise  on  Analytical  Statics.     $2.60.     M. 
Arches.     See  624.6. 
Testing  and  Strength  of  Materials.     See  620.1. 

531.3  Dynamics.     Kinetics. 

Garnett,  William — .     A  Treatise  on  Elementary  Dynamics.     6  shillings. 
GB. 
♦Lock,  J.  B. — .     Dynamics  for  Beginners.     M. 
Tait,  P.  G.— .     Dynamics.     $2.50.     M. 

531.4  Work.     Friction. 

♦Thurston,  Robt.  H. — .     Treatise  on  Friction  and  Lost  Work  in  Machinery 
and  Mill  Work.    •5th  ed.     8vo.     Cloth.     $3.00.     W. 

531.6  Energy. 

Stewart,     Balfour — .      The     Conservation     of     Energy.      12mo.     Cloth. 
$1.50.     A. 

531.7  Power. 

Flather,  J.  J. — .     Dynamometers,  and  the  Measurement  of  Power.     394 
pp.     12mo.     Cloth.     $3.00.     W. 

532  Hydraulics. 

Bovey,  Henry  T.— .    A  Treatise  on  Hydraulics.    8vo.     Cloth.    $5.Q0.    W. 

**Coffin,  Freeman  C. — .     The  Graphical  Solution  of  Hydraulic  Problems. 

80  pp.      16mo.     Morocco.     $2.50.     W. 

Fidler,  T.  Claxton — .    Calculations  in  Hydraulic  Engineering.     167  pp. 

8vo.     $2.50.  ,LG. 

*Merriman,    Mansfield — .       A    Treatise    on    Hydraulics.       8vo.       Cloth. 

$5.00.     W. 
*Unwin,  W.  C— .     Hydromechanics.     Encyclopedia  Britannica. 
Hydraulic  Engineering.     See  626  and  627. 
Hvdraulic  Motors.     See  621.2. 
Waterworks.     See  628.1. 
See  also  under  531. 

532.5  Fluids  in  Motion. 

Bazin,   H. — .     Experiments  upon  the  Contraction  of  the  Liquid  Vein 
Issuing  from  an  Orifice.     Translated  by  John  C.  Trautwine,  Jr.     8vo. 
Cloth.     $2.00.     W. 
*Flynn,   P.   J. — .     Flow  of  Water   in  Open  Channels,   Pipes,    Conduits, 
Sewers,  etc.     With  Tables.     18mo.     $0.50.     V  N. 
**rrancis,    Jas.    B. — .      Lowell    Hydraulic   Experiments.      4to.       Cloth. 
$15.00.     V  N. 
Ganguillet,  E. —  and  W.  R.  Kutter.     A  General  Formula  for  the  Uniform 
Flow  of  Water  in  Rivers  and  Other  Channels.     Translated  by  Rudolph 
Hering  and  John  C.  Trautwine,  Jr.     2d  ed.     8vo.     Cloth.     $4.00.     W. 
*Herschel,   Clemens — .     One  Hundred  and  Fifteen   Experiments  on  the 
Carrying  Capacity  of  Large,  Riveted,  Metal  Conduits.     8vo.     Cloth. 
$2.00.     W. 
Moore,  C.  S. — .     New  Tables  for  the  Complete  Solution  of  Ganguillet  and 
Kutter's  Formula.     6"  X  9'',  231  pp.     15  shillings.     B. 
♦Weston,  Edmund  B. — .     Tables  Showing  Loss  of  Head  Due  to  Friction  of 
Water  in  Pipes.     170  pp.     $1.50.     V  N  and  E  N. 
See  also  under  628.1,  Waterworks. 

533  Pneumatics. 

Blowers,  Pumps.     See  621.6. 
Meteorology.     See  551.5. 
See  also  under  531  and  532. 


*     **  Believed  to  be  specially  useful. 


BIBLIOGRAPHY.  1013 

535  Light. 

Tait,  P.  G.— .     Light.     $2.00.     M. 

536  Heat. 

Barr,  Wm.  M. — .     A  Practical  Treatise  on  the  Combustion  of  Coal.     307 

pp.     8vo.     Cloth.     $2.50.     B. 
Garnett,  William — .     An  Elementary  Treatise  on  Heat.     3  s.  6  d.     G  B. 
Maxwell,  J.  Clerk — .     Theory  of  Heat.     357  pp.     12mo.     $1.50.     L  G. 
Tait,  P.  G.— .     Heat.     $2.50.     M. 
♦Tyndall,  John — .     Heat  Considered  as  a  Mode  of  Motion. 
Steam.     See  621.1. 
Heating  and  Ventilating.     See  697. 

537-8  Electricity  and  Magnetism. 

Jenkin,     Fleeming — .     Electricity    and     Magnetism.     415     pp.     12mo. 

$1.25.     L  G. 
Electrical  Machines  and  Instruments.     See  621.3. 

540  Chemistry. 

Phillips,      Joshua — .     Engineering     Chemistry.     2d    ed.      8vo.     Cloth. 

$4.00.     V  N. 
Remsen,  Ira — .     Chemistry.     Henry  Holt  &  Co.,  New  York. 
Sexton,  A.    H. — .     Chemistry  of  the  Materials  of  Engineering.     12mo. 

Cloth.     $2.50.     V  N. 
Potable  Waters.     See  628.1. 

550  Geology. 

Merrill,  G.  P. — .     Rocks,  Rock  Weathering,  and  Soils.     8vo.     $4.00.     M. 

Stockbridge,  Horace  Edward — .  Rocks  and  Soils.  Their  Origin,  Compo- 
sition, and  Characteristics;  Chemical,  Geological,  and  Agricultural. 
With  15  full-page  plates.     2d  ed.     8vo.     Cloth.     $2.50.     W. 

Tarr,  R.  S. — .  Economic  Geology  of  the  United  States.  8vo.  Cloth. 
$3.50.     M. 

Tillman,  S.  E. — .  A  Text-Book  of  Important  Minerals  and  Rocks.  8vo. 
Cloth.     $2.00.     W. 

Building  Stones.     See  691.2. 

Mining.     See  622. 

551.5  Meteorology. 

Ferrel,   William — *.    A  Popular  Treatise  on  the  Winds.     2d  ed.     8vo. 
Cloth.     $4.00.     W. 
*Russell,  Thomas — .     Meteorology.     8vo.     Cloth.     $4.00.     M. 
Williamson,  R.  S. — .    Practical  Tables  in  Meteorology  and  Hypsometry  in 
Connection  with  the  Use  of  the  Barometer.     4to.     Cloth.    $2.50.     V  N. 

600        Useful  Arts. 

620  Engineering. 

Brooks,  Robt.  C. — .     A  Bibliography  of  Municipal  Problems  and  City 

Conditions.     346  pp.     Cloth.     $1.50.     Reform  Club. 
Carpenter,  R.  C. — .     Experimental  Engineering.     5th  ed.     Revised  1897. 

8vo.     Cloth.     $6.00.     W. 
Crehore,  Wm.  W. — .     Tables  and  Diagrams  for  Engineers  and  Architects. 

$0.25  to  $0.50  each.     Set,  $7.50.     E  N. 
Cross,  C.  S.— .     Engineers'  Field  Book.     4th  ed.     166  pp.     $1.00.     E  N. 
Dawson,   Philip — .     The  "Engineering"  and  Electric  Traction  Pocket- 

Book.      1056  pages.      1300  illus.     16mo.     Morocco  flap.     $5.00.     W. 
Goodhue,  W.  F. — .     Municipal  Improvements.     3d  ed.     12mo.     Cloth. 

$1.75.     W. 
Haupt,    L.    M. — .      Engineering   Specifications   and   Contracts.       J.    M. 

Stoddard  &  Co.,  Phila. 
Hurst.     Tables  and  Memoranda  for    Engineers.       Vest-pocket   edition. 

64mo.     Roan.     $0.50.     S  C. 
**  Johnson,  J.  B. — .    Engineering  Contracts  and  Specifications.    $3.00.  EN. 
Kempe,   H.   R. — .     The  Engmeer's  Year-Book.     670  pp.     Crown  8vo. 

Leather.     $3.00.     C  L. 

*     **  Believed  to  be  specially  useful. 


1014  BIBLIOGRAPHY. 

♦Molesworth  and  Hurst.     The   Pocket   Book  of  Pocket   Books.     32mo. 

Russia.     $5.00.     S  C. 
Philbrick,    P.    H. — .     Field    Manual    for    Engineers.     16mo.     Morocco. 

$3.00.     W. 
Rankine,  W.  J.  M. — .     Useful  Rules  and  Tables  for  Engineers  and  Others. 

8vo.     Cloth.     S4.00.     V  N. 
*Shunk,    W.    F.— .     The    Field    Engineer.     11th    ed.     12mo,     Morocco, 

tucks.     $2.50.     V  N. 
Smart,  Richard  A. — .     A  Hand-Book  of  Engineering  Laboratory  Practice. 

290  pp.      12mo.     Cloth.     $2.50.     W. 
Wait,    John    Cassan — .     Engineering   and   Architectural   Jurisprudence. 

985  pp.     8vo.     Cloth,  $6.00;  sheep,  $6.50.     W. 
Wait,  John  Cassan — .     The  Law  of  Operations  Preliminary  to  Construc- 
tion in   Engineering  and  Architecture.     720  pp.     8vo.     Cloth,  $5.00; 

sheep,  $5.50.     W. 
Weale,  John — .     A  Dictionary  of  Terms  Used  in  Architecture,  Building, 

Engineering,  Mining,  Metallurgy,  Archaeology,  the  Fine  Arts,  etc.     5th 

ed.     12mo.     Cloth.     $2.50.     V  N. 
Retaining  Walls,  etc.     See  721.1. 
River  Embankments.     See  627. 
Railroad  Earthwork.     See  625.  lac. 

620.C  Civil  Engineering. 

Butts,  Edward — .     The  Civil  Engineers'  Field  Book.     16mo.     Morocco. 

$2.50.     W. 
Mahan,  D.  H. — .     A  Treatise  on  Civil  Engineering.     Revised  by  De  Vol- 

son  Wood.     New  chapter  on  River  Improvements  by  F.  A.  Mahan. 

8vo.     Cloth.     $5.00.     W. 
♦Patton,  W.  M. — .     A  Treatise  on  Civil  Engineering.     8vo.     Half  leather. 

$7.50.     W. 
Rankine,  W.  J.  M. — .     A  Manual  of  Civil  Engineering.     20th  ed.     8vo. 

Cloth.    $6.50.     CG. 
Trautwine,  John  C. — .     The  Civil  Engineer's  Pocket-Book.     Revised  and 

enlarged  by  John  C.  Trautwine,  Jr.,  and  John  C.  Trautwine  3d.      18th 

ed.,  70th  thousand.      16mo.     Morocco,  gilt  edges.     $5.00.     W,  C  H. 
Wheeler,  J.  B. — .     An  Elementary  Course  of  Civil  Engineering.     8vo. 

Cloth.     $4.00.     W. 

620.i  Indexes. 

*Assn.  Eng.  Soc,  John  C.  Trautwine,  Jr.,  Sec'y.  Descftptive  Index  of  Cur- 
rent Engineering  Literature.  Vol.  I,  1884-1891.  $5.00.  257  S.  4th 
St.,  Phila. 

Engineering  Magazine.  Descriptive  Index  of  Current  Engineering  Litera- 
ture. *Vol.  II,  1892-1895,  $5.00.  **Vol.  Ill,  1896-1900.  120  Liberty 
St.,  New  York. 

Galloupe,  F.  E. — .  Index  to  Engineering  Periodicals.  1888-1892. 
$3.00.     RG. 

620.1  Strength  of  Materials. 

Baker,  B. — .     On  the  Strengths  of  Beams,  Columns,  and  Arches.     V  N. 
Bovey,  Henry  T. — .     Theory  of  Structure  and  Strength  of  Materials.     3d 

ed.     830  pp.     8vo.     Cloth.     $7.50.     W. 
Burr,  W.  H. — .     Elasticity  and  Resistance  of  Materials  of  Engineering. 

6th  ed..  Re-written.     772  pp.     8vo.     Cloth.     $7.50.     W. 
Johnson,  J.  B. — .     The  Materials  of  Construction.     Large  8vo.     810  pp. 

Cloth.     $6.00.     W. 
Martens,  Adolph — .    Hand-Book  on  Testing  Materials.    Part  I :  Meth9ds, 

Machines  and  Auxiliary  Apparatus.     Translated  by  Gus.  C.  Henning, 

M.'E.     2  vols.     8vo.     Cloth.     $7.50.     W. 
♦Merriman,    Mansfield — .     The   Strength   of   Materials.     2d   ed.      12mo. 

Cloth.     $1.00.     W. 
Moore  and  Kid  well.     Tables  of  Safe  Loads  for  Wooden  Beams  and  Col- 
umns.    57  pp.     $0.50.     EN. 
Spangenburs:,  Ludwig — .     The  Fatigue  of  Metals  under  Repeated  Strains. 

Translated.     18mo.     $0.50.     V  N. 

*     **  Believed  to  be  specially  useful. 


BIBLIOGRAPHY.  1015 

Thurston,  Robert  H. — .     A  Text-Book  of  the  Materials  of  Construction 

8vo.     785  pp.     Cloth.     $5.00.     W. 
Unwin.  W.  C. — .     The  Testing  of  Materials  of  Construction.     500  pp. 

Svo.     $6.00.     L  G. 
Wood,  De  Volson — .     A  Treatise  on  the  Resistance  of  Materials,  and  an 

Appendix    on    the    Preservation    of    Timber.     7th    ed.     Svo.     Cloth. 

$2.00.     W. 
See  also  721,   Architectural  Construction;  624,  Bridges;  691,  etc.;  532, 

Mechanics;  532.2,  Statics. 
631  Mechanical  Engineering. 

Clark,    D.    Kinnear — .      The    Mechanical    Engineer's    Pocket-Book    of 

Tables,  Formulae,  Rules,  and  Data.     3d  ed.     700  pp.     Svo.     Leather. 

6  shillings.     C  L. 

Clark,  D.  Kinnear — .     A  Manual  of  Rules,  Tables,  and  Data  for  Mechani- 
cal Engineers.     1012  pp.     6th  ed.     Svo.     Cloth.     $5.00.     V  N. 

Kennedy,  Alex.  B.  W. — .     The  Mechanics  of  Machiner5\     M. 
**Kent,   William — .     The  Mechanical  Engineer'^  Pocket-Book.     1100  pp. 
6th  ed.      16mo.     Morocco.     $5.00.     W. 

Lockwood's  Dictionary  of  Terms  Used  in  the  Practice  of  Mechanical  Engi- 
neering.    Edited  by  Joseph  G.  Horner,     6000  definitions.     Svo.    Cloth. 

7  s.  6d.     CL. 

Rankine,  W.  J.  M. — .     A  Manual  of  Machinery  and  Millwork.     5S0  pp. 
C  G. 
**Reuleaux,  F. — .     The  Constructor.     Translated  by  H.  H.  Suplee.     $7.50. 
312  pp.     1200  illustrations.     Eng.  Mag.,  120  Liberty  St.,  New  York. 
*Unwin,  W.  C. — .     Elements  of  Machine  Design.     Part  I,  General  Princi- 
ples, 476  pp.     $2.00.     Part  II,  Engine  Details,  306  pp.     $1.50.     L  G. 
Weisbach,  Dr.  Julius —  and  Prof.  Gustav  Herrmann.     The  Mechanics  of 
Hoisting   Machimftry,    including   Accumulators,    Excavators   and    Pile 
Drivers.     Translated  by  Karl  P.  Dahlstrom.      Svo.     Cloth.     332  pp. 
177  illus.     $3.75.     M. 
631.1  Steam  Engineering. 

*Holmes,  Geo.  C.  V.—.     The  Steam  Engine.     12mo.     $2.00.     L  G. 
*Pray,   Thomas —  Jr.     Steam   Tables  and   Engine  Constants.     Compiled 
from  Regnault,  Rankine,  and  Dixon.     Svo.     Cloth.     $2.00.     V  N. 
Rankine,  W.  J.  M. — .     A  Manual  of  the  Steam  Engine  and  other  Prime 

Movers.     C  G. 
Spangler,  H.  W.— ;  Greene,  W.  M.— ;  Marshall,  S.  M.— .     The  Elements 

of  Steam  Engineering.     Svo.     Cloth.     273  pp.     $3.00.     W. 
Thurston,  Robert  H. — .    Handy  Tables.    From  the  "Steam-Engine  Man- 
ual."    Svo.     Cloth.     $1.50.     W. 
Thurston,  Robert  H. — .     A  Manual  of  the  Steam-Engine.     Part  I,  Struc- 
ture and   Theory;  Part   II,    Design,    Construction,    Operation.     Each 
part,  Svo,  1000  pp.,  $6.00.     Two  parts,  $10.00.     W. 
Whitham,  Jay  M. — .     Steam-Engine  Design.     Svo.     Cloth.     $5.00.     W. 
631.13  Locomotives. 

**Forney,  Mathias  N. — .     Catechism  of  the  Locomotive.     $3.50.     V  N    & 
RG. 
Meyer,    J.    G.    A. — .     Modern    Locomotive    Construction.     4to.     Cloth, 

$10.00.     W, 
Reagan,  H.  C. — .    Locomotives:  Simple,  Compound  and  Electric.  12mo. 

617  pp.     Cloth.     $2.50.     W. 
"Railroad  Gazette."     Modern  Locomotives.     $7.00.     R  G. 

631.18  Steam  Generation. 

Barr,  Wm.  M. — ,     A  Practical  Treatise  on  High  Pressure  Steam  Boilers. 
456  pp.     Svo.     Cloth.     $3.00.     B. 
*Heine  Safety  Boiler  Co.     "Helios."     (Tables.)     Published  by  author. 
Peabody,  Cecil  H. —   and  Miller,   Edward  F. — .     Steam  Boilers.     Svo 

384  pp.     Cloth.     $4.00.     W. 
Thurston,  R.  H. — .    A  Manual  of  Steam-Boilers :  Their  Designs,  Construc- 
tion, and  Operation.     5th  ed.     Svo.     Cloth.     $5.00.     W.' 
*  Wilson,  Robert — .     A  Treatise  on  Steam-Boilers.     Enlarged  and  Illus^ 
trated  by  J.  J.  Flather.     3d  ed.     12mo.     Cloth,     $2.50.     W. 
Steam  Heating.     See  697. 

*     **  Believed  to  be  specially  useful. 


1016  BIBLIOGRAPHY. 

631.3  Water  Engines  and  Motors. 

**Francis,     Jas.     B. — .     Lowell     Hydraulic     Experiments.     4to.     Cloth. 
$15.00.     V  N. 
*Frizell,  J.  P.—.     Water  Power.     584  pp.     8vo.     Cloth.     $5.00.     W. 
Weisbach,  Julius — .     Hydraulics  and  Hydraulic  Motors.     Translated  by 

A.  J.  Du  Bois.     2d  ed.     Illus.     8vo.     Cloth.     $5.00.     W. 
Wood,  De  Volson— .     Turbines.     Old  ed.,  8vo,  cloth,  $1.00;  2d  ed.,   re- 
vised and  enlarged,  8vo,  cloth,  $2.50.     W. 
See  also  532.     Pumps,  621.6. 

631.3  Electrical  Engineering. 

♦Foster,  H.  A. — .     Electrical  Engineer's  Pocket-Book.     V  N. 

Haupt,  Herman — .  Street  Railway  Motors.  213  pp.  12mo.  Cloth. 
$1.75.     B. 

Rosenberg,  E. — ,  Haldane  Gee,  W.  W. — ,  Kinzbrunner,  Carl — .  Electri- 
cal Engineering.     Svo.     275  pp.     Cloth.     $1.50  net.     W. 

See  also  537. 

631.4  Air,  Gas,  and  Other  Motors. 

Goldingham,  A.  H. — .     The  Design  and  Construction  of  Oil  Engines.     196 

pp.     $2.00.     SC. 
♦Kennedy,  A.  B.  W. —    and  W.  C.  Unwin.     Transmission  by  Air-Power. 

18mo.     $0.50.     V  N. 
Richards,  Frank — .     Compressed  Air,     12mo.     Cloth.     $1.50.     W. 
Saunders,  W.  L. — .     Compressed  Air  Production.     58  pp.     $1.00.     EN. 
Wolff,  A.  R. — .     The  Windmill  as  a  Prime  Mover.     2d  ed.     Svo.     Cloth. 

$3.00.     W. 
631.6  Blowing  and  Pumping  Engines. 

Barr,  Wm.  H. — .     Pumps.     L. 
Weisbach,  Julius —  and  Gustave  Hermann.     The  Mechanics  of  Pumping 

Machinery.     Translated  by  K.  P.  Dahlstrom.     Svo.     $3.75.     M. 

631.8  Transmission  Mechanism. 

Cooper,  John  H. — .     A  Treatise  on  the  Use  of  Belting  for  the  Transmission 

of  Power.     Svo.     Cloth.     $3.50.     B. 
Flather,  J.  J. — .     Rope  Driving.     12mo.     Cloth.     $2.00.     W. 
Kerr,    E.    W. — .       Power   and    Power    Transmission.       Svo.       368  pp. 

Cloth.     $2.00.     W. 
Stahl,   Albert  W. — .     Transmission  of  Power  by  Wire  Ropes.     ISmo. 

$0.50.     V  N.  , 

Principles   of    Mechanism.      See   also   531,    Mechanics.      See   also  621, 

Mechanical  Engineering. 

633  Mining. 

Bowie,  Aug.  J. —  Jr.     A  Practical  Treatise  on  Hydraulic  Mining  in  Cali- 
fornia.    5th  ed.     Small  4to.     Cloth.     $5.00.     V  N. 
♦Drinker,    Henry    S. — .     Tunneling,    Explosive    Compounds,    and    Rock 
Drills.      1000  Illus.     3d  ed.     4to.     Half-bound.     $25.00.     W. 

Hermann,  E.  A. — .  Steam  Shovels  and  Steam  Shovel  Work.  60  pp. 
$1.00.     E  N. 

Ihlseng,  M.  C— .    A  Manual  of  Mining.    Svo.    585  pp.     Cloth.    $4.00.    W. 

Prelini,  Charles— .     Tunneling.     311pp.     6'' X  9i''.     $3.00.     V  N. 

Simms,  W.  F.— .  Practical  Tunneling.  4th  ed.  Svo.  Cloth.  $12.00. 
VN. 

Wilson,  E.  B.— .  Hydraulic  and  Placer  Mining.  12mo.  Cloth.  $2.00. 
W. 

Retaining  Walls,  etc.     See  721.1. 

Explosives.     See  also  660. 

634  Bridges  and  Roofs. 

Baker,    B. — .     Long-Span    Railway    Bridges.     97    pp.     12mo.     Cloth. 

$1.00.     B. 
Bender,   Charles   E. — .     Proportions  of  Pins   Used  in   Bridges.      18mo. 

$0.50.     V  N. 
Boiler,  A.  P. — .     Practical  Treatise  on  the  Construction  of  Iron  Highway 

Bridges.      (Written    in    popular   language.)      4th    ed.      Svo.     Cloth. 

$2.00.     W. 

* .   **  Believed  to  be  specially  useful. 


BIBLIOGRAPHY.  1017 

Boiler,  A.  P. — .  The  Thames  River  Bridge.  Limited  edition.  Illus.  4to. 
Paper.     S5.00.     W. 

Burr,  W.  H. — .  Stresses  in  Bridges  and  Roof  Trusses,  Arched  Ribs,  and 
Suspension  Bridges.  9th  ed.  Revised.  Plates.  8vo.  Cloth.  $3.50. 
W. 

Chanute,  O. —  and  George  S.  Morison.     The  Kansas  City  Bridge.     4to. 
Cloth.     $6.00.     V  N. 
*Fidler,  T.  Claxton — .     A  Practical  Treatise  on  Bridge-Construction.     2d 

ed.  8vo.  30  shillings.  C  G. 
*Green,  Chas.  E. — .  Graphics  for  Engineers,  Architects,  and  Builders. 
Part  I:  Roof  Trusses.  Diagrams.  New  revised  ed.  8vo.  Cloth. 
$1.25.  Part  II:  Bridge  Trusses.  New  revised  ed.  Svo.  Cloth.  $2.50. 
Part  III:  Arches  in  Wood,  Iron,  and  Stone.  3d  ed.  Svo.  Cloth.  $2.50. 
W. 

Johnson,  J.  B. — ,  Bryan,  C.  W. — ,  Turneaure.  F.  E. — .  The  Theory  and 
Practice  of  Modern  Framed  Structures.  Small  4to.  538  pp.  Cloth. 
$10.00.     W. 

McMaster,  John  B. — .  Bridge  and  Tunnel  Centers.  18mo.  $0.50.  V  N. 
**Merriman,  Mansfield —  and  Henry  S.  Jacoby.  A  Text-Book  on  Roofs  and 
Bridges.  Part  I:  Stresses  in  Simple  Trusses.  5th  ed.,  revised  and  en- 
larged. 8vo.  Cloth.  S2.50.  Part  II:  Graphic  Statics.  3d  ed.,  en- 
larged. With  5  folding  plates.  Svo.  Cloth.  $2.50.  Part  III*. 
Bridge  Design.  3d  ed.  Svo.  Cloth.  $2.50.  Part  IV:  Cantilever, 
Continuous,  Draw,  Suspension  and  Arch  Bridges.  2d  ed.  Svo. 
Cloth.     $2.50.     W. 

Morison,  George  S. — .     The  Memphis  Bridge.     $10.00.     W. 

Ritter,  August — .     Iron  Bridges  and  Roofs.     Translated  by  H.  R.  San- 
key.     S  C. 
*Waddell,  J.  A.  L. — .     De  Pontibus.     A  Pocket-Book  for  Bridge  Engi- 
neers.    416  pp.      16mo.     Morocco.     $3.00.     W. 

Whipple,  S. — .  An  Elementary  and  Practical  Treatise  on  Bridge  Build- 
ing.    Svo.     Cloth.     $3.00.     V  N. 

Wood,  De  Volson — .  A  Treatise  on  the  Theory  of  the  Construction  of 
Bridges  and  Roofs.  Illus.  6th  ed.  Revised  and  corrected.  Svo. 
Cloth.     $2.00.     W. 

Retaining  Walls,  Foundations,  etc.  See  721.1.  See  also  531.2,  Statics. 
Architectural  Construction.     Strength  of  Materials,  see  620.1. 

624.0s  Specifications  for  Bridges. 

*Bouscaren,  G. — .     Specifications  for  Railway  Bridges  and  Viaducts  of 

Iron  and  Steel.     9  pp.     $0.25.     E  N. 
♦Cooper,  Theodore — .     Specifications  for  Steel  Highway  Bridges.     25  pp. 

$0.25.     E  N. 
*Cooper,  Theodore — .     Specifications  for  Steel  Railroad  Bridges.     24  pp. 

$0.25.     E  N. 
*Osborn   Co.     Specifications  for  Metal   Highway  Bridge  Superstructure 

12  pp.     $0.25.     E  N. 
*Osborn    Co.      General    Specifications    for    Railway    Bridges.     10    pp. 

$0.25.     E  N. 
♦Thacher,  Edwin — .     General  Specifications  for  Highway  Bridges.     8  pp. 

$0.25.     EN.  .  . 

♦Thomson,    G.    H. — .      Standard   Specifications  for   Structural   Steel  for 

Modern  Railroad  Bridges.     $0.10.     E  N. 
*Waddell,  J.  A.  L. — .     Specifications  for  Steel  Bridges  (from  "De  Ponti- 
bus").    12mo.     Cloth.     $1.25.     W. 

634.a  Trestles.     Viaducts. 

♦Foster,  Wolcott   C. — .     A  Treatise  on  Wooden   Trestle   Bridges.     4to. 
Cloth.     271  pp.     $5.00.     W. 
Katte,    W. — .     Specifications   for    Standard    Pile    and    Timber    Trestle 
Bridges.     $0.05.     E  N. 

634.3  Girders. 

Birkmire,  Wm.  H. — .  Compound  Riveted  Girders  as  Applied  in  Build- 
ings.    Svo.     Cloth.     $2.00.     W. 

*     **  Believed  to  be  specially  useful. 


1018  BIBLIOGRAPHY. 

Philbrick,  P.  H. — .     Beams  and  Girders.     Practical  Formulas  for  'Their 
Resistance.     18mo.     $0.50.     V  N. 
*Stoney,  Bindon  B. — .     The  Theory  of  Stresses  in   Girders  and  Similar 
Structures.     777  pp.     8vo.     $12.50.     V  N. 

634.3  Trusses. 

Ricker,  N.  C. — .  Elementary  Graphic  Statics  and  the  Construction  of 
Trussed  Roofs.     4th  ed.     8vo.     Cloth.     $2.00.     E  N. 

624.6  Arches. 

Buck,  G.  W. — .     Oblique  Bridges  (Arches).     Revised  by  J.  H.  W.  Buck. 

With  plates.     C  L. 
Cain,   William — .     Voussoir  Arches  Applied  to  Stone  Bridges,  Tunnels, 

Culverts,  and  Domes.     ISmo.     $0.50.     V  N. 
Howe,   Malverd   A. — .     A  Treatise  on  Arches.     371   pp.     Svo.     Cloth. 

$4.00.     W. 
Woodbury,  D.  P.—.     Stability  of  Arches.     V  N. 

624.7  Compound  Bridges. 

Bender,  Charles — .  Practical  Treatise  on  the  Properties  of  Continuous 
Bridges.     ISmo.     $0.50.     V  N. 

624.8  Draw  Bridges. 

Wright,  Chas.  H. — .  The  Designing  of  Draw-Spans.  Part  I:  Plate  Gir- 
der Draws.  Part  II:  Riveted-Truss  and  Pin-Connected  Long-Span 
Draws.     Illus.     Svo.     Cloth.     $3.50.     W. 

625  Roads  and  Railroads. 

Berg,   Walter  G. — .     Building  and  Structures  of  American  Railroads. 

Large  4to.     Cloth.     $5.00.     V  N. 
Cleemann,    Thos.    M. — .     The    Railroad    Engineer's    Practice.     4th   ed. 

12mo.     Cloth.     $1.50.     VN. 
Dredge,  James — .     History  of  the  Pennsylvania  Railroad.     Engravings, 

Map,  Plates,  etc.     Folio.     Half-morocco,  $10.00.     Paper,  $5.00.     W. 
Godwin,    H.    C. — .     Railroad   Engineer's   Field   Book.     (An    Explorer's 

Guide.)     2ded.     16mo.     Morocco.     $2.50.     W. 
*Henck,  John  B.— .    Field  Book  for  Railroad  Engineers.    1896.    $2.50.    A. 
Nagle,  J.  C. — .     A  Field  Manual  for  Railroad  Engineers.     2d  ed.     16mo. 

Morocco.     $3.00.     W. 
Paine,  Charles— .     The  Elements  of  Railroading.     $1.00.     R  G. 
*Paine,  George  H. — .     The  New  Roadmaster's  Assistant.     $1.50.     R  G. 
Vose,  G.  L. — .     Manual  for  Railroad  Engineers.     Two  vols.,  text  and 

plates.     Lee  &  Shepard,  New  York,  1873. 
Bridges.     See  624. 
Tunnels.     See  622. 
Electric  Railways.     See  621.3. 
Locomotives.     See  621.13. 

625.1  Route,  Track.     Fixed  Equipment. 

Katte,  W. — .     General  Specifications  for  Cross  Ties.     $0.05.     E  N. 

Katte,  W. — .     Specifications  for  Track-laying.     $0.10.     E  N. 
*Parsons,  W.  B. —  Jr.     Track.     A  Complete   Manual  of  Maintenance  of 
Way.     Svo.     Cloth.     $2.00.     EN. 

Pratt,  Mason  D. —  and  C.  A.  Alden.  Street-Railway  Roadbed.  Svo. 
Cloth.     $2.00.     W. 

Railroad  Gazette.     Block  SignaUng.     $2.00.     R  G. 
*Tratman,  E.  E.  R. — .     Railway  Track  and  Track  Work.     502  pp.     200 

illus.     $3.00.     1901.     E  N. 
*Tratman,  E.  E.  Russell — .     Metal  Railroad  Ties.     Report  on  the  Use  of. 
Preservative  Processes  and  Metal  Tie-Plates  for  Wooden  Ties.     Pub- 
lished by  U.  S.  Department  of  Agriculture. 

Webb,  Walter  Loring — .  Railroad  Construction.  16mo.  691  pp. 
Morocco.     $5.00.     W. 

Railroad  Stations.     See  725. 

Bridges.     See  624. 

*     **  Believed  to  be  specially  useful. 


BIBLIOGRAPHY.  1019 

625.1a  R.  R.  Surveying. 

Allen,  C.  Frank — .     Railroad  Curves  and  Earthwork.     194  pp.     Pocket- 
book  Form.     $2.00.     S  C. 
Brooks,    John   P. — .     Hand-Book  of  Street  Railroad  Location.     16mo. 
Morocco.     $1.50.     W. 
*Gribble,  T.  G. — .     Preliminary  Survey  and  Estimates.     480  pp.     12mo. 
$2.50.     L  G. 
**Searles,  Wm.  H. — .     Field  Engineering.     Railway  Surveying,  Location, 
and  Construction.      16th  ed.     16mo.     Morocco.     $3.00.     W. 
*Shunk,  William  F. — .    A  Practical  Treatise  on  Railwa^'^  Curves  and  Loca- 
tion, for  Young  Engineers.     12mo.     Cloth,  tucks.     $2.00.     B. 
*  Welling  ton,  A.  M. — .     Economic  Theory  of  Railway  Location.     980  pp. 
$5.00.     W  &EN. 
See  also  526.9. 

635.131;  R.  R.  Curves. 

Clark,  Jacob  M. — .      A  New  System  of  Laying  Out  Railway  Turnouts 

Instantly,  by  Inspection    from   Tables.     12mo.     Leatherette.     $1.00. 

VN.  I 

♦Crandall,   Chas.   L. — .     The  Transition  Curve.     Revised  and  enlarged. 

16mo.     Morocco.     $1.50.     W. 

Fox,  Walter  G. — .     Transition  Curves.     ISmo.     $0.50.     V  N. 

Gieseler,  E.  A. — .     Scales  for  Turnouts.     Stiff  cardboard.     $0.25.     R  G. 

Howard,    Conway    R. — .     The    Transition    Curve    Field    Book.     16mo. 
Morocco.     $1.50.     W. 
**Searles,  Wm.  H. — .     The  Railroad  Spiral.     The  Theory  of  the  Compound 
Transition  Curve  Reduced  to  Practical  Formulas  and  Rules  for  Applica- 
tion in  Field  Work.     6th  ed.     16mo.     Morocco.     $1.50.     W. 

Torrey,  A. — ,     Switch  Layouts  and  Curve  Easements.     $1.00.     R  G. 

Trautwine,  John  C. — .  The  Field  Practice  of  Laying  out  Circular  Curves 
for  Railroads.  Revised  by  John  C.  Trautwine,  Jr.  13th  ed.  12mo. 
Morocco.     $2.50.     W,  C  H. 

R.  R.  Earthwork. 

Tables    for    Earthwork    Computation.     8vo.     Cloth. 

L. — .  Railway  and  Other  Earthwork  Tables.  8vo. 
W. 
♦Hudson,  J.  R. — .  Tables  for  Calculating  the  Cubic  Contents  of  Excava- 
tions and  Embankments  by  an  Improved  Method  of  Diagonals  and  Side 
Triangles.  New  edition  with  additional  tables.  8vo.  Cloth.  $1.00. 
W. 

Johnson,  J.  B. — .  Stadia  and  Earthwork  Tables.  8vo.  Cloth.  $1.25. 
W. 

Katte,  W. — .  Specifications  for  Grading  and  Masonry.  16  pp.  $0.25. 
EN. 

Taylor,  Thomas  V. — .  Prismoidal  Formulae  and  Earthwork.  Svo. 
Cloth.     $1.50.     W. 

Trautwine,  John  C. — .  A  Method  of  Calculating  the  Cubic  Contents  of 
Excavations  and  Embankments  by  the  Aid  of  Diagrams.  Revised  and 
enlarged  by  John  C.  Trautwine,  Jr.     9th  ed.     8vo.     Cloth.    $2.00.     W. 

Trautwine,  John  C. —  Jr.  Cross-Section  Sheet.  To  be  Used  with  Traut- 
wine's  Excavations.     Sheet  form.     $0.25.     W. 

Trautwine,  JohnC. —  Jr.,  and  Woodson,  D.  Meade — .  Cross-Section  Sheet. 
$0.50.     Williams,  Brown  &  Earle,  Phila. 

Earth  Handling.     See  622. 

Foundations.     See  721.1. 

635.2  Trains.     Rolling  Equipment. 

♦Railroad  Gazette.     Car-Builder's  Dictionary.     $5.00.     R  G. 
Locomotives.     See  621.13. 

635.8  Roads  and  Pavements. 

Aitken,  Thomas — .  Road  Making  and  Maintenance.  Cloth,  7"  x  9". 
440  pp.     139illus.     $6.00.     L. 

*     **  Believed  to  be  specially  useful. 


635.1ae 

Allen,    C. 
$1.50. 
♦Crandall, 
Cloth. 

F.— 

VN. 
Chas. 
$1.50. 

1020  BIBLIOGRAPHY. 

Baker,  Ira  O. — .     A  Treatise  on  Roads  and  Pavements.     8vo.     663  pp 

Cloth.     $5.00.     W. 
Byrne,  Austin  T. — .     Highway  Construction.     8vo.     Cloth.     $5.00.     W. 
Gillmore,  Q.  A. — .     Practical  Treatise  on  the  Construction  of  Roads, 

Streets,  and  Pavements.      12mo.     Cloth.     $2.00.     V  N. 
Herschel,  Clemens —  and  E.  P.  North.     Road  Making  and   Maintenance. 

156  pp.     $0.50.     E  N. 
Spalding,  Fred.  P. — .     A  Text-Book  on  Roads  and  Pavements.     12mo. 

Cloth.     $2.00.     W. 
Stone,  Gen.  Roy — .     New  Roads  and  Road  Laws  in  the  United  States. 

200  pp.     12mo.     Cloth.     $1.00.     V  N. 
Tillson,  George  W. — .     Street  Pavements  and  Paving  Materials.     500  pp.  . 

8vo.     Cloth.     W. 

626-7  Hydraulic  Engineering. 

Flynn,    P.   J. — .     Irrigation   Canals,   and  Other   Irrigation   Works,   etc. 

Author,  San  Francisco,  Cal. 
Hewson,  Wm. — .     Principles  and  Practice  of  Embanking  Lands  from 
River  Floods,  as  Applied  to  the  Levees  of  the  Mississippi.     Svo.    Cloth. 
$2.00.     V  N. 
Hill,  C.  S. — .     Chicago  Main  Drainage  Channel.     129  pp.     $1.50.     EN. 
Newell,  Frederick  Haynes — .     Irrigation  in  the  United  States.     $2.00. 
Thos.  Y.  Crowell  &  Co.,  New  York. 
♦Starling,  Wm. — .     Floods  of  the  Mississippi  River.     57  pp.     $0.50.    E  N. 
*U.  S.  Geological  Survey.     Water  Supply  and  Irrigation  Papers.     About 
50  pamphlets  have  been  issued,  and  more  are  to  follow.     6''x9".     U.S. 
Geol.  Surv.,  Wash.,  D.  C. 
Wilson,    Herbert    M. — .     Manual    of    Irrigation    Engineering.     3d    ed. 

Small  Svo.     Cloth.  ,  $4.00.     W. 
Retaining  Walls,  etc.     See  721.1. 
637.8  Dams. 

Gould,  E.  Sherman — .     Specifications  for  Dams  and  Reservoirs.     11  pp. 

$0.25.     E  N. 
Gould,  E.  Sherman — .     High  Masonry  Dams.     18mo.     $0.50.     V  N. 
Leffel,  James —  &  Co.     The  Construction  of  Mill  Dams.     312  pp.     Svo. 
Cloth.     $2.50.     B. 
♦Wegmann,  Edward — .     The  Design  and  Construction  of  Dams.     4th  ed., 
revised  and  enlarged.     4to.     Cloth.     $5.00.     W. 
Reservoirs.     See  628.13. 
628  Sanitary  Engineering. 

♦Adams,  J.  W. — ,     Sewers  and  Drains  for  Populous  Districts.     5th  ed. 

Svo.     Cloth.     $2.50.     VN. 
♦Baker,    M.    N. — .     Sewerage   and   Sewage   Purification.     ISmo.      $0.50. 

VN. 

♦Baumeister,  R. — .     The  Cleaning  and  Sewerage  of  Cities.     Adapted  from 

the  German  by  J.  M.  Goodell.     2d  ed.     291  pp.     Svo.     Cloth.     $2.00. 

VN. 

♦Folwell,  A.  Prescott — .     Sewerage.     The  Designing,   Construction,  and 

Maintenance  of  Sewerage  Systems.     445  pp.     Svo.     Cloth.    $3.00.     W. 

♦Kiersted,   Wynkoop — .     Sewage   Disposal.      12mo.      Cloth.     $1.25.     W. 

♦Merriman,    Mansfield — .     Elements    of    Sanitary    Engineering.     2d    ed. 

Svo.     Cloth.     $2.00.     W. 
♦Ogden,  H.  N.— .     Sewer  Design.     234  pp.     12mo.     Cloth.     $2.00.     W. 
♦Rafter,   G.   W. —  and  M.   N.   Baker.     Sewage    Disposal    in  the  United 
States.     598  pp.     $6.00.     V  N  &  E  N. 
Rideal,   Samuel — .     Sewage  and  the  Bacterial  Purification  of  Sewage. 

Svo.     Cloth.     $3.50.     W. 
Sedgwick,   William  T. — .     Sanitary    Science  and   the    Public   Health. 

Cloth.     Svo.     $3.00.     M. 
Swaab,  S.  M. — .     Tables  and  Diagrams  for  Making  Estimates  for  Sewerage 

Work.     20  pp.     $0.50.     EN. 
Waring,  Geo.  E. —  Jr.     Modern  Methods  of  Sewage  Disposal  for  Towns, 
Public  Institutions,  and  Isolated  Houses.     2d  ed.     260  pp.     Cloth. 
$2.00.     V  N. 
Waring,  Geo.  E. —  Jr.     Sewerage  and  Land  Drainage.     3d  ed.      Quarto. 

Cloth.,    $6.00.     V  N. 
Ventilation  and  Heating.     See  697. 

♦     ♦♦  Believed  to  be  specially  useful. 


BIBLIOGRAPHY.  •  1021 

628.1  Water  Works. 

**Baker,  M.  N. — .     Manual  of  American  Water- Works.     (Descriptive  list  of 

works,  with  names  of  officers.)     700  pp.     $3.00.     EN. 
*Baker,  M.  N. — .     Potable  Water  and  Methods  of  Detecting  Impurities. 

18mo.     $0.50.     V  N. 
Croes,  J.  J.  R. — .     Statistical  Tables  of  American  Water  Works.     $2.00. 

EN. 
Fanning,  J.  T. — .     A  Practical  Treatise  on  Hydraulic  and  Water-Supply 

Engineering.     14th  ed.     8vo.     Cloth.     $5.00.     V  N. 
*Folwell,  A.  Prescott — .     Water-Supply  Engineering.     Water-Supply  Sys- 
tems.    562  pp.     8vo.     Cloth.     $4.00.     W. 
*Fuertes,   James  H. — .     Water  and  Public  Health.     70  figures.      12mo. 

Cloth.     $1.50.     W. 
*Fuertes,    James    H. — .      Water   Filtration    Works.      300    pp.      Cloth. 

$2.50.      W. 
Goodell,  John — .     Water  Works  for  Small  Cities  and  Towns.     300  pp. 

Engineering  Record. 
Gould,  E.  Sherman — .     The  Elements  of  Water  Supply  Engineering.    168 

pp.     $2.00.     EN. 
Mason.  William  P. — .     Water  Supplv.     With  special  reference  to  health- 
fulness  of.     8vo.     Cloth.     $5.00.     W. 
McPherson,   J.   A. — .     Water  Works  Distribution.     6"  x  8".     154  pp. 

Illus.     C-loth.     $2.50.     V  N. 
Nichols,    Wm.    Ripley — .     Water   Supply.     Considered    mainly   from   a 

chemical    and    sanitary    standpoint.      With    plates.      4th    ed.      8vo. 

Cloth.     $2.50.     W. 
*Turneaure,  F.  E. —  and  Russell,  H.  L. — .     Public  Water  Supplies.     8vo. 

760  pp.     $5.00.     W. 
Turner,  J.   H.  Tudsbery —  and  A.  W.  Brightmore.     The  Principles  of 

Waterworks  Engineering.     Large  8vo.     Cloth.     $10.00.     S  &  C. 
Wegmann,  Edward —  Jr.     The  Water  Supply  of  the  City  of  New  York 

from  1658-1895.     4to.     Cloth.     $10.00-     W. 
Pumps.     See  621.6. 
Dams.     See  627.8. 
Flow  in  Pipes,  Channels,  etc.     See  532. 

638.13-13  Stand  Pipes.     Tanks.     Reservoirs. 

Hazlehurst.  J.  N. — .     Towers  and  Tanks  for  Water- Works.    8vo.     Cloth. 
$2.50.     W. 

Jacob,  Arthur — .  On  the  Designing  and  Construction  of  Storage  Reser- 
voirs. 18mo.  $0.50.  V  N. 
*Pence,  W.  G. — .  Standpipe  Accidents  and  Failures.  195  pp.  $1.00. 
EN. 
**Schuyler,  James  Dix — .  Reservoirs  for  Irrigation,  Water-Power  and 
Domestic-  Water  Supply.  Revised,  1901.  432  pp.  Large  octavo. 
Cloth.     $5.00.     W. 

Retaining  Walls,  etc.     721.1. 

See  also  627.8,  Dams. 

638.15  Pipes. 

Barstow,  C.  D.— .     Cost  of  Laying  Water  Pipe.     16  pp.     $0.10.     EN. 
Weston,  Edmund  B. — .     Tables  for  Estimating  the  Cost  of  Laying  Cast- 
iron  Water  Pipe.      12  pp.     $0.25.     E  N. 
Flow  in  Pipes.     See  532. 

638.16  Purification. 

*Fuertes,  James  H. — .     Water  Filtration  Works.     Cloth,     h"  x  8''.     Illus. 

$2.50.     W. 
**Hazen,  Allen—.     The  Filtration  of  Public  Water  Supplies.     333  pp.    8vo. 

Cloth.     $3.00.     W. 
*Hill,  John  W.— .     The  Purification  of  Public  Water  Supplies.     304  pp. 

8vo.     Cloth.     $3.00.     V  N.  _  . 

*Kirkwood,  Jas.  P. — .     Report  on  the  Filtration  of  River  Waters  for  the 

Supply  of  Cities,  as  Practised  in  Europe,  made  to  the  Board  of  Water 

Commissioners  of  the  city  of  St.  Louis.     4to.     Cloth.     $7.50.     V  N. 
Rideal,  Samuel — .     Water  and  Water  Purification.     Crown  8vo.     7s.  6d. 

CL. 

*     **  Believed  to  be  specially  useful. 


1022  BIBLIOGRAPHY. 

638.17  Use  and  Waste.     Meters. 

Browne,  Ross  E. — .     Water  Meters:  Comparative  Tests  of  Accuracy,  De- 
livery, etc.     18mo.     $0.50.     V  N. 
Kent,  W.  G.— .     The  Water  Meter.     8vo.     Cloth.     S  C. 

630  Agriculture,  Forestry. 

Green,  Samuel  B. — .     Principles  of  American  Forestry.     12mo.      Cloth, 

$1.50.     W. 
Pinchot,  Gifford — .     A  Primer  of  Forestry.     Bulletin  24,  Div.  Forestry, 

U.  S.  Dept.  Agr. 

660  Chemical  Technology.     Explosives. 

*Eissler,   Manuel — .     The   Modern   High   Explosives — Nitro-glycerin  and 
Dynamite.     3d  ed.     Plates.     8vo.     Cloth.     $4.00.     W. 
Sanford,  P.  Gerald—.     Nitro-Explosives.     270  pp.     Svo.     Cloth.    $3.00- 

VN. 
Wisser,  John  P.—.     Explosive  Materials.     ISmo.     $0.50.     V  N. 
Metallurgy.     See  670,  Manufactures. 

See  691,  Building  Materials. 
Explosives.     See  also  622,  Mining. 

670  Manufactures.     Iron  and  Steel. 

*Bauerman,  H. — .     A  Treatise  on  the  Metallurgy  of  Iron.     515  pp.     12mo. 

Cloth.     $2.00.     B. 
Bolland,  Simpson — ,     The  Encyclopedia  of  Founding  and  Dictionary  of 

Foundry  Terms   Used  in   the   Practice  of   Moulding.      12mo.     Cloth. 

$3.00.     W. 
Bolland,  Simpson — .    "The  Iron  Founder."    Supplement.    400  pp.    12mo. 

Cloth.     $2.50.     W. 
Campbell.   H.   H. — .     Manufacture  and  Properties  of  Structural  Steel. 

The  Scientific  Pub.  Co.,  N.  Y.  and  London. 
Overman,    Frederick — .     The    Manufacture   of   Steel.     285    pp.     12mo. 

Cloth.     $1.50.     B. 
West,  Thomas    D. — .     American    Foundry  Practice.     10th    ed.     12mo. 

Cloth.     $2.50.     W. 
West,  Thos.  D. — .     Moulder's  Text-Book,   Being  Part  II  of  American 

Foundry  Practice.     7th  ed.     12mo.     Cloth.     $2.50.     W. 
Iron  and  Steel.     See  also  691.7,  Building  Materials. 

690  Building. 

See  721,  Architectural  Construction. 

691  Materials  and  Preservatives. 

Byrne,  Austin  T. — .  Inspection  of  the  Materials  and  Workmanship  Em- 
ployed in  Construction.     556  pp.      16mo.     Cloth.     $3.00.     W. 

*  Johnson,  J.  B. — .     The  Materials  of  Construction.     795  pp.     3d  ed.     Svo. 

Cloth.     $6.00.     W. 
Terry,  George — .     Pigments,  Paint,  and  Painting.     12mo.     Cloth.    $3.00. 

SC. 
*Thurston,  Robt.  H. — .     Materials  of  Construction.     6th  ed.     8vo.    Cloth. 

$5.00.     W. 
Strength  of  Materials.     See  620.1. 
See  also  Tratman,  under  625.1. 

691.1  Wood. 

Boulton,  S.  B. — .  The  Preservation  of  Timber  by  the  Use  of  Antiseptics. 
18mo.     $0.50.     V  N. 

Snow,  Chas.  H. — .  The  Principal  Species  of  Wood.  Their  Character- 
istic Properties.     Large  8vo.     214  pp.     Cloth.     $3.50.'     W. 

691.3  *    Natural  Stone. 

*  Merrill,  George  P. — .     Stones  for  Building  and  Decoration.     Illus.     2d  ed. 

8vo.     Cloth.     $5.00.     W. 
See  also  693,  Masonry. 

*     **  Believed  to  be  specially  useful. 


BIBLIOGRAPHY.  1023 

691.3-5  «       Artificial  Stone,  Concrete,  Cement. 

Butler,  D.  B. — .     Portland  Cement.     360  pp.     $6.00.     EN. 
*Gillmore,   Gen.  Q.  A. — .     Treatise  on  Limes,   Hydraulic  Cements,  and 

Mortars.     8vo.     Cloth.     S4.00.     V  N. 
*Jameson,  Charles  D.— .     Portland  Cement.     8vo.     Cloth.     $1.50.     V  N. 
♦Newman,  John — ,     Notes  on  Concrete  and  Works  in  Concrete.     12mo. 
Cloth.     S2.50.     S  C. 
Spalding,   Frederick  P. — .     Hydraulic  Cement.     12mo.     Cloth.     $2.00. 
W. 
691.7  Iron  and  SteeL 

Birkmire,    Wm.    H. — .     Architectural    Iron    and    Steel.     3d    ed.     8vo. 

Cloth.     $3.50.     W. 
Davies,    James — .     Galvanized    Iron:    Its  Manufacture  and  Use.     Svo. 

Cloth.     $2.00.     S  C. 
Greenwood,  W.  H. — .    Steel  and  Iron.    536  pp.    12mo.   Cloth.    $1.75.    B. 
Keep,  William  J.—.     Cast  Iron.     Svo.     238  pp.     Cloth.     $2.50.     W. 
*Metcalf,  William—.     Steel.     12mo.     Cloth.     $2.00.     W. 
Thurston,  Robt.  H. — .    Iron  and  Steel.     6th  ed.    8vo.    Cloth.    $3.50.    W. 
See  also  670,  Manufactures. 
693  Masonry. 

*Baker,  Ira  O. — .     A  Treatise  on  Masonry  Construction.     9th  ed.     Svo. 
Cloth.     $5.00.     W. 
Maginnis,  Owen  B. — .     Bricklaying.     Svo.     Cloth.     $2.00.     V  N. 
Siebert,  Jno.  S. —  and  F.  C.  Biggin.     Modern  Stone  Cutting  and  Masonry. 

Svo.     Cloth.     $1.50.     W. 
Dams.     See  627.8. 
See  also  691.2,  Stone. 
697  Heating  and  Ventilation. 

*Carpenter,  Rolla  C. — .     The  Heating  and  Ventilating  of  Buildings.     400 
pp.     Svo.     Cloth.     4th  ed.     $4.00.     W. 
700        Fine  Arts. 

720  Architecture. 

721  Architectural  Construction. 

Birkmire,  Wm.  H. — .     Skeleton  Construction  in  Buildings.     2d  ed.     Svo. 

Cloth.     $3.00.     W. 
Birkmire,  Wm.  H. — .     The  Planning  and  Construction  of  High  Office 

Buildings.     Svo.     Cloth.     $3.50.     W. 
Black,    W.    M.— .     The    United    States    Public    Works.     Summary    of 

Methods  of  Construction,  Materials,  and  Plants  under  War  and  Treasury 

Departments.     Ob.  4to.     Cloth.     $5.00.     W. 
Bovey,  Henry  T. — .     Theory  of  Structures  and  Strength  of  Materials. 

Svo.     Cloth.     830  pp.     $7.50.     W. 
Christie,  W.  Wallace — .     Chimney  Design  and  Theory.     12mo.     Cloth. 

$3.00.     V  N. 
Fowler,  Chas.  E. — .     General  Specifications  for  Steel  Roofs  and  Buildings. 

12  pp.     $0.25.     E  N. 
Freitag,  Joseph  K. — .     Architectural  Engineering.     High  Building  Con- 
struction.    Svo.     Cloth.     2d  ed.     Re-written.     $3.50.     W. 
Freitag,  Joseph  K. — .     The  Fireproofing  of  Steel  Buildings.     Svo.     Cloth. 

S2.50.     W. 
**Johnson,  J.  B. — ,  W.  H.  Bryan  and  F.  E.  Turneaure.     The  Theory  and 

Practice  of  Modern   Framed  Structures.     7th  ed.     Revised  and  en- 
larged.    Small  4to.     Cloth.     $10.00.     W. 
♦Kidder,  F.  E.— .     The  Architect's  and  Builder's  Pocket-Book.     1030  pp. 

500    engravings.     13th    ed.     Revised    and    greatly    enlarged.     16mo. 

Morocco.     $4.00.     W. 
Weyrauch,  J.  J. — .     Strength  and  Calculations  of  Dimensions  of  Iron  and 

Steel  Construction,  with  reference  to  the  Latest  Experiments.     12mo. 

Cloth.     $1.00.     VN. 
Winslow,  Benj.  E. — .     Diagrams  for  Calculating  the  Strength  of  Wood, 

Steel,  and  Cast-iron  Beams  and  Columns.     19  plates,      llf  x  9i''. 

$2.00.     E  N. 
Bridges,  624. 
Strength  of  Materials,  620.1. 

*     **  Believed  to  be  specially  useful. 


1024  BIBLIOGRAPHY. 

731.1  Foundations. 

Baker,  Benjamin — .     The  Actual  Lateral  Pressure  of  Earth- Work.    18mo. 

$0.50.     V  N. 
Fowler,  Charles  Evan — .     The  Coffer-Dam   Process  for    Piers.     173  pp. 

100  illus.     8vo.     Cloth.     $2.50.     W. 
Howe,  Malverd  A. — .     Retaining  Walls  for  Earth.     3d  ed.,  rewritten  and 

enlarged.     12mo.     Cloth.     $1.25.     W. 
Jacob,     Arthur — .     Practical    Designing    of    Retaining-Walls.     2d    ed. 

18mo.     $0.50.     V  N. 
Newman,  John — .     Earthwork  Slips  and  Subsidences  Upon  Public  Works. 

234  pp.     12mo.     Cloth.     $3.00.     S  C. 
Osborn   Co.     General   Specifications   for   Bridge   Substructure.     10   pp. 

$0.25.     E  N. 
*Patton,  W,  M. — .     Practical  Treatise  on  Foundations.     429  pp.     8vo. 

Cloth.     $5.00.     W. 
Wellington,  A.  M. — .     Piles  and  Pile-Driving.     $1.00.     E  N. 
Embankments.     See  627. 

725  Public  Buildings. 

Dillenbeck,  Clark — .  Standard  Specifications  for  Railroad  Structures, 
Brick  Passenger  Stations,  Brick  Freight  Houses,  Frame  Passenger  Sta- 
tions, Frame  Freight  Houses.     $0.40  each.     E  N. 

740  Drawing. 

*Jacoby,  H.  S.— .     Text-Book  on  Plain  Lettering.     82  pp.     $3.00.    E  N. 

Mahan,  D.  H. — .     Industrial  Drawing.     Revised  and  enlarged  by  D.  F. 
Thompson.     30  plates.     8vo.     Cloth.     $3.50.     W. 
*Reinhardt,  C.  W. — .     Lettering  for  Draughtsmen,  Engineers,  and  Stu- 
dents.    32  pp.     $1.00.     E  N. 
*Reinhardt,  C.  W. — .     Technic  of  Mechanical  Drafting.     36  pp.     $1.00. 
E  N. 

Smith,  R.  S. — .  Manual  of  Topographical  Drawing.  Revised  and  en- 
larged by  Chas.  McMillan.  12  folding  plates.  3d  ed.  8vo.  Cloth. 
$2.50.     W. 

Warren,  S.  Edward — .  Drafting  Instruments  and  Operations.  Thor- 
oughly revised,  with  additions.     12mo.     Cloth.     $1.25.     W. 

Warren,  S.  Edward — .  Elements  of  Plane  and  Solid  Free  Hand  Geomet- 
rical Drawing.     Plates  and  wood-cuts.     12mo.    Cloth.     $1.00.     W. 

Descriptive  Geometry,  515. 

Blue  Printing,  etc.     See  770. 

Topographical  Drawing.     See  also  526.98. 

770  Photography. 

*Leitze,  Ernest — .     Modern  Heliographic  Processes.     2d  ed.     8vo.    Cloth. 
VN. 
Pettit,    James   S. — .     Modern  Reproductive  Graphic  Processes.     18mo. 

$0.50.     V  N. 
Photographic  Surveying.     See  526.9. 

*     **  Believed  to  be  specially  useful. 


GLOSSARY   OF   TERMS.  1025 

GLOSSAEY  or  TERMS. 


Abacus;  the  flat  square  member  on  top  of  a  column. 

Absciss  or  abscissa ;  any  portion  of  the  axis  of  a  curve,  from  the  vertex  to  any  point  from  which 
a  line  leaves  the  axis  at  right  angles,  and  extends  to  meet  the  curve  itself;  said  line  being  called  an 
wdinate.     An  absciss  and  ordinate  together  are  called  co-ordinates. 

Acclivity  ;  an  upward  slope,  or  ascent  of  ground,  &c. 

Adit ;  a  horizontal  passage  into  a  mine,  <fec. 

Adze;  a  well-known  curved  cutting  instrument,  for  dressing  or  chipping  horizontal  surfaces. 

Alternating  motion;  up  and  down,  or  backward  and  forward,  instead  of  revolving.  &c. 

Angle-bead,  or  plaster  bead:  a  bead  nailed  to  projecting  angles  in  rooms,  to  protect  the  plaster  on 
their  edges  from  injury. 

Angle-block;  a  triangular  block  against  which  the  ends  of  the  braces  and  counters  abut  in  a  Howe 
bridge. 

Anneal;  to  toughen  some  of  the  metals,  glass,  &c,  by  first  heating  them,  and  then  causing  them  to 
Bool  very  slowly.    This  process  however  lessens  the  tensile  strength. 

Anticlinal  axis;  in  geology  ;  a  line  from  which  the  strata  of  rocks  slope  away  downward  in.oppo* 
Bite  directions,  like  the  slates  on  the  roof  of  a  house;  the  ridge  of  the  roof  representing  the  axis. 

Apex;  a  point  in  either  chord  of  a  truss,  where  two  web  members  meet. 

Apron;  a  covering  of  timber,  stone,  or  metal,  to  protect  a  surface  against  the  action  of  water  flow, 
ing  over  it.     Has  many  other  meanings. 

Arbor.    See  Journal. 

Architrave ;  that  part  of  an  entablature  which  is  next  above  the  columns.  Applies  also  when  there 
are  no  columns.  Also,  the  mouldings  around  the  sides  and  tops  of  doors  and  windows,  attached  to 
either  the  inner  or  outer  face  of  the  wall. 

Arris;  a  sharp  edge  formed  by  any  two  surfaces  which  meet  at  an  angle.  The  edges  of  a  brick  are 
arrises. 

Ashler ;  a  facing  of  cut  stone,  applied  to  a  backing  of  rubble  or  rough  masonry,  or  brickwork. 

Astragal ;  a  small  moulding,  about  semi-circular  or  semi-elliptic,  and  either  plain  or  ornamented  by 
earving. 

Axis ;  an  imaginary  line  passing  through  a  body,  which  may  be  supposed  to  revolve  around  it ;  as 
the  diam  of  a  sphere.  Any  piece  that  passes  through  and  supports  a  body  which  revolves  ;  in  which 
case  it  is  called  an  axle,  or  shaft. 

Axle-box.     See  Journal-box. 

Axletree;  an  axle  which  remains  flxed  while  the  wheel  revolves  around  it,  as  in  wagons,  &c. 

Azimuth.  The  azimuth  of  a  body  is  that  arc  of  the  horizon  that  is  included  between  the  meridian 
circle  at  the  given  place,  and  another  great  circle  passing  through  the  body. 

Backing  ;  the  rough  masonry  of  a  wall  faced  with  finer  work.  Earth  deposited  behind  a  retaining- 
wall,  &c. 

Balance-beams ;  the  long  top  beams  of  lock-gates,  by  which  they  are  pushed  open  or  shut. 

Balk ;  a  large  beam  of  timber. 

Ballast;  broken  stone,  sand  or  gravel,  &c,  on  which  railroad  cross-ties  are  laid. 

Ball-cock;  a  cistern  valve  at  one  end  of  a  lever,  at  the  other  end  of  which  is  a  floating  ball.  The 
ball  rises  and  falls  with  the  water  in  the  cistern ;  and  thus  opens  or  shuts  the  valve. 

Ball-valve.    See  Valve, 

Bargeboards ;  boards  nailed  against  the  outer  face  of  a  wall,  along  the  slopes  of  a  gable  end  of  a 
house,  to  hide  the  rafters,  &c  ;  and  to  make  a  neat  finish.       f> 

Bascule  bridge;  a  hinged  lit't-bridge  furnished  with  a  counterpoise. 

Batter,  (sometimes  affectedly  batir,)  or  taXus ;  the  sloping  backward  of  a  face  of  masonry. 

Bay  ;  on  bridges,  &c,  sometimes  a  panel ;  sometimes  a  span. 

Bead  ;  an  ornament  either  composed  of  a  straight  cylindrical  rod  ;  or  carved  or  cast  in  that  shape 
on  any  surface- 

Bearing  ;  the  course  by  a  compass.  The  span  or  length  in  the  clear  between  the  points  of  support 
of  a  beam,  <fec.     The  points  of  support  themselves  of  a  beam,  shaft,  axle,  pivot,  &c. 

Bed-mouldings ;  ornamental  mouldings  on  the  lower  face  of  a  projecting  cornice,  &c. 

Bed-plate ;  a  large  plate  of  iron  laid  as  a  foundation  for  something  to  rest  on. 

Beetle ;  a  heavy  wooden  rammer,  such  as  pavers  use. 

Bell-crank.     See  Crank. 

Bench-mark ;  a  level  mark  cut  at  the  foot  of  a  tree  for  future  reference,  as  being  more  permanent 
than  a  stake. 

Berm,  or  herme:  a  horizontal  surface,  as  if  for  a  pathway,  and  forming  a  kind  of  step  along  the  f&ce 
•f  sloping  ground.  In  canals,  the  level  top  of  the  embankment  opposite  and  corresponding  to  the 
towpath  is  called  the  berm. 

Bessemer  steel  is  formed  by  forcing  air  into  a  mass  of  melted  cast  iron ;  by  which  means  the  excess 
of  carbon  in  the  iron  is  separated  from  it,  until  only  enough  remains  to  constitute  cast  steel.  The 
carbon  is  chemically  united  with  the  steel,  but  mechanically  with  the  iron. 

Beton;  concrete  of  hydraulic  cement,  with  broken  stone  and  bricks,  gravel,  &c. 

Bevel;  the  slope  formed  by  trimming  away  a  sharp  edge,  as  of  a  board,  &c.  Edges  of  common 
drawing  rulers  and  scales  are  usually  bevelled. 

Bevel  gear ;  cog-wheels  with  teeth  so  formed  that  the  wheels  can  work  into  each  other  at  an  angle. 

Bilge  ;  the  nearly  flat  part  of  the  bottom  of  a  ship  on  each  side  of  the  keel.  Also,  the  swelled  part 
of  a  barrel,  &c.    To  bilge  is  to  spring  a  leak  in  the  bilge,  or  to  be  broken  there. 

Bitts ;  the  small  boring  points  used  with  a  brace. 

Blast-pipes  ;  in  a  locomotive ;  those  through  which  the  waste  steam  passes  from  the  cylinder  into 
the  smoke-pipe,  and  thus  creates  an  artificial  draft  in  the  chimney,  or  smoke-pipe. 

Boasting :  dressing  stone  with  a  broad  chisel  called  a  boaster,  and  mallet.  The  boaster  gives  a 
smoother  surface  after  the  use  of  the  point,  or  the  narrow  chisel  called  a  tool. 

Body;  the  thickness  of  a  lubricant  or  other  liquid.  Also,  the  measure  ot  tuat  thickness,  expressed 
in  the  number  of  seconds  in  which  a  given  quantity  of  the  oil,  at  a  given  temperature,  flows  througk 
a  given  aperture. 

65 


1026  GLOSSARY   OF   TERMS. 


Bolster;  a  timber,  or  a  thick  iron  plate  placed  between  the  end  of  a  bridge  audits  seat  on  th| 
abutment. 

Bond;  the  disposing  of  the  blocks  of  stciie  or  brickwork  so  as  to  form  the  whole  into  a  firm  struc- 
ture, by  a  judicious  overlapping  of  each  other,  so  as  to  break  joint.  Applies  also  to  timber,  &c,  ii 
various  ways. 

Bonnet ;  a  cap  over  the  end  of  a  pipe,  &c.  A  cast-iron  plate  bolted  down  as  a  covering  over  an 
aperture. 

Bore  ;  inner  diameter  of  a  hollow  cylinder. 

Borrow-pit ;  a  pit  dug  in  order  to  obtain  material  for  an  embankment. 

M»tt;  an  increase  of  the  diameter  at  any  part  of  a  shaft  for  any  purpose.  A  projection  in  shape 
•f  a  segment  of  a  sphere,  or  somewhat  so,  whether  for  use  or  for  ornament ;  often  carved,  or  cast. 

Box-drain;  a  square  or  rectaugular  drain  of  masonry  or  timber,  under  a  railroad,  &c. 

Brace ;  a  kind  of  curved  handle  used  for  boring  holes  with  bitts.  The  head  of  the  brace  remains 
stationary,  being  pressed  against  by  the  body  of  the  person  using  it,  while  the  other  part  with  the 
bitt  is  turned  round  by  his  hand.     Also,  an  inclined  beam,  bar,  or  strut,  for  sustaining  conipression. 

Bracket;  a  projectiug  piece  of  board.  &c,  frequently  triangular,  the  vertical  leg  attached  to  the 
face  of  a  wall,  and  the  horizontal  one  supporting  a  shelf,  &c.  Often  made  in  ornamental  shapes  for 
supporting  busts,  clocks,  &c.  Also,  the  supports  for  shafting ;  as  pendent,  wall,  and  pedestal  brackets. 

Brake;  an  arrangement  for  preventing  or  diminishing  motion  by  means  of  friction.  The  friction 
is  usually  applied  at  the  circumference  of  a  revolving  wheel,  by  means  of  levers.  On  railroads,  the 
car- brakes  should  be  worked  by  steam,  as  those  of  Loughridge,  Westinghouse,  and  Creamer.  Also, 
such  a  handle  as  that  of  a  common  pump. 

Brass  is  composed  of  copper  and  zinc. 

Brasses;  fittings  of  brass  in  many  plummer-blocks,  and  in  other  positions,  for  diminishing  the 
friction  of  revolving  journals  which  rest  upon  them. 

Braze;  to  unite  pieces  of  iron,  copper,  or  brass,  by  means  of  a  hard  solder,  called  spelter  solder, 
and  composed,  like  bra.os,  of  copper  and  zinc,  but  in  other  proportions. 

Break  joint;  to  so  overlap  pieces  that  the  joints  shall  not  occur  at  the  same  place,  and  thus  pro- 
duce a  bad  bond. 

Breast-summer :  a  beam  of  wood,  iron,  or  stone,  supporting  a  wall  over  a  door  or  other  opening; 
a  kind  of  lintel. 

Breast-wall ;  one  built  to  prevent  the  falling  of  a  vertical  face  cut  into  the  natural  soil;  in  dis- 
tinction to  a  retaining-wall  or  revetment,  which  is  built  to  sustain  earth  deposited  behind  it. 

Breech;  the  hind  part  of  a  cannon.  &c. 

Bridge,  or  hridge-piece,  or  bridge  bar ;  a  narrow  strip  placed  across  an  opening,  for  supporting 
something  without  closing  too  much  of  the  opening. 

Bronze  is  composed  of  copper  and  tin. 

Bulkhead ;  on  ships,  &c,  the  timber  partitions  across  them.     Also,  a  long  face  of  wharf  parallel 

Buoy ;  a  floating  body,  fastened  by  a  chain  or  rope  to  some  sunk  body,  as  a  guide  for  finding  the 
latter.     Sometimes  also  used  to  indicate  channels,  shoals,  rocks,  &c. 

Burnish  ;  to  polish  by  rubbing  ;  chiefly  applies  to  metals. 

Bu/>h;  to  line  a  circular  hole  by  a  ring  of  metal,  to  prevent  the  hole  from  wearing  larger.  Also, 
when  a  piece  is  cut  out,  and  another  piece  neatly  inserted  into  the  cavity,  the  last  piece  is  sometimes 
said  to  be  bushed  inj  sometimes  it  is  called  a  plug. 

Butt-joint;  one  in  which  the  ends  of  the  two  pieces  abut  together  without  overlapping,  and  are 
joined  by  one  or  more  separate  pieces  called  covers  or  welts,  which  reach  across  the  joint  and  are 
fastened'to  both  pieces. 

Buttress;  a  vertical  projecting  piece  of  brickwork  or  masonry,  built  in  front  of  a  wall  t« 
strengthen  it.  ^  ^        ,  * 

Caisson ;  a  large  wooden  box  with  sides  that  may  be  detached  and  floated  away. 

Caliber;  the  inner  diameter,  or  bore.r 

Calipers ;  compasses  or  dividers  with  curved  legs,  for  measuring  outside  and  inside  diameters. 

Calk,  or  caulk:  to  fill  seams  or  joints  with  something  to  prevent  leaking. 

Calking  iron,  a  tool  for  forcing  calking  into  a  joint. 

Ca7nb,  or  cam,  or  tviper:  a  piece  fixed  upon  a  revolving  shaft  in  such  a  manner  as  to  produce  aa 
alteraating  or  reciprocating  motion  in  something  in  contact  with  th6  cam.     An  eccentric. 

Camber ;  a  slight  upward  curve  given  to  abeam  or  truss,  to  allow  for  settling. 

Camel;  a  kind  of  barges  or  hollow  floating  vessels,  which,  when  filled  with  water,  are  fastened  t© 
the  sides  of  a  ship  ;  and  the  water  being  then  pumped  out,  they  rise  by  their  buoyancy  ;  and  lift  the 
shin  so  that  she  can  float  in  shallower  water. 

Cantilevers ;  projecting  pieces  for  supporting  an  upper  balcony,  &c. 

Cants,  rims,  or  shro7(.dings ;  the  pieces  forming  the  ends  of  the  buckets  of  water-wheels,  to  prevent 
the  water  from  spilling  endwise. 

Capstan:  a  long  hollow  rope-drum  surrounding  a  strong  vertical  pivot,  upon  the  head  of  which  it 
rests,  and  around  which  it  turns,  ^ts  top  is  a  thick  projecting  circular  piece,  having  holes  around  its 
outer  edge  or  circumference,  for  the  insertion  of  the  ends  of  levers  ;  or  capstan-bars.  It  is  a  kind  of 
vertical  windlass. 

Case-harden ;  to  convert  the  outer  surface  of  wrought  iron  into  steel,  by  heating  it  while  in  contact 
with  charcoal. 

Casemate;  in  fortification;  the  small  jipaj-tment  in  which  a  cannon  stands. 

Castors  ;  rollers  usually  combined  with  swivels ;  as  those  used  under  heavy  furniture,  &c. 

Causeway  ;  a  raised  footway  or  roadway. 

Cavetto;  a  moulding  consisting  of  a  receding  quadrant  of  a  circle. 

Cementation ;  the  process  of  converting  wrought  iron  into  steel,  by  heating  it  in  contact  with  char- 
coal. This  process  produces  blisters  on  the  steel  bars;  heucQ  blister  steel.  These  are  removed,  and 
the  steel  compacted,  by  reheating  it,  and  then  subjecting  it  to  a  tilt-hammer.  It  is  then  tilted  steel, 
or  shear  steel.  Or  if  the  blister  steel  is  broken  up,  remelted  in  a  crucible,  and  then  run  into  ingots 
or  blocks,  it  is  called  crucible,  cast,  or  ingot  steel ;  which  is  harder  and  closer-grained  than  tilted  steel. 
It  may  be  softened,  and  thus  become  less  brittle,  by  annealing.  The  ingots  may  be  converted  inte 
bars  by  either  rolling  or  hammering,  the  same  as  shear  and  blister  steel. 

Center ;  the  supports  of  an  arch  while  being  built. 


GLOSSARY   OP   TERMS.  1027 


Center  of  percussion,  in  a  moving  body,  is  tliat  point  which  would  strike  an  opposing  boaywitb 
■greater  force  than  any  other  point  would.  If  the  opposing  body  is  immovable,  it  will  receive  all  thi 
,Oi-ce  of  ;i  rigid  moving  body  which  strikes  with  its  center  of  percussion.     See  Pendulum,  page  365. 

Cesspool;  a  sUallow  well  for  receiving  waste  water,  filth,  &c. 

Cfiam/er ;  means  much  the  same  as  bevel ;  but  applies  more  especially  when  two  edges  are  cut  away 
so  as  to  form  either  a  chamfer-groove,  (see  14,  p  735,  of  Trusses,)  or  a  projecting  sharp  edge. 

Cheeks ;  two  flat  parallel  pieces  confining  something  between  them. 

Chilling,  chill-hardening,  or  chill- casting ;  giving  great  hardness  to  the  outside  of  cast-iron,  by 
pouring  it  into  a  mould  made  of  iron  instead  of  wood.  The  iron  mould  causes  the  outside  or  skin  of 
the  casting  to  cool  very  rapidly  ;  and  this  for  some  unknown  reason  increases  its  hardness.  This  pro- 
cess is  frequently  confounded  with  case-hardening. 

Chock  ;  any  piece  used  for  filling  up  a  chance  hole,  or  vacancy. 

Chuck;  the  arrangement  attached  to  the  revolving  shaft,  arbor,  or  mandril  of  a  lathe,  for  holdin|5 
the  thing  to  be  turned. 

Churn-drill ;  a  long  iron  bar,  with  a  cutting  end  of  steel ;  much  used  in  quarrying,  and  worked  by 
raising  it  and  letting  it  fall.     "When  worked  by  blows  of  a  hammer  or  sledge  it  is  called  a  jumper. 

Cima,  or  cyma ;  a  moulding  nearly  in  shape  of  an  S.  "When  the  upper  part  is  concave,  it  is  called 
a  cima  recta ;  when  convex,  a  cima  reversa. 

Clack  valve.    See  "Valves. 

Clamp  ;  a  piece  fastened  by  tongue  and  groove,  transversely  along  the  end  of  others,  to  keep  then 
from  warping.  A  kind  of  open  collar,  which,  being  closed  by  a  clamp-screw,  holds  tight  what  it  sup- 
rounds.    See  Cramp. 

Clap  board* ;  short  thin  boards,  shingle-shaped,  and  used  instead  of  shingles. 

Claw,  a  split  provided  at  the  end  of  an  iron  bar,  or  of  a  hammer,  &c,  to  take  hold  of  the  heads  of 
nails  or  spikes  for  drawing  them  out ;  as  in  a  common  claw-hammer. 

Cleat;  a  piece  merely  bolted  to  another  to  serve  as  a  support  for  something  else  ;  as  at  7,  8,  10, 
&c,  p.  735,  of  Trusses.  Often  used  on  shipboard  for  fastening  ropes  to,  as  at  11.  Also  a  piece  of 
board  nailed  across  two  or  more  other  boards,  for  holding  them  together,  as  is  often  done  in  tempo- 
rary doors,  Ac. 

Clevis.     See  Shackle. 

Click.    See  Ratchet. 

Cl^;  a  fastening  like  that  on  the  tops  of  the  Y's  of  a  spirit  level ;  being  a  kind  of  half  collar  opening 
by  a  hinge. 

Clutch ;  applied  to  various  arrangements  at  the  ends  of  separate  shafts,  and  which  by  clutching  Of 
catching  into  each  other  cause  both  shafts  to  revolve  together.     A  kind  of  coupling. 

Cock;  a  kind  of  valve  for  the  discharge  of  liauids,  air,  steam,  &c. 

Coefficient;  or  a  Constant  of  friction,  safety,  or  strength,  Ac,  may  usually  be  taken  to  be  a  num- 
ber which  shows  the  proportion  (or  rather  the  ratio)  which  friction,  safety,  tensile  strength,  &c,  bear 
to  a  certain  something  else  which  !•  not  generally  expressed  at  the  time,  but  Is  well  understood.  Thus, 
when  we  say  that  the  coeff  of  friction  of  one  body  upon  another  is  -j-^,  &c,  it  is  understood  that  the 
friction  is  in  the  proportion  of  y^^th  of  the  pressure  which  produces  it.  A  coefTof  safety  of  3,  means 
that  the  safety  has  a  proportion  or  ratio  of  3  to  1  to  the  theoreti^M  breaking  load.  A  coefl  of  500  lbs, 
or  of  20  tons,  &c,  of  tensile  strength  of  any  material,  denotes  that  said  strength  is  in  the  proportion 
of  500  lbs,  or  of  20  tons,  &c,  to  each  square  inch  of  transverse  section,  &c.     Same  as  Modulus. 

Coffer-dam  ;  an  enclosure  built  in  the  water,  and  then  pumped  dry,  so  as  to  permit  masonry  or 
other  work  to  be  carried  on  inside  of  it. 

Cog  ;  the  tooth  of  a  cog-wheel. 

Collar;  a  flat  ring  surrounding  anything  closely. 

CoUar-beam ;  a  horizontal  timber  stretching  from  one  to  another  of  two  rafters  which  meet  at  top; 
but  above  the  main  tie-beam. 

Concrete ;  artificial  stone  formed  by  mixing  broken  stone,  gravel,  &c,  with  common  lime.  "When 
hydraulic  cement  is  used  instead  of  lime,  the  mixture  is  called  beton.  The  terms  "  lime  concrete" 
and  "  cement  concrete  "  would  be  convenient. 

Connecting-rod ;  a  piece  which  connects  a  crank  with  something  which  moves  it,  or  to  which  it 
fives  motion. 

Console ;  a  kind  of  ornamental  bracket,  somewhat  in  shape  of  an  S ;  much  used  in  cornices,  &0, 
for  supporting  ornamental  mouldings  above  it. 

Coping  ;  flat  plates  of  stone,  iron,  &c,  placed  on  the  tops  of  walls  exposed  to  the  weather. 

Corbel ;  a  horizontal  projecting  piece  which  assists  in  supporting  one  resting  upon  it  which  proiectt 
S4U1  farther. 

Core;  anything  serving  as  a  mould  for  anything  else  to  be  formed  around.  A  term  much  aged  In 
foundries. 

Cornice ;  the  ornamental  projection  at  the  eaves  of  a  building,  or  at  the  top  of  a  pier,  or  of  any  other 
structure. 

Cotter-bolt,  or  key-bolt ;  a  bolt  which,  instead  of  a  screw  and  nut  at  one  end,  has  a  slot  cut  through 
it  near  that  end,  for  the  insertion  of  a  wedge-shaped  key  or  cotter,  for  keeping  it  in  its  place.  Some- 
times the  ends  of  these  keys  are  split,  so  as  to  spread  open  after  being  inserted,  so  as  not  to  be  jolted 
out  of  place. 

Counterfort;  vertical  projections  of  masonry  or  brickwork  built  at  intervals  along  the  hack  of  a  wail 
to  strengthen  it ;  and  generally  of  very  little  use. 

Counter-shaft ;  a  secondary  shaft  or  axle  which  receives  motion  from  the  principal  one. 

Countersunk.     See  Ream. 

Counter-weight ;  or  counter-balance;  any  weight  used  to  balance  another. 

Couplings ;  a  term  of  very  general  application  to  arrangements  for  connecting  two  shafts  so  that 
they  shall  revolve  together. 

Cover;  see  "  butt-joint." 

Cover;  in  rp-rolling  iron  and  steel  from  piles  of  small  pieces,  a  large  bar  or  slab,  called  a  cover,  of 
the  same  width  and  length  as  the  pile,  is  employed  to  form  the  bottom  of  the  pile,  and  a  similar  slab 
fer  the  top.  The  covers  serve  to  hold  the  pile  together;  and,  after  rolling,  thejr  form  unbroken  top 
and  bottom  surfaces  of  the  finished  plate,  bar,  rail,  I  beam,  &c. 


1028 


GLOSSARY   OF   TERMS. 


drub;  a  short  shaft  or  axle,  which  serves  as  a  rope-drum  in  raising  weights  ;  and  is  revolved  either 
hs  cog-whee'a,  a  winch,  or  by  levers  or  handspikes,  inserted  In  holes  around  its  circumference  like  a 
windlass,  or  capstan,  of  which  it  is  a  variety.  It  may  be  either  vertical  or  horizontal.  It  is  often 
let  in  a  frame,  to  he  carried  from  place  to  place.  Also  the  whole  machine  is  called  a  crab. 
Cradle  ;  applied  to  various  kinds  of  timber  supports,  which  partly  enclose  th^  mass  snstainea. 
tramp;  a  short  bar  ot  metal,  having  its  two  ends  bent  downward  at  right  angles  for  insertion int* 
two  adjoining  pieces  of  stone,  wood,  &c,  to  hold  them  together.  Much  used  at  the  ends  of  coping-stones. 
Also  a  similar  bent  piece,  with  a  set-screw  passing  through  one  of  the  bent  ends,  for  holding  thiug» 
tight  between  it  and  the  other  end.     This  last  is  also  called  a  clamp. 

Crane;  a  hoisting  machine  consisting  of  a  revolving  vertical  post  or  stalk;  a  projecting  n6  ;  and 
&8tay  for  sustaining  the  outer  end  of  the  jib.  The  stay  may  be  either  a  strut  or  a  tie.  There  are 
also  cog-wheels,  a  rope  drum  or  barrel,  with  a  winch,  ropes,  puheys.  &c.  In  a  crane  the  post,  jib, 
and  stay  do  not  change  their  relative  positions,  as  they  do  in  a  derrick. 

Crank;  a,  double  bend  at  right  angles,  somewhat  like  a  Z,  at  the  end  of  a  shaft  or  axle,  and  forminf; 
a  kind  of  handle  by  which  the  axle  may  be  made  to  revolve.  Sometimes,  as  in  common  grindstones, 
this  crank  is  formed  of  a  separate  piece  removable  at  pleasure.  That  part  of  this  piece  which  has  the 
square  opening  in  it  for  fitting  it  to  the  square  end  of  the  axle,  is  called  the  crank-arm;  and  the  other 
part  the  crank-handle.  A  bell-crank  consists  of  4  bends  at  right  angles  at  the  center  of  an  axle,  form- 
ing in  it  a  kind  of  U.  A  double  crank  consists  of  two  bell  cranks  arranged  thus,  tj^.  The  bend  in 
the  U  forms  the  crank-wrist.  The  term  bell-crank  is  applied  also  to  those  used  in  fixing  common  dwell- 
ing house  bells :  and  to  larger  ones  on  the  same  principle.  A  crankpin  is  a  pin  projecting  from  a  re- 
volving wheel,  disk,  or  other  body,  and  serving  as  a  crank-handle.  A  crank-shaft  is  a  shaft  which 
has  a  crank  in  It,  or  at  its  end.  A  cranked  shaft  has  it  in  it  only.  A  ship  or  other  vessel  is  said  to 
be  crank  when  its  breadth  is  so  small  in  proportion  to  its  depth  as  to  make  it  liable  to  upset  easily ;  or 
when  the  same  liability  is  caused  by  want  of  sufiBcient  ballast. 
Crest ;  that  top  part  of  a  dam  over  which  the  water  pours. 
Cross-cut  saw  ;  a  large  horizontal  saw  worked  by  two  men,  one  at  each  end. 

Cross-head  ;  a  piece  attached  across  the  end  (or  near  it)  of  another  piece,  and  at  right  angles  to  it, 
CO  as  to  form  a  kind  of  T  or  cross.     Often  seen  on  piston  rods,  which  they  serve  to  keep  in  place  by 
resting  on  the  slides,  or  guides. 
Crowbar  ;  a  bar  of  iron  used  as  a  lever  for  various  purposes  ;  often  pointed  at  one  end. 
Crown,  or  contrate  wheel ;  a  cog-wheel  in  which  the  teeth  stand  not  upon  its  outer  circumference  as 
vsual.  but  upon  the  plane  of  its  circle. 

Curb  ;  a  broad  flat  circular  ring  of  wood,  iron,  or  stone,  placed  under  the  bottoms  of  circular  walls, 
as  in  a  well,  or  shaft,  to  prevent  unequal  settlement;  or  built  into  the  walls  at  intervals,  for  the  same 
purpose.     Has  many  other  meanings. 

Cut-off;  an  arrangement  for  cutting  ofiF  the  steam  from  a  cylinder  before  the  piston  has  made  its 
full  stroke.     Also  a  channel  cut  through  a  narrow  neck  of  land,  to  straighten  the  course  of  a  river. 

Cutwater,  or  starling ;  the  projecting  ends  of  a  bridge  pier,  &c,  usually  so  shaped  as  to  allow  water, 
ice,  &c,  to  strike  them  with  but  little  injury. 
Damper ;  a  door  or  valve  to  regulate  the  admission  of  air  to  a  furnace,  stove,  &c. 
Dead  load;  the  cars,  engine,  &c,  in  a  train  ;  non-paying  load. 

Dead-load;  in  a  bridge,  the  weight  of  the  bridge  itself,  with  flooring,  roof,  &c;  as  distinguished 
from  the  live  load  of  passing  trains,  vehicles,  pedestrians,  &c. 

Dead  points;  those  two  points  in  the  revolution  of  a  crank,  where  the  crank  arm  is  parallel  with 
Ihe  rod  which  connects  it  with  the  moving  power;  and  at  which  said  rod  exerts  no  tendency  to  turn 
the  crank. 

Declination,  of  the  sun,  or  of  a  star,  is  its  angle  north  or  south  of  the  earth's  equator  at  the  time 
of  observation. 

Declivity;  a  downward  slope  or  descent  of  ground,  &c. 

Dentils ;  blocks  constituting  ornaments  in  a  cornice ;  placed  at  short  intervals  apart,  they  resembU 
teeth.  "When,  instead  of  mere  blocks,  they  are  handsomely  carved  in  various  shapes,  they  are  called 
modillions. 

Derrick;  a  kind  of  crane,  differing  from  common  ones,  chiefly  in  the  fact  that  the  rope  or  chain 
which  forms  the  stay  may  be  let  out  or  hauled  in  at  pleasure,  thus  raising  or  lowering  the  inclination 
of  a  jib;  thereby  enabling  the  raised  load  to  be  placed  vertically  at  the  required  spot.     This  cannot 
be  done  with  a  crane,  which,  therefore,  is  not  as  well  adapted  for  laying  heavy  masonry,  especially 
at  great  heights. 
Diaphram ;  a  thin  plate  or  partition  placed  across  a  tube  or  other  hollow  body. 
Die;  that  part  of  a  stamp  that  gives  the  impression.   Dies  are  also  two  flat  plates  of  hardened  steel, 
on  an  edge  of  each  of  which  is  hollowed  out  a  semicircular  half  of  a  short  female  screw.     "When  these 
plates  are  put  in  contact  they  form  a  complete  female  screw,  like  that  in  a  nut;  and  being  strongly 
held  together  by  an  iron  bo  dng  called  the  die-stocks,  which  have  long  handles  for  revolving  them,  they 
constitute  a  mould  or  cutter  for  forming  threads  on  a  male  screw.     Also  the  main  body  of  a  pedestal. 
Dip ;  in  geology,  either  the  angle  which  the  slope  of  a  stratum  forms  with  a  horizontal ;  or  the 
direction  by  compass,  toward  which  it  slopes.     In  surveying,  the  inclination  at  which  an  unbalaneed 
eompass-needle  rests  on  its  pivot  after  being  magnetized. 
Disk ;  a  flat  circular  piece. 

Dock;  an  artificial  enclosure,  either  partial  or  total,  in  which  ships  and  other  vessels  are  placed 
for  being  loaded  or  unloaded,  or  repaired.    The  first  is  a  wet  dock ;  the  last  a  dry  one. 

Dog-iron;  a  short  bar  of  iron,  forming  a  kind  of  cramp,  with  its  ends  bent  down  at  right  angles 
•ad  pointed,  so  as  to  hold  together  two  pieces  into  which  they  are  driven.  Often  used  for  temporary 
parposes.    It  is  also  railed  a  dog-iron  when  only  one  end  is  bent  down  and  pointed  for  driving,  thfl 


GLOSSARY   OF    TEEMS.  1029 

•tber  end  being  formed  into  an  eye  or  a  handle  by  which  the  piece  into  which  the  other  end  is  driven 
may  be  hauled  or  towed  away. 

Donkey-engine  ;  a  small  steam  engine  attached  to  a  large  one,  and  fed  from  the  same  boiler.  It  is 
Bsed  for  pumping  water  into  the  boiler. 

Double  crank.    See  Crank. 

Dovetail;  a  joint  like  20.  page  735 ;    it  is  a  poor  one  for  timber  when  there  is  much  strain, 

being  then  apt  to  draw  out  more  or  less. 

Dowel;  a  straight'  pin  of  wood  or  metal,  inserted  part  way  into  each  of  two  faces  which  it  unites. 

Draft;  the  depth  to  which  a  floating  vessel  sinks  in  the  water;  in  other  words  the  water  it  draws. 

Draught;  a  drawing.  A  narrow  level  stripe  which  a  stonecutter  first  cuts  around  the  edges  of  a 
rough  stone,  to  guide  him  in  dressing  off  the  face  thus  enclosed  by  the  draught. 

Draw-plate ;  a  plate  of  very  hard  steel,  pierced  with  small  circular  holes  of  different  diameters, 
through  which  in  succession  rods  of  iron  are  drawn,  and  thus  lengthened  out  into  wire.  Sometimes 
the  holes  are  drilled  through  diamond  or  ruby,  &c,  instead  of  steel. 

Drift;  a  horizontal  or  inclined  passage-way,  or  small  tunnel,  in  mines,  &c.  To  float  away  with  a 
•urrent.     Trees,  &c,  carried  along  by  freshets. 

Drip  ;  a  small  channel  cut  under  the  lower  projecting  edge  of  coping,  &c,  so  that  rain  when  it 
reaches  that  point  will  drip  or  fall  off,  instead  of  finding  its  way  horizontally  beneath  to  the  wall, 
which  it  would  make  damp. 

Drop ;  short  pieces  of  nearly  complete  cylinders,  placed  at  small  distances  apart,  in  a  row  like 
teeth,  as  an  ornament  to  cornices,  kc. 

Drum;  a  revolving  cylinder  around  which  ropes  or  belts  either  travel  or  are  wound.  When  nar. 
row  and  used  with  belts  they  are  called  pulleys. 

Dry-rot ;  decay  in  such  portions  of  the  timber  of  houses,  bridges,  <fec,  as  are  exposed  to  dampness, 
•specially  in  confined  warm  situations.  The  timber  in  cellars  and  basement  stories  is  mere  liable  to 
it  than  in  other  parts,  owing  to  the  greater  dampness  absorbed  by  the  brickwork  from  the  ground. 
Contact  with  lime  or  mortar  hastens  dry  rot.  The  ends  of  girders,  joists,  &c,  resting  on  damp  walls, 
may  be  partially  protected  by  placing  pieces  of  slate  or  sheet  iron  under  them.  The  painting  or  tar- 
ring of  unseasoned  timber  expedites  internal  dry  rot.  A  thorough  soaking  of  timber  in  a  solution  of 
28  grains  of  quicklime  to  1  gallon  of  water  is  said  to  be  a  preventive  of  dry-rot ;  but  the  best  process 
for  that  purpose  is  saturation  with  creosote  or  carbolic  acid. 

Dyke ;  mounds  of  earth,  &c,  built  to  prevent  overflow  from  rivers  or  the  sea.  A  kind  of  geologicsti 
irregularity  or  disturbance,  consisting  of  a  stratum  of  rock  injected  as  it  were  by  volcanic  action,  be« 
tween  or  across  strata  of  rocks  of  another  kind.     A  levee. 

Eccentric;  a  circular  plate  or  pulley,  surrounded  by  a  loose  ring,  and  attached  to  a  revolving 
shaft,  and  moving  around  with  it,  but  not  having  the  same  center ;  for  producing  an  alternate  motion. 
Often  used  instead  of  a  crank,  as  they  do  not  weaken  the  axle  by  requiring  it  to  be  bent.  There  are 
many  modifications. 

Escarpment ;  a  neai'ly  vertical  natural  face  of  rock  or  soil. 

Escutcheon;  the  little  outside  movable  plate  that  protects  the  keyhole  of  a  lock  from  dust. 

Eye;  a  circular  hole  in  a  flat  bar,  &c,  for  receiving  a  pin,  or  for  other  purposes. 

Eye  and  strap ;  a  hinge  common  for  outside  shutters,  &c,  one  part  consisting  of  an  iron  strap  one 
end  of  which  is  forged  into  a  pin  at  right  angles  to  it;  and  the  other  part,  of  a  spike  with  an  eye, 
through  which  the  pin  passes.  When  the  eye  is  on  the  str.ip,  and  the  pin  on  the  spike,  it  is  called  a 
hook  and  strap.     Such  hinges  are  sometimes  called  "  backfiaps." 

Eye-bolt;  a  bolt  which  has  an  eye  at  one  end. 

Face-wall;  one  built  to  sustain  a  face  cut  into  natural  earth,  in  distinction  to  a  retaining- wall, 
which  supports  earth  deposited  behind  it. 

Fall;  the  rope  used  with  pulleys  in  hoisting. 

False-works ;  the  scaffold,  center,  or  other  temporary  supports  for  a  structure  while  it  is  being 
built.  In  very  swift  streams  it  is  sometimes  necessary  to  sink  cribs  filled  with  stone,  as  a  base  for 
false-works  to  foot  upon. 

Fascines;  bundles  of  twigs  and  small  branches,  for  forming  foundations  on  soft  ground. 

Fatigue;  of  materials;  the  increase  of  weakness  produced  by  frequent  bending;  or  by  sustaining 
heavy  loads  for  a  long  time. 

Faucet;  a  short  tube  for  emptying  liquids  from  a  cask,  &c;  the  flow  is  stopped  by  a  spigot.  The 
wider  end  of  a  common  cast-iron  water  or  gas  pipe. 

Feather ;  a  slightly  projecting  narrow  rib  lengthwise  of  a  shaft,  and  which,  catching  into  a  corre- 
sponding groove  in  anything  that  surrounds  and  slides  along  the  shaft,  will  hold  it  fast  at  any  required 
part  of  the  length  of  the  feather.     Has  other  applications. 

Feather-edge  ;  when  one  edge  of  a  board,  &c,  is  thinner  than  the  other. 

Felloe,  or  felly ;  the  circular  rim  of  a  wheel,  into  which  the  outer  ends  of  the  spokes  fit;  and  which 
Is  often  surrounded  by  a  tire. 

Felt ;  a  kind  of  coarse  fabric  or  cloth  made  of  fibres  of  hair,  wool,  coarse  paper,  &c,  by  pressure, 
and  not  by  weaving. 

Fender;  a  piece  for  protecting  one  thing  from  being  broken  or  injured  by  blows  from  another; 
frequently  vertical  timbers  along  the  outer  faces  of  wharves,  to  prevent  injury  from  the  rubbing  of 
vessels. 

Fender-piles ;  piles  driven  to  ward  off  accidental  floating  bodies. 

Ferrule  ;  a  broad  metallic  ring  or  thimble  put  around  anything  to  keep  it  from  splitting  or  breaking. 
A  small  sleeve. 

Fillet;  a  plain  narrow  flat  moulding  in  a  cornice,  &c.     See  Platband. 

Fish  ;  to  join  two  beams,  &c,  by  fastening  other  long  pieces  to  their  sides. 

Flags  ;  broad  flat  stones  for  paving. 

Flange  ;  a  projecting  ledge  or  rim. 

Flashings;  broad  strips  of  sheet  lead,  copper,  tin,  &c,  with  one  edge  inserted  into  the  joints  of 
brickwork  or  masonry  an  inch  or  two  above  a  roof,  &c ;  and  projecting  out  several  inched,  so  as  to  be 
flattened  down  close  to  the  roof,  to  prevent  rain  from  leaking  through  the  joint  between  the  roof  and 
the  brick  chimney,  &c,  which  projects  above  it. 

Flasks;  upper  and  lower;  the  two  parts  of  the  box  which  contains  the  mould  into  which  meltel 
iron  is  poured  for  castings. 

Flatting  ;  causing  painting  to  have  a  dead  or  dull,  instead  of  a  glossy  finish,  by  using  turpentin* 
Instead  of  oil  iu  the  last  coat. 

Fliers ;  a  straight  flight  of  steps  in  a  stairway. 

Floodgate;  a  gate  to  let  off  excess  of  water  in  floods,  or  at  other  times. 


1030 


GLOSSARY  OF  TERMS. 


Flume;  a  ditch,  trough,  or  other  channel  of  moderate  size  for  conducting  water.  The  ditches  oi 
culverts  through  which  surplus  water  passes  from  an  upper  to  a  ]»wer  reach  of  a  canal. 

Flush,;  forming  an  even  continuous  line  or  surface.  To  clean  out  a  line  of  pipes,  sewers,  gutters. 
&c.  by  lettiug  on  a  sudden  rush  of  water.     The  splitting  of  the  edges  of  stones  under  pressure. 

Fluxes;  various  substances  used  to  prevent  the  instantaneous  formation  of  rust  when  welding  two 
pieces  of  hot  metal  together.  Such  rust  would  cause  a  weak  weld.  Borax  is  used  for  wrought  iron  ; 
^  mixture  of  borax  and  sal  ammoniac  for  steel ;  chloride  of  ziuc  for  zinc  ;  sal  ammoniac  for  copper 
or  brass ;  tallow  or  resin  for  kad. 

Fly-wheel;  a  heavy  revolving  wheel  for  equalizing  the  motion  of  machinery. 

Foaming;  an  undue  amount  of  boiling,  caused  by  grease  or  dirt  in  a  boiler. 

Follower;  any  cog-wheel  that  is  driven  by  another;  that  other  is  the  leader. 

Forceps;  any  tools  for  holding  things,  as  by  pincers,  or  pliers. 

Forebay,  or  penstock;  the  reservoir  from  which  the  water  passes  immediately  to  a  water-wheel. 

Forge;  to  work  wrought  iron  into  shape  by  first  softening  it  by  heat,  and  then  hammering  it  into 
the  required  form. 

Forge-hammer ;  a  heavy  hammer  for  forging  large  pieces;  and  worked  by  machinery. 

Foxtail;  a  thin  wedge  inserted  into  a  slit  at  the  lower  end  of  a  pin,  so  that  as  the  pin  is  driven 
down,  the  wedge  enters  it  and  causes  it  to  swell,  and  hold  more  firmly. 

Frame;  to  put  together  pieces  of  timber  or  metal  so  as  to  form  a  truss,  door,  or  other  structure. 
The  thing  so  framed. 

Friction-rollers;  hard  cylinders  placed  under  a  body,  that  it  may  be  moved  more  readily  than  bj 
sliding. 

Friction-wheels  ;  wheels  so  placed  that  the  journals  of  a  shaft  may  rest  upon  their  rims,  and  thus 
be  enabled  to  revolve  with  diminished  friction. 

Frieze;  in  architecture,  the  portion  between  the  architrave  and  cornice.  The  term  is  often  applied 
when  there  is  no  architrave. 

Fulcrum;  the  point  about  which  a  lever  turns. 

Furrings;  pieces  placed  upon  others  whicte  are  too  low,  merely  to  bring  their  upper  surfaces  up  to 
a  required  level ;  as  is  often  done  with  joists,  when  one  or  more  are  too  low  ;  a  kind  of  chock. 

Fuze,  or  fuse;  to  melt.  A  slow  match,  which,  by  burning  for  some  time  before  the  tire  reaches  tha 
powder,  gives  the  men  engaged  in  blasting,  time  to  get  out  of  the  way  of  liying  fragments  of  stone. 

Gasket;  rope-yarn  or  hemp,  used  for  stuffing  at  the  joints  of  water-pipes,  &c. 

Crearmy;  a  train  of  cog-wheels.    Now  much  supplanted  by  belts. 

Gib;  the  piece  of  metal  somewhat  of  this  shape,  I — I,  often  used  in  the  same  hole  with  a  wedge- 
shaped  key  for  confining  pieces  together.  In  common  use  for  fastening  the  strap  to  the  stub-end  of 
the  connecting-rod  of  an  engine. 

Gin;  a  revolving  vertical  axis,  usually  furnished  with  a  rope-drum,  and  hnVing  one  or  more  long 
arms  or  levers,  by  means  of  which  it  is  worked  by  horses  walking  in  a  circle  around  it.  Used  for 
hoisting.     Cotton-gin,  a  machine  for  separating  cotton  from  its  seeds. 

Girder;  a  beam  larger  than  a  common  joist,  and  used  for  a  similar  purpose. 

Glacis;  in  fortification,  an  easy  slope  of  earth. 

Gland.   See  Stuffing-box.     Also,  a  kind  of  coupling  for  shafts. 

Glue;  a  cement  for  wood,  prepared  chiefly  from  the  gelatine  furnished  by  boiling  the  parings  of 
hides.     Good  glue  will  hold  two  pieces  of  wood  together  with  a  force  of  from  400  to  750  Bis  per  sq  in. 

Governor  ;  two  balls  so  attached  to  an  upright  revolving  axis  as  to  fly  outward  by  their  centrifugal 
force,  and  thus  regulate  a  valve. 

Grapnel;  a  kind  of  compound  hook  with  several  curved  points,  for  finding  things  in  deep  water. 

Grillage;  a  kind  of  network  of  timbers  laid  crossing  each  other  at  right  angles;  frequent! v  placed 
on  the  heads  of  piles,  for  supporting  piers  of  bridges,  and  other.masoury. 

Groin ;  an  arch  formed  by  two  segmental  arches  or  vaults  intersecting  each  other  at  right  angles. 
Also,  a  kind  of  pier  built  from  the  shore  outward,  to  intercept  shingle  or  gravel. 

Groove;  a  small  channel.  A  triangular  one  is  called  a 

ehamfered  groove. 

Chround-awell ;  waves  which  continue  after  a  storm  has  ceased ;  or  caused  by  storms  at  a  distance. 

Grout ;  thin  mortar,  to  be  poured  into  the  interstices  between  stones  or  bricks. 

Gudgeons  ;  the  metal  journals  of  a  horizontal  shaft,  such  as  that  of  a  water-wheel.  For  moderate 
speeds 

Diam,  ins     \~^  Weight  in  fits  on  one  gudgeon 
if  of  cast-iron  >  20     '  ' 

For  wrought- iron,  add  one-twentieth. 

Own-metal,  or  bronze  ;  a  compound  of  copper  and  tin,  sometimes  used  for  cannon.    Also,  a  quality   • 
of  cast  iron  fit  for  the  same  purpose. 

Gussets;  plain  triangular  pieces  of  plate  iron,  riveted  by  their  vertical  and  horizontal  legs  to  the 
•ides,  tops,  and  bottoms  of  box-girders,  tubular  bridges,  <fec,  inside,  for  strengthening  their  angles. 

Guy  ;  ropes  or  chains  used  to  prevent  anything  from  swinging  or  moving  about. 

Gyrate;  to  revolve  around  a  central  axis,  or  point. 

Halving ;  to  notch  together  two  timbers  which  cross  each  other,  so  deeply  that  the  joint  thickness 
shall  equal  only  that  of  one  whole  timber. 

Hammer  dress;  to  dress  the  face  of  a  stone  by  slight  blows  of  a  hammer  with  a  cutting  edge.  Th« 
patent  hammer  for  such  purposes  has  several  such  edges  placed  parallel  to  each  other,  each  of  which 
may  be  removed  and  replaced  at  pleasure. 

Hand-lever;  in  an  engine,  a  lever  to  be  worked  by  hand  instead  of  by  steam. 

Handspike ;  a  jvooden  lever  for  working  a  capstan  or  windlass  ;  or  other  purposes. 

Hand-wheel ;  a  wheel  used  instead  of  a  spanner,  wrench,  winch,  or  lever  of  anv  kind,  for  screwing 
■uts,  or  for  raising  weights,  or  for  steering  with  a  rudder,  &c. 

Hangers,  or  pendent  brackets;  fixtures  projecting  below  a  ceiling,  to  support  the  journals  of  long 
lines  of  shafting  ;  and  for  other  purpose.    Should  be  "  self-adjusting." 

Hasp;  a  piece  of  metal  with  an  opening  for  folding  it  over  a  staple. 

Hatchway;  a  horizontal  opening  or  doorway  in  a  floor,  or  in  the  deck  of  a  vessel. 

Haunches  ;  the  parts  of  an  arch  from  the  keystone  to  the  skewback. 

Head-block  ;  a  block  on  which  a  pillow-block  rests. 

Header;  a  stone  or  brick  laid  lengthwise  at  right  angles  to  the  face  of  the  masonry. 

Heading  ;  in  tunnelling,  a  small  driftway  or  passage  excavated  in  advance  of  the  main  body  of  the 
tunnel,  but  forming  part  of  it;  for  facilitating  the  work. 

Hcadicay;  the  clear  height  overhead.     Progress. 

Heel-post;  that  on  which  a  lock  gate  turns  on  its  pivvt. 

Helve  :  the  handle  of  an  axe. 


GLOSSARY   OF   TERMS.  1031 

Binge ;  those  commonly  used  on  the  doors  of  dwellings  are  called  butts,  or  butt  hinges.  (Eye  and 
BUap,  .)    Rising  hinges  are  such  as  cause  the  door  to  rise  a  little  as  it  is  opened,  aud  thus  cause 

the  door  to  shut  itself. 

Mip  roof,  or  hipped  roof;  one  that  slopes  four  ways ;  thus  forming  angles  called  hips. 

Hoarding;  a  temporary  close  fence  of  boards,  placed  around  a  work  in  progress,  to  exclude 
Biragglers. 

Holding-plates,  or  anchors  ;  strong  broad  plates  of  iron  sunk  into  the  ground,  and  generally  sur- 
rounded by  masonry  ;  for  resisting  the  pull  of  the  cables  of  suspension  bridges ;  and  for  other  simi- 
lar purposes. 

Hook  and  strap.     See  Eye  and  strap. 

Horses;  the  slopiug  timbers  which  carry  the  steps  in  a  staircase. 

Housings;  in  roUiug  mills,  &c,  the  vertical  supports  for  the  boxes  in  which  the  journals  revolve. 

Huh,  or  nave ;  the  central  part  of  a  wheel,  through  which  the  axletree  passes,  and  from  which 
the  spokes  radiate. 

Impost;  the  upper  part  of  a  pier  from  which  an  arch  springs. 

Ingot ;  a  lump  of  cast  metal,  generally  somewhat  wedge-shaped.     A  pig  of  cast  iron  is  an  ingot. 

Invert;  an  inverted  arch  frequently  built  under  openings,  in  order  to  distribute  the  pressure  more 
evenly  over  the  foundation. 

Jack;  a  raising  instrument,  consisting  of  an  iron  rack,  in  connection  with  a  short  stout  timber 
which  supports  it,  and  worked  by  cog-wheels  And  a  winch.  A  screw-jack  is  a  large  screw  working 
in  a  strong  frame,  the  base  of  which  serves  for  it  to  staud  on  ;  and  which  is  caused  to  revolve  and 
rise,  carrying  the  load  on  top  of  it,  by  turuiug  a  nut,  or  otherwise. 

Jack-rafters,  or  common  rafters;  small  rafters  laid  on  the  purlins  of  a  roof,  for  supporting  the 
shingling  laths,  &c. 

Jag-spike;  a  spike  whose  sides  are  jagged  or  notched,  with  the  mistaken  idea  that  its  holding  power 
is  thereby  much  increased.  If  a  spike  or  bolt  is  first  put  into  its  place  loosely,  and  then  has  melted 
lead  run  around  it,  the  jaggiug  does  assist;  but  not  when  it  is  driven  into  wood.  • 

Jambs;  the  sides  of  an  opening  through  a  wall,  &c;  as  door,  window,  and  fireplace  jambs. 

Jamh -lining s  ;  the  facing  of  woodwork  with' which  jambs  are  covered  and  hidden. 

Jaw;  an  opening,  often  V-shaped,  the  inner  edges  of  which  are  for  holding  something  in  place. 

Jettie,  or  jetty  ;  a  pier,  mound,  or  mole  projecting  into  the  water;  as  a  wharf-pier,  &c. 

Jib  ;  the  upper  projecting  member  or  arm  of  a  crane,  supported  by  the  stay. 

Jig-saw;  a  very  narrow  thin  saw  worked  vertically  by  machinery,  and  used  for  sawing  curved 
ornaments  in  boards. 

Joggle  ;  a  joint  like  that  at  3  or  4,  &c,  p  735,  of  Trusses,  for  receiving  the  pressure  of  a  strut  at 
right  angles  or  nearly  so.  Also  applied  to  squared  blocks  of  stone  sometimes  inserted  between 
courses  of  masonry  to  prevent  sliding,  &c. 

Joist  ;  binding  joists  are  girders  for  .sustaining  common  joists.  The  common  ones  are  then  called 
bridging  joisits.  Ceiling  joists  are  small  ones  under  roof  trusses,  or  under  girders,  and  for  sustain- 
ing merely  the  plastered  ceiling. 

Journal-box;  a  fixture  upon  which  a  journal  rests  and  revolves,  instead  of  a  plummer-block. 

Joxirnals  ;  the  cvlindrical  supporting  ends  of  a  horizontal  revolving  shaft.  Their  length  is  usually 
about  1  to  13^  times  their  diam.     In  lines  of  shafting  4  diams.     To  find  the  diam,  see  Gudgeon. 

Jumper;  a  drill  used  for  boring  holes  in  stone  by  aid  of  blows  of  a  sledge-hammer. 

Kedge ;  a  small  anchor. 

Keepers  ;  the  pieces  of  metal  or  wood  which  keep  a  sliding  bolt  in  its  place,  and  guide  it  in  sliaing. 

Kerf;  the  opening  or  narrow  slit  made  in  sawing. 

Key-holt.     See  Cotter-bolt. 

Keystone;  the  center  stone  of  an  arch. 

Kibble;  the  bucket  used  for  raising  earth,  stone,  &c.  from  shafts  or  mines. 

King-x>ost,  king-rod  ;  the  center  post,  vertical  piece,  or  rod,  in  a  truss ;  all  those  on  each  side  of  it 
are  queen- posts,  or  queen-rods.     Frequentlv  called  simply  kings  and  queens. 

Knee  ;  a  piece  of  metal  or  wood  bent  at  a"n  angle ;  to  serve  as  a  bracket,  or  as  a  means  of  uniting 
two  surfaces  which  form  with  each  other  a  similar  angle. 

Lagging,  or  sheeting  ;  a  covering  of  loose  plank ;  as  that  placed  upon  centers,  and  supporting  the 
•rchstones.     Also,  an  outer  wooden  casing  to  locomotive  boilers  and  others. 

Landing;  the  resting-place  at  the  end  of  a  flight  of  stairs. 

Lantern  wheel.     See  Trundle.  ^.      ^         j  xu  »    r  1.1,      *». 

Lap  ;  to  place  one  piece  upon  another,  with  the  edge  of  one  reaching  beyond  that  of  the  other. 

Lap-welding;  welding  together  pieces  that  have  first  been  lapped  ;  in  distinction  to  butt-welding 

Lead,  (pronounced  l^ed  t  in  stejun-etigines,  a  certain  amount  of  opening  of  the  port-valve  befoie 
each  stroke  ©f  the  piston  begins.    The  distance  to  which  earth  is  hauled  or  wheeled. 

Leader:  a  cog-wheel  that  pives  motion  to  the  next  one  or  follower. 

ieadino-fteam;  «eadm<7-2J!i 3;  one  placed  as  a  guide  for  placing  others. 

Leading-wheels;  in  a  locomotive,  those  frequently  placed  in  front  of  the  driving-wheels. 

Leaves;  the  cogs  of  pinions.  .      .  .  .       .  .        j       •,  j 

Ledge;  a  part  projecti-ng  over  like  a  shelf;  a  rock  so  projecting.  A  narrow  strip  of  board  nailed 
across  other  boards,  to  hold  them  together,  as  in  temporary  ledge-doors.  ^  v.  ,    •        ki     1, 

Lewis;  an  arrangement  composed  of  2  or  3  pieces  of  metal  let  into  a  wedge-shaped  hole  m  a  block 
of  stone,  by  which  to  raise  the  block.  ,  ^        ^.       ^_ 

Lighter;  a  scow,  raft,  or  other  vessel,  used  for  unloading  vessels  out  from  the  shore. 

Linck-pin;  a  pin  near  the  end  of  an  axle,  to  hold  the  wheel  on. 

Link  :  one  of  the  divisions  of  a  chain :  or  a  piece  shaped  like  one.  ^ 

Lmk-motion;  a  device  for  regulating  the  movement  of  the  main  or  port  valve  in  a  steam-engine. 

Lintel;  a  horizontal  beam  across  an  opening  in  a  wall,  as  seen  in  windows,  doors,  &c.  W  hen  of 
Wide  span,  and  supporting  heavy  brickwork  or  masonry,  it  is  called  a  breast-summer,  or  bre.summer 

Lock-  those  cornmon  door  locks  which  are  entirely  concealed  within  the  thickness  of  tho  4„or.  are 
called  mortice  locks  ;  those  which  are  screwed  against  the  face  of  a  door,  rim  locks.  It  must  be  remem, 
bered  that  locks  are  "right  and  left."  „       ,     ,j       ..      «.  -/i^^  „.j*i,  k.^- 

Louvre  ;  a  kind  of  vertical  window,  frequently  at  the  tops  of  roofs  of  depots,  &c,  provided  with  hor 
izontal  slats,  which  permit  ventilation,  and  exclude  rain. 

iozenoe?  the  shape  of  a  rhomb;  often  called  diamond-shaped.  v       #— 

Xt/^-  in  casting,  small  projections  from  the  general  surface,  and  for  various  purposes,  such  aa  fo« 
Ufting  the  body ;  or  for  a  flange  for  joining  it  to  nnother ;  or  for  a  support  for  something  else. 
tIaXlet :  the  wooden  hammer  used  by  stonecutters. 


1032  GLOSSARY    OF   TERMS. 

A^^^^^rlL^.l'^'l'i  '■"'^  "l^K  *\*  core  around  which  a  flat  piece  may  be  bent  into  a  cylindrical  shaM 
Also  the  shaft  that  carries  the  chuck  of  a  lathe.  "»f« 

Manhole;  an  opening  by  which  a  man  can  enter  a  boiler,  culvert,  &c.  to  clean  or  repair  it. 

Mattock;  a  kind  of  pick  with  broad  edges  for  digging. 

Maul;  a  heavy  wooden  hammer. 

Mean,  arithmetical :  half  the  sum  of  two  numbers. 
"      ,  geometrical;  the  sq  rt  of  the  product  of  two  numbers. 

Mean-proportional ;  the  same  as  the  geometrical  mean. 

Meridian  ;  a  north  and  &outh  line.     Noon. 

Mitre-joint;  a  joint  formed  along  the  diagonal  line  where  the  ends  of  two  pieces  are  united  ataa 
angle  with  each  other. 

Mitre-till;  the  sill  against  which  the  lock  gates  of  a  canal  shut. 

Modulus ;  a  datum  serving  as  a  means  of  comparison.    Same  as  constant  or  coefficient. 

Moment ;  tendency  of  force  acting  with  leverage. 

Moment  ofrupttire,  or  of  tendings  the  tendency  which  any  load  or  force  exerts  to  break  or  bend  a 
l^ody  by  the  aid  of  leverage.  Its  amount  is  found  in  foot-pounds  by  multiplying  the  force  in  fts,  by 
the  length  of  leverage  in  feet  between  it  and  that  part  of  the  body  upon  which  the  tendency  is  exerted. 

Monkey  ;  the  hammer  or  ram  of  a  pile-driver. 

Monkey-wrench,  or  screw-wrench  ;  a  spanner,  the  gripping  end  of  which  can  be  adjusted  by  meaaa 
of  a  screw  to  fit  objects  of  different  sizes. 

Moorings;  fixtures  to  which  ships,  &c,  can  make  fast. 

Mortise;  a  hole  cut  in  one  piece,  for  receiving  the  tenon  which  projects  from  another  piece. 

Muck;  soft  surface  soil  containing  much  vegetable  matter. 

Muntins,  or  mullions ;  the  vertical  pieces  which  separate  the  panes  in  a  window-sash. 

Nailing -blocks ;  blocks  of  wood  inserted  in  walls  of  stone  or  brick,  for  nailing  washboards,  &c,  to. 

Nave ;  the  main  body  of  a  building,  having  connecting  wings  or  aisles  on  each  side  of  it.  The  hub 
jf  a  wheel. 

Newel;  the  open  space  surrounded  by  a  stairway. 

Newel-post ;  a  vertical  post  sometimes  used  for"  sustaining  the  outer  ends  of  steps.  Also  the  large 
baluster  often  placed  at  the  foot  of  a  stairway. 

Nippers  ;  pincers.     An  arrangement  of  two  curved  arms  for  catching  hold  of  anything. 

Normal;  perpendicular  to.     Accc^ding  to  rule,  or  to  correct  principles. 

Nosing  ;  the  slight  projection  oft?n  given  to  the  front  edge  of  the  tread  ot  a  step  ;  usually  rounded. 

Nut,  or  burr ;  the  short  piece  w^th  a  central  female  screw,  used  on  the  end  of  a  screw-bolt,  &c,  for 
keeping  it  in  place. 

Ogee ;  a  moulding  in  shape  o7  an  S,  the  same  as  a  cima.' 

Ordinate;  a  line  drawn  at  r'lght  angles  from  the  axis  of  a  curve,  and  extending  to  the  curve. 

Oscillate  ;  to  swing  backward  and  forward  like  a  pendulum. 

Oict  of  wind,  pronounced  ivynd ;  perfectly  straight  or  flat. 

Ovolo ;  a  projecting  con''ex  moulding  of  quarter  of  a  circle ;  when  it  is.  concave  it  is  a  cavetto,  or 
hollow. 

Packing  ;  the  material  placed  in  a  stuffing-box,  &c,  to  prevent  leaks. 

Packing-pieces ;  short  pieces  inserted  between  two  others  which  are  to  be  riveted  or  bolted  together, 
to  prevent  their  comiug  in  contact  with  each  other. 

Pall,  or  pawl.     Se-J  Ratchet. 

Parapet;  a  wall  or  any  kind  of  fence  or  railing  to  prevent  persons  from  falling  ofiF. 

Parcel;  to  wrao  canvas  or  rags  round  a  rope. 

Purge ;  to  make  the  inside  of  a  flue  smooth  by  plastering  it. 

Patent  hamn'.er  ;  a  hammer  with  several  parallel  sharp  edges  for  dressing  stone. 

Pay.     To  cover  a  surface  with  tar,  pitch,  &c.     A  ship  word. 

Pay  out.     To  slacken,  or  let  out  rope. 

Pediment ;  the  triangular  space  in  the  face  of  a  wall  that  is  included  between  the  two  sloping  sides 
of  the  roof  and  a  line  joining  the  eaves. 

Penstoch.     See  Forebay. 

Pier ;  the  support  of  two  adjacent  arches.  The  wall  space  between  windows,  &c.  A  structure  built 
out  into  the  water. 

Pierre-perdue ;  lost  stone ;  random  stone,  or  rough  stones  thrown  into  the  water,  and  let  find  their 
own  slope. 

Pilaster ;  a  thin  flat  projection  from  the  face  of  a  wall,  as  a  kind  of  ornamental  substitute  for  a 
column. 

Pileplanks;  planks  driven  like  piles. 

Pillow-block,  or  plummer- block ;  a  kind  of  metal  chair  or  support,  upon  which  the  journals  of  hor- 
izontal shafts  are  generally  made  to  rest,  and  on  which  they  revolve. 

Pinion;  a  small  cog- wheel  which  gives  motion  to  a  larger  one. 

Pintle  ;  a  vertical  projecting  pin  like  that  often  placed  at  the  tops  of  crane-posts,  and  over  ^W* 
the  holding  rings  at  the  tops  of  the  wooden  guys  fit.  Also,  such  as  is  used  for  the  hinges  of  r"  ''^- 
K  of  window-shutters  to  turn  around. 


GLOSSARY    OF   TERMS.  1033 

Pitch ;  the  slope  ol  a  roof,  &c.  The  distance  from  center  to  center  of  the  teeth  of  a  cog  wheel,  or 
the  threads  of  a  screw.     Boiled  tar.    Also  the  dist  apart  of  rivets,  &c. 

Pitman;  a  connecting-rod  for  transmitting  motion  from  a  prime  mover  to  machinery  at  a  distance, 
moved  by  it. 

Pit-i&w;  a  large  saw  worked  vertically  by  two  men,  one  of  whom  (the  pitman;  stands  in  a  pit. 

Pivot;  the  lower  end  of  a  vertical  revolving  shaft,  whether  a  part  of  the  shaft  jtself,  or  attached  to 
it.  It  should  be  flat;  and  both  it  and  the  step  or  socket  upon  which  it  rests  should  be  of  hard  steel. 
If  a  steel  pivot  has  to  revolve  rapidly  and  continuously,  it  is  well  to  proportion  its  diam,  so  as  not  to 
have  to  sustain  more  than  250  Bbs  per  sq  inch;  otherwise  it  will  wear  quickly.  Dust  and  grit  should 
for  the  same  reason  be  carefully  guarded  against.  Pivots  which  revolve  but  seldom,  and  slowly,  as 
those  of  a  railroad  turntable,  may  be  trusted  with  half  a  ton,  or  even  a  whole  ton  per  sq  inch.  As  a 
rude  rule,  cast-iron  pivots  should  not  be  loaded  with  more  than  half  as  much  as  steel  ones.  A  steel 
«ne  may  be  welded  to  the  foot  of  its  cast-iron  shaft;  or  may  be  inserted  part  way  into  it;  and  th« 
whole  strengthened  by  iron  bands  shrunk  on. 

Planish;  to  polish  metals  by  rubbing  with  a  hard  smooth  tool. 

Plant;  the  outfit  of  machinery,  &c,  necessary  for  carrying  on  any  kind  of  work. 

Plaster-bead ;  a  small  vertical  strip  of  iron  or  wood  nailed  along  projecting  angles  in  rooms,  to 
protect  the  plaster  at  those  parts. 

Platband ;  a  plain,  flat,  wide,  slightly  prqiecting  strip,  generally  for  ornament.  When  narrow  It 
is  called  a  fillet. 

pliers  ;  a  kind  of  pincers. 

Plinth ;  the  square  lowest  member  of  the  base  of  a  column  or  pillar. 

plug;  a  piece  inserted  to  stop  a  hole.     Screw-plug,  a  plug  that  is  screwed  into  a  hole. 

Plumb;  vertical. 

Plummet,  or  plumb-bob ;  a  weight  at  the  lower  end  of  a  string,  for  testing  verticality. 

Plunger  ;  a  kind  of  solid  piston,  or  one  without  a  valve. 

Point;  a  kind  of  pointed  chisel  for  dressing  stone.  To  put  a  finish  to  masonry  by  touching  up  the 
outer  mortar  joints.     To  dress  stone  with  a  point  and  mallet. 

Pole-plate ;  a  longitudinal  timber  resting  on  the  ends  of  tie-beams  of  roofs ;  and  for  supporting 
tiie  feet  of  the  common  or  jack  rafters,  when  such  are  used. 

Port;  the  opening  or  passage  controlled  by  a  valve. 

Prime ;  to  put  on  the  first  coat  of  paint.  Priming  also  is  when  water  passes  into  a  steam  cylinder 
•long  with  the  steam. 

Projection.  If  parallel  straight  lines  be  imagined  to  be  drawn  in  any  one  given  direction,  from 
every  point  in  any  surface  s.  whether  flat,  curved,  or  irregular,  then  if  all  these  lines  be  supposed  to 
be  intersected  or  cut  by  a  plane,  either  at  right  angles  to  their  direction,  or  obliquely,  the  flgure 
which  their  cross-section  thus  made  would  form  upon  said  plane,  is  called  the  projection  of  the  sur- 
face 8.  If  such  lines  be  supposed  to  be  drawn  fr'ra  a  person's  face,  in  a  direction  in  front  of  him, 
and  to  be  cut  by  a  plane  at  right  angles  to  their  direction,  their  projection  on  the  plane  would  be  the 
person's  full-face  portrait.  If  the  lines  be  drawn  sideways  from  his  face,  the  projection  will  be  his 
profile.  The  projection  of  a  globe  upon  a  flat  plane,  will  evidently  be  a  circle  if  the  plane  cuts  the 
lines  at  right  angles  ;  and  an  ellipse  if  it  cuts  them  obliquely.  Shadows  cast  by  the  sun  are  projec- 
lions. 

Puddle ;  earth  well  rammed  into  a  trench,  Ac,  to  preyent  leaking.  A  process  for  converting  cast 
Iron  into  wrought  by  a  puddling  furnace. 

Pug-mill ;  a  mill  for  tempering  clay  for  bricks  or  pottery,  &c. 

Pulley  ;  a  circular  hoop  which  carries  a  belt  in  machinery. 

Puppet;  in  machinery,  a  small  short  pedestal  or  stand.     Puppet  valve.  See  "Valves. 

Purlins;  the  horizontal  pieces  placed  on  rafters,  for  supporting  the  roof  covering. 

Put-logs,  or  put-locks;  horizontal  pieces  supporting  the  floor  of  a  scaffold;  one  end  being  inserted 
into  put-log  holes  left  for  that  purpose  in  the  masonry. 

Quay;  a  wharf. 

Quoin ;  the  hollow  into  which  a  quoin-post  of  a  canal  lock-gate  fits.  Stones,  usually  dressed,  placed 
along  the  vertical  angles  of  buildings,  chiefly  for  ornament. 

Quoin-post ;  the  vertical  post  on-  which  a  lock-gate  turns.    The  heel-post. 

Rabbet,  or  rebate ;  a  half  groove  along  the  edge  of  a  board,  &c.  See  16,  p  735,  of  Trusses,  where 
two  rabbets  are  shown  overlapping  each  other. 

Race;  the  channel  which  conducts  water  either  to  or  from  a  water-wheel :  the  first  is  a  head  race ; 
the  last  a  tail  race.  The  waves  produced  by  the  meeting  of  strong  opposing  currents  ;  also,  a  rapid 
tidewav,  or  roost. 

Rack  and  pinion  ;  the  rack  is  a  straight  row  of  cogs  on  a  bar,  and  called  a  rack-bar ;  and  the  pInioH 
is  a  small  cog-wheel  working  into  it. 

Radius  of  gyration.     See  Center  of  gyration. 

Rag-bolt.     See  Jag-spike. 

Rag-wheel,  or  Sprocket-wheel ;  one  with  teeth  or  pins  which  catch  into  the  links  of  a  chain. 

Mam ;  the  hammer  of  a  pile-driver. 

Random  stone ,'  rip  rap,  or  rough  stones  thrown  promiscuously  into  the  water,  to  form  a  founda- 
tion, &c. 

Rasp  ;  a  coarse  file. 

Ratchet  and  pall ;  the  former  is  sometimes  a  straight  bar,  at  others  awheel:  in  either  case  it  is 
furnished  with  teeth  between  which  the  pall  drops  and  prevents  backward  motion.  Used  for  safety 
in  hoisting  machinery.  &c.    The  pall  is  sometimes  called  a  click. 

Ream;  to  enlarge  the  diameter  of  a  hole  by  scraping  away  the  material  forming  its  circumfer- 
ence. 

Rebate.    See  Rabbet,  above. 

Reciprocal  of  a  number  is  the  quotient  found  by  dividing  1  by  that  number. 

Reciprocating  motion.    See  Alternating  motion. 


1034  GLOSSARY   OF   TERMS. 

Be-entering  angle  ;  an  angle  or  corner  projecting  inward.    See  Salient,  below. 

Revetment ;  steep  lacing  of  stone  to  the  sides  of  a  ditch  or  parapet  in  fortification.  A  retaining- wall. 

Bib  ;  the  curved  pieces  which  form  the  arches  of  iron  or  wooden  bridges,  &c.  Also,  those  to  which 
the  outer  planking  of  a  sailing  vessel,  &c,  are  fastened.   . 

Bidye  of  a  roof;  its  peak,  or  the  sharp  edge  along  its  very  top.     Has  various  similar  applications. 

Bidgepole,  ridye-piece,  or  ridgeplate ;  the  highest  horizontal  timber  in  a  roof,  extending  from  top 
to  top  of  the  several  pairs  of  rafters  of  the  trusses ;  for  supporting  the  heads  of  the  jack-rafters. 

Bight  and  left ;  a  lock  which  in  its  proper  position  suits  one  flap  of  a  pair  of  folding  doors,  will 
not  suit  if  fastened  to  the  other  flap  ;  nor  even  to  the  same  flap  if  required  to  open  to  the  right  in. 
stead  of  to  the  left,  or  vice  versa,  according  to  whether  it  is  a  right  or  a  left-hand  lock.  And  so  with 
many  otuer  things,  as,  for  instance,  certain  arrangements  for  working  railway  switches,  <kc.  Right 
and  left  boots  and  shoes  are  a  familiar  illustration  ;  also,  right  and  left  screws.  Therefore,  in  order- 
ing several  of  anything,  it  is  necessary  to  consider  whether  they  may  all  be  of  the  same  pattern,  or 
whether  some  must  be  right-hand,  and  others  left-hand  ones. 

Bight  shore  of  a  river;  that  which  is  on  the  right  hand  when  descending  the  river. 

Bight-solid  body  ;  one  which  has  its  axis  at  right  angles  to  its  base;  when  not  so,  it  is  oblique, 

Bing-boU ;  a  bolt  with  an  eye  and  a  ring  at  one  end. 

Bip-rap.    See  Random  stone. 

Boadstead;  anchorage  at  some  distance  from  shore. 

Bock-ahaft ;  a  shaft  which  only  rocks  or  makes  part  of  a  revolution  each  way,  instead  of  revolving 
entirely  around. 

Bockwork;  squared  masonry  in  which  the  face  is  left  rough  to  give  a  rustic  appearance. 

Bubble ;  masonry  of  rough,  undressed  stones.  Scabbled  rubble  has  only  the  roughest  irregularities 
knocked  off  by  a  hammer.  Ranged  rubble  has  the  stones  in  each  course  rudely  dressed  to  nearly  a 
uniform  height. 

Bundle,  or  round  ;  the  step  of  a  ladder. 

Bustic ;  much  the  same  as  rockwork. 

Saddle;  the  rollers  and  fixtures  on  top  of  the  piers  of  a  suspension  bridge,  to  accommodate  ex- 
pansion and  contraction  of  the  cables.  The  top  piece  of  a  stone  cornice  of  a.  pediment.  Has  many 
other  applications. 

Sag ;  to  bend  downward. 

Salient ;  projecting  outward.    See  Re-entering,  above. 

Sandbag  ;  a  bag  filled  with  sand  for  stopping  leaks. 

Scabble ,  to  dress  off  the  rougher  projections  of  stones  for  rubble  masonry,  with  a  stone-axe,  of 
ccabbliug  hammer. 

Scantling  ;  the  depth  and  breadth  of  pieces  of  timber;  thus  we  say,  a  scantling  of  8  by  10  ins,  &c 

Scarf;  the  uniting  of  two  pieces  by  a  long  joint,  aided  by  bolts,  &c. 

Scarp  ;  a  steep  slope.     In  fortification  the  inner  slope  of  a  ditch. 

Scotia  ;  a  receding  moulding  consisting  of  a  semi-circle  or  semi-ellipse,  or  similar  figure. 

Screeds  ;  long  narrow  strips  of  plaster  put  on  horizontally  along  a  wall,  and  carefully  faced  out  of 
wind,  to  serve  as  guides  for  afterward  plastering  the  wide  intervals  between  them. 

Screw-bolt ,  a  bolt  with  a  screw  cut  on  one  end  of  it. 

Screw-jack.    See  Jack. 

Screw-wrench.    See  wrench. 

Scribe ;  to  trim  off  the  edge  of  a  board,  Ac,  so  as  to  make  it  fit  closely  at  all  points,  to  an  irregular 
surface.  The  lower  edges  of  an  open  caisson  are  scribed  to  fit  the  irregularities  of  a  rocky  river  bottom. 

Scroll;  an  ornamental  form  consisting  of  volutes  or  spirals  arranged  somewhat  in  the  shape  of  S. 

Scupper  nails  ;  nails  with  broad  heads  for  nailing  down  canvas,  &c. 

Scuppers  ;  on  shipboard,  holes  for  allowing  water  to  flow  off  from  the  deck  into  the  sea. 

Scuttle  ;  a  small  hatchway.    To  make  holes  in  a  vessel  to  cause  sinking. 

Sea-wall;  a  wall  built  to  prevent  encroachment  of  the  sea. 

Secret  nailing  ;  so  nailing  down  a  floor  by  nails  along  the  edges  of  the  boards,  that  the  nail-heads 
do  not  show. 

Serve  ;  to  wrap  twine  or  yarn,  &c,  closely  round  a  rope  to  keep  it  from  rubbing. 

Set-screw,  or  tightening -screw ;  a  screw  for  merely  pressing  one  thing  tightly  against  another  at 
will ;  such  as  that  which  confines  the  movable  leg  of  a  pair  of  dividers  in  its  socket. 

Shackle,  or  clevis ;  a  link  in  a  chain  shaped  like  a  U,  and  so  arranged  that  by  drawing  out  a  bolt 
or  pin,  which  fits  into  two  holes  at  the  ends  of  the  U,  the  chain  can  be  separated  at  that  point. 

Shaft;  a  vertical  pit  like  a  well.     The  body  of  a  column.     A  large  axle. 

Shank;  the  body  of  a  bolt  exclusive  of  its  head.  The  long  straight  part  of  many  things,  as  of  an 
anchor,  a  key,  &c. 

Shears,  or  sheers;  two  tall  timbers  or  poles,  with  their  feet  some  distance  apart,  and  their  tops 
festened  together;  and  supporting  hoisting  tackle. 

Sheave;  a  wheel  or  round  block  with  a  groove  around  its  circumference  for  guiding  a  rope. 

Sheeting,  or  sheathing ;  covering  a  surface  with  boards,  sheet  iron,  felt,  &c. 

Shingle ;  the  pebbles  on  a  seashore. 

Shoes;  certain  fittings  at  the  ends  of  pieces;  as  the  pointed  iron  shoes  for  piles.  The  wall  shoes 
into  which  the  lower  ends  of  iron  rafters  generally  fit,  &c. 

Shore ;  a  prop. 

Shot ;  the  edge  of  a  board  is  said  to  be  shot  when  it  is  planed  perfectly  straight. 

Shrink.  When  an  iron  hoop  or  band  is  first  heated,  and  then  at  onoe "placed  upon  the  body  which 
it  is  intended  to  surround,  it  shrinks  or  contracts  as  it  cools,  and  therefore  clasps  the  body  more  firmly. 
This  is  called  shrinking  on  the  hoop. 

Shuttle ;  a  small  gate  for  admitting  water  to  a  water-wheel,  or  out  of  a  canal  lock,  &c. 

Siding ;  a  short  piece  of  railroad  track,  parallel  to  the  main  one,  to  serve  as  a  passing-place. 

Silt ;  soft  fine  mud  deposited  by  rivers,  &c. 

Siphon  culvert;  a  culvert  built  in  shape  of  a  U,  for  carrying  a  stream  under  an  obstncle.  and  allow- 
ing it  afterward  to  rise  again  to  its  natural  level.  The  term' is  improper,  inasmuch  as  the  principle 
of  the  siphon  is  not  involved. 

Skewbnck;  the  inclined  stone  from  which  an  arch  springs. 

Skids ;  vertical  fenders,  on  a  ship's  sides.     Two  parallel  timbers  for  rolling  things  upoa. 

Skirting;  narrow  boards  nailed  along  a  wall,  as  the  washboards  in  dwellings. 

Sledge';  a  heavy  hammer. 

Sleeper;  any  lower  or  foundation  piece  in  contact  with  the  ground. 

Sleeve;  a  hollow  cylinder  slid  over  two  pieces  to  hold  them  together. 


GLOSSARY   OF   TERMS.  1035 

SUde-hars,  or  slides;  bars  for  anything  to  slide  along;  as  those  for  the  cross-heads  of  piston-rods. 
&c.     Often  called  guides. 

Slings ;  pieces  ot  rope  or  chain  to  be  put  around  stones,  &c,  for  raising  them  by. 

Slip;  the  sliding  down  of  the  sides  of  earth-cuts  or  banks.  A  long  narrow  water  space  or  dock 
between  two  wharf  piers. 

Slope-wall;  a  wall,  generally  thin  and  of  rubble  stone,  used  to  preserve  slopes  from  the  action  of 
water  in  the  banks  ot  canals,  rivers,  reservoirs,  &c  ;  or  from  the  action  of  rain. 

Slot;  a  long  narrow  hole  cut  through  anything. 

Sluice;  a  water-channel  of  wood,  masonry,  &c;  or  a  mere  trench.  The  flow  is  usually  regulated 
hy  a  sluice-gate. 

Smoke-box;  in  locomotives,  that  space  in  front  of  the  boiler,  through  which  the  smoke  passes  to 
the  chimney. 

Snag  ;  a  lug  with  a  hole  through  it,  for  a  bolt. 

Socket;  a  cavity  made  in  one  piece  for  receiving  a  projection  from,  or  the  end  of,  another  piece;  as 
that  into  which  the  movable  leg  of  a  pair  of  dividers  fits. 

Soffit;  the  lower  or  underneath  surface  of  an  arch,  cornice,  window,  or  door-opening.  &c. 

Solder;  a  compound  of  dififerent  metals,  which  when  melted  is  used  for  uniting  pieces  of  metal  also 
heated.  Soft  solder  is  a  compound  of  lead  and  tin,  and  is  used  for  uniting  lead  or  tin.  There  are 
various  hard  solders,  suph  as  spelter  solder,  composed  of  copper  and  zinc,  for  uniting  iron,  copper,  or 
brass. 

Sole;  that  lining  around  a  water-wheel  which  forms  the  bottoms  of  the  buckets. 

Spandrel ;  the  space,  or  the  masonry,  &c,  between  the  back  or  extrados  of  an  arch  and  the  roadway. 

Spanner;  a  kind  of  wrench,  consisting  of  a  handle  or  lever  with  a  square  eve  at  one  end  of  it ;  much 
used  for  tightening  up  the  nuts  upon  screw-bolts,  Ac.  The  eye  fits  over  or  surrounds  the  nut. 

Spar;  a  beam ;  but  generally  applied  to  round  pieces  like  masts,  &c. 

Spelter;  zinc. 

Spigot ;  the  pin  or  stopper  of  a  faucet.    The  smaller  end  of  a  common  cast-iron  water  or  gas  pipe. 

Spindle ;  a  thin  delicate  shaft  or  axle. 

Splay ;  to  widen  or  flare,  like  the  jambs  of  a  common  fireplace,  or  those  of  many  windows ;  or  like 
the  wiug- walls  of  most  culverts. 

Splice;  to  unite  two  pieces  firmly  together. 

Springer;  the  lowest  stone  of  an  arch. 

SprocfCet-tvheel,  or  rag-wheel ;  one  with  teeth  or  pins  which  catch  in  the  links  of  a  chain. 

Spur-wheel ;  a  common  cog-wheel,  in  which  the  teeth  radiate  from  a  common  cen,  like  those  of  a  spur. 

Square  ;  in  roofing  ;  100  square  feet. 

Square-head ;  a  square  termination  like  that  upon  which  a  watch-key  fits  for  winding  ;  or  that 
upon  which  the  eye  of  the  handle  of  a  common  grindstone  fits  for  turning  it.  &c. 

Staging  ;  the  temporary  flooring  of  a  scaffold,  platform,  &c. 

Stanchion  ;  a  vertical  prop  or  strut. 

Standing -holt,  or  stud-holt ;  a  bolt  with  a  screw  cut  upon  each  end  ;  one  end  to  be  screwed  perma- 
nently  into  something,  and  the  other  end  to  hold  by  means  of  a  nut  something  else  that  may  be  re- 
quired to  be  removed  at  times. 

Staple ;  a  kind  of  double  pin  in  shape  of  a  U ;  its  two  sharp  points  are  driven  into  timber,  and 
curved  part  is  left  projecting,  to  receive  a  hoop,  pin.  or  hasp,  &c. 

Starlings  ;  the  projecting  up  and  down-stream  ends  or  cutwaters  of  a  bridge  pier. 

Stay ;  variously  applied  to  props,  struts,  and  ties,  for  staying  anything  or  keeping  it  in  place. 

Stay-bolts;  long  bolts  placed  across  the  inside  of  a  boiler,  &c,  to 'give  it  greater  strength. 

Steam-chest ;  the  iron  box  in  locomotive  engines  and  others,  through  which  the  steam  is  admitted 
to  the  cylinders. 

Steam-pipe ;  the  one  which  leads  steam  from  a  boiler  to  the  steam-chest. 

Step  ;  a  cavity  in  a  piece  for  receiving  the  pivot  of  an  upright  shaft ;  or  the  end  of  any  upright  piece. 

Stiles  ;  the  flat  vertical  pieces  between  and  at  the  sides  of  the  panels  in  doors,  &c. 

Stock;  the  eye  with  handles  for  turning  it,  in  which  the  dies  for  the  cutting  of  screws  are  held. 

Stove-vp,  or  staved,  or  upset;  when  a  rod  of  iron  is  heated  at  one  end,  and  then  hammered  end- 
wise so  that  that  part  becomes  of  greater  diameter  or  stouter  than  the  remainder.  The  heads  of  bolts 
are  frequently  made  in  one  piece  with  the  shank  in  this  way  ;  and  the  screw  ends  of  long  ecrew-rods 
are  often  upset,  so  that  the  cutting  of  the  threads  of  the  screw  may  not  reduce  the  strength  of  the  bar. 

Strap ;  a  long  thin  narrow  piece  of  metal  bolted  to  two  bodies  to  hold  them  together.  A  strap- 
hinge  is  a  strap  fastened  to  a  shutter,  &c,  and  having  an  eye  or  a  pin  at  one  end  for  fitting  it  to  the 
other  part  of  the  hinge  which  is  attached  to  the  wall. 

Stratum ;  a  layer,  or  bed;  as  the  natural  ones  in  rocks,  Ac. 

Stretcher ;  a  brick,  or  a  block  of  masonry  laid  lengthwise  of  a  wall.  A  frame  for  stretching  any 
thing  upon. 

Stretcher-course ;  a  course  of  masonry  all  of  stretchers,  without  any  headers. 

Strike;  an  imaginary  horizontal  line  drawn  upon  the  inclined  face  of  a  stratum  of  rocks.  Thus, 
If  the  slates  or  shingles  on  a  roof  represent  inclined  strata  of  rocks,  then  either  the  ridge  or  the  eaves 
of  the  roof,  or  any  horizontal  line  between  them,  will  represent  their  strike.  The  inclination  is 
called  the  dip  of  the  strata;  and  the  strike  is  always  at  right  angles  to  it  by  compass. 

String  ;  variously  applied  to  longitudiual  pieces. 

String-board ;  the  boarding  (often  ornamented)  at  the  outer  ends  of  steps  in  staircases.  It  hides 
the  horses,  as  the  inclined  timbers  which  carry  the  steps  are  called. 

String-course ;  a  long  horizontal  course  of  brick  or  masonry  projecting  a  little  beyond  the  others; 
and  often  introduced  for  ornament. 

Stringer;  any  longitudinal  timber  or  beam,  &«. 


1036 


GLOSSARY   OF  TERMS. 


Strut ;  a  prop.    A  piece  that  sustains  compression,  whether  vertical  or  inclined. 

Strut-tie,  or  tie-strut ;  a  piece  adapted  to  sustain  both  tension  and  compression- 

Stub-end;  a  blunt  end. 

Stud;  a  short  stout  projecting  pin.     A  prop.     The  vertical  pieces  in  a  stud  partition. 

Stud-bolt.     See  Standing-bolt. 

Stuffing-box;  a  small  boxing  on  the  end  of  a  steam  cylinder,  and  surrounding  the  piston-rod  likt 
a  collar  ;  or  in  other  positions  where  a  rod  is  required  to  move  backward  and  forward,  or  to  revolve, 
in  an  opening  through  any  kind  of  partition,  without  allowing  the  escape  of  steam,  air,  or  water,  &c, 
as  the  ease  may  be.  The  box  is  filled  with  greased  hemp  or  other  packing,  which  is  kept  pressed  close 
around  the  moving  rod  by  means  of  a  top-piece  or  kind  of  cover  called  the  gland,  which  may  be 
screwed  down  more  or  less  tightly  upon  it  at  pleasure.     The  rod  passes  through  the  gland  ai*o. 

Sumpt,  or  stimp ;  a  draining  well  into  which  rain  or  other  water  may  be  led  by  little  ditches  from 
dififereuf  parts  of  a  work  to  which  it  would  do  injury. 

Surhuse  ;  the  inside  horizontal  mouldings  just  under  a  window-sill.  Also  those  around  the  top  of  a 
pedestal,  or  of  wainscoting,  «fcc. 

S'v  ige.  or  swedge;  a  kind  of  hammer,  on  the  face  of  which  is  a  semi-cylindrical,  or  other  shaped 
groove  or  indentation  ;  and  which,  being  held  upon  a  piece  of  hot  iron  and  struck  by  a  heavy  hammerv 
leaves  the  shape  of  the  indentation  upon  the  iron. 

Sivitch ;  the  movable  tongue  or  rail  by  which  a  train  is  directed  from  one  track  to  another. 

Swivels ;  devices  for  permitting  one  piece  to  turn  readily  in  various  .directions  upon  another,  with- 
out danger  of  entanglement  or  separation. 

Stfnclinal  axis ;  in  geology,  a  valley  axis,  or  one  toward  which  the  strata  of  rocks  slope  downward 
from  opposite  directions.     The  line  of  the  gutter  in  a  valley  roof  may  represent  such  an  axis. 

Ts;  pieces  of  metal  in  that  shape,  whether  to  serve  as  straps,  or  for  other  purposes.  So  also  with 
L's.  S's,  W's,  -|-*s,  &c.     See  figs  to  Welded  Iron  Tubes. 

Tackle;  a  combination  of  ropes  and  pulleys. 

Tnlus  ;  the  same  as  batter. 

T'lmp;  to  fill  up  with  sand  or  earth,  &c,  the  remainder  of  the  hole  in  which  the  powder  has  been 
poured  for  blasting  rock.     To  compact  earth  generally,  as  under  cross-ties,  &c. 

Tap;  a  kind  of  screw  made  of  hard  steel,  and  having  a  square  head  which  maybe  grasped  by  a 
wrench  for  turning  it  around,  and  thus  forcing  it  through  a  hole  around  the  inside  of  which  itcuts'au 
interior  screw.     To  strike  with  moderate  force.     To  make  an  opening  in  the  side  of  any  vessel. 

Tappet;  a  pin  or  short  arm  projecting  from  a  revolving  shaft;  or  from  an  alternating  bar,  and  in- 
tended to  come  into  contact  with,  or  tap,  something  at  each  revolution  or  stroke. 

Teeth ;  or  cogs  of  wheels. 

Temper ;  to  change  the  hardness  of  metals  by  first  heating,  and  then  plunging  them  into  water,  oil, 
<fec.     To  mix  mortar,  or  to  prepare  clay  for  bricks,  &c. 

Templet;  the  outline  of  a  moulding  or  other  article,  cut  out  of  sheet  metal  or  thin  wood,  to  serve 
as  a  pattern  for  stonecutters,  carpenters,  &c. 

Tenon ;  a  projecting  tongue  fitting  into  a  corresponding  cavity  called  a  mortise. 

Terra  cotta ;  baked  clay.     Brick  is  a  coarse  kind. 

Thimble;  an  iron  ring  with  its  outer  face  curved  into  a  continuous  groove.  A  rope  being  doubled 
around  this  and  tied,  the  thimble  acts  as  an  eye  for  it,  and  prevents  that  part  of  the  rope  from  wear- 
ing.   Also,  a  short  piece  of  tube  slid  over  another  piece,  or  over  a  rod,  &c,  to  strengthen  a  joint,  &c. 

Thread ;  the  continuous  spiral  projection  or  worm  of  a  screw. 

Through-stone;  a  stone  that  extends  entirely  through  a  wall. 

Throw;  the  radius,  or  distance  to  which  a  crank  "  throws  out "  its  arm.  Applies  in  the  same  way 
to  lathes.  Some  use  it  to  express  the  diameter  instead  of  the  radius.  To  avoid  mistakes,  the  terms 
"  single  "  and  "  double  "  throw  might  be  used. 

Tie  ;  any  piece  that  sustains  tension  or  pull. 

Tie-strut ;  a  piece  adapted  to  sustain  either  tension  or  compression. 

Tightning-screw.     See  Set- screw. 

Tire ;  the  iron  ring  placed  around  the  outer  circumference  of  the  felloe  of  a  wheel. 

Tongue;  a  long  slightly  projecting  strip  to  be  inserted  into  a  corresponding  groove,  as  in  tongued 
and  grooved  floors. 

Tooling  ;  dressing  stone  by  means  of  a  tool  and  mallet;  the  tool  being  a  chisel  with  a  cutting  edge 
•f  I  to  2  inches  wide.     Tooling  is  generally  done  in  parallel  stripes  across  the  stone. 

Torus;  a  projecting  semi-circular,  or  semi-elliptic  moulding;  often  used  in  the  bases  of  columns. 
It  is  the  reverse  of  a  scotia. 

Trailing -wheels  ;  in  a  locomotive,  those  sometimes  placed  behind  the  driving-wheels. 

Train;  a  number  of  cog-wheels  working  into  each  other. 


GLOSSARY   OF   TERMS.  1037 


Transom ;  a  beam  across  the  opening  for  a  door,  &c.  Also,  a  horizontal  piece  dividing  a  high 
window  into  two  stories,  &c,  &c.     Also,  an  opening  above  a  door,  for  ventilation  or  light. 

Tread ;  the  horizontal  part  of  a  step. 

Treadle;  a  kind  of  foot-lever,  tor  turning  a  lathe,  grindstone,  &c,  by  the  foot. 

Treenail;  a  long  wooden  pin. 

Trimmer;  a  short  cross-timber  framed  into  two  joists  so  as  to  sustain  the  ends  of  intermediate 
{oists,  to  prevent  the  latter  from  entering  a  chimue}  -tiue.  or  interfering  with  a  window,  &c. 

Trip-hammer,  OT  tilt-hammer ;  a  large  hammer  worked  by  camb  machinery,  and  used  for  heavy 
Iron  work,  especially  for  hammering  irregular  masses  into  the  shape  of  bars,  &c. 

Truck;  a  kind  of  small  wagon  consisting  of  a  platform  on  two  or  more  low  wheels.  Also,  those 
frames  and  wheels  usually  placed  under  railroad  cars  and  engines,  and  which,  by  means  of  a  pintle 
connecting  the  two,  allow   them  to  vibrate  or  move  laterally  to  some  extent  independently  of  each  other. 

Truttdle,  lantern-wheel,  or  w allow er  ;  used  instead  of  a  cog-wheel,  and  consisting  of  two  parallel 
circular  pieces  some  distance  apart,  and  united  by  a  central  axis,  and  by  cylindrical  rods  placed 
around  and  parallel  to  the  axis,  to  serve  instead  of  cogs  or  teeth. 

Trunk;  a  long  wooden  boxing  forming  a  water  channel. 

Trunnions  ;  cylindrical  projections,  as  at  the  sides  of  a  cannon,  forming  as  it  were  an  interrupted 
axle  or  shaft  for  supporting  the  cannon  on  its  carriage;  and  allowing  it  to  revolve  vertically  through 
some  distance. 

Tumbler;  a  kind  of  spring  catch,  which  at  the  proper  moment  falls  or  tumbles  into  a  notch  or 
hole  prepared  for  it  in  a  piece ;  thus  holding  the  piece  in  position  until  the  tumbler  is  iifted  out  of  the 
notch. 

Tumhlinghay ;  see  "  waste- weir." 

Tumbling-shaft ;  in  locomotives,  a  shaft  used  in  the  "  link  motion.'* 

Turntable ;  the  well-known  arrangement  for  turning  loeomotires  at  rest. 

Undermine ;  to  excavate  beneath  anything. 

Underpin;  to  add  to  the  height  of  a  wall  already  constructed,  by  excavating  and  building  beneath 
it.     Also,  to  introduce  additional  support  of  any  kind  beneath  anything  already  completed. 

Upset.    See  Stove-up. 

Valves;  various  devices  for  permitting  or  stopping  at  pleasure  the  flow  of  water,  steam,  gas,  &c. 
A  SAFETY  VALVE  is  oue  SO  balanced  as  to  open  of  itself  when  the  pressure  becomes  too  great  for 
safety.  A  slide  valve  is  one  that  slides  backward  and  forward  over  the  opening  through  which  the 
flow  takes  place.  A  ball  valve,  or  spherical  valve,  is  a  sphere,  which  in  any  position  fits  the  open- 
ing. When  the  pressure  below  it  raises  it  ofif  from  its  seat,  it  is  prevented  from  rolling  away  by 
•»»,eaus  of  a  kind  of  open  caging  which  surrounds  it.  A  conical  or  puppet  valve  is  a  horizontal  slice 
tyf  a  cone,  which  fits  into  a  corresponding  conical  seat  made  in  the  opening.  In  rising  and  falling  it 
is  kept  in  position  by  a  vertical  valve-stem  or  spindle,  which  passes  through  its  center,  and  which 
plays  through  guide-holes  in  bridge-pieces  placed  above  and  below  the  valve.  A  tkap.  clack,  flap, 
or  DOOR  VALVE,  is  a  plate  with  hinges  like  a  door.  When  two  such  valves  are  used,  with  their  hinged 
edges  adjacent  to  each  other,  so  that  in  opening  and  shutting  they  flap  like  the  wings  of  a  butterfly, 
they  constitute  a  butterfly  valve.  A  throttle  valve  is  one  which  when  closed  forms  a  partition 
across  a  pipe ;  and  opens  by  partially  revolving  upon  an  axis  placed  along  its  diameter.  A  rotary 
VALVE  works  like  a  common  stopcock.  A  sniftins  valve  is  one  which  lets  out  steam  under  water  ;  and 
is  so  called  from  the  snifting  noise  thereby  produced.  The  port  valve  is  the  sliding  one  which  ad- 
mits steam  from  the  steam-chest  into  the  cylinders.  A  double  seat,  or  double  beat  valve  is  a  pe- 
culiar one  wich  two  seats,  one  above  the  other  :  and  so  arranged  that  the  pressure  of  steam  or  water 
against  it  when  shut,  does  not  oppose  its  being  opened.  A  cup  valve  is  in  shape  of  an  inverted 
cylindrical  cup,  with  a  length  somewhat  greater  than  its  diameter.  Its  lower  or  open  edge  is  ground 
to  fit  the  seat  over  which  it  rests.  As  this  cup  rises  and  falls,  it  is  kept  in  place  by  a  cylindrical 
caging  closed  at  top,  and  having  for  its  sides  four  or  more  vertical  pieces,  against  the  inner  sides  of 
which  the  sides  of  the  cup  play.  A  check  valve  is  any  kind  so  placed  as  to  check  or  prevent  the 
return  of  the  fluid  after  its  passage  through  the  valve  into  the  pipe  or  vessel  beyond  it. 

Vault ;  an  arch  long  in  comparison  with  its  span.     The  space  covered  by  sucli  an  arch. 

Veneer;  a  very  thin  sheet  of  ornamental  wood  glued  over  a  more  common  variety. 

Wainscot ;  a  wooden  facing  to  walls  in  rooms,  instead  of  plaster,  or  over  a  facing  of  plaster ;  usually 
&ot  more  than  3  or  4  feet  high  above  the  floor. 

Wales ;  long  longitudinal  timbers  in  the  sides  of  a  ship,  coffer-dam,  caisson,  &c. 

Wallow;  a  water-wheel,  &c,  is  said  to  wallow  when  it  does  not  revolve  evenly  on  its  journals. 

Wallotver.     See  Trundle. 

Wall-plate,  or  raising-plate ;  a  timber  laid  along  the  tops  of  walls  for  the  roof  trusses  or  rafters  to 
rest  on,  so  as  to  distribute  their  weight  more  equally  upon  the  wall. 

Warped;  twisted,  as  a  board,  or  the  face  of  a  stone,  &c,  which  is  not  perfectly  flat.  To  warp;  to 
haul  a  vessel  ahead  by  means  of  an  anchor  dropped  some  distance  ahead.  To  flood  an  extent  of 
ground  with  water  for  a  short  time  to  increase  its  fertility. 

Washboards ;  boards  nailed  around  the  walls  of  rooms  at  the  floor,  so  as  to  prevent  injury  to  the 
plaster  when  washing  the  floors. 

Washers ;  broad  pieces  of  metal  surrounding  a  bolt  and  placed  between  the  faces  of  the  timber 
through  which  the  bolt  passes,  and  the  head  and  nut  of  the  bolt,  so  as  to  distribute  the  pressure  over 
a  larger  surface,  and  prevent  the  timber  from  being  crushed  when  the  bolt  is  tightly  screwed  up. 

Waste-weir ;  an  overfall  provided  along  a  canal,  &c,  at  which  the  water  may  discharge  itself  in 
case  of  becoming  too  high  by  rain,  &c.     Sometimes  called  a  tumbling-bay. 

Watch-tackle ;  ropes  running  in  different  directions  from  a  boat,  and  ased  is  bringing  it  into  a 
desired  position. 


1038 


GLOSSARY   OF   TERMS. 


Watmr-shed  ;  the  sloping  ground  from  which  rain-water  descends  into  a  stream. 

Water-table ;  a  slight  projection  of  the  lower  masonry  or  brickwork  on  the  outside  of  a  wall,  and 
reaching  to  a  few  feet  above  the  ground  surface,  as  a  partial  protection  against  rain,  or  as  ornamenU 

Ways ;  the  inclined  timbers  along  which  a  vessel  glides  when  being  launched. 

Weather-boards  :  boards  used  instead  of  bricks  or  masonry  for  the  outsides  of  a  building,  or  bridge, 
&c.  They  are  nailed  to  vertical  and  inclined  indoor  timbers ;  and  may  be  either  vertical  or  hor. 
When  hor,  they  are  so  placed  that  the  lower  edge  of  one  overlaps  the  upper  edge  of  the  one  below. 
Whea  vert,  their  edges  should  be  tongued  and  grooved  ;  and  narrow  slips  be  nailed  over  the  vert  joints, 
to  keep  out  rain,  A;c. 

Weir,  or  wier;  a  dam,  or  an  overfall. 

Weld;  to  join  two  pieces  of  metal  together  by  first  softening  them  by  heat,  and  then  hammering 
them  in  contact  with  each  other.    la  this  operation  fluxes  are  used. 

TTeJt;  see  "  butt-joint." 

Wharf;  a  level  space  upon  which  ressels  lying  along  its  sides  can  discharge  their  cargoes  ;  or  from 
which  they  can  receive  them. 

Wheel-base ;  the  distance  from  center  to  center  from  the  extreme  front  wheels,  to  the  extreme  hind 
ones  in  a  locomotive,  car,  <fec. 

Wicket ;  a  small  door  or  gate  made  In  a  larger  one ;  as  the  shuttle  or  valve  in  a  lock-gate,  for  letting 
out  the  water. 

Winch;  a  handle  bent  at  right  angles,  and  used  for  turning  an  axis  ;  that  of  a  common  grindstone. 

Wind.     See  Out  of  wind. 

Winders;  those  steps  (often  triangular)  in  a  staircase  by  which  we  wind,  or  turn  angles. 

Windlass ;  the  wheel  and  axle,  or  winch  and  drum,  as  often  used  in  common  wells.  Also,  a  hori* 
Bontal  shaft  on  shipboard,  by  which  the  anchor  is  raised ;  the  windlass  being  revolved  by  means  of 
wooden  levers  called  handspikes. 

Wing-dam ;  a  projection  carried  out  part  way  across  a  shallow  stream,  so  as  to  force  all  the  water 
to  flow  deeper  through  the  channel  thus  contracted. 

Wings  ;  applied  in  many  ways  to  projections.  The  flanges  which  radiate  out  from  a  gudgeon  ;  and 
by  which  it  is  fastened  to  the  shaft.  Small  buildings  projecting  from  a  main  one.  The  wings  or 
flaring  wing-walls  of  a  culvert  or  bridge. 

Wing-walls ;  the  retaining-walls  which  flare  out  from  the  ends  of  bridges,  culverts,  &c. 

Wiper.    See  Camb. 

Working-beam,  or  walking-beam ;  a  beam  vibrating  vertically  on  a  rock-shaft  at  its  center,  as  seeu 
in  some  steam-engines ;  one  end  of  it  having  a  connection  with  the  piston-rod ;  and  the  other  end  with 
a  crank,  or  with  a  pump-rod.  &c. 

Worm;  the  so-called  endless  screw,  which  by  revolving  without  advancing  gives  motion  to  a  cog- 
wheel (worm-wheel),  the  teeth  of  which  catch  in  the  thread  of  the  screw. 

Wrench;  a  long  handle- having  at  one  end  an  eye  or  jaw  which  may  catch  hold  of  anything  to  b« 
twisted  or  turned  around,  as  a  screw-nut,  &c.  When  it  has  a  jaw  which  by  means  of  a  screw  is 
•d«ptable  to  nuts,  &c,  of  differeBt  sizes,  it  is  a  monkey- wreneh,  or  screw- wr«Boh. 


INDEX. 


The  numbers  refer  to  the  pag^es.  In  the  alphabetical  arrangement, 
minor  ^rords,  as  "and,"  "between,"  "in,"  "on,"  "through,"  etc.,  are 
neglected.  See  also  Glossary,  pp.  1025,  etc.,  and  Table  of  Contents,  pp. 
XXV,  etc. 


Abacus— Amortization. 


Abacus,  defined,  1025. 
Abrasion 

of  cements,  937. 

by  streams,  577,  582. 
Abscissa,  defined,  1025. 
Absorbent  bodies, 

specific  gravity  of — ,  211. 
Absorbents  for  nitro-glycerine,  949, 

951. 
Absorption 

by  bricks,  927. 

by  earth,  etc.,  329. 
Abutment,   Abutments, 

of  arch,  617. 

batter,  619. 

courses  in — ,  inclination  of — ,620. 

of  dams,  645. 

foundations  for — ,  582. 

masonry,  623. 

piers,    619. 

to  proportion — ,   617,  618. 
Accelerated  tests  for   cements,  941. 
Acceleration,  334. 

of  gravity.  335,  336,  348,  350,  539. 
equivalents  of — ,  250. 

on  inclined  planes,  349. 

units  of — ,  conversion  of — ,  250. 
Acid  fumes,  effect  of- —  on  roofs,  970. 
Acre,  Acres, 

area  of—,  222. 

equivalents  of — ,  233. 

-foot,  equivalents  of — ,   235. 

required  for  railroads,  254. 
Action, 

line  of—,  359. 

and  reaction,  333. 
Addition  of  fractions,  36. 
Adhesion 

of  cement,  934. 

of  glue,  922,   1030. 

of  locomotives,  413,  860. 

of  mortar,  926. 

of  nails  and  spikes,  818. 
Adit,  defined,  1025. 
Adjustable  counters,  721. 


Adjustment.      See    the   several   in- 
struments. 
Adjutages,  flow  through — ,  540. 
Admiralty  knot,  220. 
Age,  effect  of —  on  cements,  932. 
Agonic  line,  301. 
Air,  320. 

buoyancy,  513. 
chambers,  663. 
compressed—,  320,  597,  681. 
breathing—,  320,  597,  etc. 
in  diving  bells,  321. 
in  foundations,  596. 
in  rock  drills,   681. 
compressors,  681. 

manufacturers,  991. 
density.  320. 
locks,  597. 
pressure,   320,   502. 

barometer,  levelling  by — ,  312. 
of  compressed — ,  320,  597,  etc. 
on  water  surface,  502. 
in  siphons,  521. 
slacking,   925. 
in  tunnels,  812. 
valves,   662. 
ventilation,      quantity      required 

for—    320. 
vessel,  663. 
volumes    of    unit 
conversion  of — , 
weight,  320. 
weights    of    unit    volumes 

conversion  of — ,  242. 
wind,   321. 
Alcohol,  weight  of—,  212. 
Alioth,  285. 
Allardyce  process,  955. 
Alligation,  40. 
Alphabet,  Greek—,  34. 
Alternating  stresses,  465.  761. 
Alternation  of  ratios,  38. 
Altitude.     See  Height. 

of  the  pole,  284. 
Alumina,  in  cement,  930. 
Aluminum 

oxide,  in  cement,  930. 
weight,  212. 
Amortization,  43. 


weights    of — , 
242. 

of—. 


1039 


1040 


INDEX. 


Amount— Arch. 


Amount,  in  interest,  etc.,  41. 
Anchorage,  Anchorages. 

of  suspension  bridges,  770. 

wind —  in  bridges,  759. 
Angle,  Angles,  92. 

arc,  angle    subtended  by — ,  181, 
780,  etc. 

blocks,  736. 

in  built  sections,  723. 

chords  subtending — ,  143,  780,  etc. 

circular  measure  of — ,  34. 

complement  and  supplement,  94. 

co-secants  of — ,  97. 

cosines  of — ,  97. 

co-versed  sines  of — ,  97. 

deflection—,  780,  784-789,  840. 

degrees  in — ,  decimals  of — ,  95. 

of  direction,   765. 

of  friction,  409. 
in  arches,  432. 
in  dams,  433. 

frog—,   835,   839. 

hour—,  285. 

iron,  497,  896-899,  912. 

of  maximum  pressure,  607. 

to  measure 

with  the  hand,  etc.,  96. 
with  the  sextant,  152. 
with  the  tape  line,  152. 
with  the  two-foot  rule,  etc.,  96. 

minutes  and  seconds  in  decimals 
of  a  degree,  95. 

plates  for  rail  joints,  820-823. 

in  polygons,   148. 

rule,  2  ft. — ,  to  measure —  by,  96. 

secant  of^-,  97. 

seconds  in  decimals  of  a  degree, 
95. 

sines  of — ,  97. 
table,  98. 

of  slope,  255-257. 

on  sloping  ground,  151. 

steel,  896,  898. 
test,  753. 

subtended  by  arc,  181,    780,  etc. 

supplement  and  complement,  94. 

switch—,  827. 

symbol  for — ,  33. 

tangent  of — ,  97. 

tangential — ,  tables,  784-786. 

in  triangles,    148. 

versed  sines  of — ,  97. 
Angular  velocity,  351. 
Animal  power,  685. 
Anneal,  defined,  1025. 
Annual 

earnings  of  railroads,  867,  etc. 

expenses  of  railroads,  867,  etc. 

magnetic  variations,  301. 
Annuity,  Annuities,  43. 

equations,  44. 

required  to  redeem  $1000,  46. 
Antecedent,  38. 
Anthracite, 

heat  from — ,  317. 

space  occupied  by — ,  215. 

weight,  212,  215. 


Anti-bursting  device,   665. 
Anticlinal  axis,  1025. 
Anti-component,    362. 
Anti-friction  rollers,  417,  725,  751, 

846. 
Antilogarithms,    71. 
Antimony, 

strength,  920. 
weight,  212. 
Anti-resultant,  362. 
Apertures, 

contiguous — ,  flow  through — ,  542. 
shape  of — ,  effect  on  flow,  541. 
in  thin  partition,  541. 
Apex,  apex  distance,  780. 
Apothecaries' 

measure,  223,  224. 
weight,  220. 
Apparent  solar  time,  265. 
Application  of  force,  332,  347. 

point  of—,  333,  359. 
Applied  and  imparted  forces,  372. 
Approach,  velocity  of — ,  556. 
Aqueduct,  Aqueducts, 
flow  in—,  560. 
Kutter's  formula,  563. 
Arc,  Arcs, 

circular — ,   179. 

angles   subtended    by — ?     181, 

780,  etc. 
center  of  gravity,  391. 
chords,    143,   179,  780,  etc. 
co-secant,  97. 
cosine,  97. 
cosine,  table,  98. 
co-versed  sine,  97. 
graduated — ,  292. 
large — ,  to  draw — ,  181. 
ordinates,  180,  784,  817,  840. 
radii,    180,  781,  etc. 
rise,  180. 
secant,  97. 
sine,  97. 

table,  98. 
tables,  183,  185. 
tangent,  97. 
table,  98. 
time  equivalents  of — ,  265,  290. 
versed  sine,  97. 
elliptic—,  189. 
parabolic — ,    192. 
semi-elliptic — ,   189. 
Arch,  Arches,  424,  430,  etc.,  613,  740, 
abutments,   617. 
angle  of  friction  in — ,  432. 
brick—,  629,  632. 
bridges,   613. 
centers  for — ,  631. 
concrete — ,  615,  616. 
design  of—,  431,  432.  613,  etc. 
elliptic—,  616. 

joint  in — ,  to  draw — ,  190. 
keystone,   613,   615. 
line  of  pressure,  430. 
line  of  resistance,  430. 
line  of  thrust,  430. 
mechanics  of — ,  430,  432. 


I 


INDEX. 


1041 


Arch— Beam. 


Arch,  Arches — continued. 

moments  in — ,  424. 

practical  considerations,  432. 

pressure  in — ,  430,   614. 

pressure,  line  of — ,  430. 

radius,  to  find — ,  614. 

resistance  line,  430. 

roofs,  740,  742. 

rubble—,  616. 

settlement  of, —  432. 

statics  of—,  430,  432. 

stones,   613. 

chamfering,  634. 

pressure  in — ,  430,  etc.,  614. 

pressure  of — ,  on  centers,  633. 

theory  of—,  430,  432. 

thrust  line,  430,  432. 
Archimedes  screw,  687. 
Architrave,  defined,  1025. 
Area,    Areas, 

of  a  circle,  to  find — ,  161. 

of  a  circle,  table,    163-178. 

contraction  of — in  iron  and  steel, 
752,  754,  873. 

crippling — ,  of  rivet,  775. 

of  pipes,  526. 

reduction  of—,  752,  754,  873. 

of  sections  of   beams,   468,   892, 
etc. 

of  surfaces.     See  the  surface  in 
question. 

unit — ,  equivalents  of — ,  233. 
Arithmetic,  35. 

Arithmetical  complements,  71. 
Arithmetical  progression,  39. 
Arm,  lever — ,  360. 
Arris,  defined,  1025. 
Arroba,   227. 
Artesian  wells,  671. 
Artificial 

horizon,  298. 

stone,  concrete,  943. 
Ascent.      See  Grade,  Height,  Slope. 

effect    of —  on   power  of    horses, 
683. 

effect  of —  on  power  of  locomo- 
tives,  860. 
Ash  wood,  strength,  459,  476,  957, 
958. 

weight,  212. 
Ashlar  masonry,  cost,    601,  602. 
Asphaltum,  weight,  212. 
Atlas  powder,  951. 
Atmosphere,  See  Air,  320. 

buoyancy,  513. 

(unit    of    pressure),    equivalents 
of—,  240. 

weight,  212,  320. 
Augers  for  earth  and  sand,   670. 
Avoirdupois  weight,  220. 
Axis,    See    the    given     surface     or 
solid. 

of  buoyancy,  514. 

of  equilibrium,  514. 

of  flotation,  514. 

neutral — ,   466. 

of  symmetry,  514. 


Axle,  friction,  416,  417. 
Azimuth,  Azimuths,  284-290. 


B. 

Back,  Backs, 

of  arch,  defined,  613. 

of  retaining  walls,  603. 
Backing  of  walls,  603. 
Backstays   for   suspension   bridges, 

766. 
Bag  scoop  or  spoon,  581. 
Baggage  cars,  865. 
Bailing  by  bucket,  day's  work,  686. 
Balk,  defined,  1025. 
Ballast, 

for  railroads,  815. 
cost  of — ,  855. 
Balloon,  principle  of — ,  513. 
Balls,  weight,  874,  877,  879,  918. 
Baltimore  truss,  694. 
Bar,  Bars, 

in  built-up  sections,   723. 

dredging,  580. 

iron—,  weight,  877,  878. 
Barbed  fence,  854. 
Bargeboards,  defined,  1025. 
Barometer,  312,  320. 
Barrel,  contents,  223. 
Barrow.    See  Wheelbarrow. 
Barschall  process,  955. 
Bascule  or  lift  bridges,   697. 
Base,  Bases, 

of  logarithmic  systems,  72. 

wheel —  of  locomotives,  etc.,  856. 
Batten  plates,   724. 
Batter, 

of  abutments,  619. 

defined,    1025. 

of  retaining  walls,  603,   etc. 
Bazin's  formula,  552. 
Beam,   Beams, 

breaking  loads,  coefficients,  476. 

channel—,  497,  894,  912. 

coefficients    for    breaking    loads, 
476. 

concrete — ,  strength  of — ,   945. 

continuous — ,  489. 

curved — ,  446. 

deflections,  481,  959. 
loads  for  given — ,  480. 
under  sudden  loading,  460,  959. 

elastic  limit,  482. 

elasticity,  modulus  of — >  457. 

end  reactions,  439.* 

equilibrium  of — ,  437. 

floor—,  720,  749. 

forces  acting  upon — ,  437. 

granite — ,    924. 

I —  and  channel — ,  892,  etc.,  912, 
in  fire-proof  floors,  894. 
as   pillars,    497,    902,  912, 
separators  for — ,  900. 
tables,  892.  etc. 

inclined — ,  445. 


1042 


INDEX. 


Beam— Boxed  Timber. 


Beam,  Beams — continued. 

iron — ,  safe  loads  and  deflections, 

960,  962. 
limit  of  elasticity,  482. 
loads,  437,  etc.,  466,  etc.,  476,  760, 
762,  892,  etc. 
for  given  deflection,  480. 
within  limit  of  elasticity,  482. 
suddenly  applied — ,  460,  959. 
mechanics   of — ,    437,  etc.,   466, 
etc. 
<  modulus    of  elasticity,  457. 
moments,  440,  443,  445. 
reactions,  439. 
rolled — .        See      Beams,       I — ; 

Beams,  Channel — ;  etc. 
shear,  446. 
steel  and  iron — ,  loads,  892,  etc., 

960,  962. 
stone—,  476,  924. 
strength.     See  Beams,  Loads, 
suddenly  loaded,  460,  959. 
timber — ,   See  Beams,  Wooden — . 
and  trusses,  comparison,  689. 
of  uniform  strength,  486. 
wooden—,  760,  762,  764. 
deflections,  table,  959. 
loads,  tables,  959,  962. 
Bearing,  Bearings, 
of  piles,  590. 
power  of  soils,  583,  593. 
and  reverse  bearing,  277. 
stresses,  permissible — .  762. 
in  trusses,  721,  725,  750,  751. 
Beats,  of  clocks  and  watches,  266. 
Beaume  hydrometer,  211. 
Bed  plates,  721,  750. 
Beech-wood,  strength,  459,  476,  957, 

958. 
Beetle,  defined,  1025. 
Bell, 

diving — ,  pressure  in — ,  321,  597. 
joint  for  pipes,  660. 
Belting,  cost  and  mfrs.,  987. 
Belts,  leather — ,  strength,  922. 
Bench  mark,  1025. 
Bending 

of    beams.      See  Beams,    Deflec- 
tions, 
and  compression,  combined,  724. 
moments,  466,   etc. 
stresses  in  bridge  members,  per- 
missible— ,  762. 
tests,  871. 

bridge  steel,   752. 
iron  and  s^eel,  873. 
steel  castings,  754. 
Bends  in  water  pipes,  537. 
Bents  in  trestles,  814. 
Berm,  defined,   1025. 
Beton,  defined,  1025. 

See  Concrete. 
Beveled  joints  for  rails,  819. 
Bibliography,  1008. 
Bilge,  defined,  1025. 
Birch,  strength,  459,  476,  957,  958. 
Birmingham  gauges.  887.  890. 


Bismuth, 

strength,   920. 

weight,   212. 
Bitumen,  weight,  212. 
Bituminous  coal, 

space  occupied  by — ,  215. 

weight  of—,   212,   222. 
Black-line  print.s,  982. 
Blasting,  600,  953. 

apparatus,  952. 
Bled  timber,  strength,  957. 
Blocks,  angle—,  736. 
Bloom  ton,  216. 
Blue-line  prints,  982. 
Blue  prints,   979. 
Board  measure,  table,  269. 
Boasting,  defined,   1025. 
Boat,  canal — ,  684. 
Body,    Bodies.     See    the    body   in 
question. 

defined,  330. 

expansion  of —  by  heat,  317. 

falling—,    348,   539. 

floating — ,  513. 

rigid—,  force  in—,  330,  358. 
Boiler,  Boilers, 

cost  and  mfrs.,  988. 

incrustation,  327. 

iron—    872, 

thickness,  511. 

tubes,  882. 
Boiling  point,  326. 

levelmg  by — ,  314. 
Boiling  tests  for  cement,  941. 
Bollman  truss,  695. 
Bolster,  defined,  1026. 
Bolts,  883,  etc. 

cost  and  mfrs.,  986. 

expansion — ,  884. 

iron—,  table,  886. 

strength    and     weight,     table, 
886. 

stresses,  permissible — ,  762. 
Bonnet,  defined,  1026. 
Bonzano  rail-joint,   823. 
Books,  1008. 
Boring, 

artesian  wells,  671. 

augers  for  earth — ,  670,  671. 

test—,  582,  670. 

wells,  671. 
Borrow  pit, 

defined,    1026. 

to  measure — ,  195,   196. 
Boss,  defined,  1026. 
Bottom, 

heading,  812. 

of  stream,   scour  on — ,  577. 

velocity,  560. 
Bowstring, 

centers,  637,  638. 

truss,   695,  699. 
Box 

cars,  865. 

drains,   627. 

sextant,  297. 
Boxed  timber,  strength,  957. 


INDEX. 


1043 


Bracing:— Camel. 


Bracing. 

in  bridges,  691,  710,  748,  749. 

counter — ,  See  Counter-bracing. 

cross — ,  691. 

for  dams,  502. 
Brad  spikes,  818. 
Brake  friction,  412. 
Branches  in  pipes,  661. 
Brass, 

balls,  weight,  9i8. 

ductility,  etc.,  459. 

effect  of  mortar,  etc.,  on — ,  936, 

effect  of  water  on — ,  327. 

expansion,  by  heat,  317. 

friction,  411,  415. 

pipes  and  tubes,  seamless — ,  919. 

strength,  476,  920,  921. 

tubes,  seamless — ,  919. 

weight,  212,  875-878,  887. 

wire,  887. 
Brasses,  defined,  1026. 
Braze,  defined,  1026. 
Breast-wall,  defined,  1026. 
Breathing, 

air  consumed  in — ,  320. 

in  diving-bells,  320. 
Brick,  Bricks,  926,  927. 

arches,   629,   632.  ^ 

cost  and  mfrs.,  985. 

cylinders,  sinking  of — ,  599. 

dust,  925. 

friction,  4JL1. 

incrustations,  929,  936. 

laying,  927. 

strength,  476,  922,  923. 

weight,  212,   213. 

work,  927. 

mortar  required  for — ,  931. 
weight  of — ,  213. 
Bridge,    Bridges.      See    also    Arch, 
Beam,    Girder,    Trestle,    Truss, 
etc. 

arch—,   613. 

brick—,  629,  632. 

Brooklyn — ,  foundations,  598. 

camber,  726,  746. 

centers,    631. 

clearance,    746. 

combination — ,  specifications,  763. 

cross-section,  746. 

design,  720,   745. 

electric   railway — ,  specifications, 
745. 

erection  of—,  743,  763. 

friction  rollers,  725,  751. 

gage  on — ,  746. 

headway  on — ,  746. 

highway — ,  specifications,  745. 

joints  (connections),  774. 

manufacturers,  988. 

painting,  763,  764. 

protection,  763,  764. 

railroad — ,  specifications,  745. 

roadways,  drainage,  628. 

specifications,  digest  of — ,  745. 
stone — ,  613.     See  also  Arch. 

stone-^,  centers,  631. 


Bridge,  Bridges — continued. 

suspension — ,  765. 

cables  of—  765,  891,   976,  977. 

test  of  completed — ,  753. 

trusses,  689. 

weight  of—,  731,  738. 

Wissahickon — ,  Phila.,  640. 

wooden — ,     specifications    for — i 
763. 
Briggs  logarithms,  70,  78,  80,  etc. 
Briquettes,  cement — ,  938,  etc. 
British 

Imperial  measure,  223,  224. 

rod  of  brickwork,   222,   928. 
Broken 

bubble  tube,  to  replace — ,  296. 

cross-hairs,  to  replace — ,  296. 

joints,  819. 

stone.     See  also  Rubble,  583. 
for  concrete,  943,  945. 
foundations,  583. 
voids  in—,  688,  943. 
Bronze, 

phosphor —  wire,  strength,  920. 

weight,  212. 
Brooklyn  bridge,  foundations,  598. 
Bubble  tube,  to  replace — ,  296. 
Buckle  plates,  750,  885. 
Builder's  level,  to  adjust — ,  311. 
Building,  Buildings, 

Specifications  for — ,  764. 
Built  beams,  479,  734. 
Built  sections,  truss  members,  722. 
Bulk,  increase  of — ,  broken  stone, 

etc.,  688,  943. 
Buoyancy, 

of  air,  513. 

of  liquids,  210,  513,  515. 
Burkli-Ziegler  formula,  575. 
Burned  clay,  925. 
Burnettizing,  955. 
Burr  truss,  695. 
Bursting, 

anti —  device,  665. 

•of  pipes,  513,  663,  665,   668. 
Bush,  defined,   1026. 
Bushel,  equivalents  of — ,  234. 

volume  of—,  223. 
Business  directory,  983,  996. 
Butt-joint,  773. 
Buttresses,  612. 


Cable,  Cables, 

number  of  wires  in — ,  891. 

stays,  766. 
Caisson,  585. 

Brooklyn  bridge,  598. 

work  in — .     See  Diving-bell,  321; 
Calcium  carbonate,  in  cement,  930. 
Calcium  oxide,  in  cement,  930, 
Calking,  660. 
Camber,  696,  726.  746. 
Camel,  defined.  1026. 


1044 


INDEX. 


Canal— Cliuek. 


Canal,  Canals, 

boats,  684. 

flow  in — ,  560. 

Kutter's  formula,  523,  563. 

leakage  from—,  329,  561. 

traction  on — ,  683. 
Cantara,  227. 

Cantilever,    Cantilevers.      See    also 
Beams. 

bridges,   696. 

end  reactions,  439. 

equilibrium  of — ,  437. 

forces  acting  upon — ,  437. 

moments,  440,  442,  445. 

reactions,  439. 

of  uniform  strength,  486. 
Cap,   for  blasting,   952. 
Capitalization,  41,  44. 
Car,  Cars,  865. 

derrick—,   808. 

earthwork,  807. 

friction,  417. 

mfrs.,  990. 

pressed  steel — ,  865. 

resistance,  417. 

wrecking,  808. 
Carat,  219,  220. 
Carbon, 

dioxide,  in  cement,  930. 

in  steel,  872. 

in  steel  castings,  754. 
Carbonic  acid,  in  cement,  930. 
Carnegie  beams,  channels,  etc.,  892, 

etc. 
Cart,  Carts, 

earthwork,  800,  802,  805. 

excavating     (wheeled     scrapers), 
805. 

road,  repairs  of — ,  801. 

rock,  removing,  810. 

traction,  683. 
Cartridge, 

dynamite — ,   950. 

rack-a-rock,  951. 
Cassiopeia,  285. 
Castellano,  227. 
Castelli's  quadrant,  561. 
Casting,  Castings, 

safety—,  824,  833. 

steel — ,  requirements,  754,  872. 

weight  of — ,   by  size  of  pattern, 
875. 
Cast-iron.     See  Iron,  Cast — . 
Cattle  cars,  865. 
Cedar,  strength,  476,  957,  958. 
Cement,  930-942. 

brick-dust—,  925. 

concrete,   943. 

cost  and  mfrs.,  985. 

and  iron  pipes,  657. 

for  leaks,  971,  973. 

moisture,  effect  on — ,  929. 

slag—,  940. 

strength,  923,  932,   etc. 

weight,  212,  934,  etc. 
Center,  Centers, 

for  arches,  631. 


Center,  Centers — continued. 

of  buoyancy,  514. 

of  circle,  to  find — ,  161. 

for  fire-proof  floors,  895. 

of  force,  399. 

of  gravity,  386. 

of  gyration,  496. 

of  moments,  361. 

of  oscillation,  351. 

of  percussion,  351. 

of  pressure,    399,  501,   506,   514. 
Centigrade  thermometer,  318. 
Centigram,  equivalents  of — ,  236. 
Centiliter,   235. 
Centimeter, 

centigram,  etc.,  definitions,  231. 

cubic — ,  weights,  212. 

equivalents  of — ,  233. 
Central  forces,   354. 
Centre.     See  Center. 
Centrifugal  force,  354,  711,  758. 
Centripetal  force,  354. 
Chain,  Chains, 

cost  and  mfrs.,  987. 

equivalents  of — ,  232. 

iron — ,  915 

loaded — ,  428. 

pump,  687. 

riveting,  774. 

strength  of — ,  915. 

surveying,  274,  282. 

of  suspension  bridges^  765. 

weight  of — ,  915. 
Chaining,  282. 
Chalk, 

strength,   923. 

weight,  212. 
Chamfer,  defined,  1027. 
Chamfering  arch  stones,  634. 
Channel,  Channels, 

flow  in—,  523,  560. 

Kutter's  formula,  523,  563. 

steel— as  pillars,  497,  912. 
table,  894. 
Channeling  in  rock,  681. 
Characteristics,  70-79. 
Charcoal,  weight,  212. 
Chart,  Charts, 

isogonic^.  U.  S.,  300. 

logarithmic — ,  73. 
Checking  of  cement,  938. 
Cherry  wood,  weight,  212. 
Chestnut.     See  Wood. 
Chilling,  defined,   1027. 
Chock,   defined,    1027. 
Chord,   Chords, 

of  arcs,  to  find — ,  179. 

in  circles,   162,   179. 

in  curves,  780,  etc. 

increments,  701. 

long—,  table,  787. 

members.     See  Trusses. 

to  radius  1,  table,  143. 

stresses,  701,  709. 

of  trusses,  689. 
Chronometers,  behavior,  266. 
Chuck,  defined,   1027. 


INDEX. 


1045 


Cbarn  Drilling— Component. 


Churn  drilling,  600. 
Cippoletti  trapezoidal  notch,  559. 
Circle,  Circles,    161,  etc.     See    also 
Circular. 

angles  in — ,  94. 

chords,    162,   179. 

great—,  184,  284. 

radius,  161,  180. 

tables,  163-178. 

vertical—,  284. 
Circular 

ar-cs.     See  Arc,  Circular — . 
tables,  183,  185. 

curves  for  railroads,  780. 

inch,  222,  889. 

lune,  186. 

measure  of  angles,  34. 

measure  of  wires,  etc.,  889. 

motion,  351. 

ordinates,  180. 

tables,  784,  786,  817,  840. 

plates,  strength,  493. 

rings,   186. 

sector,  186. 

center  of  gravity,  393. 

segment,   186. 

center  of  gravity,  394. 
table,  187. 

spindle,   209. 

zone,  186. 
Circulating  decimals,  38. 
Circumference 

of  circle,  to  find—,  161,  171,  178. 

of  ellipse,  189. 
Cisterns,  512,  851,  854. 
Civil  day,  month,  time,  year,  266. 
Clamp,  Clamps, 

pouring — ,   for  pipe   joints,   660. 

rod,  switch,  825. 
Clay, 

effect  on  mortar,  925,  926,  935,936. 

in  foundations,  583. 

loosening  of — ,  800. 

swelling  of —  by  absorption,  583. 
Clearance, 

in  highway  bridges,  726. 

in  railroad  bridges,  746. 
Clearing,  cost,  855. 
Cleat,  defined,  1027. 
Climate,  effect  of —  on  rainfall,  322. 
Clinometer,  256,  311. 
Clips,  for  cement  briquettes,  939. 
Clock, 

to  regulate —  by  star,  266. 

-wise,  360. 
Close  piles,  590. 
Cloth,  tracing—,  978. 
Coal, 

cars,  865. 

consumption  of —  by  locomotives, 
860,  etc. 

corrosive  fumes  from — ,  880,  916. 

for  locomotives,  856. 

oil,  weight  (petroleum),  214. 

space  occupied  by  ton  of — ,  215. 

ton  of — ,  volume,  215,  222. 

weight,  212,  222. 


Cocks,  corporation — ,  664. 
Coefficient,  Coefficients, 

of  contraction,  542. 

defined,   1027. 

of  deflection,  483. 

of  friction,  408. 

of  kinetic  friction,  409. 

for    loads    within    elastic     limit, 
482. 

of  roughness,  523,  564,  565. 
for  pipes,  523. 

of  safety.  See  Safety,  Factor  of — . 

of  stability,  423. 
Coffer-dam,  585,  586. 

defined,   1027. 
Cohesion,  957. 
Coin,  Coins,  218,  219. 
Coke,  weight,  212. 
Cold, 

effects  of — , 

on  cement,  932. 

on  explosives,  948,  950. 

on  iron,  819,  874. 

on  mortar,  928. 

-rolled  iron,  920. 
Colinear  forces,  361,  363. 
Collision,  impact,  347. 

posts,  695. 
Cologarithm,  71. 
Colors, 

draughtsmen's,  978. 
Column,    Columns    (pillars). 

See  Pillars. 

Gray—,  905. 

Phcenix- ,  497,  904,  912,  913. 

water — ,  852. 

Z-Bar— ,  901-903. 
Combination 

bridges,  738. 

specifications,  763. 

and  permutation,  40. 
Combined  stresses,  493,  724. 
Commercial 

measures,  size  of —  by  weight  of 
water,  224. 

weight,  220. 
Common 

denominator,  35. 
to  reduce  to — ,  36. 

divisor,  35. 

factor,  35. 

fraction,  36,  37. 

logarithms,  70-91. 

measure,   35. 

multiple,  35. 
Compass, 

to  adjust,  298. 

declination,  301. 

variation,  301. 
Compensating  reservoir,  653. 
Compensation,  water,  653. 
Complement  and  supplement,  94. 
Component,  Components. 

equations  for — ,  365. 

of  force,  362. 

normal — -,  370. 

rectangular — ,  369. 


1046 


INDEX. 


Coinponeiit— Co-tangrent. 


Component,  Components — cont'd, 
stress — ,  371. 
summation  of — ,  466. 
tangential — ,  369. 
Composition 

of  couples,  405. 
of  forces,  362,  etc. 
of  ratios,  38. 
of  steel,  753. 
Compound 
interest,  42. 
equations,  44. 
table,  43. 
levers,  420. 
locomotives,   856. 
stresses,  762. 
Compressed 

air,  320,  597,  681. 
gun-cotton,  951. 
Compressibility 
of  air,  320. 
of  liquids,  326. 
of  sand,  935. 
Compression 

and  bending,  combined,  493,  724. 
members,  721,  722,  732,  747. 
and  tension,  359. 
Compressive  strength 
of  cements,  934. 
of  concrete.  943. 
of  metals,  921. 
of  stone,  etc.,  923. 
of  timber,  958. 
Compressors,  air — ,  681,  991. 
Concentrated 

excess  loads,  705. 
loads,  deflections,  484. 
loads,  moments  due  to — ,  444. 
loads,  shears  in  beams,  446. 
Concrete,  943. 

beams,  strength  of — ,  945. 
defined,    1027. 

-metal  construction,  cost,  989. 
mixers,  943. 

cost  and  mfrs.,  992. 
strength,      compressive — ,      923, 

943,  etc. 
strength,  transverse,  945. 
Concretions  in  pipes,  655. 
Concurrent  forces,  361,  364,  380 
Cone,  Cones,  200. 

center  of  gravity  of — ,  395,  397. 
frustum,  201,  395,  397. 
Conical  rollers,  846. 
Connections,  pin  and  riveted,  721. 
Conoid, 

frustum  of—,  209. 
parabolic — ,  209. 
Consequent.  38. 

Consolidation  locomotives,  857. 
Constants.     See  Coefficients. 
Construction, 
bridge—,  720. 
railroad — ,  cost,  855. 
Consumption 

of  fuel  by  locomotives,  860. 
of  water,  649. 


Contiguous  openings,  flow  thro — , 

542. 
Continued  proportion,  38. 
Continuous  beams,  489. 
Contour  lines,  302. 
Contracted  vein,  541. 
Contraction, 

of    area,     iron    and    steel,     752, 
754,  873. 

coefficient  of — ,  542. 

by  cold,  317,  819. 

incomplete—,  540,  544. 

of  outflow,  541. 

of  rails,  819. 

of  waterway,  623. 
Contractor's  profit,  801. 
Conversion      tables      of     units      of 

weights  and  measures,  etc.,  228. 
Conveying     machinery,     cost     and 

mfrs.,  991. 
Cooper's  standard  loading,  755. 
Co-ordinates  in  resolution  of  forces, 

372. 
Coping, 

defined,  1027. 
Coplanar  forces,  361. 
Copper, 

balls,  w^eight,  918. 

compressibility,  etc.,  459. 

cost,  919,  987. 

effect    of    cement,     mortar,  etc., 
on—,  ,926,  936. 

effect  of  water  on — ,  327. 

expansion  of —  by  heat,   317. 

pipes,  seamless — ,  919. 

roofs,   918. 

sheets.  918. 

strength,  499,  500,  920,  921. 

sulphate,  for  wood,  955. 

tubes,  seamless — ,  919. 

weight,   212,  875,  878,  886,   887. 
918. 
Corbel,  defined,  1027. 
Cord, 

loaded—,  428. 

mechanics  of  the — ,  425. 

polygon,  377,  428. 

of  wood,  222,  234. 
Cork, 

weight,  212. 
Corporation  cocks  or  stops,   664. 
Corrections  for  tapes,  283. 
Corrosion, 

by  acid  fumes,  970. 

by  coal  fumes,  880,  916. 

by  water,  327,  594. 
Corrugated 

flooring,  914. 

sheet-iron,  880. 
Co-secants,    97. 
Cosines,  97,  98. 

logarithmic — ,  72. 
Cost,   Costs.     See    articles  in  ques- 
tion, 
price  list,  983. 

of  operation  of  railroads,   867. 
Co-tangent,  97,  etc. 


INDEX. 


lt)47 


Cotter-bolt—Dam. 


Cotter-bolt,  defined,   1027. 
Counter,  Counters, 

adjustable — ,  721. 

-bracing,   690,  705,  712,  721,  738, 
746. 
of  centers,  634. 

-clockwise,  360. 

-scarp  revetment,  612. 

sloping  revetment,  612. 
Counterforts,  612,  1027. 
Couples,  404. 
Couplings  for  pipes  and  tubes,  657, 

882. 
Courses 

of  masonry,  603,  620. 
Cover 

in  a  butt-joint,  773. 

plate,  723. 
Co-versed  sines,  97. 
Crab,  defined,  1028. 
Cradle,  defined,  1028. 
Cramp,  defined,   1028. 
Crane,  defined,  1028. 
Creeping 

of  rails,  819,  820. 
Creo-resinate  process,  955. 
Creosote,  815,  954.  984. 
Crescent  truss,  695. 
Crib,  Cribs, 

coffer-dam,  645. 

dams,  cost,  645. 

foundations,   584,   585. 
Criterion, 

for  maximum  chord  stresses,  709. 

for  maximum  web  stresses,  706. 

bracing.  691,  710,  748. 
girts,  turntable,  846. 
-hairs,  in  level,  306. 
-hairs,  to  replace — ,  296. 
section  of  bridge,  746.  • 
section  paper,  978. 
logarithmic,    73. 
-shaped  beam,  492. 
ties,  815,  855. 
Crowd,  weight  of — ,  726. 
Crown, 

of  arch,  defined,  613. 
(coin),  value,  218. 
Crushers,  stone—,  943,  992. 
Crushing 

loads,  923,  934,  943,  958. 
in  timber  construction,  732. 
Cube,  Cubes,  55,  194.  195. 

roots,  54,  etc.     See  also  Powers, 
of  decimals,  to  find — ,  67. 
of    large    numbers,    to    find — , 

66. 
tables,   54,  etc. 
tables,  55. 
Cubic 

centimeter,  foot,   inch,   etc.     See 

Conversion  Tables, 
measure,  222. 
metric,  225. 
meter,  etc.     See  Meter,  etc. 
Culmination  of  Polaris,  287,  288. 


Culmination 

of  a  star,    284. 
Culvert,  Culverts, 

arches  for — ,  613. 

box—,  627. 

foundations  of — ,  627. 

lengths  of—,  622. 

quantity  of  masonry  in — ,  622. 
Curbs 

in  highway  bridges,  750. 
Current  meters,  562.  cost,  etc.,  993. 
Curvature  of  the  earth,    table,  153. 
Curve,    Curves.       See    Arc,    Circle, 
Ellipse,  Parabola,  etc. 

effect  of —  on  distribution  of  live 
load  on  bridge,  712,  756. 

elastic—,  483. 

railroad — ,  780. 
gauge  on — ,  789. 
tables  of—,  784-789.    • 
in  tunnels,  812. 
in  turnouts,  840. 

in  water  pipes,  537. 
Curved 

beams,  446. 

chords  in  trusses,  695. 
Curvilinear  motion,  351. 
Cuttings, 

level—,  790. 
Cutwater,  defined,  1028. 
Cycloid,  194. 

center  of  gravity,  394. 
Cylinder,  Cylinders,  196. 

contents,  table,   197,   223,  525. 

in     foundations,     .593,    594,    596, 
599.      See  also  Foundations. 

of  locomotives,  856,  861. 

pneumatic  process,  596. 

pressure  in — ,  511. 
locomotive — ,  861. 

strength  of — ,  511. 
Cylindrical 

beams,  deflections,  485. 

pillars,  497,  912,  913. 

ungula,  199. 

center  of  gravity  of — ,  397. 
Cyma,  to  draw — ,  191. 


Dam,  Dams,  400,  etc.,  430,  etc.,  433. 

etc.,  502,  576,  642. 
center  of  pressure  against — ,  400. 
coffer—,  585,  586. 
construction,  585,  642. 
danger  in — ,  436. 
deflection,  436. 
discharge  over — ,  547. 
height  of  water,  554. 
leakage  through—,  329,  651. 
masonry — ,   400,  etc.,     430,  etc., 

433,  etc. 
practical  considerations,  436. 
stability,  433,  508. 


10'48 


INDEX. 


Dam— Dischargee. 


Dam,  Dams — continued. 

stone,  400,  etc.,  430,  etc.,  433,  etc. 

timber,  642. 

trembling,  648. 

walls,  508,  611. 

wooden,  642. 
Day,  Days,  236,  265,  266. 
Dead  load.  Dead  loads, 

for  bridges,  690,  755. 

in  cars.  865. 

defined,   1028. 

in  roof  trusses,  713. 

stresses,   graphic  method,   703. 
Dead  point,  defined,  1028. 
Decagon,  148. 
Decay  of  timber,  954. 
Decigram,    equivalents    of — ,     231, 

236. 
Deciliter,  equivalents  of — ,  235. 
Decimal,  Decimals,  37. 

of  a  degree,  mins.  and  sees,  in — , 
95. 

of  a  foot,  inches  expressed  m — , 
table,  221. 

fractions,  37. 

logarithms,   70. 

roots  of — ,  67. 
Decimeter,  equivalents  of — ,  233. 
Deck  trusses,  692. 
Declination,  284. 

magnetic — ,  301. 

of  a  star,  formula,  290. 
Deepened  beams,  479,  734. 
Deflection,   Deflections, 

angle,  780.  784-789,  840. 

of  arch,  436. 

of  beams,  480,  483,  486. 
of  uniform  strength,  486. 

of  cantilevers  of  uniform  strength, 
486._ 

coefficient,  483. 

of  cross-shaped  beams,  492. 

of  dam,  436. 

distance,  781. 
tables,  784-786. 

and  fiber  stress,  481. 

of  suspension  bridges,  765. 

of  trusses,  718. 
Degree,  Degrees, 

of  latitude,  length  of—,  184,  220. 

of  longitude,  length  of—,  221. 

mins.  and  sees,  in  decimals  of — , 
95. 
Dekagram,    equivalents    of — ,    231, 

236. 
Dekaliter  or  centistere,  equivalents 

of—    235. 
Dekameter,  Dekameters,  225. 

chord  of    2 — ,    curve,  table,  786. 

equivalents  of — ,  233. 
Dekastere  or  hectoliter,  equivalents 

of—    235. 
Delta  Cassiopeia,  285. 
Demagnetization  of  compass  needle, 

302. 
Demi-revetment,   612. 
Denominator,  35. 


Density 

of  air,  320. 

relative — ,  210. 
Departures  and  latitudes,  274. 
Depreciation,  43,  864. 
Depth,  Depths, 

for  beams,  477. 

conversion  table,  239. 

on  dams,  554. 

effective — ,  759. 

equivalents  of — ,  in  volumes  pel 
surface,  250. 

flow  at  different — ,  560. 

for  given  deflection,  483. 

hydraulic  mean — ,  564. 

of  keystone,  613. 

for  permissible  deflection,  485. 

pressure  at  different — ,  501,  648.    . 
Derrick, 

car,  808. 

defined,   1028. 
Detrusion,  499. 
Dew-point,  321. 
Diagonal,  Diagonals, 

counterbraces,  712. 

of  parallelogram,  95,  157. 

of  trapezoid,  etc.,  158. 

in  trusses,  689. 
Diagram,  Diagrams, 

for  dead  load  stresses,  703. 

influence —  for  beams,  449. 
for  pressure  distribution,  403. 

for  Kutter's  formula,  570. 

for  live  load  stresses,  706. 

moment  and  shear,  449. 

for  shear  in  trusses,  702. 

truss — ,   707. 

wheel—,  706. 
Dial,  to  make—,  268. 
Diameter,    Diameters, 

of  bolts,  883. 

of  circle,  to  find,  161. 

of  pipes,  524,  882,  918. 

actual  and  nominal — ,526,  882. 

of  rivets  for  safety,  775. 

sq.  roots  of — ,  526. 

of  wire,  887-891. 
Diamond  drill,  675. 
Dike,  defined,   1029. 
Dimensions.       See    the    article    in 
question. 

for  beams,  477. 

limiting — ,    in    bridge    members, 
762. 

in  roof  trusses,  764. 

in  wooden  bridges,  764. 
Dip,  defined,  1028. 
Dipper,  (constellation),   285. 
Direction,  angle  of — ,  765. 
Directory,  business — ,  996. 
Discharge,  Discharges, 

through  adjutages,  540. 

through  channels,  560. 

through  contiguous  openings,  542. 

over  dams,  547. 

head   for  a  given — ,   to   find — , 
527. 


INDEX. 


1049 


Discbarge— £artli. 


Discharge,  Discharges — continued. 

through  notches,  559. 

through  orifices,  539,  546. 

through  pipes,  516,  etc. 

through  sewers,  574. 

through  short  tubes,  540. 

tables  of—,  261-265. 

through  thin  partition,  541. 

units    of   rates    of — ,    conversion 
of—,  243. 

over  weirs,  547,  etc. 
Disks,  centrifugal  force  in — ,  355. 
Distance,  Distances, 

deflection — ,    tangential — ,    781. 

frog—,  839. 

polar—,  284. 

by  sound,  316. 
Distributing  reservoirs,  653. 
Distribution  of  pressure,  400. 
Diurnal  magnetic  variations,  301. 
Diving 

apparatus,   cost  and  mfrs.,   992. 

-bell,  321. 

dress,  992. 
Division 

of  decimals,  37. 

of  fractions,  36. 

by  logarithms,  71. 

by    logarithmic    chart,    or    slide 
rule,  75. 

of  a  modified  logarithm,  72. 

of  ratios,  38. 
Divisor,  common,  35. 
Dodecagon,  148. 
Dodecahedron,   194. 
Dog-iron,  defined,  1028. 
Dollar, 

U.  S.— ,  weight,  etc.,  219. 

value  of—,  218. 
Dolomitic  limestones,  931. 
Dome,  pneumatic — ,  665. 
Donkey  engine,  defined,  1029. 
Double 

float,  561. 

intersection  trusses,  694. 

riveting,  772. 

rul^  of  three,  39. 

shear,  499,  774. 
Dovetail,  defined,  1029. 
Dowel,  defined,  1029. 
Draft 

of  horses,  683,  685. 

of  locomotives,  860. 

of  vessels,  515. 
Drag 

scrapers,  earthwork  by — ,  805. 

of  train  on  bridge,  711,  758. 
Drain,  Drains, 

area  drained  by — ,  575. 

box—,  627. 

foundations  of — ,  627. 

pipe,  575. 
Drainage 

of  roadways  of  bridges,  628. 

sewers,  574. 

of  tunnels,  812. 
Draw-bridges,  696. 


Drawing 

instruments,  cost  and  mfrs.,  993. 

materials,    978. 
Drawn  pipes  and  tubes,  919. 
Dredge,  Dredges,  580. 

land—,  808. 

mfrs.,  992. 
Dredging,    580. 

by  screw-pan,  596. 
Dress,  diving — ,  992. 
Dressing  of  stone,  601. 
Drift,  defined,  1029. 
Drifting  test,  752. 
Drill,  Drills, 

cost  and  mfrs.,  989. 

rock—,  600,  675. 
Drilling, 

artesian  well — ,  671. 

rock—,  500,  670,  675. 

tunnel—,  812. 
Driving 

wheels,  856. 

weights     on — ,   705,  etc.,  755, 
etc.,  856,  etc. 
Drop 

tests,  871. 

timbers,  644. 
Drowned  or  submerged  weirs,  554. 
Dry 

drains,  627. 

measure,  223. 

rot,  defined,  954,  1029. 
Dualin,  952. 
Dubuat's  formula,  555. 
Ducat,  value  of — ,  218. 
Ductility,  455,  459. 
Dump-cars,  865. 

Duodecimals,     duodenal     or     duo- 
denary notation,  47. 
Duplicate  ratio,  38. 
Dyke,  defined,  1029. 
Dynamics,  330, 
Dynamite,  949,  984. 


E. 

E  and  W  line,  to  run — ,  277. 
Earnings  of  railroads,  867. 
Earth, 

augers,  670. 

bearing  power,  583. 

blasting,  950. 

boring,  670. 

cars  (dump-cars),  865. 

curvature,  table,   153. 

friction,  612,  683. 

hauling,  801. 

heat  of—,  320. 

leakage  through — ,  329,  651. 

leveling  of — ,  801. 

loosening  of — ,  800. 

natural  slope,  419,  607,  610. 

pressure,  607. 

resistance  of — ,  583. 

shoveling  of — ,  800. 

shrinkage,  799. 


1050 


INDEX. 


Eartb— Expansion. 


Earth — continued. 

slope  of — ,  natural — ,  607,  610. 
supporting  power,  583. 
weight,  212. 
-work,  790-811. 

cost,  800,  855,  988. 
in  tunnels,  812. 
volume  of — ,  790,  etc. 
East  and  west  line,  to  run — ,  277. 
Eastern  elongation,  284. 
Easting,  274. 
Eccentric, 

defined,  1029. 
loads,  712. 

deflections,  484. 
Efflorescence,  929,  936. 
Effort,  total—  of  force,  371. 
Elastic 

curve,  483. 

deflection,  trusses,  718. 
limit,  458,  459,   482. 
bridge  steel,  752. 
cast  iron,  874. 
iron  and  steel,  873. 
steel  castings,  754. 
ratio,  458,  461. 
Elasticity, 

limit  of—,  458,  459,  482. 

in  beams,  482. 
modulus  of — ,  456,  459. 
cast  iron,  874. 
Electric 

blasting  machine,  952. 
railroad  bridges,  loads  for — ,  757. 
Electricity  in  compass  box,  302. 
Elevation  of  outer  rail  on  curves, 

787. 
Ellipse,  189,  190. 

false — ,  to  draw — ,  191. 
ordinate,   189. 

tangent  to — ,  to  draw — ,   190. 
Ellipsoid,  208 
Elliptic 
arc,  189. 

ordinates.    189. 
table,  190 
arch,  616. 

joints  in — ,  to  draw — ,   190. 
Elm  wood, 

strength,  459,  476,  957,  958. 
weight,  212. 
Elongation, 

bridge  steel,  752. 

by  heat,  317. 

polar  distances  and  azimuths  of 

Polaris  at — ,  table  of — ,  290. 
of  Polaris,    location   of   meridian 

by—,  286. 
of  Polaris,  times  of — ,  288. 
required,  iron  and  steel,  873. 
of  a  star,  284. 
in  steel  castings,  754. 
under  tension,  455. 
of  truss  members,  718. 
Embankment,    790-811. 
cost,  800. 
shrinkage,  799. 


End 

post,  design,  723. 

reactions,  360,  439,  699,  702,  714. 
Energy,  343. 

kinetic—,    343. 

potential — ,  346. 
Engine,  Engines, 

cost  and  mfrs.,  990. 

locomotive — ,  856. 
dimensions,  856. 
performance,  860. 
weight,  856. 

pumping — ,  852. 

wheel  loads,  705. 
Entry  head,  516. 
Equal 

altitudes,  location  of  meridian  by 
any  star  at — ,  287. 

shadows  from  the  sun,  location  of 
meridian  by — ,  288. 
Equality  of  ratios,  38. 
Equation 

of  payments,  42. 

of  time,  265. 
Equilibrium,  358. 

of  beams  and  trusses,   437,   466, 
698. 

of  couples,  405. 

of  floating  bodies, 
axis  of — ,  514. 

indifferent—,  387,  514. 

in  levers,  419. 

of  moments,  360. 

polygon,  trusses,  707. 

stable—,  387,  514.  ' 

unstable — ,  387,  514. 

vertical  of — ,  514. 
Equipment,    railroad — ,    cost,    855, 

867. 
Equivalence  of  work, 

in  trusses,  718. 
Equivalents.  See  Conversion  Tables, 

230,  231,  etc. 
Erection  of  bridges,  743,  763. 
Erie    R.  R.     locomotive   standard, 

858. 
Establishment  of  a  port,  328. 
Evaporation,  329,  561. 

by  locomotives,  854. 
Even  joints,  819. 
Evolution  by  logarithms,  71. 
Excavating  carts  (wheeled  scrapers), 

805. 
Excavation,  790-811. 

cost  of—,  800,  855,  988. 

in  tunnels,  812. 

volume,  790. 
Excavators, 

mfrs.,  992. 

steam —  (land  dredge),  808. 
Excess  loads,  concentrated — ,  705. 
Expansion 

bearings,  721,  725,  751. 

bolts,  884. 

of  cement,  937. 

by  heat,  317.    See  Heat, 
of  rails,  819. 


INDEX. 


1051 


Expense— Flume. 


Expense,  Expenses, 

locomotive  running — ,  864. 

railroad — ,  867. 
Exploder,  Exploders,  952. 
Explosive,  Explosives,  948. 

cost  and  mfrs.,  984. 
Express  cars,  865. 
Extrados,    613. 
Extreme  fiber  stress,  permissible — , 

759. 
Extremes  of  ratio  and  proportion, 

38 
Eye-'bars,  721,  747. 

design,  722. 

full  size — ,  test  of — ,  753. 


Face  of  arch,  613. 
Face  wall,  603. 
Facing  switch,  824. 
Factor,   Factors, 

common — ,  35. 

and  multiples,  35. 

safety — , 

for  piles,  593. 

for  pillars,  909,  912. 

See  also  Safety,  factor  of — . 
Fahrenheit  thermometer,  318. 
Fall,  Falls, 

defined,    1029. 

required    for   a   given    discharge, 
527,  566,  573. 

in  sewers,  574. 
Falling 

bodies,   348,   539. 

water,  horse  power,  578. 
False 

ellipse,  to  draw — ,  191. 

-works,  743. 
defined,   1029. 
Fanega,  227. 
Fascines,  599. 

defined,   1029. 
Fathom,  232. 
Fatigue  of  materials,  465. 

defined,   1029. 
Faucet  in  pipe  joint,  660. 
Feather,  defined,  1029. 
Feet.     See  Foot. 
Felloe  or  Felly,  defined,  1029. 
Fence,   854. 
Fencing,  987. 
Ferris-Pitot  meter,  536. 
Ferrule, 

defined,   1029. 

for  water  pipe,  664. 
Fiber 

reactions,  466. 

stress,  466,  467,  etc. 
and  deflection,  481. 
permissible — ,  759. 
Field  tests  for  cements,  942. 
Fifth  powers  and  roots,  67-69. 
Figure,  Figures,  148. 

areas  of — ,  160. 


Figure,  Figures — continued. 

defined,  92. 

to  draw — ,  159. 

to  enlarge — ,  160. 

irregular — ,  to  find  area  of — ,  160. 
Filler  in  pin  joints,  725. 
Filling,  spandrel — ,  613. 
Filters,  mfrs.,  994. 
Fineness, 

of  cement,  938,  940. 

of  sand  and  cement,  937. 
Finish,  hard—,  968. 
Fink  truss,  695. 
Fir,  strength,  957,  etc. 
Fire,  Fires, 

heat  of—,  317. 

hydrant  (fire-plug),  669. 

-proof  floors,  894. 

-proofing,  cost,  989. 

protection,  water  for—,  650. 
Firing,  simultaneous — of  blasts,  952. 
Fish-plates,  820. 
Fittings  for  pipes,  656,  882. 
Flagging, 

strength  of — ,  476. 
Flashings,  defined,  l629. 
Flasks,  casting — ,   defined,    1029. 
Flats,  in  built-up  sections,  723. 
Flexible  joints  for  pipes,  661. 
Floating 

bodies,  513.  ' 

mills,   578. 
Floats,  560,  561. 
Floor,  Floors, 

beams,  720,  749. 
connections,   730. 

bridge—,  720,  749. 

fire-proof—,  894. 

glass — ,  974. 

sections,  rolled — ,  914. 

svstems  of  bridges,  720,  749. 

trough-,  750,  914. 

wooden — ■  in  bridges,  750. 
Flooring, 

corrugated — ,  914. 

Z-bar—   914. 
Florin,  value,  218. 
Flotation,   513. 
Flow, 

through  adjutages,  540. 

in  channels,  560. 

through  contiguous  openings,  642. 

full—   540. 

Kutter's  formula,  523,  563,  564. 

obstructions  to — ,  537,  576,  578. 

through  orifices,  539,  546. 

in  pipes,  516. 

in  sewers,   574. 

through  short  tubes,  540. 

in  streams,  560. 

in  syphon,  520. 

through  thin  partition,  541. 

in  trough,  544. 

over  weirs,  formula,  549. 
Fluid,  Fluids, 

friction  of — ,  415. 
Flume,  defined,  1030. 


1052 


INDEX. 


Flasb— Friction. 


Flush,  defined,  1030. 
Fluxes,  defined,  1030. 
Fly-wheels,  centrifugal  force,  355. 
Follower,  in  pile  driving,  594. 
Foot,  Feet, 

cubic — , 

equivalents  of — ,  222,  234. 

of  substances,  weight  of — ,  212. 

equivalents  of — ,  232. 

of    mercury    (pressure),    equiva- 
lents of — ,  241. 

per  mile,  equivalents  of — ,  237. 

per  second,  equivalents  of — ,  242. 

-pound,   341. 

equivalents  of — ,  237. 
Forbay,  defined,  1030. 
Force,  Forces,  330,  332,  358. 

acting  upon  beams  and  trusses, 
437. 

application  of — ,  point  of — ,  333. 

applied  and  imparted — ,  372. 

center  of—,  399,  506,  514. 

centrifugal — ,  354. 
on  bridges,  758. 

centripetal — ,   354. 

classification,  361. 

colinear — ,  363. 

component,  362. 

composition  of — ,  362,  364. 

defined,  332. 

diffusion  of —  through  liquids, 506. 

on  inclined  planes,  349. 

internal —  in  beams,  466. 

living—,  343. 

measure  of — ,  338. 

paraUel— ,  382. 
couples,  404. 
resultant  of—.  399. 

parallelogram,  364. 

parallelopiped,   380. 

point  of  application  of — ,  333. 

polygon,  374,  377. 

resolution  of—,  362,  364. 

resultant  of — ,  362. 

in  rigid  bodies,    330,   358. 

total  effort,  371. 

transmission,  358. 

triangle,  367. 

units  of — ,  358. 

conversion  of — ,  235. 
Forcite,  952. 
Foreign 

coins,  218. 

explosives,  952. 
Forgings,  steel — ,  requirements,  872. 
Formula.     See  also  the  given  prob- 
lem. 

Gordon's — ,  495. 

Kutter's— ,  523,  563,  564. 

prismoidal — ,  203. 
Foundations,  582. 

of  arches,  613. 

artificial  islands,  600. 

brick  cylinders,  599. 

caissons,  585. 

for  centers,  631. 

in  clay,  583. 


Foundations — continued. 

close  piles,  590. 

coffer-dams,  585,  586. 

crib—,  585. 

of  culverts,  627. 

cylinders,  594,  596,  597.  599,  600. 

of  drains,  627. 

fascines,  599. 

on  gravel,  583. 

grillage,  590. 

iron  piles,  594. 

islands,  artificial — ,  600. 

loads  for—,  583. 

masonry — ,  cylinders,  599. 

Nasmyth  pile-drivers,  591. 

Pierre  perdue,  583. 

pile — .     See  Pile,  Piles. 

plenum  process,  597. 

pneumatic  process,  596. 

random  stone,  583. 

resistance  of — ,  583,  592 

of  retaining  walls,  612. 

rip-rap,  583. 

on  sand,  582. 

sand  piles,  599,  670. 

sand  pump,  599. 

screw  piles,  594. 

sheet  piles,  590. 

sustaining  power,  583,  592. 

for  trestles,  814. 

for  turntables,  845.     • 

vacuum  process,  596. 
Four-way  stop-valve,  667. 
Fourth  proportional;  38. 
Fractions,   35. 

logarithms  of — ,  72.  ^ 

Frames,  blue-print — ,  980. 
Framing,  timber — ,  734. 
Framework,   steel — ,   specifications^ 

764. 
Franc,  value  of — ,  218. 
Francis's  formula,  550. 
Franklin  Institute  standard  dimen- 
sions of  bolts,  etc.,  883. 
Free  end  reaction,  roof  trusses,  715. 
Freezing 

of  dynamite,  950. 

effect  of—  on  cement,  932. 

of  explosives,  948. 

of  mercury,  318. 

of  mortar,  928. 

of  nitro-glycerine,  948. 

in  pipes,  656,  665. 

behind  retaining  walls,   604. 

in  stand  pipes,  663. 

in    track   tank,    how    prevented* 
853 

of  water,  326. 
Freight, 

cars,  865. 

earnings,  867. 

locomotives,  856. 

ton-mile,  867. 
Friction,  407. 

angle  of — ,  409. 
in  arch,  432. 
in  dams,  433. 


INDEX. 


1053 


Friction— Orade. 


Friction — continued. 

axle—,  416. 

of  cars,  417. 

coefficient  of — ,  408. 

of  earth,  612. 

head,  516,  527. 

on  inclined  planes,  350. 

of  iron  cylinders,  593. 

journal — ,  416. 

kinetic — ,  coefficient,  409. 

launching — ,  415. 

longitudinal —  of  revolving  shafts, 
419. 

of  masonry,  411,  612. 

Morin's  laws,  410. 

of  piles,  593. 

rollers,  417,  725,  751,  846. 
defined,   1030. 

of  walls,  608. 

of  water,  415. 
Frictional  stability,  409. 
Frog,  Frogs,  834-840. 

angle  of—,  835,  839. 

distance,  839. 

graphic  method,  842. 

length,  835. 

number,  835,  840. 

point,  835. 
Frost 

jacket  in  fire  hydrant,  669. 

-proof  tank,  852. 
Frustum, 

of  cone,  201. 

of  parabola,   192. 

of  paraboloid,  209. 

of  prism,  195. 

of  pyramid,  201. 
Fteley  and  Stearns's  formula,  552. 
Fuel    consumption   of  locomotives, 

861. 
Fulcrum,  419. 
Full 

flow,  540. 

size  eye-bars,  tests  of — ,  753. 
Fumes,  acid — ,  effect  on  roofs,  970. 

coal — ,  effect  on  iron,  880 
Funds,  sinking — ,  43. 
Funicular  machine,  427. 
Furlong,  equivalents  of — ,  232. 
Furrings,  defined,    1030. 
Fuse,  defined,  1030. 


G.  C.  D.,  35. 
Gage,  Gages, 

Birmingham—,  887,  890. 

hook — ,  548. 

narrow —  cars,  865. 

narrow —  locomotives,   857. 

railroad—,  827, 
on  bridges,  746. 
on  curves,  787,  789. 

rain — ,  324. 

stubs—,  890. 

stuff,  968. 

wire—,  887-891. 


Gaging  of  streams,  560. 
Gallon,  223,  224,  234. 
Galton's  experiments,  412. 
Galvanic  action  in  water  pipes,  656. 
Galvanized 

iron,  880. 

pipes,  664. 
Gas 

engines,  mfrs.,  990. 

weight,  211. 
Gasket,    660. 

defined,   1030. 

to   prevent  washing —  into  pipe, 
661. 
Gate  valves,  666,  cost,  etc.,  995. 
Gates  for  water  pipes,  666. 
Gauge,  Gauges.     See  Gage,  Gages. 
Gauging  of  streams,  560. 
Gauthsy's  pressure  plate,  561. 
Gearing,  ratio  of  power  and  weight, 

420. 
Gelatine,  explo^ve — ,  952. 
Geographical  mile,  220. 
Geometrical  progression,  39. 
Geometrical  similarity,  92. 
Geometry,  92. 
Giant  powder,  951. 
Gib,  defined,  1030. 
Gin,  686. 

defined,   1030. 
Girders, 

details,  728. 

erection,  743. 

plate—,  731.  747. 
bracing,  749. 

and  trusses,  comparison,  689. 
Glass,  973. 

compressibility,  etc.,   459. 

cost  and  mfrs.,   974,   985. 

dimensions,   etc.,   973. 

expansion  by  heat,  317. 

friction,  411. 

strength,  476,  922,  923,  974. 

weight,  212. 
Glazing,  973. 
Globe,  204,  205. 
Glossary  of  terms,  1025. 
Glue, 

adhesion  of — ,  922. 

defined,   1030. 
Glycerine,  nitro — ,  948. 
Gneiss,  weight,  213. 
Gold, 

strength,  920. 

value—,  219. 

weight,  213,  219. 
Gondola  cars,  865. 
Gordon's  formula,  495. 
Grade,  Grades,   255-257. 

contour  lines,  300. 

etc.,  conversion  of — ,  237. 

defined,  255,  256. 

effect  on  horses,  683. 

effect  on  locomotives,  860. 

hydraulic — ,  519,  521. 

percentage,  255.     * 

resistance,  683,  860. 


1054 


INDEX. 


Orad  e— Hec  tomete  r. 


Grade,  Grades — continued. 

of  roads,  255,  683. 

of  sewers,  574. 

tables,  255-257. 

traction  on — ,  683. 

in  tunnels,  812. 

on  turnpikes,  255. 

of  water-pipes,  653. 
Gradient,  hydraulic — ,  519,  521. 
Grading,  cost,  800,  855. 
Grain,  equivalents  of — ,  235 
Gram,  or  Gramme,  217,  226. 

equivalents  of — ,  236. 
Granite, 

beams,  924. 

cost  of —  blocks,  601. 

expansion  by  heat,  317. 

rubble,  cost,  602. 

strength,  476,  923,  924. 

weight,  212. 
Graphic 

method,  truss  stresses,  703,  706. 

representation  of  couples,  405. 

statics,  428-431,  435. 
Gravel, 

boring  in — ,  670. 

in  concrete,  943. 

dredging  fn — ,  580. 

for  foundations,  582. 

natural  slope  of — ,  610. 

weight,  213. 
Gravity, 

acceleration  of—,  335,  336,  348, 
349,  539. 

center  of — ,  386. 

on  inclined  planes,  349. 

line  of—,  389. 

plane  of—,  389. 

specific — ,  210. 
Gray  column,  905. 
Great 

bear,  constellation,  285. 

circle,  284. 
Greatest  common  divisor,  35. 
Grillage,  590. 

defined,    1030. 
Groin,  defined,  1030. 
Gross  ton,  216. 
Ground  lever,  826. 
Grout,  926. 

defined,   1030. 
Grubbing,  cost,  855. 
Guard,  Guards, 

rails,  750,  828,  833,  835. 

wheel—,  750. 
Gudgeon,  416. 

defined,    1030. 
Guide-rails,  828,  833,  835. 
Guldinus  theorem,  194. 
Gun 

-cotton,  compressed,  951. 

metal,  strength,  920. 

-powder,  953. 
pile-drivers,  591. 
weight  (under  Powder),  214. 
Gunter's  chain,  "220,  232,  282. 
Gusset,  defined,  1030. 


Gutta-percha 

pipe,  657. 

weight,  213. 
Gypsum,  weight,  213. 
Gyration, 

center  of — ,  496. 

radius  of—,  352,  496,  892,  etc. 


H. 

H.  C.  F.,  35. 

H.  P.     See  Horse-power. 
Hair, 

cross — ,  to  replace — ,  296. 

stadia — ,  293. 
Half-section,  equivalents  of — ,  233, 
Hand 

level,   310. 

spike,  defined,  1030. 
Hard  finish,  968. 
Hardening  of  cement,  930. 

rate  of—,  932. 
Hasselmann  process,  955. 
Haul,  mean — ,  801. 
HauUng,  683,  685,  801,  805. 
Haunches,  defined,  1030. 
Head,  Heads, 

block,  825. 

of. bolts,  883. 

diie  to  a  given  velocity,  539. 

entry — ,  516. 

friction—,  516,  527. 

for  a  given   velocity,    to   find — , 
527. 

for  piles,  593. 

plate,  825. 

pressure — ,  258,  etc.,  518. 

theoretical — ,  539. 

tripod—,  292. 

velocity — ,  516,  539. 

of  water,  516. 

for  water  supply,  654. 
Header,  defined,  1030. 
Heading,  812. 

defined,   1030. 
Headway  in  bridges,  746. 
Heat, 

of  the  air,  320. 

conduction  of — ,  by  air,  320. 

expansion  of  air  by — ,    320. 

expansion  of  rails  by — ,  819. 

expansion  of  solids  by—,   317. 

expansion  of  surv.   chains    by — , 
274,  283. 

of  fires,  317. 

subterranean — ,  320. 

thermometer,   318. 

and  work,  units  of — ,  conversion 
of—,  237. 
Hectare, 

equivalents  of — ,  234. 
Hectogram,  equivalents  of — ,  236, 
Hectoliter,  equivalents  of — ,  235. 
Hectometer, 

equivalents  of — ,  233. 


INDEX. 


1055 


Heel— Inclined. 


Heel  of  frog,  835. 

of  switch,  825,  828,  839. 
Height, 

effect  on  temperature,  320. 

effect  on  weight,  336,  348. 

to  find —  by  barometer,  312. 

to  find —  by  boiling  point,  314. 

to  find —  by  trigonometry,  151. 

of  locomotive  smoke-stack,  856. 
Heliography,  979. 
Helve,  defined,  1030. 
Hemlock, 

strength,  476,  499,  958,  965. 

weight,  213. 
Heptagon,   148. 
Hexagon,   148,   159. 
Hickory, 

strength,  476,  957.  958. 

weight,  213. 
High  explosives,  948. 
Highest  common  factor,  35. 
Highway  bridges,  745,  etc. 
Hip 

roof,  defined,  1031. 

suspender,  stress  in — ,  709. 
Hoisting 

engines,  cost  and  mfrs.,  990. 

machinery,  cost  and  mfrs.  of — ,  991. 
Holes, 

for  blasting,   600. 

boring —  in  earth,  670. 

boring—  in  rock,   600,  670,  675. 
Homogeneity,  tests  for — ,  871. 
Hook-head  spikes,  818. 
Hopkins's  pneumatic  dome,  665. 
Horizon,  artificial — ,  298. 
Horizontal, 

defined,    153. 

forces,  summation  of — ,  466. 

loads  in  trusses,  710. 

shear  in  beams,  478. 
Horse,  Horses, 

power  of—,  683,  852. 

-power,  342,  685. 

equivalents  of — ,  244. 
of  falling  water,  578. 
-hour,  equiA'^alents  of — ,  237. 
metric — .      See    Metric    horse- 
power, 
of  running  streams,  578. 
'  pumping,  day's  work,  852. 

weight,  685. 
Hose, 

cost,  995. 
Hour,   hours, 

angle,  285. 

defined,   265. 

equivalents  of — ,  236. 
House,  engine — ,  cost,  850. 
Howe  truss,  692,  736,  738. 
H.  P.     See  Horse-power. 
Hydrant,  Hydrants. 

cost  and  mfrs.,  995. 

fire  (fire-plug),   669. 
Hydraulic,    Hydraulics,    516.      See 
also  Water,  Flow,  Velocity,  Dis- 
charge, etc. 


Hydraulic,  Hydraulics — continued. 

cement.      See   Cement. 

grade  line,  519,  521. 

index,  930. 

lime,   930. 

mean  depth,  523,  564. 

radius,  523,  564. 

rain,  578. 

cost  and  mfrs.,  991. 
Hydraulicity  of  cement,  930. 
Hydrogen,  specific  gravity — ,  213.' 
Hydrometers.  211. 
Hydrometric  pendulum,  561. 
Hydrostatic,  Hydrostatics,  501. 

paradox,  501. 

press,  506. 
Hyperbolic  logarithms,  72. 


I. 


I-beams.    See  Beams,  I — . 

in  fire-proof  floor,  894. 

as  pillars,  497,  912. 

separators  for — ,  900. 

table,  892. 
Ice, 

adhesion  to  piles,  594. 

blasting  of — ,  950. 

in  stand  pipes,  663. 

strength,  compressive — ,  923. 

weight,  213,   326. 
Icosahedron,  194. 
Illumination 

of  cross-hairs,  286. 

of  stake,  surveying,  286. 
Impact,  347. 

of  trains  on  bridges,  711,  758. 
Imperial 

gallon.     See  Gallon. 

measure,  British,  224. 
Impost,  defined,  1031.   . 
Impulse,  337. 
Inch,  Inches, 

equivalents  of—,  221,  232. 

circular — ,  222. 

cubic — ,  equivalents,  222,  234. 

in  decimals  of  a  foot,  221. 

per  foot,  equivalents  of — ,  237. 

of    mercury     (pressure),    equiva- 
lents of — ,  241. 

miner's — .  546. 

spherical — ,  equivalents  of — ,  222. 

square — ,  equivalents  of — ,  233. 
Inclination.     See  Grade. 

of  courses  in  masonry,  603,  620. 

tables  of—,  255-257. 

in  tunnels,  812. 
Inclined 

beams,  445,  485. 

plane,  349,  369. 
descent  on — ,  349. 
ropes  for— ,  976-977. 
stability  on — ,  424. 
tables,  255-257. 
velocity  on — ,  349. 


1056 


INDEX. 


Incomplete — Iron . 


Incomplete  contraction,  544. 
Increments,  chord — ,  701. 
Incrustation, 
of  boilers,  327. 
of  walls,  929,  936. 
Indeterminate  stresses,  720. 
Index,  logarithms,  70. 
India  rubber,  weight,  213. 
Indifferent  equilibrium,  387,  514. 
Inertia,  338. 

moment  of — ,  351,  468. 
Infinity,  symbol  for — ,  33. 
Influence  diagrams,  403,  449,  702. 
Ingot,  defined,  1031. 
Instability,  514. 
Interest,  40. 

Internal  forces  in  beams,  466. 
Internat'l  metric  screw  thread,  883. 
Intersections, 

in  railroad  curves,  780. 
in  trusses,  694. 
Intrados,  defined,,  613. 
Inverse  proportion,  39. 
Inversion  of  ratios,  38. 
Invert,  defined,  1031. 
Involution.  54-69. 

by  logarithms,  71. 
Iron, 

balls,  weight,   874,  875,  877,  879, 

918 
bars,  weight,  877,  878. 
beams.     See  Beams,  iron — . 
bending  tests,  873. 
blasting  of—,  950. 
bolts,  883,  886. 
in  bridges,  requirements,  754. 
cast — , 

balls,  weight  of—,  918. 

cohesive  strength,  920, 

compressive  strength,  874,  921. 

elastic  limit,  459,  874. 

expansion  by  heat,  317, 

friction,  411, 

malleable — ,  strength,  874, 

modulus  of  elasticity,  459,  874, 

pillars,  495, 

pipes,  flow  in — ,  522, 

pipes,  weight,  656,  876, 

requirements,  874. 

salt  water  on — ,  327,  594, 

shearing  strength,  499. 

strength,    476,    499,    500,    874, 

920,   921. 
tensile  strength,  874,  920. 
torsional  strength,  500, 
transverse  strength,  476,  874. 
weight,  213,  875,  918. 
casting,  weight,  875. 
and  cement,  pipes  of — ,  657. 
chains,  915, 

channels,  as  pillars,  497,  912. 
cohesive  strength  of — ,  920, 
cold,  effect  on—,  274,  819,  874, 
cold-rolled-,  920, 
columns.  See  also  Pillars,  iron — . 

495. 
compressive  strength,  921, 


Iron — continued. 

contraction  of —  by  cold,  274,  819. 

corrosion  of —  by  coal  fumes,  880. 

corrugated  sheet — ,  880. 

cost,  986. 

crushing  strength,  921, 

cylinders,  bursting  pressure  in — , 

511,  512. 
cylinders,    foundations,  etc.     See 

also  Foundations,  593-598. 
ductility  of—,  459. 
effect  of  cement  on — ,  936. 
effect  of  cold  on—,  274,  819,  874. 
effect  of  heat  on—,  274,  317,  819. 
effect  of  mortar  on — ,  926,  936. 
effect  of  water  on — ,  327,  594. 
elastic  limit,  459,  872,  874. 
expansion  of —  by  heat,  274,  317, 

819, 
friction  of — ,  411. 
galvanized — ,  880. 
heat,  effect  on—,  274,  317,  819. 
limit  of  elasticity,  459,  872,  874. 
malleable  cast — ,  strength,  874. 
manufacture,  870. 
manufacturers,  986. 
modulus  of  elasticity,  459,  874. 
net,  774. 

paints  for  preserving — ,  763,  972. 
piles,  594,  See  also  Foundations, 
pillars,   495,    497,   901-913.     See 

also  Pillars, 
pipes, 

cast — ,  weight,  656. 

fittings  for—,  882. 

flow  in — ,  522. 

galvanized — ,  664. 

joints  for — ,  656,  660. 

thickness,  512,  656. 

wrought — ,  656. 

diams,  actual  and  nominal — , 
526.  882, 
plates,  buckled — ,  885. 
porosity  of — ,  512, 
prices — ,  986. 
re-rolled—,  920, 
rolled — ,     See  Iron,  wrought — . 
rolled —  for  bridges,  requirements, 

754, 
roofs.     See  Roofs, 
salt  water,  effect  on — ,  327,  594. 
shearing  strength,  499. 
sheet—,  880. 
specific  gravity,  213. 
specifications,  870. 
spikes,  818, 
strength,  476,  499,  500,  870,  872, 

874,  907,  920,  921. 
stretch  of — ,  459, 
T— ,  497,  898,  912, 
tensile  strength,  920, 
tests,  bending — ,  873. 
torsional  strength,  500. 
transverse  strength,  476,  874. 
tubes,  882, 
water,  effect  on—,  327,  594, 

salt—,  effect  on—,  327,  594. 


INDEX. 


1057 


Iron— liap. 


Iron — continued. 

weight,    213,  875-882.     See  also 
Iron,  cast — ;  Iron,  wrought — . 
wire,  891. 

rope,  976,  977. 
-wood  (Canadian),  strength,  476. 
wrought — , 

bars,  weight,  877,  878. 
cohesive  strength,  873,  920. 
compressive  strength,  921. 
elastic  limit,  4.59,  872. 
expansion   by  heat,    274,    317, 

.819.  , 

friction  of — ,  411. 
pillars.     See  also    Pillars,  495. 
pipes,  656,  657. 

diams,  actual  and  nominal — , 

526,  882, 
fittings  for—,  882. 
joints  for—,  656,  660. 
weight,  656,  882. 
prices,  986. 

shearing  strength,  499. 
strength,    476,    499,    500,    872, 

920,  921. 
tensile  strength,  872,  920. 
torsional  strength,  500. 
transverse  strength,  476. 
tubes,  weight,  882. 
water  pipes,  656. 
weight,  213,  877-882, 
Island,  artificial — ,  for  foundations, 

600. 
Isogenic  chart  and  lines,  300,  301. 


Jack,  defined,  1031. 

rafters,  defined,  1031. 
Jag-spike,  818.  defined,  1031, 
Jaw-plate,  724. 
Jet 

pile  driving,  595. 
Jetty,  defined,  1031. 
Jig-saw,  defined,  1031, 
Joint,  Joints, 

in  arches,  629. 

bell —  for  pipes,  660. 

in  bridges,  724. 

butt—,  773. 

in  chimneys,  etc.,  cement  for — , 
971,  973. 

distribution  of  pressure  in — ,  400. 

end — ,  roof  trusses,  733. 

flexible —  for  pipe,  661. 

lap—,  773,  778. 

masonry — , 

distribution    of   pressure   in — , 

400. 
inclination  of—,  603,  620. 
lead  in — ,  634,  921. 

net—,  774. 

pin — ,  747. 

pin  and  riveted — ,  721. 

for  pipes,  656,  660,  882. 

67 


Joint,  Joints — continued. 

rail-,  819. 

cost  and  mfrs.,  994. 

riveted-,  721,  749,  772. 

in  roofs,  733,  916,  971. 

timber — ,  734. 

toggle — ,   427. 
Joule, 

equivalents  of — ,  237. 

per  second,  equivalents  of — ,  245. 
Journal  friction,    416. 
Jumper, 

defined,   1031. 

drill,  600, 

K. 

Key  frog,  836, 
Keystone,  613,  615. 

pressure  on — ,  614. 
Kieselguhr,  949. 
Kilogram, 

centigrade,  equivalents  of — ,  237. 

equivalents  of — ,  236. 
Kilogrammeter,    equivalents    of — , 
237. 

per  second,  equivalents  of — ,  245, 
Kiloliter,  equivalents  of — ,  235. 
Kilometer, 

equivalents  of — ,  233. 

per  hour,  etc.,  equivalents  of — , 
243. 
Kilowatt,  equivalents  of — ,  245. 
Kinetic 

energy,  343. 

friction,  407. 

coefficient  of — ,  409. 
King, 

post,  defined,  1031. 

truss,   691. 
Knife-edge,  strength,  921. 
Knot  (nautical),  220. 
Kutter's  formula,  523,  563,  564. 
Kyanizing,  955. 


L.  C.  D.,  35. 
L.  C.  M.,  35. 
Lacing,  722. 
Lagging, 

for  centers,  631,  639. 

defined,   1031. 
Laitance,  947. 
Land, 

dredge.  808. 

measure,   222,  233. 
metric — ,  225. 

required  for  railroads,  254. 

section  of—,  area  of — ,  222,  233. 

surveying,  274. 

ties,   612. 
Lap 

joint.  773,  778. 

welded  boiler  tubes,  882. 

welded  pipe,  656,  882. 


1058 


INDEX. 


liard— liiving  Force. 


Lard, 

as  a  lubricant,  415. 

weight,  213. 
Lateral 

bracing,  691,  720. 
timber  trusses,  737. 
Laths,  968,  969. 
Latitude,  Latitudes, 

astronomical — ,   284. 

degree  of — ,  length,  220. 

and  departures,  274. 

effect  of —  on  barometer,  312,  314. 

effect  of—  on  gravity,  336,  348. 
Lattice 

bars,  747. 

truss,   694. 
Latticing,    722. 
Launching,  friction  of — ,  415. 
Laying 

bricks,   927. 

out  of  turnouts,  839. 

pipe,  cost,  658. 

track,  cost,  855. 
Lead, 

balls,  weight,  918. 

defined,   1031. 

effect  of  cement,  mortar,  etc.,  on 
— ,  926,  936. 

elasticity,  etc.,  459. 

expansion  by  heat,  31*5. 

in  masonry  joints,  634,  921. 

paint,  971. 

pencils,    978. 

pipe,   513,  918. 

for  pipe-joints,  658-661. 

roofs,  918. 

sheets,  918. 

strength,  920,  921. 

weight,  213,  875-878,  887,  918. 

white —  cement  for  leaks,  971. 

white —  paint,  971. 
Leaded  tin,  916. 

Leak  in  roof,  to  stop — ,  971,  973, 
Leakage,  329,  561,  642,  650,  651. 
Leap  year, 

defined,  266. 

equivalents  of — ,  236. 
Least 

common  denominator,  35. 

common  multiple,  35. 
Leather, 

friction,  415. 

strength,   922. 
Legua,  227. 
Length, 

per   time — ,    units    of — ,    conver- 
sion of—,  242. 

units  of — ,  conversion  of — ,  232. 
Level,  Levels,  306. 

builder's — ,  to  adjust — ,  311. 

cost  and  mfrs.,  993. 

cuttings,    790. 

engineer's — ,  306. 

hand — ,  Locke — ,  310. 

lines,  defined,   153. 

note-book,  form  of — ,  309. 

Y— ,  306. 


Levelling, 

by  barometer,  312. 

by  boiling  point,  314. 

of  earth  on  embankment,  801. 

screws,  292. 
Lever,  Levers,  419. 

switch—,  826,  830. 

tumbling—,  826. 
Leverage,  360,  419. 
Libra,  227. 
Life, 

average — , 
,        of  cars,  865. 

of  shingles,  971. 
of  ties,  815. 
Lift  bridges,  696. 
Lignum  vitae, 

strength,  476,  957. 

weight,  213. 
Lime,  925. 

in  cement,  930. 

hydraulic—,   930. 

paste,  926. 

quick — ,  930. 

weight,  213. 
Limestone,  213,  923,  930. 
Limit, 

elastic—,  458,  459,  482. 
cast  iron,  874. 
iron  and  steel,  752,  873. 

of   elasticity.       See   Limit,   elas- 
tic— . 
Limnoria,  954. 
Linch  pin,  defined,  1031. 
Line,  Lines,  92. 

of  action,  359. 

agonic — ,  301. 

center  of  gravity  of — ,   391. 

contour — ,  302. 

of  gravity,  389. 

hydraulic  grade — ,  519. 

isogonic — ,  301. 

of  no  variation,  301. 

parallel — ,  to  draw — ,  94. 

of  pressure,  399,  430. 

resistance—.  430,  432,  434-436. 

of  resultants,  430,  432,  etc. 

thrust-,  430,  432,  434-436. 
Lining  of  tunnels,  812. 
Link,  equivalents  of — ,  232. 
Liquid,  Liquids.      See  Water. 

buoyancy   of — ,    513,    514. 

compressibility,  326. 

flow,  516. 

friction,   415. 

measure,   223. 

pressure,   500,  518. 

transmission  of — ,  506. 

specific  gravity,  211. 
Liter, 

equivalents  of — ,  225,  235. 
Lithofracteur,  952. 
Little  bear,  constellation,  285,  286. 
Live  load.  Live  loads.     See  Loads 

live — . 
Living  force,  343. 


INDEX. 


1059 


liOad— Masonry. 


Load,  Loads, 

on  bridges,  726,  755. 

cart — ,  of  earth,  800. 

chord  stresses,  709. 

on  columns.    495,  etc.,   901,  etc., 
963,  etc. 

dead—,   690. 

on  driving-wheels,  705,  etc.,  755, 
etc.,  856-859,  861. 

on  earth,  safe — ,  583. 

for  given  deflection,  480,  481,  483. 

line,  707. 

live—    690,    705,   709,   726,   755, 
856-859,  861. 

locomotive — .       See    Loads     on 
driving  wheels. 

moving.      See  Loads  on  driving 
wheels. 

permissible —  on  beams,  473. 

for  permissible  deflections,  485. 

on  piles,  592. 

on  pillars,  495,  901,  etc.,  963,  etc. 

on  roofs,  321,  713. 

on  roof -trusses,  713,  764. 

on  sand,  582. 

stresses,  705. 

graphic  method,  706. 

suddenly  applied—,  460,  486,  959. 

on  wooden  bridges,  764. 
Loaded 

chain,  428. 

cord,    428. 
Loading, 

standard — ,  705,  755. 

of  trusses,  690. 
Local  time,  287. 
Location  of  the  meridian,   284. 
Lock, 

air—,  597. 

gates,  spacing  of  cross-bars,  506. 

nut,  821,  885. 
Locke  level,   310. 
Locomotive,  Locomotives,  856. 

adhesion  of — ,  413. 

house,  cost,  850. 

mfrs.,  990. 

statistics,  867. 

tonnage  rating  of — ,  862. 

turntables  for — ,  845. 

water  for—,  327,  851. 

wheel-load,    705,   etc.,   755,  etc., 
856-861. 
Locust,  strength,  476,  957,  958. 
Logarithmic 

chart,  73. 

plotting,  74. 

sines,  tangents,  etc.,  72. 
Logarithms,  70-91. 
Long 

chords,   table,  787. 

measure,  220,  225,  232. 

ton,  216. 
Longitude,  degree  of — ,  length,  220, 

221. 
Longitudinal  and  transverse  stresses 

combined,  493,  724. 


Lower 

chord,     design,    timber    trusses, 
733. 

chord  splice,  736. 

culmination,  284. 
Lowering  of  centers,  631,  etc. 
Lowest  terms,  36. 
Lubricants,  415. 
Lubrication  of  turntables,  846. 
Lumber.     See  also  Wood,  Timber. 

cost  and  mfrs.,  984. 
Lunation,  266. 
Lune,  circular — ,  186. 


M, 

Machine, 

drill,  675. 

funicular — ,  427. 

riveting,  775. 

for  tapping  pipes,  657,  664. 
Magnesia  in  cements,  931. 
Magnetic 

declination,  301. 

variation,   301. 
Magneto-electric  blasting,  952. 
Mahogany, 

strength,  459,  476,  957,  958. 

weight,  213. 
Mail 

cars,  865. 

earnings,  867. 
Man  power,  686. 
Mandrel,  defined,  1032. 
Manganese,  in  steel,  872. 
Mantissa,  logarithmis,  70. 
Manufacture  of  iron  and  steel,  870. 
Manufacturers,  list  of,  996. 
Map,  to  reduce  or  enlarge — ,  160. 
Maple-wood, 

strength,  476,  957. 

weight,  213. 
Marble, 

cost,  602. 

expansion  by  heat,  317. 

strength,  476,  923. 

weight,  213. 

German — ,  218,  246,  etc. 
Spanish—,  227. 
Masonry, 

in  abutments,  quantity,  623. 
adhesion  of  mortar  to — ,   926. 
inarches,  quantity,  622-628. 
and  concrete,  945. 
cost,  601,  988. 
courses, 

inclination  of — ,  603,  620. 

lead  between,  634,  921. 
dam,  433. 

foundations,  loads  on — ,  750. 
friction  of — ,  411,  603,  620. 
incrustation  of — ,  929,  936. 
joints,     distributipn    of    pressure 

on—,  400. 
mortar  required  for — ,  931. 


1060 


INDEX. 


Masonry— Mikron. 


Masonry — continued, 
in  piers,  quantity,  628. 
quantity 

in  arches,  622-628. 
in  piers,  628. 
in  retaining  walls,  610. 
in  walls  of  wells,  198. 
in  wing-walls,  624. 
railroad — ,  cost,  855. 
in  retaining  walls,  603. 
strength,  compressive — ,  923. 
weight,  213. 
Mass,  334,  336. 
Material,   Materials, 
fatigue,  465. 
particle,  358. 
point,  358. 
strength,  454. 
weight,  210,  etc. 
Mathematical  symbols,  33. 
Mathematics,  33. 
Matter,  defined,  330. 
Mattock,  defined,  1032. 
Maximum 

intensity  of  rainfall  at  points  in 

U.  S.,  table,  323. 
and  min.  stresses  in  truss   mem- 
bers, 712. 
pressure, 

angle,  etc.,  of — ,  607. 
velocity,  560. 

depth,  hydraulic—.  523,  564. 

haul,  801. 

proportional,  38. 

radius,  523,  564. 

of  ratio  and  proportion,  38. 

solar  time,  defined,  265. 

sun,  265. 

velocity,  522,  560. 
Means,  defined,   1032. 
Measure,  Measures,  216. 

apothecaries',  223,  224. 

circular —  of  angles,  34. 

circular —  of  wires,  889. 

commercial — ,  size  of — ,  by  weight 
of  water,  224. 

common — ,  35. 

conversion  tables,  228,  etc. 

cubic—,  222,  234. 

fluid—,  223,  224. 

long—  220,  232. 

metric—,    217,    225,     etc.,     228, 
etc. 

Russian — ,  227. 

Spanish — ,  227. 

square — ,  222,  233. 

weights,  etc.,  conversion  tables  of 
units  of—,  228. 

wine—,  223,  224. 
Measuring  weirs,  547,  646. 
Mechanics,  330. 

of  arch,  430-432. 

of  beam,  437,  etc.,  466,  etc. 

of  masonry  dam,  430,  433-436. 

of  trusses,  698,  etc. 
Melting  points,  317.        , 


Mercury, 

barometer,  312,  320. 

foot  of — ,  etc.  (pressure),  equiva* 
lents  of—,  241. 

freezing-point,  318. 

thermometer,  318. 

weight,  213. 
Meridian, 

location  of — ,  284. 

of  longitude,  220,  221. 

variation  of  compass,  296,  301. 
Metacenter,  514. 

Metal,  Metals.     See  also  Iron,  and 
Steel. 

blasting  of — ,  950. 

cohesive  strength,  920. 

compressibility,  459. 

compressive  strength,  921. 

ductility,  459. 

effect  of  cement  on — ,  926,  936. 

effect  of  heat  on — ,  317,  819. 

effect  of  lime  on—,  926,  936. 

effect  of  mortar  on—,  926,  936. 

effect  of  water  on—,  327,  594. 

elastic  limit,  459. 

expansion  by  heat,  317,  819. 

limit  of  elasticity,  459. 

modulus  of  elasticity,  459. 

preservation  of —  by  cement,  936. 

roof  trusses,  740. 

shearing  strength,  499. 

sheet-,  880,  881,  887,  916-919. 

strength,  454,  476,  499,  500,  920, 
921. 

stretch  of — ,  459. 

tensile  strength,  920. 

torsional  strength,  500. 

transverse  strength,  476. 

weight,  210,  etc.,  etc. 

equivalents  of — ,  225,  233. 
Ferris-Pitot— ,  536. 
length,  217,  225,  233,  etc. 
Pitot— ,  536,  561,  562. 
radii,  etc.,  of  curves  in — ,  786. 
Venturi— ,  532,  etc. 
water — ,  649,  cost  and  mfrs.,  994. 
wheel,  562. 
Metric, 

atmosphere,  240,  320. 
horse-power,     equivalents     of — , 

245. 
horse-power     hour,     equivalents 

of—,  237. 
measures,  217,  225,  etc.,  228,  etc. 
railroad  curves,  tables,  786. 
screw     thread,      international — , 

883. 
ton,  equivalents  of — ,  236. 

weights.} See  Metric  measures. 
Mica,  weight,  213. 
Middle 

ordinates,  180,  784,  786,  788,  817. 
840. 

third,  402. 
Mikron,  equivalents  of — ,  232. 


INDEX. 


1061 


Mil— N  arro  w-g'ang'e. 


Mil, 

equivalents  of — ,  232. 
Mile,  Miles, 

equivalents  of — ,  232. 

freight  ton-mile,  867. 

geographical — ,  220. 

per  hour,  etc.,  equivalentsof— ,242. 

land  and  sea—,  220,  233. 

nautical — ,  220. 

passenger — ,  867. 

sea—,  220. 

square—  (section),  222,  23a 

ton—,  867. 
Milligram, 

equivalents  of — ,  236. 
Milliliter,  equivalents  of—,  235. 
Millimeter,  equivalents  of — ,  233. 
Mills,  floating-,  578. 
Miner's  friend  powder,  951. 
Miner's  inch,  546. 
Minim,  223,  224. 
Minimum 

and  maximum  stresses  in  truss 
members,  712. 

sections,  722. 
Minute,  Minutes, 

in  decimals  of  a  degree,  95. 

equivalents  of — ,  236. 

of  time,  265. 
Mitred  joints  for  rails,  819. 
Mitre-joint,  defined,  1032. 
Mixing  cement  for  briquettes,  938. 
Mizar,  285,  287. 
Models, 

of  beams, strength  and  weight,  478. 

in  force  composition,  380. 
Modern  explosives,  948. 
Modified  logarithms,  72. 
Modulus.    See  Coefficient,  Strength. 

defined,   1032. 

of  elasticity,  456,  459.    . 

of  flow,  540. 

of  resilience,  460. 

of  rupture,  468. 

section — ,  467-8,  473,  892  to  898. 
Mogul  locomotives,  856. 
Moisture, 

effect  of — , 

on  cement,  934. 
on  sound,   316. 
on  zinc,  917. 
Molded  concrete,  945. 
Molds  for  cement  briquettes,  939. 
Molecular  action,  358. 
Moment,  Moments,  360. 

in  arches,  424. 

in  beams,  440,  443. 

in  cantilevers,  440,  442. 

in  continuous  beams,  489. 

of  couple,  405. 

defined,  360,  1032. 

diagrams,  479. 
trusses,  707. 

of  inertia,  351. 
in  beams,  468. 

influence  diagrams,  449. 

in  levers,  419. 


Moment,  Moments — continued. 

live  load—,  709. 

maximum  bending — ,  474. 

of  non-coplanar  forces,  381. 

resisting — ,  467. 

and    shear,    relation    between — ■; 
452 

of  stability,  422,  508,  514,  608. 

summation  of — ,  466. 

in  trusses,  440,  701. 
Momentum,  338,  345. 
Money,  218. 
Monkey-switch,  826. 
Mont  Cenis  tunnel,  812. 
Month,  civil — ,  sidereal — ,  synodic— j 

266. 
Morin's  laws  of  friction,  410. 
Mortar, 

adhesion  of — ,  926. 

in  arches,  616,  629,  633. 

bricks,  etc.,  925. 

cement — ,  931. 

clay,  effect  on—,  926,  935,  936. 

effect  on  iron,  770,  926,  936. 

effect  on  wood,  926. 

grout,  926. 

in  retaining  walls,  604. 

rubble — ,  cost,  602. 

weight.      See    under  Masonry, 
213 

salt,  effect  on — ,  926,  936. 

sand  for — ,  925,  926,  935,  936. 

strength,  tensile — ,  933. 

weight,  213,  926. 
Mortise,  defined,  1032. 
Motion,   331. 

circular — ,  351. 

quantity    of — ,     338. 

relative—,  331,   358. 
Mould.     See  Mold. 
Movable  bridges,  696. 
Moving  load.     See  Load,  live — . 
Muck,  defined,  1032. 
Mud, 

penetrability,  593. 

in  reservoirs,  651. 

weight,  213,  581. 
Multiple,  common — ,  35. 
Multiples  and  factors,  35. 
Multiplication 

of  decimals,  37. 

of  fractions,  36. 

by  logarithms,  71. 
chart  or  slide  rule,  75. 
Muskrats,  651. 
Myriagram,  equivalents  of — ,  236. 

N. 

Nails, 

cost  and  mfrs.,  986. 

shingling,  971. 

slating,  970. 
Napierian  logarithms,  72. 
Narrow-gauge 

locomotives,  857. 

railroad  oars,  865. 


1062 


INDEX. 


Nasmytli— Parabolic. 


Nasmyth  pile-driver,  591. 
Natural 

cements,  930,  937,  940. 

logarithms,  72. 

sines,  97,  98. 

slope,  419,  604,  606,  610. 
Nautical  mile,   220. 
Neat  cement  tests,  941. 
Needle, 

compass—,  293,  299. 
variation  of — ,  301. 
Negative 

characteristics,  logarithms,  72. 

exponents,  logarithmic  chart,  76. 

numbers,  logarithms,  72. 
Net 

earnings  of  railroads,  867. 

iron,  net  plate,  net  joint,  774. 

section  of  tension  members,  759. 

ton,  216. 
Neutral 

axis,  466; 

surface,   466. 
Newel,  defined,  1032. 
Niagara  cantilever,  696. 
Nicholson  hydrometer,  211. 
Nickel  steel,  872. 
Nicking  test,  752,  871. 
Nitro-glycerine,  948. 
Nonagon,  148. 
Non-concurrent  forces,  375. 
Non-coplanar  forces,  380. 
Normal  component,  370. 
North 

point,  etc.,  284. 

star,  285. 
Northing,  274. 
Number,  Numbers, 

and  equivalents  in  common  use, 
conversion  tables,  231. 

of  frog,  835,  840. 

prime — ,  35. 

by  wire  gage,  887-891. 
Numerus  logarithmi,  71. 
Nut,  Nuts,  883. 

locks,  821,  885. 


Oak, 

strength,  459,  476,  499,  957,  958. 

weight,  214. 
Oblique,  Obliques, 

lines,  92. 

pillars,  498. 

pressure,  372,  504,  607. 
Obstacles  in  surveying,   to  pass — , 

281. 
Obstruction,  Obstructions, 

to  flow,  575. 

by  piers,  575. 

in  pipes,  to  prevent — ,  655. 
Octagon, 

area,  148. 

to  draw — ,  159. 
Octahedron,  194. 
Ogee,  defined,  1032. 


Oil,  Oils, 

coal —  (petroleum),  weight,  214. 

olive — ,  415. 

weight,  214. 

wells,  nitro-glycerine,  948. 
Open  channels,  flow  in — ,  523,  560. 
Open    hearth    steel,    requirements, 

872. 
Openings, 

flow  through — ,  540-542. 
Ordinate,  Ordinates, 

defined,  1032. 

elliptic — ,  189. 

to  find—,  180. 

middle—   180,  784-789,  817,  840. 

parabolic — ,  192. 

tables,  784-789,  817,  840. 
Orifices,  flow  through — ,  539,  546. 
Oscillation,  center  of — ,  351. 
Ounce,  220,  235. 

equivalents  of — ,  235. 

fluid—,  223,  224,  235. 
Outer  rail,  elevation  of — ,  787. 
Outflow,   velocity  of — ,   theoretical 

— ,  539. 
Outlet  valves,  653. 
Oval,  to  draw — ,  191. 
Overfall 

dams,  642. 

discharge  over — ,  547. 

for  reservoir,  652. 
Overturning, 

effect  of  wind,  710. 

work  of—,  422. 

P. 

Packing, 

defined,   1032. 

of  eye-bars,  722. 

piece,  775. 
Paint,  Paints,  971. 

cost,  984. 

for  iron,  763,  972. 

on  zinc,  880. 
Painting,  971. 

of  bridges,  763,  764. 
Panel,  Panels, 

diagonal  of — ,  to  find  length  of—, 
160. 

points,  692. 

reactions,  702. 

in  trusses,  692. 
Paper,  978. 
Parabola,  192,  193.     See  Parabolic. 

center  of  gravity,  394. 

to  draw-,  193. 

ordinates,  192. 

semi — ,    center   of  gravity   of — , 
394. 

tangent  to — ,  to  draw — ,   193. 
Parabolic 

arc,  192,  193. 

conoid,  209. 

frustum  of—,  209. 

curve,  192,  193. 

frustum,  192. 


INDEX. 


1063 


Parabolic— Pillar. 


Parabolic — continued. 

ordinates,  192, 

zone,  192. 
Paraboloid,  209. 

center  of  gravity  of — ,  398. 

frustum,  209. 
Paradox,  hydrostatic — ,  501. 
ParaUel 

forces,  382,  514. 
couples,  404. 
defined,  361. 
resultant  of—,  382,  399. 

lines,  to  draw — ,  94. 

plates,  292. 
Parallelogram,    Parallelograms,     95, 
157. 

angles  in — ,  95. 

force—,  364. 
Parallelopiped,  195. 

force—,  380. 
Parlor  cars,  865. 
Partial  contraction,  540,  544. 
Particle,  material — ,  358. 
Partition,   thin — ,   flow  through — , 

541. 
Passenger 

cars,  865. 

earnings,  865,  867. 

locomotives,  856,  etc. 

mile,  867. 
Paste,  lime — ,  926. 
Patterns, 

weight  of  casting,  875. 
Paving, 

Belgian—,  602. 

brick—   927. 

cost,  989. 
Payments,  equation  of — ,  42. 
P.  C,  P.  I.,  P.  T.,  780. 
Pedestals,  bridge—,  721,  750. 
Pencils,  lead,  978. 
Pendulum,  Pendulums,  350. 

hydrometric,  561. 

seconds—,  216,  351. 
Pennsylvania  R.  R. 

locomotives,  857,  859. 

track-tank,  853. 
Penstock,  defined,  1032.     See  Fore- 
bay. 
Pentagon,   148. 
Per,  Percentage,  etc.,  40. 

of  grade,  255. 

interest,  annuities,  40. 
Perch,  222. 

equivalents  of — ,  235. 
Percussion,  center  of — ,  351. 

drills,  676. 
Perimeter.    See  also  Circumference, 

wet—,  523,    563. 
Permanent  way,  815. 
Permutation,  40. 
Perpendicular,  to  draw — ,  93. 
Perpetual  snow,  -limit  of — ,  324. 
Persian  wheel,  687. 
Petroleum,  weight,  214. 
PhcBnix      segment-columns,       497, 

904,  912,  913. 


Phosphor  bronze, 

permissible  load,  762. 

requirements,  754. 

wire,  strength,  920. 
Phosphorus, 

in  steel,  753,  754,  872. 
Pi,  symbol  and  value,  34. 
Picks,  wear,  801. 
Pier,  Piers, 

abutment — ,  619. 

foundations,  582. 

masonry,  quantity  in — ,  628. 

obstructions  by — ,  575,  etc, 

of  suspension  bridges,  768. 
Pierre  perdue,  583. 
Piezometer,  518. 
Pig  iron  ton,  216. 
Pile,  Piles,  589,  etc. 

adhesion  of  ice,  594. 

bearing—,   590. 

blasting  of — ,  950. 

cost,  984. 

in  cylinders,  600. 

drivers,  590,  591,  687. 
gunpowder — ,  590. 
mfrs.,  992. 
steam — ,  591. 

driving,  590,  etc. 
by  jets,  595. 

factor  of  safety,  593. 

foundations,  589,  etc. 

friction,  593. 

grillage,  590. 

head  for — ,  594. 

hollow — ,  596. 

ice,  adhesion  to — ,  594. 

iron — ,  594. 

jet  driving,  595. 

loads  for—,  592. 

resistance  of — ,  592. 

sand—,  599,  670. 

screw — ,  594. 

sheet — ,  590. 

shoes  for — ,  593. 

sustaining  power,  592. 

water  jet  for  driving — ,  595. 

withdrawal,  594. 
Pillar,  Pillars,   495,   etc.,  901,  etc., 
907,  etc.,  963,  etc. 

of  angle-iron,  497,  912. 

capitals  of — ,  shapes  of — ,  498. 

Carnegie  Z-bar — ,  901-903. 

of  channel-iron,  497,  912. 

ends  of — ,  shapes,  495,  498. 

factor  of  safety,  909,  912. 

Gordon's  formula,  495. 

hinged  ends,  495. 

of  I-beams,  497,  912. 

iron—,  495,  etc.,  760,  901-913. 
factor  of  safety,  909,  912. 

masonry — ,  strength,  923. 

oblique—,  498. 

Phoenix  segment,  497,  904,  912. 

pin-ended — ,  495. 

radius  of  gyratipn,  496. 

with  rounded  ends,  495. 

safety  factor  of—,  909,  912. 


1064 


INDEX. 


Pillar—Plaster. 


Pillar,  Pillars — continued, 
segment — .    See  Phoenix — . 
steel — .     See  Pillar,  iron — . 
strength— ,495, 760,  901-913,  963, 

T  iron—,  497,  912. 
wooden — ,  761,  764,  963,  etc. 
Z-bar—,  901-903. 
Pillow-block,  defined,  1032. 
Pin,  Pins, 

connections,  721,   724,   725,    747, 

762. 
-end  columns,  495. 
surveying — ,  282. 
Pine, 

pillars,  963,  etc. 

strength,  459,  476,  499,  957,  958, 

963. 
weight,  214. 
Pinions  and  wheels,  420. 
Pintle,  defined,  1032. 
Pipe,  Pipes, 

air-valves  for — ,  662. 

areas  and  contents — ,  197,  526. 

bends  in — ,  537. 

branches,  661. 

brass  and  seamless — ,  919. 

bursting  of — ,  513,  663,    665,  668. 

thickness      required      to    pre- 
vent—, 511,  513. 
bursting  pressure  in — ,  518. 
cast-iron — ,  653,  658,  662,  876. 

cost  of —  and  laying,  658. 

weight,  656,  658,  876. 
cement  and  iron — ,  657. 
concretions    in — ,    to    prevent — , 

655. 
contents  and  areas,  197,  526. 
copper  seamless — ,  919. 
cost  of — ,  658. 
cost  of  laying — ,   658. 
cost  and  mfrs- — ,  994. 
couplings  for—,  656,  660,  882. 
cracks  in — ,  661. 
curves  in — ,  537. 
diameter   of—,     524,     653,     654, 
656. 

actual    and    nominal — ,      526, 
882. 

for  water-supply,  653. 

square  roots  of — ,  526. 
discharge  from — ,  516,  522. 
drain — ,  575. 
drawn  brass — ,  919. 
ferrules  for — ,  664. 
flexible  joints  for — ,  661. 
flow  in — ,  516,  etc. 

Kutter's  formula,  523,  563,  564. 
galvanic  action  in — ,  656. 
galvanized — ,  664. 
gates  for — .  666. 
gutta-percha — ,  657. 
iron — , 

cast—,  653-656,  658-662,  876. 
weight,  656,  658,  876. 

and  cement,  657. 

fittings  for-,  661,  882. 


Pipe,  Pipes — continued, 
iron — ,  continued. 

joints  for—,  660,  661,  882. 
flexible—,   661. 

laying — ,  658,  etc.,  660. 
lead—,  664.  918. 

thicknesses  of — ,  513. 
material  of — ,  effect  on  velocity, 

523. 
to  mend — ,  661. 
obstructions  in — ,   to  prevent — , 

655. 
pressure  of  water  in — ,  511,  518. 
seamless — ,  919. 
service—,  657,  664,  918. 
sleeves  for — ,  661. 
stand — ,  663. 
steam — ,  882. 
stop-valves  for — ,  666. 
street—,  653. 
tapping  of—,  657,  664. 
terra-cotta — ,  575. 
thickness  required,  511,  513,  656. 
valves  for — ,  666. 
of    varying    diameter,    discharge 

through — ,  531. 
velocity  in — ,  522-524.  ' 

water—,  653,  657,  876. 

cost    of —    and   laying,  658. 

freezing,    anti-bursting  device, 
665. 
weight,  656,  658,  876,  882. 

of  water  in — ,  525. 
wooden — ,   657. 
wrought  iron—,  656,  657,  882. 

diam,     actual   and   nominal — , 
526,  882. 

weight,  656,  882. 
Pitch,    . 

defined,  1033. 

effect  of — ,  on  wind  pressure,  714. 
of  rivets,  776. 
of  roofs,  970. 
of  screw,  436. 
weight — ,  214. 
Pitman,  defined,  1033. 
Pitot's  tube,  536,  561. 
Plane,  Planes,  148. 
of  couple,  404. 
of  flotation,  514, 
of  gravity,  389. 
inclined — . 

See  Inclined  Plane, 
of  moment,  360. 
surfaces,  148. 
trigonometry,  150. 
Plank, 

board  measure  table,  269. 

in  foundations,  582. 

sheet  piling,  590. 

thickness    for    a    given    pressure, 

586,  648. 
Plaster  of  Paris,  968. 
effect  on  metals,  936. 
price — ,  985. 
strength,  922,  923. 
weight,  214. 


INDEX. 


1065 


Plastering— Pressure. 


Plastering,  968. 
Plate,  Plates, 

bed—   750. 

buckle—,  750,  885. 

fish—,  820. 

frog—,  837. 

girders,  747. 

bracing  in — ,  749. 
details,  728. 

glass — ,  974. 

iron — ,  prices,  986.     . 

net—,  774. 

parallel—,  292. 

resistance  of — ,  492. 

steel — ,  tinned,  916. 

strength,  492. 

terne— ,  916. 

tie—,  816. 

tin—,  916. 

tinned  steel — ,  916. 
Platform, 

cars,  865. 

revolving — ,  850. 
Platinum,  214,  920. 
Plenum  process,  597. 
Plows,  cost  and  mfrs.,  992. 
Plug,  fire—  (fire-hydrant),  669. 
Plumb  level,  to  adjust — ,  311. 
Plumbago  as  a  lubricant,  415. 
Plummet,  defined,  1033. 
Plunger,  defined,  1033. 
Plus,  33,  782. 
Pneumatic 

dome,   Hopkins' — ,   665. 

foundations,  596. 
Pocket  sextant,  297. 
Point,  Points, 

of  application  of  force,  333,  359. 

boiling—,  314,  326. 
levelling  by — ,  314. 

of  curve,  780. 

freezing — ,  326. 

frog—,  835-843. 

of  intersection,  780. 

material — ,  358. 

melting — ,  317. 

position  of — ,  to  find — ,  156. 

switch—,  828,  830. 

of  tangent,  780. 
Pointers — ,  astronomy,  285. 
Pointing  with  cement,  936. 
.  Polar  distance,  284. 

and    azimuth    of    Polaris,   table, 
290. 
Polaris,   285-290. 
Pole,  north—,  284. 
Polygon,  Polygons,  148. 

cord—,  377,  428. 

force—,  374,  377. 

irregular — ,  to  find  area  of — ,  160. 

to  reduce  to  a  triangle,  159. 

regular — ,  to  draw — ,  159. 
Polyhedron,  Polyhedrons,  regular — , 

194. 
Pond,  discharge  of — ,  time  required 

for — ,  545. 
Pony  trusses,  692. 


Pood,  227. 

Poplar,   strength,   476,   957,   958. 

Porous  bodies,  specific  gravity,  211. 

Port,  establishment  of — ,  328. 

Portal  bracing,  691. 

Portland  cement,   930,  etc. 

Posts,  359.      See  also  Pillars. 

design    of — ,    722. 
timber  trusses,  733. 

fence — ,   854. 

pivot —  in  turntables,  846. 

and  ties,  689. 
Potential  energy,  346. 
Pound 

equivalents  of — ,  235. 

Fahrenheit,  equivalents  of — ,  237. 

sterling,  value,  218. 

weight,  220. 
Pouring-clamps  for  pipe  joints,  660. 
Powder,  214,  953. 
Power,  Powers.     See  Steam,  Water, 
Wind,  Animal,  etc. 

animal — ,  685. 

defined,  342. 

fifth—   67-69. 
sq.  rts.  of — ,  69. 

finding —  by   logarithmic    chart, 
76. 

finding —  by  logarithms,  71. 

finding —  by  slide  rule,  76. 

of  horse,  683,  685,  852.     See  also 
Horse-power. 

in  levers,  419. 

of  locomotives,  860,  861. 

man — ,  686. 

second  and  third — ,  tables.  55,  etc- 

tractive — ,  683. 

units  of — ,  conversion  of — ,  244. 
Pratt  truss,  692. 
Precipitation  (rainfall),  322. 

in  the  U.  S.,  table  of  details  of—. 
325. 
Present  worth,  41,  42,  44. 
Preservation 

of  metals  by  cement,  936. 

of  timber,  954. 
Press,  Presses, 

hydrostatic — ,  506. 
Pressed 

brick,  927. 

steel  cars,  865. 
Pressure.     See  Load. 

of  air,  320,  597. 

in  arches,  614,  616. 

of  atmosphere,  320. 

barometer — ,  320. 
levelling  by — ,  312. 

center  of—,    399,   501,    506,  514, 

on  centers  of  arches,  633. 

in  dams,  648. 

distribution  of — ,  400. 

of  earth,  603,  607. 

on  foundations,  583. 

head,  258,  518. 

hydrostatic — ,  501,  etc. 

on  inclined  planes,  349. 

line  of—,  430,  etc. 


■M 


1066 


INDEX. 


Pressure — Ramming. 


Pressure — continued. 

maximum — , 

angle,  prism,  slope  of — ,  607. 

in  pipes,  511,  518. 

plate,  Gauthey's,  561. 

in  reservoirs,  651. 

on  retaining  walls,  603,  607. 

of  running  streams,  578. 

of  running  water  in  pipes,  518. 

steam  cylinder — ,  861. 

transmission  of —  through  liquids, 
506. 

unit — ,  conversion  of — ,  240. 

of  water,  501,  etc.,  516,  etc. 
in  cylinders,  511. 
in  pipes,  511,  518, 
plank  to  resist — ,  586,  648. 
running,  518,  578. 
walls  to  resist — ,  508. 

of  wind,  321. 

on  roof  trusses,  714. 
Price  list,  983. 
Prime, 

defined,   1033. 

number,  35. 
Principal,  in  interest,  41. 
Printing,  blue—,  979. 
Prints, 

black-line—,  982. 

blue—,  979. 
Prism,  Prisms,   195. 

center  of  gravity  of — ,  395. 

frustums  of — ,   195. 

center  of  gravity  of — ,   395. 

of  max.  pressure,  607. 
Prismoid,  202. 
Prismoidal  formula,  202. 
Profile,  Profiles,  304. 

paper,  978. 

transformation  of — ,  611. 
Progression,    39. 
Projection,  defined,  1033. 
Proportion, 

by  logarithms,  71. 

and  ratio,   38. 
Proportionals,  38. 
Protection  of  bridges,  763,  764. 
Protracting  by  chords,  143. 
Puddle, 

defined,   1033. 

walls,  651. 
Pug-mill,  defined,   1033. 
Pull 

and  push,  359. 

on  tapes,  surveying,  282. 
Pulley,  428. 
Pump,  Pumps,  852. 

chain — ,  687. 

cost  and  mfrs.,  991. 

day's  work  at — ,  686,  852. 

sand—,  599,  670. 
Purlins,   713. 

defined,    1033. 
Push  and  pull,  359. 
Puzzolan  cement,  940. 
Pyramid,  Pyramids,  200. 

frustum  of—,  201. 


R. 


Rabbet,  defined,  1033. 
Race,  defined,  1033. 
Rack-a-rock,  951. 
Radii,  Radius, 

to  find—,  161,  179. 
of  gyration,   352,   353,   495,   496, 
892,  etc. 

square  of — ,  496. 
mean,  523,  564. 
of  railroad  curves,  784-786. 
of  turnouts,  840. 
Rail,  Rails,  817. 

bending — ,  ordinates  for^-,  817. 
cost  and  mfrs.,  994. 
creeping  of—,  819,  820. 
elevation  of  outer — ,  787. 
expansion  by  heat,  317,  819. 
fence — ,  854. 
frog,  835. 
guard  or  guide—,  750,  828,  833, 

835. 
joints,  819. 

ordinates   for   bending — ,    817. 
outer — ,  ' 

elevation  of — ,  787. 
roads,  780-869.  * 

acres  required  for — ,  254, 

ballast,  815. 

bridges.       See    Bridge,    Truss, 
Arch,  etc. 

construction,  855. 

cost,  855. 

cross-ties,  815. 

resistance  on — ,  417. 

roadway,  815. 

shops,  cost,  850. 

slopes,  255-257. 

spikes,  818. 

switch,  824. 

ties,  815. 

time,  standard — ,  267. 

track  tank,  853. 

traction  on^ — ,  860. 

turnout,  824. 

water  stations,  851. 
safety—,  833. 
steel — ,  requirements,  872. 
stock—,  828. 
switch-length,  830. 
way.     See  Railroad. 
Rain,  322. 
fall,  322. 

depths,  equiv.  volumes,  250. 

equivalent  of  snow,  324. 
gages,  324. 

reaching  sewer,  rate,  575. 
and  snow,  322. 
water,  327. 
Rainy  days,  table  of  average  num- 
ber of—,  325. 
Ram,  Rams, 

hydraulic — ,  578. 

cost  and  mfrs.,  991. 
water—,  513,  663,  668. 
Ramming  concrete,  945. 


INDEX. 


1067 


Random — Roller. 


Random  stone,  583. 

defined,   1033. 
Ratio, 

elastic—,  458,  461. 

and  proportion,  38. 
Reaction,  333. 

end—,  360,  439. 

in  trusses,   699,  702,  714. 

of  fibers,  466. 

of  soils,  elastic — ,  593. 
Reaumur  thermometer,  318. 
Reciprocal,  Reciprocals,  48-53. 

or  inverse  proportion,  39. 

on  logarithmic  chart,  76. 

by  logarithms,  71. 

by  slide  rule,  77. 
Rectangle,  Rectangles,  157. 
Rectangular 

components,  369. 

plates,  strength  of — ,  492. 
Recurring  decimals,  38. 
Reduction 

of  area,  752,  754,  873. 

of  figures,   160.  ' 
Redundant  members,  720. 
Reflection,  to  measure  heights  by — , 

155. 
Refraction    and    curvature    tables, 

153. 
Regular  figures,  148. 
Regular  solids,  194. 
Regulation    of   time-pieces   by   the 

stars,   266. 
Reinforcing  plates,  724,  747. 
Relative  density,  210. 
Renewal  of  bridges,  743. 
Repair,  Repairs, 

of  bubble-tube,  296. 

of  cars,  865. 

of  cross-hairs,  296. 

of  pipe,  661. 

in  reservoirs,  652, 

of  road,  801. 

of  rolling  stock,  865. 
Repeating  decimals,  38. 
Reservoir,  Reservoirs,  650. 

evaporation  from — ,  329. 

for  railroads,  852. 
Resilience,  460. 
Resistance 

of  cars,  417. 

to  flow,  523,  537,  563. 

of  foundations,  583,  592. 

on  grades,  860. 

line,  430,  432,  434-436. 

of  piles,  592. 

of  plates,  492. 

on  railroads,  417. 
Resolutes,  369. 

Resolution  of  forces,  362,  etc. 
Rest,  relative — ,  358. 
Resultant, 

of  forces,  362. 

line  of—,  430,  432. 

of  moments,  360. 

of  parallel  forces,  382,  399. 

sense  of — ,  366. 


Retaining  walls,  603. 

masonry  in — ,  quantity  of — ,  610^ 
612. 

surcharged,  605. 

theory  of,  606. 

transformation  of  profile,  611. 
Reverse  bearing,  277. 
Revetment,  612. 

defined,   1034. 
Revolving  bodies,  351. 
Rhomb,  157,  195. 
Rhombic  prism,  195. 
Rhombohedron,  195. 
Rhomboid,    157. 
Rhombus,  157. 
Rhumb-line,  277. 
Ridge-pole,  defined,   1034. 
Right  angle,  to  draw,  93. 
Rigid  bodies, 

force  in—,  330,  358. 
Rigidity  in  bridges,  721. 
Ring,  Rings,  cu-cular,  186,  209. 
Rip-rap,  583. 

defined,   1034. 
Rise  of  arch,  613. 
River,  Rivers.  See  Water,  Rain,  etc. 

dams,  642. 

flow  in—,  560. 

scour  of — ,  577.  - 
Rivet,  Rivets,  772. 

stresses  in — ,  permissible — ,  762. 
Riveted 

connections,   721. 

joints,  749,  772. 
Road,  Roads, 

cart — ,  repairs,  801. 

grade,  255,  683. 
tables,  255-257. 

maintenance,  801. 

rail — .    See  Railroad. 

rollers,  mfrs.  of — ,  992. 

traction  on — ,  683. 

-way,  acres  required  for — ,  254. 
drainage  of — ,  in  arches,  628. 
Rock,  Rocks, 

blasting,  948,  etc. 

broken,  voids  in — ,  688,  810,  943. 

channeling,  681. 

crushers,  cost  and  mfrs.  of — ,  992. 

drill,  600. 
hand—,  681. 
machine — ,  675. 

removal,  810,  811. 

weight,  212,  etc. 

work  in  tunnels,  812. 
Rocker  bearings,  725,  730. 
Rod,   Rods, 

of  brickwork,  928. 

cost  and  mfrs.,  993. 

equivalents  of — ,  232. 

upset—,  886. 
Rolled  iron.     See  Iron,  wrought—, 
and  Steel, 

cost  and  mfrs.,  986. 
Roller,  Rollers, 

anti-friction—.  417,  751,  846. 

bearings,  725-728. 


1068 


INDEX. 


Rolling— Scrapers. 


Rolling 

friction,  414. 

lift  bridges,  697. 

load.  See  Load,  live — . 

stock,  856,  865,  867. 
resistance  of — ,  417. 
Roof,   Roofs, 

acid  fumes  on — ,  880,  970. 

copper — ,  918. 

iron  for—,  880. 

lead—,  918. 

leak  in—,  to  stop — ,  971,  973. 

painting,  764,  880,  972. 

pitch  of—,  916,  970. 

sheet-iron — ,  880. 

shingle — ,    971. 

slate — ,  weight,  970. 

tin—,  916. 

trusses,    698,   713.     See  Truss, 
loads  on — ,  764. 
metal — ,  740. 
specifications  for — ,  764. 

wind  on—,  321,  713. 

zinc—,  916. 
Roofing, 

cost  of—,  989. 
Root,  Roots, 

cube  and  square — ,  tables,  54. 

of  decimals,  to  find — ,  67. 

fifth—,  67,  68. 

finding —  by  logarithmic  chart,  76. 

finding —  by  logarithms,  71. 

finding —  by  slide  rule,  76. 

of  large  numbers,  to  calculate — , 
66. 

square — ,  tables,  54. 
of  diameters,  526. 
of  fifth  powers,  69. 
Rope,  Ropes,  975. 

cost  and  mfrs.,  987. 

strength,   922,  975. 

wire—,  976,  977. 
Rosendale  cement,  931. 
Rosin,  weight,  214. 
Rot,  dry—,  954. 
Rotary 

drills,  675. 

motion,   351. 
Rotating  bodies,  351. 
Rotundity  of  the  earth,  153. 
Rough  casting,  973. 
Roughness, 

coefficients  of—,  523,  564,  565. 
Rubble, 

adhesion  to  mortar,  926. 

arches,  616. 

cost,  602, 

defined,   1034. 

proportion  of  mortar  in — ,  213. 

retaining  walls,   610. 

strength,  923. 

voids  in—,  688,  799. 

weight,  213. 
Rule,  Rules, 

of  three,   39. 

two-foot — ,     to    measure    angles 
by—,  96. 


Run-off,  equivalents  of — ,  251,  252. 
Rupture,  modulus  of — ,  468. 
Russian     weights     and     measures, 

227. 
Riitger's  process,  955. 


Sachine,  227. 
Safety, 

castings,  824,  833. 
factor  of — , 

for  beams,  959. 
for  piles,  593.  • 

for  pillars,  495,  909,  912,  913. 
for  retaining  walls,  605. 
for  suspension  bridges,  767. 
rail,  833. 
Sag, 

of  tape,  correction  for — ,  282. 
in  trusses,  718. 
Salt, 

effect  of—  on  mortar,  926,  936. 

effect  of —  on  iron,  327,  594. 
weight,  326. 

weight,  214. 
Sand, 

augers,  670. 

blasting  of — ,  950. 

cement,   937. 

in  cement,  effect,  931,  etc. 

for  cement,  quantity,  935. 

for  centers,  striking,  633. 

in  concrete,  943,  etc. 

cost,  985. 

dredging,  580. 

effect  of —  in  cement,  932,  etc. 

excavating  in — ,   800. 

for  foundations,  582. 

natural  slope,  419,  610. 

penetrability,  593. 

piles,  599,  670. 

in  plaster,  968. 

pressure,   603. 

price  of — ,  985. 

pump,  599,  670. 

required  in  mortars,  931. 

retaining  walls  for — ,  603. 

slope  of — ,   natural — ,  419,  610, 

specific  gravity,  211,  214. 

stone,  expansion  by  heat,  317. 
strength,  476,  922,  923. 
weight,  214. 

sustaining  power,  583,  593. 

voids  in—,  214,  935. 

weight,  211,  214. 
Sandage,   216. 
Sap,  954. 

Scabble,  defined,   1034. 
Scale,  Scales,  track — ,  854. 
Scantling,  defined,   1034. 
Scour  of  streams,  577. 
Scrapers, 

cost  and  mfrs.,  992. 

earthwork  by — ,   805. 


INDEX. 


1069 


Screeding:— IShillingr. 


Screeding,  968. 
Screw,  Screws,  436. 

Archimedes,    687. 

cylinders,  594. 

levelling—,  292,  307. 

piles,    594. 

standard  dimensions,   883. 

for  striking  centers,  633. 

thread,  metric — ,  883. 
Scribe,  defined,  1034. 
Sea, 

mile,  220. 

tides,   328. 

water,  326,  328,  594. 

worms,  954. 
Seamless, 

pipes  and  tubes,  919. 
Seasoning,  954. 
Secant,  97. 
Second,    Seconds, 

in  decimal  of  a  degree,  95. 

equivalents  of — ,  236. 

to  estimate — ,  266. 

pendulum,  216. 

of  time,  defined,  265. 
Section, 
■     equivalents  of—,  222,  233. 

of  land,  area,  222,  233. 

of  members,  minimum — ,  722. 
in  timber  trusses,  733. 

method  by — ,  stresses  in  trusses, 
700. 

modulus,  467,  468,  473,  892,  894, 
896,  898. 

net —  in  tension  members,  759. 
Sector, 

center  of  gravity,  393. 

circular,   186. 

spherical — ,     center    of    gravity, 
396. 
Secular  magnetic  variation,  301. 
Sediment  in  reservoirs,  651. 
Segment,  Segments, 

circular—,    186,    187,    394. 

columns.     Phoenix — ,    497,  *  904, 
912,  913. 

spherical — ,  208. 

center  of  gravity,  395. 
Sellers'     standard     dimensions     of 

bolts,  etc.,  883. 
Semi- 
circle,  center  of  gravity,    391. 

parabola,      center      of      gravity, 
394. 
Sense, 

of  force,  359. 

of  moment,  360. 

of  resultant,  366. 
Separators  for  I-beams,  900. 
Series,    arithmetical   and   geometri- 
cal— ,  39. 
Service  pipe,   657,   664,   918. 
Set-screw,  defined,  1034. 
Setting 

of  cement,  930,  etc. 
Settling 

of  arch,  432. 


Settling — continued. 

of  backing,  604. 

of  centers,  633,  640. 

of  embankment,   799. 
Sewer,  Sewers, 

cost,  989. 

flow  in — ,  574. 

Kutter's  formula,  523,  563,  564. 

rain   water,    rate   of   reaching — , 
575. 

velocities  in — ,  574,  575. 
Sextant, 

angles  measured  by — ,  152. 

box  or  pocket — ,  297. 

center  of  gravity  of — ,   393. 
Shackle,    defined,    1034. 
Shadows,    equal —   from    the    sun, 

location  of  meridian  by — ,  288. 
Shaft, 

revolving — ,  longitudinal  friction, 
419. 

of  tunnel,  812. 
Shafting, 

friction,  416. 

strength,  500. 
Shale,  weight — ,  214. 
Shapes,    structural — ,    tables,    892- 
898. 

T— ,    898. 
Sharpening  tools,  cost,  801. 
Shear,   Shears, 

in  beams,  446. 

in  continuous  beams,  489. 

defined,   1034. 

diagrams,    479. 
trusses,  702,  706. 

double  and  single — ,  499,  774. 

horizontal —  in  beams,  478. 

influence      diagrams     for — ,      in 
beams,  450. 

and  moments,  relation  between — , 
452. 

in  trusses,  702. 
web  stresses,  706. 
Shearing, 

of  rivets,  774. 

strength,  499. 
of  cements,  9.34. 

stresses,  permissible — ,  762. 

in  timber  construction,  732. 
Sheet,  Sheets, 

copper — ,  918. 

iron—,  880. 

corrugated — ,  880. 

galvanized — ,  880. 

roof,  880. 

and  steel,  cost  and  mfrs.,    986 

lead,  918. 

metals,  thickness,  887-890. 

piles,  590. 

zinc,  916. 
Sheeting  of  centers,  631,  etc.,  639. 
Shell, 

-lime,  926. 

spherical — ,  208. 
weight,  875,  877. 
Shilling,  value,  218. 


1070 


INDEX. 


Staing^les— ISpberical. 


Shingles,  971. 
Shoes, 

bridge — ,  721. 

for  piles,  593. 
Shops,  railroad — ,  cost,  850. 
Shore,  defined,  1034. 
Short  ton,  216. 
Shoveling  earth,  800. 
Shovels,  wear  of — ,  801. 
Shrinkage  of  embankment,   799. 
Sidereal,    day,    month,    time,    year, 

266. 
Sieves,  for  cement  tests,  938. 
Signal  target,  826,  829,  833. 
Silica  cement,  937. 
Silicate  of  alumina,  930. 
Silver, 

coins,  etc.,  218. 

strength,  920. 

weight,  214,  219. 
Similarity,  geometrical — ,  92. 
Simple  interest,  41. 
Sine,  Sines,  97. 

logarithmic — ,  72. 

natural — ,  defined,  97. 
table,  98. 

by  slide  rule,  77. 
Single 

riveting,  772. 

rule  of  three,  39. 

shear,  499,  774. 
Sinking  fund,  43. 
Siphon,   520. 
Skew 

back,   613. 
defined,   1034. 

bridge,  697. 
Skidding  of  wheels,  413. 
Slacking  of  lime,  925,  926,  930. 
Slag  cement,  940. 
Slaking,  925,  926,  930. 
Slate,  969. 

compressibility,  etc.,  459. 

expansion  by  heat,  317. 

roofs,  weight,  970. 

strength, 

compressive — ,  923. 
tensile — ,  922. 
transverse — ,  476. 

weight,  214. 
Sleeping  cars,  865. 
Sleeves  for  pipes,  661. 
Slide  rule,  73. 
Slope,   Slopes, 

angle  of—,  255,  256. 

description  of — ,  255,  256. 

earthwork,  description  of — ,  256. 

hydraulic—,  523,  564. 

instrument,  256,  311. 

Kutter's  formula,  523,  563,  564. 

of  maximum  pressure,  607. 

natural—,  419,   604,   606,   610. 

railroad—,  255,  256. 

structures  built  upon — ,  424. 

tables,  255-257. 

of  tapes  and  chains,  corrections 
for—,  283. 


Slope,  Slopes — continued, 

in  tunnels,  812. 
Sloping  weirs,  558. 
Sluice,  defined,  1035. 
Sluices  in  dams,  645. 
Snow,  214,  323. 

load,  713. 

rainfall  equivalent,  324. 
Soakage,  loss  by — ,  329,  561,  651. 
Soap, 

as  a  lubricant,  415. 

stone,  weight,  214. 

wash  for  walls,  928. 
Soffit,  defined,  613,  1035. 
Soil,  Soils, 

boring  in — ,   670. 

dredging  in — ,  580. 

excavation  of — ,   800. 

leakage  through — ,  329,  561,  651. 

penetrability,  593. 

pressure  of — ,  603. 

reaction  of — ,  elastic — ,  593. 

scour,  577. 

sustaining  power  of — ,  583,  593.    ' 

weight,  212.     See  under  Earth. 
Solar  time,  mean  and  apparent — ; 

265. 
Solid,  Solids,   194. 

center  of  gravity,  396. 

defined,  92. 

expansion  by  heat,  317. 

floors,  721,  750,  914. 

measure,  222,  234. 
metric—   225»  234. 

mensuration  of — ,  194. 

specific  gravity,  210. 

surface  of — , 

center  of  gravity,  395. 
Sound,  316. 

Soundness  of  cement,  940. 
Southing,  274. 
Sovereign,  218. 
Span,  613,  759. 
Spandrel,  613-618. 

defined,   1035. 
Spanish-    weights    and    measures, 

227. 
Spanner,  defined,  1035. 
Specific  gravity,  210,  etc. 
Specifications, 

for  bridges  and  buildings,  745. 

for  combination  bridges,  763. 

for  iron  and  steel,  870. 

for  roof  trusses,  steel  framework 
and  buildings,   764. 

for  wooden  bridges,  763. 
Speed,  Speeds, 

of  teams,  801,  806. 

of  trains,  to  estimate — ,  866. 
Spelter.     See  Zinc. 
Sphere,     Spheres.       See    Spherical, 
204,  205,  222,  396,  875,  877,  918. 

volume  of — ,  222. 
Spherical 

sector,  center  of  gravity,  396. 

segment,  208. 

center  of  gravity,  395. 


INDEX. 


1071 


Spberical— steel. 


Spherical — continued. 

shell,  208,  875,  877. 

zone,    208. 

center  of  gravity,  396. 
Sphericity  of  the  earth,  153. 
Spheroid,  208. 

center  of  gravity,  395. 
Spigot,  defined,   10.15. 

in  pipe  joint,  660. 
Spikes,  818. 
Spindle, 

circular — ,  209. 

torsional  stress  in — ,  500. 
Splice  bars,  requirements,  872. 
Splice,  timber — ,  736. 
Split  switch,  828-830. 
Spreading  of  earth,  801. 
Spring,  Springs, 

of  arch,  613. 

in  foundations,  583. 

frog,  838. 
Spruce, 

strength,    459,     476,    499,     957, 
958. 

weight,  214. 
Spudding,  672. 
Spur-wheel,  defined,  1035. 
Square,  Squares.     See  also  Powers. 

area,  157. 

equivalents  of —  in  circles,  161. 

center  of  gravity,  391. 

measure,  222,  233. 

conversion  table,  233. 
metric — ,  225. 

mensuration  of — ,   157. 

of  radius  of  gyration,  496. 

of  roofing,  970,  1035. 

roots,  54. 

of  decimals,  to  find — ,  67. 

of  diameters,  526. 

of  fifth  powers,  69. 

of  large  numbers,  to  calculate 

— ,   66. 
tables,  54. 

sides  of—,  157,  161. 

tables  of — ,  55. 
Stability,  422,  514. 

of  arches,  430,  432,  620. 

of  dams,  433,  510. 

frictional — ,  409. 

on  inclined  planes,  424. 

of  retaining  walls,  603. 
Stable  equilibrium,  387,  514. 
Stadia  hairs,  293. 
Stand,  switch—,  826. 
Stand  pipes,  663. 

for  railroad  water-station,  852. 

for  water-works,   663. 
Standard 

railway  time,  267. 

wheel  loads,  705,  etc.,  755,  etc. 
Starlings,  defined,  1035. 
Stars,  to    regulate    a    watch,   etc., 

by—   266. 
Static  friction,  407. 
Statics,  330,  358. 

of  arch,  430,  432. 


Statics — continued. 

of  beam,  437,  etc.,  466,  etc. 

graphic—,  428-431,  435. 

of  masonry  dam,  430,  433-436. 

of,  trusses,  698,  etc. 
Station,  Stations, 

in  surveys,  309. 

water — ,    851. 

way — ,  cost,  854. 
Stationary  engines,  cost  and  mfrs. 

of—    990. 
Stays,  cable,  766. 
Steam, 

dredges,  580. 

engines, 

locomotives,  856. 
pumps — ,   852. 

excavator,  808. 

pile  drivers,  590,  591. 

pipes,  882. 

rock-drill,    675. 
Steel, 

angles,  896,  898. 

beams,  476,  892. 

bending  tests,  873. 

in  bridges,  requirements,  751. 

cars,  865. 

castings  in  bridges,  754. 
requirements,  872. 

channels,  894. 

cohesive  strength  of — ,  920. 

columns.  See  Pillars,  iron — . 

composition  of — ,  753,  872. 

compressibili+y,   459. 

compressive  strength,  921. 

cost  and  mfrs.,  986. 

ductility,  459. 

elastic  limit,  459. 

expansion  by  heat,  317. 

forgings,  requirements,  872. 

framework,  specifications,  764. 

friction,   411. 

I-beams,  892. 

manipulation,  751. 

manufacture,  751,  870. 

modulus  of  elasticity,  459. 

open  hearth — ,  requirements,  872. 

pillars.    See  Pillars,  iron — . 

plates,  tinned,  916. 

price,  986. 

rails,  frogs  of — ,  835. 

requirements,  751,  872. 

roof  trusses,  740. 

rope,  976,  977. 

shearing  strength,  499. 

shop  work  on — ,  751. 

specifications,  870. 

strength,  476,  499,  500,  870,  920, 
921. 

stresses  in — ,  permissible — ,  760. 

stretch,    459. 

structural — ,  requirements,  872. 

tensile  strength,  920. 

tests,  bending — ,  873. 

torsional  strength,  500. 

transverse  strength,  476. 

weight,  214,  877,  878. 


1072 


INDEX. 


Steel— Structural. 


Steel — continued. 

wire,  891. 

rope,  976,  977. 

yard,    383. 
Stere,  equivalents  of — ,  235. 
Stiffeners,    748. 
Stiffness  in  bridges,  721. 
Stirrups,  timber  framing,  734. 
Stock  rails,  828. 
Stone,  Stones, 

arch — ,    613. 

in  arches,  quantity,  622. 

artificial — ,   943. 

ballast,  815,  855. 

beams,  476,  924. 

bridges,  613. 

centers  for — ,  631. 

broken — ,  voids  in — ,  688,  943. 

cohesive  strength,  922. 

compressive  strength,  923. 

cost,  985. 

crushers,  943. 

cutter's  day's  work,  601. 

dams,  400,   etc.,    430,    etc.,    433, 
etc.,  508,  510. 

dressing,    601. 

drilling,   600. 

expansion  by  heat,  317. 

friction,   411. 

key—   613. 

quantity    of — ,    in    arches,    etc., 
622. 

quarrying,  600,  601. 

random — ,  583. 

strength,  476,  922,  923. 

tensile  strength,  922. 

transverse  strength,  476. 

weight,  212,  etc. 

work,  600,  809. 

mortar  required  for — ,  931. 
strength,   923. 
weight,   213. 
Stop,  Stops, 

corporation —  for  pipes,  657,  664. 

valves  for  water  pipes,   666. 
Storage  reservoirs,  652. 
Stove-up,  defined,   1035. 
Strain,  Strains,  454.  455. 
Straps,  timber  framing,  735. 
Stratum,  defined,  1035. 
Stream,  Streams, 

abrasion  by — ,  577,  578. 

flow  in — ,   560. 

-flow  and    precipitation,   relation 
between — ,   323. 

to  gauge — ,  560. 

horse-power  of — ,  578. 

pressure  of  running — ,  578. 

scour  of — ,  577,  578. 

virtual  head,  578. 
Strength,      Strengths.        See      also 
article  in  question. 

of  arches,  368,  430,  etc.,  613. 

of  beams,  466,  473,  476,  478,  892. 

of  cast  iron,  874. 

of  cement,  932,  etc. 

tests  for—,  939,  940,  etc. 


Strength,  Strengths — continued. 

of  chains,  915. 

of  channels,  894. 

cohesive—,  454,  920,  922,  957. 

compressive — ,  454,  921,  923,  958. 

of  concrete,  944. 

of  cylinders,  511, 

of  iron,  476,  499,    500,   870,    907, 
etc.,  920. 

of  materials,  454. 

of  piles,  592. 

of    pillars,     495,    etc.,    901,    etc., 
907,  etc.,  963,  etc. 

of  plates,  492. 

of  retaining  walls,  603. 

of  riveted  joints,  772,  etc. 

of  shafting,  500. 

shearing — ,   499. 

of  steel,  476,  499,  500,    870,   920, 
etc.  ' 

tensile—,  454,  920,  922,  957. 

of    timber,    476,   499,    500,     764, 
957,  958. 

torsional — ,  499. 

transverse — ,  466,  473,   476.   478, 
892, 

uniform — ,     beams     and     canti- 
levers of — ,  486. 

of  wood,  476,  499,  500,  764,  957, 
958. 
Stress,  Stresses,  359,  454,  etc. 

alternating — ,  761. 

bearing — ,  permissible — ,  762. 

bending —  in  bridge  members,  per- 
missible— ,  762. 

in  bridges,   permissible — ,   759. 

combined  longitudinal  and  trans- 
«      verse—,  493,  724,  762. 

components,   371. 

compound — ,  493,  724,  762. 

fiber—,  466,  467,  etc. 
and  deflection,  481. 
permissible — ,  762. 

in  bridge  trusses,  759,  764. 
in  roof  trusses,  764. 

range  of — ,  465. 

repeated — ,  465. 

shearing — ,  permissible — ,  762. 

in  truss  members,  698. 

in  trusses,  graphic  method,  703. 

unit — ,    456.       See    also    Stress, 
fiber — . 
in  beams,  467. 

wind —  in  bridges,  710,  758. 
Stretch, 

of  materials,  459. 

of  tape,   correction  for — ,   282. 

of  truss  members,  718. 

unit—,  456. 
Stretcher,  defined,  1035. 
Strike,  defined,  1035. 
Striking  of  centers,  631,  633,  640. 
Stringers, 

in  trusses,  713,  720,  749. 
Structural 

shapes,  tables,  892,  etc.,  986. 

steel,  872. 


INDEX. 


1073 


stmt— Tensile. 


Strut,  Struts,  689. 

design,  trusses,  722,  733. 

and  ties,  eriterion  for — ,  359,  699. 
Stub  switch,  824,  825. 
Stubs  gauge,  890. 
Stucco,  968. 

Stuffing  box,  defined,  1036. 
Stumps, 

blasting  of — ,  950. 
Sub-delivery, 

cost,  855. 
Submerged  weirs,  554. 
Subterranean  temperature,  320. 
Subtraction  of  fractions,  36. 
Sub-verticals,  694. 
Suddenli^  applied  loads,  460,  486. 
Sulphur, 

in  steel,  753,  S72. 

weight,  214. 
Summation 

of  deflections  in  trusses,  720. 

of  forces,  466. 
Sump,  defined,  1036. 
Sun, 

dial,  to  make — ,  268. 

equal   shadows   from — ,    location 
of  meridian  by — ,  288. 

mean — ,  265. 
Superelevation,  787. 
Supplement  of  angle.  94. 
Supported  joints,  819. 
Surcharged  walls,  605,  609. 
Surface,  Surfaces, 

neutral — ,  466. 

per  length, 

conversion  of — ,  238. 

pressure  of  water  against — ,   501, 
etc. 

units  of — , 

conversion  of—,  233,  238. 

velocity,  560. 
Surveying,  274. 
Suspended  joints,  819. 
Suspender, 

hip — ,  stress  in — ,  709. 
Suspenders   of   suspension   bridges, 

770. 
Suspension  bridges,  765. 
Swage,   defined,    1036. 
Sway  bracing,  691,  710,  749. 
Swing  bridges,  696. 
Switch,  Switches,  824. 
Swivels,  defined,   1036. 
Sycamore, 

strength,  459,  476,  957,  958. 

weight,  214. 
Symbols, 

mathematical — ,  33. 
Symmetry, 

axis  of — ,  514. 
Svnclinal  axis,  defined,  1036. 
Synodic  month,  266. 
Syphon,  520. 
System,  metric — ,  225. 
Systfeme, 

ancien,  226. 

usuel,  226. 


T,  Ts, 

defined,   1036. 
iron,  896,  898,  912. 
rails,  817. 

shapes,  896,  898,  912. 
Table,  Tables.     See   the   article   in 
question, 
conversion —  of  units  of  measures, 

weights,  etc.,  228. 
turn—,  844. 
Tackle,  defined,  1036. 
Tallow,  214,  415. 
Talus,  612. 
Tamp,  defined,  1036. 
Tamping,  nitro-glycerine,  948. 
Tangent,    Tangents,    97,    98. 
to  circles,  to  draw — ,  162. 
to  an  ellipse,  to  draw — ,   190. 
logarithmic — ,  72. 
natural—,   97,   98. 
to  a  parabola,  to  draw — ,  193. 
screw,  293,  307. 
by  slide  rule,  77. 
Tangential 

angles,  table,  784-786. 
component,  369. 
distance,  table,   784-786. 
Tank, 

of  tender,  capacity,  856,  860. 
thickness,  506,  854. 
track — ,  853. 
water—,   851,  856,  860. 
Tapes, 

surveying — ,  282. 
cost  and  mfrs.,  993. 
Tapping, 

of  pipes,  657,  664. 
of  trees,  effect  on  timber,  957. 
Tar,  weight,  214. 
Target,    signal—,    826,    829,   833. 
Tarpaulin,  946. 
Teams,  speed  of—,  801,  806. 
Temperature.     See  Heat,  317. 
of  air,  320. 

altitude,  effect  on — ,  320. 
corrections  for  tapes,  283. 
effect  of —  on 
cement,  932. 
evaporation,  329. 
metals,  etc.,   317. 
rails,  317,  819. 
strength  of  iron,  874. 
surveying  chains,  274,  283. 
velocity  of  sound,  316. 
weight  of  water,  326. 
subterranean — ,  320. 
thermometers,  318. 
Templet,  defined,  1036. 
Tender,  Tenders,  853,  856. 

scoop,  853. 
Tensile 

strength,  454,  920,  922,  932,  957. 
of  cement,  932,  etc.,  939. 
of  chains,  915. 
of  riveted  joints,  772,  etc. 


1074 


INDEX. 


Tension— Track. 


Tension, 

and  compression,  359. 

members,  722,  732,  746. 
flexible  and  rigid — ,  721. 
net  section,  759. 

in  tapes,  282. 
Tents,  cost  and  mfrs.  of — ,  993. 
Teredo,  954. 
Terne  plates,   916. 
Terra-cotta  pipes,  575. 
Test,  Tests.     See  Requirements. 

bending — ,  iron  and  steel,  873. 

borings,  582,  670,  etc. 

of  cement. 

Am.  Soc.  C.  E.,   937. 
U.  S.  Eng'rs,  940. 

of  completed  bridges,  753. 

of  full-size   eye-bars,    753. 

-pieces,   iron  and  steel,   870. 

of    surveying    instruments,    293, 
etc. 
Testing  machine  for  cements,   941. 
Tetrahedron,  194. 
Thawing,    effect    of —  on   cement, 

932. 
Theodolite,   296. 
Thermometers,  318. 
Thimble,   defined,    1036. 
Thin  partition,  flow  through — ,  541. 
Third, 

middle—,    402. 

proportional,  38. 
Three, 

rule  of—,  39. 

-throw  switch,  830. 
Throat  of  frog,  835. 
Through  trusses,  692. 
Throw, 

defined,    1036. 

of  switch,   827. 
Thrust, 

in  arch,  430,  432. 

line,  430,  432,  434-436. 
Tides,  328. 
Tie,  Ties, 

cost  of—,  994. 

cross — ,  815,  855. 

land—   612. 

plates,   816. 

and  struts,    criterion   for — ,  359, 
699. 

in  trusses,  689. 
Timber.     See  also  Wood,  Wooden, 

beams,  760,  762,  764,  959,  962. 
bled — ,  strength,  957. 
board  measure,  table,  269. 
bridges,    732-740. 
cohesive  strength,  957. 
columns,  963,  etc. 
compressibility,   459. 
compressive  strength,  958. 
cost,  984. 

crushing  strength,  958. 
dams,  642. 
decay  of — ,  954. 
ductility,  459. 


Timber — continued. 

elastic  limit,  459. 

friction,  411. 

joints,  733,  etc. 

limit,  elastic — ,  459. 

modulus  of  elasticity,  459. 

pillars,  761,  764,  963,  etc. 

preservation,  954. 

requirements,   754,  760,  764. 

roof  trusses,  716,  732,  742. 

shearing  strength,  499. 

strength,  476,  499,  500,  760.  764, 
957,  958. 

stresses  in — ,  760,  764. 

stretch,    459. 

tensile  strength,  957. 

for  ties,  815. 

torsional  strength,  500. 

transverse    strength,  476. 

trestles,  813. 

turntables,  848. 

weight,  212. 
Time,  265. 

effect    of — ,    on    strength    of   ce- 
ments, 932. 

local—,    287. 

-piece,  to  regulate —  by  stars,  266. 

standard  railway — ,  267. 

units  of — ,  conversion  of — ,  236. 
Tin,  916. 

elastic  limit,  etc.,  459. 

expansion  by  heat,  317. 

leaded—,  916. 

roofing — ,    916. 

strength,  920,  921. 

weight,  215,  877. 
Tinned  steel-plates,  916. 
Toe  of  switch,  825,  828,  839. 
Toggle  joint,  427. 
Ton,  216,  220. 

of  coal,  volume  of—,  215,  222. 

(2240  lbs.),  equivalents  of—,  236. 

-mile,  867. 

net—,  216. 
Tonelada,    227. 
Tongue, 

of  frog,  834. 

switch,   828. 
Tonite,  951. 

Tonnage  rating  of  locomotives,  862. 
Tonne,   or  metric  ton,   equivalents 

of—,  236. 
Tools,  wear  of — ,  801. 
Top  heading,  812. 
Torpedoes,   nitro-glycerine — ,   948. 
Torsion,  499. 
Towers, 

of  suspension  bridges,  768,  770.  • 

valve—,   652. 
Towne  lattice  truss,  694. 
Tracing  cloth  and  paper,  978. 
Track.     See  Rail. 

gauge,  827. 

laying,  cost,  855.  / 

scales,    854. 

tank,    853. 

trough,  853. 


INDEX. 


1075 


Traction— llnit. 


Traction,  683. 

of  cars,  860. 

on  grades,  860. 

of  horses,  683,  685. 

of  locomotives,  860. 
Trailing  switch,  824. 
Train, 

centrifugal  force  of — ,  758. 

drag  of —  on  bridge,  758. 

earthwork  by — ,   807. 

-shed  roof.  Broad  St.,  Phila.,  740. 
Transformation  of  profile,  611. 
Transit,  Transits, 

the  engineer's,  291. 
cost  and  mfrs.,  993. 
Transmission, 

of  force.  358. 

of  pressure  in  liquids,  506. 
Transportation    of    bridges,    743. 
Transverse 

and    longitudinal    stresses    com- 
bined, 493,   724,  762. 

strength,   466. 
of  concrete,  945. 
Trap  rock,  weight,  214. 
Trapezium,   158. 

center  of  gravity,  392. 
Trapezoid,   158. 

center  .of  gravity,  392. 
Trapezoidal  notch,  559. 
Tread 

-wheel,  590,  686. 

of  wheel,  821. 
Trees,  blasting  of—,  950. 
Trembling  of  dams,   648. 
Tremie.    946. 

Trenching  machine,  mfrs.  of — ,  992. 
Trenton  wire  gauge,  891. 
Trestles,  813. 
Triangle,  Triangles,  148. 

in  or  about  a  circle,  161. 

element  of  truss,  690. 

force—,  367. 

mensuration  of — ,  148. 

right-angled — ,  150. 
Triangular  truss,  692. 
Trigonometric  functions,  97,  98,  etc. 

logarithmic,    72. 
Trigonometry,   plane — ,   150. 
Trimmer,  defined,   1037. 
Trip-hammer,  defined,   1037. 
Tripod,  292. 
Trough 

floors,  721,  750,  914. 

flow  through — ,  544. 

track—,  853. 
Troy  weight,    220. 
True  or  apparent  solar  time,  265. 
Trundle,  defined,  1037. 
Trunnion,  friction  of — ,  416. 
Truss.  Trusses,  689. 

and  beams,  comparison,  689. 

bracing  in — ,  691,  710,  748. 

for  centers,  636,  etc. 

counterb racing,  690,  705,  712,  721, 
738,  746. 

diagonals,  to  find  lengths  of — ,160. 


Truss,  Trusses — continued. 

end  reactions,  439,  699,  702,  714 
equilibrium  of — ,  437. 
forces  acting  upon — ,  437. 
loads  on — ,   moving — , 

See  Loads,  live — ■. 
members,  stresses  in — ,  698. 
mornents  in — ,  440,  443. 
moving  loads  on — , 

See  Loads,  live — . 
rafters  of—,  691,  713,  etc. 
reactions,  439. 
roof — , 

loads  on—,  764. 

specifications  for — ,  764. 
specifications  for — ,  745. 
in  suspension  bridges,  765. 
weights  of—,  731,  738. 
Tube,  Tubes.     See  also  Pipes,  Flow, 

boiler—,  882. 

brass  seamless  drawn — ,  919. 

bubble — ,  to  replace — ,  296. 

copper  seamless  drawn — ,  919. 

flow  in — ,   516. 

iron—,  882. 

Pitot's— ,  536,  561. 

pressure  of  water  in — ,  511,  518. 

seamless — ,  919. 

short — ,  flow  through — ,  540. 

welded—,    882. 
Tumbling  lever,  826,  830,  833. 
Tun,  216. 
Tunnel,   812. 
Turbines,  mfrs.,  990. 
Turf,  weight,  215. 
Turnbuckle,  Turn  buckles,  986. 
Turnouts,  824,  839. 
Turnpike,  grades  on — ,  255. 
Turntables,  844. 
Turpentine,  957,  972. 
Twaddell  hydrometer,  211. 
Tympan,  687. 
Typical  wheel  loads,  705,  etc.,  755, 

etc. 

u. 

U.  S.     See  United  States. 
Undecagon,  148. 
Underpin,   defined,    1037. 
Ungula,   cylindric— ,    199,   397. 
Uniform, 

live  load,  705. 
loads, 

deflections,   485. 
influence  diagram,  703. 
moments  due  to — ,  444. 
shears  in  beams  due  to,  447. 
strength,   beams  and  cantilevers; 

486. 
velocity,    331. 
Unit,  Units, 

of  force,  338,  358. 
of  measures,    weights,   etc.,   con- 
version tables  of — ,  228. 


1076 


INDEX. 


Unit — Voiissoir. 


Unit.  Units — continued. 

of  moment  of  inertia,  468. 

pressures,  conversion  of — ,  240. 

of  rate  of  work,  342. 

stress.     See  Stress,  unit — . 

stretch.     See  Stretch,  unit — . 

of  work,    341. 
United  States, 

average  precipitation  in — ,  322. 

coins,  219. 

gallon.     See  Gallon. 

measures,  223. 

railroad  statistics,  867. 

standard  dimensions  of  bolts,  etc., 
883. 
Unstable  equilibrium,  387,  514. 
Unsymmetrical  loading,  690. 
Upper  chord,  723,  733. 
Upper  culmination,  284. 
Upset  rods,  886. 
Ursa  minor  and  major,  285. 


V. 

Vacuum        process       for       sinking 

cylinders,  596. 
Value, 

per  length,  surface,  time,  volume, 
weight,  work,  etc.,  conversion 
of  units  of — .  See  Conversion 
tables,   246,  etc. 

present,  42,  44. 
Valve,  Valves, 

air—,  662. 

cost  and  mfrs.,  995. 

defined,    1037. 

four-way — ,  667. 

outlet—,   653. 

stop — ,  666. 

tower — ,  652. 

for  water-pipes,  666. 
Vara,   227. 
Variation, 

of  compass,  301. 

line  of  no—.  300. 

magnetic,  301. 

vernier,    296. 
Vegetation 

in  reser\'X)irs,  652. 
Vehicles,  friction,  414. 
Vein, 

contracted — ,  541. 
Velocity,  Velocities, 

of  abrasion,  577. 

accelerated—,  331. 

through  adjutages,  540. 

affected  by  material  of  pipes, 
523. 

angular — ,  351. 

of  approach,  556. 

in  channels,  560. 

Kutter's  formula,  563,  564. 

defined,   331. 

due  to  a  given  head,  539. 

effect  of —  on  friction,  412. 


Velocity,  Velocities — continued. 

equivalent  to  discharge  per  sur- 
face   253 

of  falling  bodies,  348,  539. 

head,   516. 

for  a  given  velocitv,  to  find^ — , 
527. 

on  inclined  planes.  349. 

Kutter's  formula,  523,  563,  564. 

material    of    pipe,     effect    on — , 
523 

mean—,  522,  560. 

through  orifices,  539,  546. 

of  outflow,  539. 

in  pipes,  516,  etc. 

retarded — ,  331. 

in  rivers,  660. 

in  sewers,  574. 

in  short  tubes,  540. 

of  sound,  316. 

theoretical — ,   of  outflow,   539. 

uniform — ,  331. 

units  of — ,  conversion  of — ,  242. 

of  wind,  321. 
Vena  contracta,  541. 
Ventilation, 

air,  quantity  required,  320. 

of  tunnels,  812. 
Venturi  meter,  532,  etc. 
Vernier,  293. 

variation — ,  296. 
Versed  sines,  97. 
Verst,  227. 
Vertical,  Verticals, 

of  buoyancy,  514. 

circle,  astronomy,  284. 

defined,  92. 

of  equilibrium,  514. 

of  flotation,  514. 
Vessel,  Vessels, 

air—,  663. 

contents  of — ,  198,  223. 

floating—,  514,  515.  v 

metallic — ,  effect  of  water  on — , 
327,  917. 
Viaduct,  Viaducts,       See    Trestles, 
and  Trusses. 

erection,  743. 
Vibrating  bodies,  350. 
Vibration,  350. 
Vinculum,   symbol,   33. 
Virtual  head.  539,  578. 
Vis  viva,  343. 
Voids, 

in  broken  stone,  688,  943. 

in  concrete,  943. 

in  rubble,  799,  925. 

in  sand,  214,  935. 
Volume,  Volumes, 

of   air,   weights  of — ,  conA^ersion 
of—,  242. 

equivalent  depths,  251. 

increase  of —  of  stone,  943. 

occupied  by  coal,  215. 

unit — ,  conversion  of — ,  234. 

of  water,  weights  of — ,  241. 
Voussoir,  613. 


INDEX. 


1077 


l¥agons— Watt. 


w. 


Wagons,  friction,  414. 
Wales,  defined,  1037. 
Wall,  Walls, 

backing  of—,  603. 

battered,  605. 

bricks,  number  in  a  sq.  ft.  of — , 
925-927. 

cost,  601,  602. 

dams,  508. 

face—,  603. 

foundations  for — ,  582. 

incrustation  of — ,  929,  936. 

to  resist  water  pressure,  508. 

retaining — ,  603. 

soap-wash  for — ,  928. 

spandrel — ,  613. 

stability  of—,  508,  606. 

surcharged — ,  605,  etc. 

water, 

to  render  impervious  to,  928. 
to  resist  pressure  of — ,   508. 

wharf—,  515,  611. 

wing — ,   624. 
Walnut, 

strength,  476,  957,  958. 

weight,  215. 
Warp,  defined,   1037. 
Warren  truss,  692. 
Washers,  883. 

defined,   1037. 

lock-nut  or  nut  lock,  885. 
Washes  for  walls,  928,  972. 
Waste  of  water,  649. 
Waste-weir, 

defined,    1037. 

for  reservoirs,  652. 
Watch,  to  regulate — ,  by  the  stars, 

266. 
Water,  326.     See  also  Pipes,  Flow, 
etc. 

for  boilers,  327. 

boiling — ,     to     measure     heights 
by—    314. 

brick  work,  to  render  impervious 
to — ,  928. 

buoyancy,  513. 

in  cement,  932.  936,  938,  941. 

cisterns,  512,  851-854. 

column,   852. 

compensation,    653. 

composition  of — ,  326. 

compressibility,    326. 

in  concrete,  944. 

concrete  under — ,   946. 

consumption  of — ,  649. 

corrosion  by — ,  327,  594. 

dams  for—,  508,  642. 

discharge.     See  Discharge. 

effect  on 

cement,   932,  936,  938,  941. 
dynamite,  950. 
iron,  327,  594. 
lime,  925,  926. 

effect  of  zinc  on—,  328,  917. 

evaporation,  329. 


Water — continued. 

for  fire  protection,  650. 
flow  of — .     See  Flow, 
foot  of — ,  etc.  (pressure),  equiva- 
lents of—,  240. 
foundations  in — ,  583. 
freezing  of—,  326,  328. 
gates,  666. 

head  of—   258-260,  516. 
horse-power,  578. 
jet  for  pile-driving,  595. 
leakage,   329,   561,   642,  649,  651. 
for  locomotives,   327,   852. 
masonry,    to    render    impervious 

to—,  928. 
meters,  532,  536,  562,  649. 

cost  and  mfrs.,  994. 
motors,   mfrs.,   990. 
pipe,  prevention  of  bursting  of — , 

by  freezing,  665.    * 
in    pipes.      See    Pipes,    Velocity, 

Flow,  Discharge,  Pressure,  etc. 
pipes,  653,  etc.,  etc. 

cost  of — ,  658. 
power,   578. 
pressure,  501,  etc.,   518. 

in  cylinders,  511. 

in  pipes,  511,  518. 

plank,  to  resist — ,  586,  648. 

running — ,  578. 

still — ,  501,  etc. 

on  surfaces,  501,  etc. 

wall  to  resist — ,  508. 
rain—,  322,  327. 
ram,  513,  663,  668. 
resistance  to  moving  bodies,  578. 
running — ,  pressure,  578. 
salt — ,  effect  on  metals,  327,  594. 
scouring  action,  577. 
shed,  defined,  1038. 
size  of  commercial  measures  by 

weight  of—,  224. 
stations,  851. 
storage  of — ,  650,  etc. 
supply—,  322,  649. 
tank,  thicknesses,  854. 
traction  on — ,  683. 
in  tubes,  flow  of — ,  516. 
velocity.     See  Velocity, 
volumes  of  unit  weight  of — ,  con- 
version of — ,  242. 
walls,  to  render  impervious  to — , 

928. 
walls  to  resist  pressure  of — ,  508, 

515. 
waste  of — ,  649. 

way,  contraction  of — ,  575,  623. 
weight,  241,  326. 

in  pipes,  525. 

size    of    commercial    measures 
by—,  224. 
wheel,  578. 
works,  649. 

depreciation,  45. 
Watt. 

equivalents  of — ,  245. 

-hour,    equivalents   of — ,    237. 


1078 


INDEX. 


Wax— Wood. 


Wax,  weight,  215. 
Way, 

permanent — ,  815. 

station,  cost,  854. 
Wear, 

of  cars,  865. 

of  locomotives,  864. 

of  ties,  815. 

of  tools.  801. 
Web,  Webs, 

members,  689. 

in  plate  girders,  748. 

plates,  permissible  shear,  762. 

stresses,  702. 
live  load — ,  706. 
Wedge,  Wedges, 

mensuration  of — ,  203. 

striking — ,  for  centers,  631,  632, 
640. 
•Week,  236, -265. 

Weight,  Weights.     See  also   article 
in  question,  212. 

of  air,  320. 

conversion    of    unit     volumes, 
242. 

of  beams, 

compared,  478. 
as  load,  477. 

of  bridges,  731,  738. 

of  cement,  939. 

of  centers  for  arches.  639. 

on  driving  wheels,  705,  etc.,  755, 
etc.,  856. 

French—,  old—,  226. 

in  levers,  419. 

and  measures,  216. 

conversion  tables  of  units  of — , 
228. 

metric—,  217,  226,  228,  etc. 

of  roof  covering,  713. 

of  roof  trusses,  713. 

Russian — ,  227. 

of  snow,  323. 

Spanish—,  227. 

of  steel  railroad  bridges,  731. 

of    substances,    table,    212.      See 
also  the  article  in  question. 

unit — ,  conversion  of — ,  235. 

of  water  in  pipes,  525. 

of  wooden  bridges,  738. 
Weir,   547. 

discharge,   formulae  for — ,   549. 

measuring — ,  547,  646. 

submerged — ,  554. 
Well,  Wells, 

artesian — ,  671. 

boring,  670. 

contents,   197. 

machinery,  mfrs.,  992. 

masonry,  quantity  in  walls,   198. 
Wellhouse  process,  955. 
Western  elongation,  284. 
Westing,  274. 
Wet  perimeter,  523,  563. 
Wheel,  Wheels, 

barrows,     earthwork    by — ,    803, 
810. 


Wheel,  Wheels — continued. 

base,  787,  789,  856,   1038. 

centrifugal  force,  355. 

diagram,   706. 

driving — ,   856. 

loads  on — ,  705,  etc., -755,  etc., 
856-859. 

guards,  750. 

loads,    705,  etc.,    755,   etc.,  856- 
859. 

of  locomotives,  856. 

loads  on — ,  705,  etc.,  755,  etc., 
856-859. 

meters,  562. 

Persian — ,  687. 

and  pinions,  420. 

skidding  of — ,  413. 

tread—,  590.  686. 

tread  of — ,  821. 

water — ,  578. 
Wheeled  scrapers,  805. 
Whipple  truss,  694. 
White 

effervescence    on  walls,  929,  936. 

lead  paint,  971. 

wash,  973. 
Whitworth  screw  thread,  etc.,  883. 
Winch,  686. 
Wind,  321. 

effect  on  suspension  bridges,  766. 

loads,  710. 

mills,   852. 

pressure  on  roofs,  321. 

on  roof  trusses,  713. 

stresses,  710,  758. 
Wine  measure,  223,  224. 
Wing, 

dam,  defined,  1038. 

defined,    1038. 

of  frog,  834. 

walls,  624. 
Wire,  887-891.  " 

circular  measurement  of — ,  889. 

cost  and  mfrs.,  987. 

fence,  854. 

gauges,  887-891. 

iron—,  891. 

rope,  976.  977. 

steel — ,  891. 

strength,   920. 
Wohler's  law,  465. 
Wood. 

See  Timber,  Wooden,  etc. 

board  measure,  table,  269. 

cohesive  strength,  957. 

compressibility,   459. 

compressive  strength,  958. 

cost,  984. 

crushing  strength,  958. 

ductility,  459. 

effect  of  lime  on — ,  926. 

effect  of  mortar  on — ,  926. 

elastic  limit  of — ,  459. 

friction,  411, 

limit  of  elasticity,  459. 

modulus  of  elasticity,  459. 

pipe,  cost,  995. 


INDEX. 


1079 


Tl'ood— Zone. 


Wood — continued. 

shearing  strength,  499. 

shingles,  971. 

specific  gravity,  212. 
•    strength,  476,  499,  500,  760,  764. 
957,  958. 

stretch  of — ,  459. 

tensile  strength,  957. 

torsional  strength,  500. 

transverse  strength,  476. 

weight,  211. 
Wooden.     See  also  Wood. 

beams.  See  Beams, wooden — ,  476. 

bridges,   732,  etc. 

specifications  for — ,  763. 

dams,  642,  etc. 

floors  in  bridges,  750. 

joints,  734. 

pillars,  761,  764,  963. 

pi])es,  657. 

roof  trusses,  716,  742. 

trestles,   813. 

turntables,  848. 
Work,  341. 

equivalence  of — ,  trusses,  718. 

of  friction,  418. 

and    heat,     conversion    of    units 
of—,  237. 

of  overturning,  422. 

per   time    [power],    conversion  of 
units  of — ,  244. 

units  of — ,   341. 

useful—,   342. 
W^orking  beam,  defined,  1038. 
Worm, 

defined.    1038. 

fence,  854. 

sea — ,  954. 
Worth,  present — ,  42,  44. 
Wrecking  car,  808. 
Wrought  iron.    See  Iron,  wrought — . 

pillars,   breaking  loads,   910. 


Yard,  Yards,  216,  220. 

cubic — , 

of  earthwork,  790. 
equivalents  of — ,  222. 

equivalents  of — ,  232. 
Year, 

civil—,  266. 

equivalents  of — ,  236. 

sidereal—,  266. 
Yield  point,  459,  871,  873. 


z. 

Z-bars,  901-903. 

flooring,   914. 
Zenith,  284. 
Zigzag  riveting,  774, 
Zinc,  916. 

cost,  987. 

effect    of    cement,    mortar,  etc., 
on—,  926,  936. 

effect  of —  on  water,  917. 

effect  of  water  on — ,  328. 

expansion  of —  by  heat,  317. 

paint,  971. 

paint  on — ,  880. 

price,  987. 

roofing,    916. 

sheets,  916. 

strength,  920. 

compressive — ,  921. 

weight,  215,  875,  877,  878,  887. 
Zone,  Zones, 

circular — ,  186. 

of  circular  spindle,  209. 

parabolic — ,  192. 

spherical — ,  208. 

center  of  gravity  of — ^  396. 


THE  END. 


ADVERTISEMENTS. 


INDEX  TO  ADVERTISEMENTS. 


PAGE 

Association  of  Engineering  Societies, 15 

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Builders  Iron  Foundry,      17 

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Engineering  Record,    . 3 

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Municipal  Engineering  and  Contracting  Company, 4,  20 

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National  Tube  Company, S 

Railroad  Gazette, 13 

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Railway  and  Engineering  Review, 11 

Railw^ay  Engineering  and  Maintenance  of  Way, IC 

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Abrasive  Material  Co.,           "  S.  Morgan  Smith  Co.,  York,  Penna. 

France  Metallic  Packing  Co.,"  Birdsboro  Steel  Foundry  and  Machine 

The    Green   Fuel    Economizer    Co.,  Co.,  Birdsboro,  Pa. 

Matteawan,  N.  Y.  Shepherd  Engineering  Co.,  Franklin, 

Frick  Co.,  Waynesboro,  Penna.  Pa.             and  others. 
Taking  entire  charge  of  all  Advertisements,  relieving  the  Advertiser  from 
trouble  and  annoyance,  besides  making  a  great  saving  in  cost. 

ADVERTISEMENTS    INSERTED    IN    ALL   THE    LEADING    INDUSTRIAL    PAPERS. 

7 


(IJ^STRUMEJ^TS  OF  PEECISIOJ^.) 


Messrs.  HELLER  &  BRIGHTLY  publish  a  book  containing  tables  an< 
maps  useful  to  Civil  Engineers  and  Surveyors ;  also  contains  much  valuabi 
information  respecting  the  selection,  proper  care,  and  use  of  field  instru 
ments.  A  copy  of  this  book  they  send  by  mail,  post-paid,  to  any  Civi 
Engineer  or  Surveyor  in  any  part  of  the  world,  on  receipt  of  a  postal  car* 
containing  the  name  and  post-office  address  of  the  applicant. 

8  ^ 

in     0  6  6  5 


THIS  BOOK  IS  DUE  ON  THE  LAST  DATE 
STAMPED  BELOW 


RENEWED  BOOKS  ARE  SUBJECT  TO  IMMEDIATE 
RECALL 


LIBRARY,  UNIVERSITY  OF  CALIFORNIA,  DAVIS 

Book  Slip-50m-5,'70(N6725s8) 458— A-31/5 


J 


22^69 


Trau twine,  J.C. 

The  civil  engineer's 
pocket-book. 


Call  Number: 

TAl^l 

T7 

1902 


22569 

Trau twine,   J.C. 

The  civil  engineer's 
pocket-book. 


TAl^l 

T7 

1902 


LIBRARY 

UNIVERSITY  OF   CALIFORNIA 

DAVIS 


